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■^ »% 7
/■" 4ar'.' ' /17. A. 3/1
/
/
3 2044 097 044 374
Gift of Tin PMpIe of the URited States
Through the Victery Book Campaign
(A.LA. — A. R.C. — U. S.C.)
To the Armed Forces art! ?."crc
t w ^ •
^.— I —
\
f
DE8CAKTE3
DURELL'S
SCHOOL ALGEBRA
BT
FLETCHER DUKELL, Ph.D.
■BAD Of THE HATBBHATIOAL DEPARTMENT IN TOM
IJIWKEIICKVIL1.E SCHOOL
NEW YOEK
CHARLES E. MERRILL COMPANY
t dcui / /i^^j. /^.3^^
HARVARD^
IBRAP>/
DURELPS MATHEMATICAL SERIES
ARITHMETIC
Two Book Series
Eleimentary Arithmetic, 48 cents
Teachers' Edition, 56 cents
Advanced Arithmetic, 72 cents
Three Book Series
Book One, 56 cents
Book Two, 60 cents
Book Three, 64 cents
ALGEBRA
Two Book Course
Book One, $1.00
Book Two, 96 cents
Book Two with Advanced Work, $1 .00
Introductory Algebra, 60 cents
School Algebra, $1.25
GEOMETRY
Plane Geometry, 88 cents
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WITH Surveying and Tables, $1.75
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[81
Copyright, 1911, 1916,
By CHARLES E. MERRILL CO.
PREFACE
The main object in writing this Schooi^ Algebra
has been to simplify principles and give them interesty
by showing more plainly, if possible, than has been
done heretofore, the practical or commonsense reason
tor each step or process. For instance, at the outset it
is shown that new symbols are introduced into algebra
not arbitrarily, but because of definite advantages
in representing numbers. Each successive process is
taken up for the sake of the economy or new power
which it gives as compared with previous processes.
This treatment should not only make each prin
ciple clearer to the pupil, but should give increased
unity to the subject as a whole. We believe also that
this treatment of algebra is better adapted to the
practical American spirit, and gives the study of the
subject a larger educational value.
Among the special features of this School Algebra,
the following may be mentioned :
A large number of written problems are given in the
early part of the book, and these are grouped in types
which correspond in a measure to the groups used in
treating original exercises in the author's Geometry.
Many inform^ational facts are used in the written
problems. The central and permanent numerical
facts in various departments of knowledge have been
collected and tabulated on pages 496504 for use in
making problems. Similarly the most important
3
4 PREFACE
formulas in arithmetic, geometry, physics, and engi
neering have been tabulated for use by teacher and
pupU ^p. 496, 497).
The self activity of the pupU is aroused by examples
which require the pupil to invent and solve problems
of a specified kind, material for such examples being
made available in the tables of formulas and niunerical
facts.
Many of the examples in the book require a frequent
review of the principles of arilhmetiCy as of decimal
fractions and percentage.
Nmnerous and thorough reviews of the portion of
the Algebra already studied are also called for. A
unique feature is the series of spiral reviews of the
preceding part of the book by means of examples at the
end of Exercises. Oral work is called for in Uke man
ner and is also emphasized m special important Exer
cides.
The utilities in symbolism in general, apart from
technical algebra, are brought out in a special Exercise
(pp. 249, 250) and thus the direct practical value of the
study of algebra is much broadened.
The history of algebra is discussed in Chapter XXVI,
and questions on this chapter are inserted in appropri
ate places in the text.
The author wishes to express his indebtedness to
Professor WiUiam Betz of the East High School,
Rochester, New York, and to Dr. Henry A. Converse
of the Polytechnic Institute, Baltimore, Maryland,
for important aid in preparing the book. He is in
debted also to School Science and Mathematics and
the Mathematics Teacher for a few of the problems.
; . 1. .
CONTENTS
CHAPTER PAOB
I. Algebraic Symbols ; . . 7
II. Negative Numbers 28
III. Addition and Subtraction; the Equation 39
IV. Multiplication 58
V. Division . 76
VI. Equations (Continued) 93
VII. Abbreviated Multiplication and Division 109
VIII. Factoring 134
IX. Highest Common Factor and Lowest
Common Multiple * . . . 160
X. Fractions 166 
XI. Fractional and Literal Equations . . . 196
XII. Simultaneous Equations 223
XIII. Graphs . 251
XIV. Inequalities . 266
XV. Involution and Evolution 272*
XVI. Exponents 289
XVII. Radicals 305
XVIII. Imaginary Quantities 334
XIX. Quadratic Equations op One Unknown
Quantity 341
5
6 CX)NTENTS
CHAPTER PAOB
XX. Simultaneous Quadratic Equations . . . 366
XXI. Graphs of Quadratic and Higher Equa
tions 387
XXII. General Properties of Quadratic Equa
tions 397
XXIII. Ratio and PROPORTio^f 406
XXIV. The Progressions 423
XXV. The Binomial Theorem 447
XXVI. History of Elementary Algebra .... 454
Appendix
Fundamental Laws of Algebra 465
Detached CoeflScients 466
Factor Theorem 467
H. C. F. and L. C. M. obtained by Long
. Division 469
Cube and Higher Roots . ^. 472
Review Exercises 475
Logarithms 496
Material for Examples
Formulas 512
Important Numerical Facts 514
Index 521
SCHOOL ALGEBRA
CHAPTER I
ALGEBRAIC SYMBOLS
1. The Use of Letters.
Ex. Walter and Harold made $27 by gardening one sum
mer. Walter, who was older and stronger, received a
double share of the profits. How much did each receive?
SOLUTION WITHOUT THE AID OP X
1 share » Harold's part of the profits
2 shares « Walter's part of the profits
1 share + 2 shares = $27
3 shares » $27
1 share = $9, Harold! 8 pari
2 shares » $18, WaUer^s part
SOLUTION BY Am OF X
Let X s Harold's part of the profits
Then 2x = Walter's part of the profits
Hence a; + 2« = $27
ac = $27
X — $9, Harold's part
2x = $18, WaUer'a part
We see that by use of the letter x the solution is much
shortened.
^2. Algebra is that branch of mathematics which treats
of number by the extended use of symbols.
Later algebra comes to have a wider meaning.
Algebra may also be briefly described as generalized arithmetic.
7
8 . SCHOOL ALGEBRA
3. Vtility of Algebra. A more extended use of symbols
than is practiced in arithmetic (1) shortens the work of solv
ing problems; (2) enables us to solve problems which we
could not otherwise solve; and (3) gives other advantages
which will become evident as we proceed (see Art. 143 and
Exercise 76, p. 249).
EXERCISE 1
(Problems of Type I, i. e. of the form x + ax = b.)
1. Two boys together catch 84 fish. If the boy who
owns the boat which they use, receives twice as many fish
as the other boy, hpw many fish does each boy receive?
2. A man left $12,000 to his son and daughter. To his
daughter, who had taken care of him in his old age, he left a
double share. What did each receive?
3. A man and boy by working a garden one summer made
$128.80. If the man received a share of the profits three
times as large as the share received by the boy, how much
did each receive?
4. Two boys together gathered 1 bu. 4 qt. of hickory nuts.
If the boy who climbed the trees received a double share,
how many quarts did each receive?
5. Make up and work a similar example concerning two
boys who gathered chestnuts.
6. Two girls made $18.60 by sewing. The girl who sup
plied the thread and machine received twice as much as the
other girl. How much did each make?
7. Make up and work a similar example concerning two
girls who kept a refreshment stand.
ALGEBRAIC SYMBOLS 9
a Solve Ex. 1 without the use of x (see Art. 1). How
much of the labor of writing out the solution is saved by the
use of xf Is there any other advantage in the use of a; in
solving a problem?
9. The total cotton crop of the world in a certain year
was 15,000,000 bales, and the United States in that year
produced three times as much as all the rest of the
world. How many bales of cotton did the United States
produce?
10. A farm is worked on shares. As the tenant supplied
the tools and fertilizers, he received twice as large a share
of the profits as the owner of the farm. If the profits for one
year are $6000, how much does the tenant receive? The
owner?
11. If the sum of the areas of New York and Massachu
setts is 57,400 sq. mi. approximately, and New York is 6
times as large as Massachusetts, what is the area of each
state?
12. One number is 5 times as large as another and the
sum of the numbers is 240. Find the niunbers.
*13. One munber is twice as large as another and the siun
of the niunbers is 7.26. Find the numbers.
^ 14. One fraction is three times as large as another and their
siun is i. Find the fractions.
* 15. One number is 4 times as large as another and their
sum is .0045. Find the numbers.
16. Separate $120 into two parts such that one part is
three times as large as the other.
Sua. Let z <« the smaller part.
10 SCHOOL AIXIEBRA
17. Separate Si into two parts such that one part is 7
times as large as the other.
18. Make up and work an example similar to Ex. 11*.
Also one similar to Ex. 15. To Ex. 16. \
Material for examples may be obtained from the lists of
Important Numerical Facts given on pp. 514620.
19. To look well^ the middle part of a steeple should be
twice as high as the lowest part, and the top part 8 times as
high as the lowest part. If a steeple is to be 132 ft. high,
how high should each part be?
20. A man wants to save $6000 in three years. If he is
to save twice as much the second year as the firsts and three
times as much the third year as the firsts how much must
he save each year?
21. A girl has $42 to spend for a hat^ coat^ and suit. She
wants to spend twice as much for her coat as for her hat^
and three times as much for her suit as for her hat. How
much does she spend for each?
22. A man bequeathed $84,000 to his niece, daughter, and
wife. If the daughter received twice as much as the niece,
and the wife four times as much as the niece, how much did
each receive?
23. A certain kind of concrete contains twice as much sand
as cement and 5 times as much gravel as cement. How many
cubic feet of each of these materials are there in 1000 cu. yd.
of concrete?
24. Make up and work a similar example for yourself
where the materials in the concrete are as 1, 2, 4.
25. In a certain kind of fertilizer the weight of the nitrate
of soda equals. that of the ground bone, and the weight of
ALGEBRAIC SYMBOLS 11
the potash is twice as great as that of the ground bone. How
many pounds of each ,of the materials are there in a ton of
fertilizer?
26. If the amount of pota^ in a given kind of glass is 5
times as great as the amount of lime, and the amount of
sand 3 times as great as the amount of potash, how many
pounds of each will there be in 4000 lb. of glass?
27. The railroad fare for two adults and a boy traveling
for half fare was $49.50. What was the fare for each person?
Stjg. Let X = the smallest of the fares.
2a Separate 120 into three parts, such that the second part
is twice as large as the first, and the third part three times
as large as the first.
29. Separate 120 into three parts which shall be as 1, 2', 3.
30. Separate .0372 into three parts in like manner. Also 3^3.
31. Separate 240 into four parts which shall be as 1^^ 1,
2,4.
32. Separate $1800 into three parts, such that the second
is three times as large as the first, and the third 5 times as
large as the second.
33. In one kind of concrete the parts of cement, sand, and
gravel are as 1, 2, and 4; in another kind three parts are as
1, 2, and 5. How many more pounds of cement are needed
in a ton of one than of the other?
34. How many of the examples in this Exercise can you
work at sight?
To get the greatest possible benefit from the use of letters
to represent numbers, we now make further definitions and
rules.
12 SCHOOL ALOfiBBA
4. Three Olasses of Sjrmbols. Three principal kinds of
symbols are used in algebra: (1) Symbols of qiuintity, (2)
Symbols of operaMon, and (3) Symbols of relation.
5. Symbols for Known ttnantitieg. Known quantities are
represented in arithmetic by figures; as 2, 3, 27. They
are represented in the same way in algebra^ but also in
another more general way, viz.: by letters; as by a, b, c.
The advantages in the use of letters to represent known niun
bers are: (1) letters are brief to write; and (2) a letter may stand
for any known number, and thus by the use of letters we obtain
results which are true for all numbers. See Exs. 3440, p. 97.
'^ 6. Symbols for Unknown Quantities. Unknown quan
tities in algebra are usually denoted by the last letters of
the alphabet; as x, y, z, u, v, etc.
The advantages in the use of distinct symbols for imknown
quantities are numerous and will be gradually realized as we
proceed. Some of these advantages are stated in Art. 3. See
also Art. 143.
7. The Signs +, — , X, s, and = are used in algebra, as in
arithmetic, to denote addition, subtraction, multiplication,
division, and equality respectively.
In algebra, multiplication is also denoted by a dot placed
between the two quantities multiplied, or by placing the
quantities side by side without any intervening symbol
, Thus, instead of a X 6, we may write ah or ab.
8. Signs of Aggregation. The parenthesis sign, ( ), is used,
as in arithmetic, to indicate that all the quantities inclosed by
it are to be treated as a single quantity; that is, subjected to
the same operation.
Thus, 5(2a — 6 + c) means that the quantities inside the paren
thesis, viz. 2a, — h, and + c, are each to be multiplied by 5.
ALGEBRAIC SYMBOI^ 13
Again, (a + 26) (a + 26 + c) means that the sum of the quanti
ties in the Gist parenthesis is to be multiplied by the sum of those
in the second parenthesis.
Instead of the parenthesis, to prevent confusion, the fol
lowing signs are sometimes used: the brackets [ ], the braces
{}, and the vinculum .
9. The Sign of Continuation is ... . This sign is read
"and so on" or "and so on to."
Thus, 1, 3, 5, 7, .... is read "1, 3, 5, 7 and so on."
But 1, 3, 6, 7, 19 is read "1, 3, 6, 7 and so on to 19."
10. The Sign of Dednotion is .*. and it is read "therefore"
or "hence."
This sign is used to show the relation between succeeding
propositions.
EXERCISE 2
1
1
Express in words:
.
1. 5 + a.
7.
56 — a.
13.
a + 6 ^ 3.
2. d — a.
a
2a + 3c.
14.
4 + 5(a + 6).
3. a 7 b.
9.
cd — ab.
15.
(a + b){x  y).
4. ad.
10.
7(a +. 6).
16.
2a + 36  5c.
5. 2a + 36.
11.
7(o  b).
17.
a ^ (x + y).
c d
6.   .
a
12.
5a + b
X + y
la
a + 6 , c
5 d .
19. If a = 1, 6 = 2, c = 3, d = 4, find the value of
the combinations of symbols in Exs. 110.
20. Make and read an example similar to Ex. 5. To
Ex. 10. Ex. 14.
14 SCHOOL ALGEBRA
Express in symbols:
21. X plus 3. The sum of x and 3. The number which
exceeds x by 3.
22. X diminished by 3. The number 3 less than x,
23. Two times a plus three times b.
24. The sum of 4 and of 5 times x.
25. One third oi the sum of a and 6.
Answer the following in algebraic language:
26. If a boy has a cents and earns 10 cents^ how many
cents will he then have?
27. How many, if he has a cents and earns b cents? How
many, if he then spends c cents?
28. Walter has x marbles and his brother has 10 more
than Walter. How many marbles has his brother?
29. Walter has b marbles and his brother has 5 more than
twice Walter's marbles. How many has his brother?
30. If Mary is a years old now, how old will she be in 3
years? In 5 years? In x years?
31. What is the next larger niunber than 5? Than «? n?
a; + 1? a? + 2? n  1? a;  2?
32. What is the next larger even number than 6? Than
2y? 2a:? 2n + 2?
33. Taking x as the smallest number, write two consecu
tive numbers. Three consecutive numbers. Four. Five.
(The following problems are mainly of Type U, i. e. of
the form z + x + a = b.)
ALdEBRAIC SYMBOLS 16
34. If there are 214 pupils in our school^ and the number
of girls exceeds the number ot boys by 8, how many boys and
how many girls are there?
Let
X = the number of boys
Then
jc + 8 = the number of girls
Hence
a: + X + 8 = 214
Or
2a? + 8 = 214
Subtracting 8 from the
8 8
equals
gives 2x = 206
X = 103, the number of hoys
a? + 8 = 111, the number of girls
35. Walter and his brother together had 60 marbles, and
his brother had 10 more than Walter. How many marbles
had each boy?
36. Make up and work an example similar to Ex. 35.
*37. At New York on Dec. 21, the night is 5 hr. 32 min.
longer than the day. Find the length of the day.
38. Separate 28^ into two parts such that ont shall exceed
the other by 2f .
39. A baseball nine has played 62 games and won 8 more
games than it has lost. How many games has it won?
40!. In a certain election 12,784 votes were cast. If the
successful candidate had a majority of 1732, how many votes
did he receive?
41. Make up and work an example similar to Ex. 40.
"^42. The sum of two consecutive numbers is 15. Find the
niunbers.
43. The sum of three consecutive numbers is 33. Find
the numbers.
16 SCHOOL ALGEBRA
44. If 112,216 sq. mi. are added to 24 times the area of
the British Isles, the result will be 3,025,600 sq. mi. (the
area of the United States). Find the area of the British
Isles.
45; Twice the height of Mt. Washington with 1567 ft.
added equals the height of Pike's Peak, or 14,147 ft. Find
the height of Mt. Washington.
46. How many of the examples in this Exercise can you
work at sight?
47. Which of the symbols mentioned in Arts. 610 are
r
symbols of quantity? Of operation? Of relation?
4a Make up and work an example similar to Ex. 44. To
Ex. 45.
Definitions and Principles
11. The term Factors has the same meaning in algebra
as in arithmetic; that is, the factors of a numl^er are the
numbers which, multiplied together, produce the given
number.
For example, the factors of 14 are 7 and 2; the factors of abc are
a, 6, and c.
12. Coefficients. A numerical factor, if it occurs in a
product, is written first and is called a coefficient. Hence,
A coefficient is a number prefixed to a quantity to show
how many times the given quantity is taken.
For example, in ^xy^ 5 is the coefficient.
When the coeflScient is 1, the 1 is not written, but is
understood.
Thus, osy means Ixy.
DEFINITIONS AND PRINCIPLES 17
The following enlarged definition of coefficient is often
used. In the product of several factors, the coefficient of
any factor, or factors, is the product of the remaining factors.
Thus, in 5abxy, the coefficient of 2/ is 5abx; of xy, is 5ab; of 06 is
5xy. What is the coefficient of 6? Of a? xl 5a? 5?
A numerical coefficient is a coefficient composed only of
figures; as 15 in 15ab.
A literal coefficient is a coefficient composed only of letters;
as ob in abx.
What, then, is a mixed coeffijcientt Give an example of one.
13. Power and Exponent are used in the same sense in alge
bra as in arithmetic.
A power is the product of equal factors.
A power is expressed briefly by the use of an exponent.
An exponent is a small figure or letter written above and
to the right of a quantity to indicate how many times the
quantity is taken as a factor.
Thus, for ocxxXy or four a;'s multiplied together, we write a;*, the
exponent in this case being 4. The expression is read *^x Xo the
fourth power."
When the exponent is unity, it is omitted. Thus, x is
used instead of a:^, and means x to the first power.
A power is composed of two parts: (1) the base (i. e. one
of the equal factors) ; and (2) the exponent.
Thus, in the power a', the base is a and the exponent is 3.
14. Boot and Badical Sign have the same meaning in
algebra as in arithmetic.
A root of a number is one of the equal factors which, when
multiplied together, produce the given number.
18 SCHOOL ALGEBRA
The sqnare root of a number is one of two equal factors
which, multiplied together, produce the given number.
What is the cube root of a number? The fifth root?
Thus, 4 is the cube root of 64, and a of a'.
The radical sign is V> ^^d means that the root of the
quantity following it is to be found. The degree of the root
is indicated by a small figure placed above the radical sign.
The number denoting the degree of a root is the index of
the root. For the square root, the figure or index of the root
is omitted.
Thus, V9 means "square root of 9."
Va means "cube root of a."
15. Aids in Solving Problems ; Axioms. In solving prob
lems like those given in Exercise 1 and the latter part of
Exercise 2, certain principles are often important aids in
discovering the relations used and simplifying them.
The most important of these principles are as follows:
1. The whole is equal to the mm of its parts. .
2. Things equal to the same things, or equal things, are equal
to ea/ch other.
3. If equals are added to equals, the results are equal.
4. // equals are subtracted from equxds, the results are equal.
5. // equals are multiplied by equals, the results are equal.
6. // equals are divided by equals, the results are equal.
7. lAke powers, or like roots, of equals are equal.
These principles are sometimes called axioms.
ALGEBRAIC SYMBOLS 19
EXERCISE 8
Write in words:
1. 56». a« + 6* ^ 12. Vo + '5^.
1' o • '
,. 36V. •• (« + «•• "■ '^'° + '•
*• 10. a + (6 + c)2.
6. 2a2 + 36^. ^^' c a ■ 5 4 '■
16. If a = 1, 6 = 2, and c = 3, find the value of the
combinations of symbols in Exs. 1^.
17. If a = 4, 6 = 8, and c = 3, find the value of the
expressions in Exs. 912.
Write in symbols:
18. The square of the sum of a and 6. Of 2a minus 36.
19. The cube root of the siun of a and 6.
20. X plus z increased by 4 equals 14.
21. X plus twice X plus x increased by 3 equals 108.
22. Make up and work an example similar to Ex. 18. To
Ex. 20.
23. Reduce to its simplest form 5 + 6 + 6 + 6 + 6. Also
6x6x6x6x6.
If 6 = 2, what is the value of each of these results?
24. Make up and work an example similar to Ex. 23.
25. Reduce 3aaa + 76666 — bcccccc to its simplest form.
How many more symbols are used in the long form than in
the short form?
20 SCHOOL ALGEBRA
26. Find the value of 2" when n = 1. Also when n = 2.
3. 5. 7.
27. Find the value of a", when a = 3 and w = 4. Also
when a = 5 and n = 3.
28. Express the number of your greatgrandparents as a
power of 2.
(The following are miscellaneous problems of Typos I
and n.)
29. A man and boy together spade up a garden containing
6000 sq. ft. If the man spades four times as much ground as
the boy, how much does the boy spade?
30. Two boys earn $38 by taking passengers on a motor
boat. If the boy who owns the boat receives $10 more than
the other boy, how much does each receive?
31. A certain macadam road cost $1800, of which the
county paid twice as much as the state, and the township the
same amount as the county. How much did each pay?
32. The top of the Statue of Liberty in New York Harbor
is 306 ft. above the surface of the water. If the altitude of
the pedestal is 4 ft. greater than the height of the statue,
how high is each?
33. In a certain kind of gunpowder the weight of the char
coal equals that of the sulphur, and the amount of niter equals
the charcoal and sulphur combined. How many pounds of
each substance are needed to make a ton of gunpowder?
34. In a certain year in the United States 200,000,000
bushels plus three times the number of bushels in the wheat
crop equaled the corn crop, or 2,600,000,000 bushels. How
many bushels were in the wheat crop?
ALGEBRAIC EXPRESSIONS 21
35. Point out the problems among Exs. 2934 which belong
to Type I. Also those which belong to Type II.
36. Make up and work an example similar to Ex. 29. To
Ex. 31.
37. How many of the examples in this Exercise can you
work at sight?
Algebraic Expressions
16. An Algebraic Expression is an algebraic symbol or
combination of symbols representing some quantity; as
bQ?y — 6ab + 7Vax.
17. A Term is a part of an algebraic expression which
does not contain a plus or minus sign. (Signs occurring
inside a parenthesis are not considered in fixing the terms.)
Ex. 1. 5x^  6a6 + 7^^.
This algebraic expression contains three terms: viz. 5x^, — 6a6,
and 7^ax,
. Ex. 2. 5a; f a 5 6 + c.
This expression also contains thi*ee terms: 5x, a i b, and c.
Ex. 3. 7ax^ + 5(a + &)  c».
Since the parenthesis, (a + b), is treated as a single quantity,
three terms occur in this expression: 7ax^, 5(a + 6), and — c*.
18. A Monomial is an algebraic expression of only one
term; as 5xh/ or c.
19. A Polynomial is an algebraic expression containing
more than one term; as Sab — c + 2x + by^.
A monomial is sometimes called a simple expression, and a
polynomial a compound expression,
20. A Binomial is an algebraic expression of two terms;
as 2a — 36.
22 SCHOOL ALGEBRA
A Trinomial is an algebraic expression of three terms; as
2a  36 + 5c.
Evaluation op Algebraic Expressions
21. The Order of Operation in obtaining numerical values
is the same in algebra as in arithmetic.
I. In a series of operations involving addition, subtract
tion, multiplication, division, and root extraction, the mvUi
plications, divisions, and root extractions are to be performed
before any of the additions and subtractions.
Ex. 1. Find the value of 4 + 12 X 3.
4 + 12 X 3 = 4 + 36 = 40 Ans,
(hence 4 + 12 X 3 does not equal 16 X 3, etc.)
Ex. 2. What is the value of608■^23x7?
608^2+3x7604+21 =77 Ans.
II. If a given expression contains one or more parentheses
(or other signs of aggregation), ea/:h parenthesis is to be re^
duced to a single number before the. operations of the expression
as a whole are to be performed.
Ex. 1. 5 + 4(6  2)  5 + 4 X 4 = 5 + 16 = 21 Ans.
(hence 5 + 4(6  2) does not equal 9(6  2) or 9 X 4, etc.)
Note that in an expression like V 16 + 9 the bar above
the 16 + 9 is a vinculum, or sign of aggregation.
Ex. 2. Vl6 + 9 = V25  5 Ana.
(hence a/16 + 9 does not equal Vie + V9^ etc.)
22. The Numerical Value of an Algebraic Expression is
obtained thus:
SubsUtvie for each letter in the expression the number which
the letter stands for;
Perform the operations indicated.
ALGEBRAIC EXPRESSIONS 23
Thus, if a = 1, 6 = 2, c = 3:
Ex. 1. Find the numerical value of lab — A
7a6  c2 = 7 X 1 X 2  3«
= 149
= 5 Ans.
96
Ex. 2. Find numerical value of baJt^ + 7(a» + 26)« + 3c».
c
The given expression
= ^^  5 X 1 X 22 +7(1» + 2 X 2)2 + 3 X 3«
= 3X25X4+ 7(1 +4)2 +3X9
= 620 + 175+27
= 188 Ana.
EXERCISE 4
In each of the following examples, state the order of
operations before working the example. Wherever possi
ble, use cancellation. When a = 5, 6 = 3, c = 1, and
a; = 6, find the numerical value of
1. 2 + 3a. 13. a^ — bx^.
2. a; — 2c. 14. 2(2a — c).
3. 46 — 2x. 15. x{a — 6).
4. a + 3a:. 16. 4(a — 3c)\
5. 5a  3a:. 17. 2a:(2a  36)^.
6. '3(a + c). 18. 3 + 2(a:  a).
7. a + 3c — a:. 19. 5a: — 3(26 + c).
8. 5a: — 26 + a. 20. 2(q? — a^) + 3ac.
9. a + a: ^ 6 — c. 21. 3a:(a: — 3)^ — 9a:.
10. 5 a:4 6 — c. 22. (a: — 1) (a: — 3) + a: (x—a).
11. 36  X. 23. 3 (2a:  5c)  a(262  3a:).
12. 2x — 46c. 24. (56 + a:) (x — 6 + a — 5c^).
24
SCHOOL ALGEBRA
25.
26.
a + 7c
^^^——^—^ •
X
a + 2c
27.
5a«
X
 + ?£,
(a;  1) (6 + 1) (5c  b)
abx
If a = i, 6 '^= I, « = 2, y = f , find the value of
34. 3a6^
35. X — 26.
36. 2x + 5y,
37. 6a6 — 6*y.
43. 5x(6y — a?) — 6x.
44. 6(a + 6)2 + 10(y  of.
45. X 4 VSa.
46. VSa + VSfc.
30. 6a. 32. abx.
31. hy. 33. aV.
38. fe(10y36).
39. 3x(4a + 36).
40. ax + 5a;(36 — y).
"^ 41. 3a + d(3a;  lOy).
42. Sx — 3(62/ — a6).  47. hy — VOoa:.
48. Does a;2 + X = 12, if X = 2? If x = 3? 4? 5? 1?
49. Does 3x*  4a;  4, if X = 1? If x = 2? 3? ? 0?
50. Doesx^  5x + 6 = 0,if X = 1? If x = 2? 3? 4? 5?
51. Doesx» ix  2 = 0, if X = 1? If x = 2? 3? 4? J?
52. Show that (o — 26)* = c* — 4a6 + 46*, when a = 3
and 6 = 1.
53.
That
= o* + 06 + 6*, when a = 2 and 6 =« 1.
2f»
2
— 6
54. Find the value of 2x* when x = 1. When
X = 2. 5. i. 1.5;
Suo. The results may be conveniently arranged as in
the following tabulation:
Find the value of each of the following and
tabulate results:
X
T
2
5
\
1.6
8
50
4.6
95. 2x + 1, when x = 1. When x == 2. 3. 5. \. J. 1.5.
ALGEBRAIC EXPRESSIONS 25
56. a? + 2, when x — 1. When x — 2. 3. J. 1. 5.
57. a:(x + 1), when x = 1. When a; = 2. 3. .2. J. i.
58. In Exs. 410 state which of the expressions used are
monomials. Also which are binomials. Trinomials. State
the same for Exs. 3540.
EXERCISE 5
1. If ^ = Iw, find the value of A when / = 12 and w; = 5j.
Also when / = 10.4 and w = 5.8.
Do you know what use is made of the formula A = Iw in arith
metic in finding areas?
2. If F = Iwh, find V when I = 12, w = 5, and h = 3.
Also when I = 10.4, w = 5.8, and h = 3.05.
Do you know what use is made of the formula V Iwh in arith
metic in finding volumes?
3. If p ='br, find p when b = 350 and r = 1.07. Also
when b = 7.68 and r = .045. Also when 6 = 84,000 and
r = .00.
What does the formula p = 6r mean in arithmetic in connec
tion with the subject of percentage?
4. If i = prt, find i when p = $300, r = .05, and t = 2.
Also when p = $9327.50, r = .06, and < = 3f .
What is the meaning in arithmetic of the formula i = prt?
5. If A = ttJ?^, find the value of A when ir = 3.1416 and
R = 10.
Do you know of any use that is made of the formula A = ^/2*
in arithmetic?
6. If A = 'v a^ + b^^ find the value of h when a = 8 and
6 = 6.
Do you kn ow of any use that is made of the formula
h « Va' + 6* in arithmetic?
26 SCHOOL ALGEBRA
7. If 8 = ^gP, find * when g = 32.16 and ^ = 4. Also
when g = 32.16 and t = 2^.
Can you find out the meaning of the formula 8 ^ i gP?
8. A stone dropped from the top of a precipice reaches its
base in 5 seconds. How high is the precipice?
9. If C = %{¥  32), find C when F = 95°. Also when
F = 100°.
Do you know the meaning of the formula used in this example?
10. If iron melts at a temperature of 2700° F., at what
temperature does it melt on the centigrade scale?
11. If .4 = tt/P  Trr*, TT = 3.1416, R = 13, and r = 12,
find A in the shortest way.
12. If 1 orange costs 3 cents, how many oranges can be
bought for 12 cents? For x cents? For a; + y cents?
13. If 1 orange costs a cents, how many oranges can be
bought for 25 cents? For x cents? For a; + y cents?
14. If 1 acre of land costs x dollars, what will one half an
acre cost?  of an acre?  of aii acre?
(The following problems are variations of Type I.)
15. If a 12yearold boy and a 16yearold boy together
earn $48 in mowing lawns, and the younger boy receives only
half as much as the other, how much does each boy receive?
Let X — no. dollars received by 16yearold boy
Then Jx = no. dollars received by 12yearold boy
Hence re + Jx = $48
or x = $48
Multiplying these equal numbers by 2 (Art. 15, 5)
3x = $96
Dividing equals by 3 (Art. 15, 6)
x = $32, share of older boy
}x = $16, share of younger boy
ALGEBRAIC EXPRESSIONS 27
16. A man left $24,000 to his son and daughter. As his
daughter had cared for him in his old age, he left his son only
f as much as he left his daughter. How much did each
receive?
* 17. A man and boy together made $124.80 by working a
garden one summer. If the boy received ^ as much as the
man, how much did he receive?
18. A farm is worked on shares. As the owner of the farm
supplies the tools and fertilizers, the tenant receives only f
as large a share of the profits as the owner. If the profits for
one year are $4410, hdw much does each receive?
19. Two men manage a store, and as one of them owns
the building, the other receives only f as large a share of the
profits as the owner of the store. If the profits for one year
are $6600, what does each receive?
20. Separate 126 into tWo parts such that one of them is
i as large as the other, f as large.
21. Separate .028 in the same manner as in Ex. 20.
22. A macadam road cost $18,000. The county paid i as
much of the cost as the township, and the state paid J as
much as the township. How much did each pay?
23. A certain kind of concrete contained J as much sand
as gravel and ^ as much cement as sand. How many pounds
of each material were there in If tons of concrete?
24. Make up and work an example similar to Ex. 16. To
Ex.20.
25. How many of the examples in this Exercise can you
work at sight?
CHAPTER n
NEGATIVE NUMBERS
23. Positive and ITegative Quantity. Negative quantity is
quantity exactly opposite in quality or condition to quantity
taken as positive.
If distance east of a certain point is ta^^en as positive, distance
west of that point is called negative.
If north latitude is positive, south latitude is negative.
If temperature above zero is taken as positive, temperatiu*e
below zero is negative.
If in business matters a man's assets are his positive possessions,
his debts' are negative quantity.
Positive and negative quantity are distinguished by the signs +
and — placed before them.
Thus, $50 assets are denoted by + $50, and $30 debts by  $30.
We denote 12** above zero by + 12^, and 10° below zero by  10°.
The use of the signs + and — for this purpose, as well as to indi
cate the operations of addition and subtraction, will be explained
in Art. 26.
24. Algebraic ITiinibers is a general name for both positive
and negative numbers.
The absolute value of a number is the value of the number
considered without regard to its sign.
Thus, if one man travels 5 miles east and another man travels' 5
miles west, the absolute distance traveled by the two men is the
same, viz*. : 5 miles. The two distances traveled, however,, are dif
ferent algebraic numbers, one distance being + 5 miles and the other
distance being — 5 miles.
In general the absolute value of both + 5 and — 5 is 5; and of
both + a and — a is a.
28
NEGATIVE NUMBERS 29
25. The XTtility of Hegative Ifumber lies in the fact that
the use of negative number enables us to use two opposite
or contrasted kinds of quantity in working a given problem.
Also by the use of negative quantity we are often able to
choose an advantageous starting point in solving a problem.
The full meaning of these utilities and other advantages in the
use of negative quantity will appear as we advance in the study of
algebra.
EXERCISE 6
1. What is meant by a temperature of — 8°? By a latitude
of  23°? By the date  776? (Dates after the birth of
Christ are taken as positive.)
2. If the temperature was 17° at noon and — 8° at mid
nighty how many degrees did it fall?
3. If in a given time the temperature should fall from
— 5° to — 12°, how many degrees would it fall?
4. If the temperature were 15° at a given time, what would
it become after a fall of 10°? Of 28°?  15°?
5. If the temperature were — 8° at a given time, what
would it become after a rise of 4°? Of 15°?  8°?
6. Make up and work an example similar to Ex. 3. Also
to Ex. 5.
7. If a traveler is in latitude — 4° and travels north 7°,
what does his latitude become? What does it become if
instead he travels south 7°?
8. If a man's property is — $7000 and he saves $2000 a
year for 8 years, what does his property become?
9. If a vessel, at latitude 3°, sails south 345 miles, what
does her latitude become if 60 miles equal l°t
30 SCHOOL ALGEBRA
10. If a man fought a horse for S150 and sold it for $200i
what was his gain? What would his gain have been if he
had sold it for $125? For $100?
11. What is meant by saving — $10? By a distance — 10
miles north?
12. What is the absolute value of — 4 miles? Of + 4
miles? 5 inches?  3°? $4200?
13. Make up an example for yourself showing the meaning
of absolute value.
(The following problems are variations of Type n, or are
of Type in, viz.: x + ax + b = c.)
14. Walter and his brother together had 90 marbles, and
his brother had 10 less than Walter. How many marbles had
each boy?
Let X s no. of marbles Walter had
Then a; — 10 « no. of marbles his brother had
re + a;  10 = 90
2a;  10 = 90
Adding 10 to each of these equals (Art. 15, 3)
2a; = 100
X = 50, no. of marbles WaUer had
re — 10 = 40, no, of marhUs his brother had
15. A basket ball team has played 27 games and has lost
3 less than it has won. How many games has it won?
16. In a certain election 12,420 votes were cast, and the
defeated candidate had 210 less votes than the winning can
didate. How many votes had each candidate?
17. Make up and work a similar exaniple for yourself.
18. Walter and his brother together have 83 marbles. If
his brother has 7 less than twice the number Walter has,
how many has each boy?
NEGATIVE NUMBERS 31
19. One number exceeds 4 times another nmnber by 5,
and the sum of the numbers is 100. Find the numbers.
•
20. One number exceeds 3 times another number by .12,
and the sum of the numbers is 4.4. Find the numbers.
21. One fraction exceeds 5 times another fraction by i,
and the sum of the fractions is V« T^d the fractions.
22. The distance from New York to Chicago is 912 miles.
If this is 24 miles less than 4 times the distance from New
York to Boston, what is the latter distance?
23. The Eiflfel Tower is 984 ft. high. If this is 126 ft. less
than twice the height of the Washington Moniunent, what
is the height of the Washington Monument?
24. How many of the examples in this Exercise can you
work at sight?
25. Which of Exs. 1423 are of type x + x — a =^ b, and
which are of type x + ax =^ b = c^
26. Make up and work an example similar to Ex. 19. To
Ex. 23.
26. Doable Use of + and — Signs. The signs + and —
are employed for two purposes (see Arts. 7 and 23) : first,
to indicate the operations of addition and subtraction; and
second, to express positive and negative quantity. We are
able to make this double use of these signs because, in each
use, the signs are governed by the same laws.
—7 6 5 4 8 2 1 . +1 +2 +8 +4 +5 +6 +7 +8
w I I I I I I I I I I I I I I js
^ K B O A F ^
A person walks from toward E a distance of 5 miles (to F) and
then walks back toward W a distance of .3 miles (to A). If the dis
32 SCHOOL ALGEBRA
tance to the right of is regarded as positive, and therefore the dis
tance to the left of is negative, the distance from the starting
point to the destination may be expressed as the sum of a positive
quantity and a negative quantity; that is,
(positive distance OF) + (negative distance FA),
or, +5 + (3)  5 3 = 2.
The position arrived at may be determined in another way — viz.
by deducting 3 miles from 5 miles. We obtain
5 (43) =5 3 =2.
From this example we see that adding negative quantity is
the same in effect as subtracting positive quantity.
Therefore, in the expression 5 — Z, the minus sign may
be considered either a sign of the quality of 3, or as a sign of
operation to be performed on 3. Hence, we are able to use
the signs + and — to cover two meanings.
27. Laws for the Use of + and — Signs. Whichever of
the two meanings of + and — named in Art. 26 is assigned,
we see that + (— 3) = — 3; also, — (+ 3) = — 3.
The signs + and — applied in succession to a quantity are
equivalent to the single sign — •
Or in symbols,
+ (— a) = — a; and _ (+ a) = — a.
Ex. Find the value of 8 + 4 — 11 + 3 — 6. On squared
paper show the meaning of the numbers involved.
81411 +36 = 15 17 2 Ana.
Taking the distances to the right of OP as positive, we have the
diagram on p. 33 showing the meaning of the numbers involved.
Note that the above process holds true whether a number pre
ceded by a minus sign is regarded ad the subtraction of a positive
number or the addition of a negative number.
If in the illustration on p. 31 a person walks in the nega
tive direction from (i. e. toward W) a distance of 4 miles
NEGATIVE NUMBERS
33
to Kf and then reverses his direction and goes 2 miles, he
will be at B, Or stated in another way, diminishing the
+
8
+
4
U
+3
2
*
^^^"
—
i
distance traveled west by 2 miles, brings him to the same
place as walking the full direction west and then walking 2
miles east.
It may be well to study another illustration of this principle.
If a man owes two notes of $500 and $100 respectively, removing
the note for $100 is the same in effect as annexing $100 in money to
the debts as they are. That is,
 $500  $100  ( $100)   $500  $100 + $100 $500
Hence:
The sign — applied twice to a given positive quantUy gives
a + resvU.
Or in symbols, — (— a) = + a.
These laws enable us to iLse
negative quantity with as great
freedom as we use positive quan
tity, and hence are an important
source of power, as "will become
more evident later.
Ex. On squared paper show
the meaning of — 5 — (— 3). Also of —5 + 3. Hence,
show that  5  ( 3) =  5 + 3.
 A
6
Yl
Tl
riiih
D
J
J
13
^
34 SCHOOL ALGEBRA
On the lower diagram on p. 33 — 5 — ( — 3) means OA — DA,
or OD. Also 5+3 means OA + BC, or OD.
Hence, — 5 — ( — 3) and —513 ^ve the same result; or we
may say  5  ( 3)  513.
28. The Algebraic Sum of two or more algebraic numbers
is the result of combining the given algebraic numbers into
a single number.
Thus, the algebraic sum of 4 and — 7 is — 3.
Find the value of each of the following and verify the
result on squared paper:
1. 5  2. 6.  4. u. 5  ( 8).
2. 6  8. . 7. 8  6  4. 12.  7 + ( 2).
3. 5  5. 8. 7  5 + 4. 13.  (5).
4.4 + 2. 9. 3 + 1  5. 14. + ( 5).
5.42. 10.  4  ( 3). 15.  4  ( 1.5).
16. 4 + 5  12 + 3  6. ,
17. 3 + 862 + 21.
18. At 6 A. M. a thermometer read 57°. It then made
successive changes as follows: + 7°, — 2°, + 5°, — 3°, — 2°.
What was the final reading?
19. In a certain football game, taking a distance toward
the north goal as positive, during the first seven plays the ball
started at the middle of the field and shifted its position
in yards as follows: +5010155 + 10520.
Find the final position of the ball with reference to the middle
of the field. On squared paper show {he changes in the posi«
tion of the ball, letting 5 yd. equal one space on the paperr
. NEGATIVE NUMBERS 36
20. State in the language of debts and credits the mean
ing of
 $700  $200 1 (  $200) =  $700  $200 + $200
SuG. If a man has debts of $700 and $200, the removal of the
$200 debt is the same as leavmg his debts unchanged and adding
$200 to his possessions. He becomes worth — $700 in either case.
21. State in the language of distance traveled east and
west the meaning of
— 10 mi. — 2 mi. — ( — 2 mi.) = — 10 mi. — 2 mi. + 2 mi.
(The following are miscellaneous problems of Type II and
Type m.)
22. A man and a boy together catch 320 fish, and the man
receives three times as many fish as the boy. How many fish
does each have?
23. A man has $3220 in two banks and the amount in one
bank exceeds that in the other by $540. How much has he
in each bank?
24. Two girls make $24.60 by sewing, and the younger
girl receives only one half as much as the older. How much
does each receive?
25. Separate $12.68 into two parts one of which shall be
smaller than the other by $5.
26. A given piece of bronze weighs 4600 lb. It contains
twice as much tin as zinc, and 8^ times as much copper as
zinc. How many pounds of each metal does the bronze
contain?
27. The distance from the mouth of the Mississippi River
to the source of the Missouri River is 4500 miles. The dis
tance between the mouth of the Mississippi and the mouth of
36 SCHOOL ALGEBRA
the Missouri is 1700 miles less than the length of the Missouri.
What is the length of the Missouri?
28. A farmer obtained 2720 pounds of cream in one month
by the use of a separator. This is ^ more than he would
have obtained if his milk had been skimmed by hand. How
much would he have obtained by the latter process?
29. The cost of a macadam road was $24,000. Thecoimty
paid twice as much as the state, and the township three times
as much as the state. How much did each pay?
30. Three partners divided $14,000, the second partner
receiving $2000 more than the first, and the third partner re
ceiving twice as much as the first. How much did each
receive?
31. Mt. Washington is 6290 ft. high. This is 170 ft. more
than 10 times the height of the Singer Building (N. Y.).
How high is the latter?
32. Make up and work an example similar to Ex. 18.
Ex.21. Ex.24. Ex.29.
33. How many of the examples in this Exercise can you
work at sight?
34. Which of Exs. 2233 of this Exercise are of Type I?
OfTVpeH? TypeHI?
29. Oraphs. A set of numerical facts may often be com
bined as a geometrical picture called a graph. The meaning
and use of negative numbers are often well illustrated on a
graph.
Ex. On a given day the following were the temperatures
at a given place:
NEGATIVB NUMBERS
37
9 A.M. 2°
Noon 10"
3 P.M. 15°
Temperatures
afr
6 p. M. 10°
9 P. M. OP
Midnight  10°
HSoun
Midnight  15°
3 A.M. 20°
6 A. M.  10°
Graph these facts.
We draw a horizon
tal line and on it mark
off spaces to represent
hours, • as in the dia
gram. Perpendicular to
this we draw a line and
on it mark off spaces to
represent temperatures.
Above or below each
point which represents
an hour, a point is lo
cated which represents
the temperature at that
hour. Through the points thus located a continuous line ABCD
is drawn. This is the required graph.
EXERCISE 8
Graph each of the following sets of temperatures:
Mid
night
3 A.M.
6 A.M.
9 A.M.
12 M.
3 P.M.
6 P.M.
9 P.M.
1.
20**
30**
20**
10**
0**
10**
10**
0**
2.
10**
20**
10**
0**
10**
20**
10**
0**
3.
10**
15**
 5**
10**
15**
25**
15**
5**
4.
0**
10**
 5**
15°
25**
30**
15**
5**
5. Make up and work a similar example for yourself.
Graph each of the following sets of temperatures:
6.
7.
8.
Jan. 1
Feb. 1
Mar. 1
Apr. 1.
May 1
June 1
New York
New York
London
31**
1**C.
37**
31**
1**C.
38**
35**
1**C.
40**
42**
6**C.
45**
54**
12** C.
50**
64**
. 18** C.
57**
38
SCHOOL ALGEBRA
July 1
Aug. 1
Sept. 1
Oct. 1
Nov.l
Dec. 1
71**
22** C.
62**
73**
23** C.
62**
69**
21** C.
59**
61**
16** C.
54**
49**
9**C.
46**
39**
4**C.
41**
9. Convert the temperatures given for London in Ex. 8
to temperatures on the Centigrade scale and graph them
(see Ex. 9, p. 26).
10. Collect and graph sets of numerical facts similar to
those given in the preceding examples.
CHAPTER III
I
ADDITION AND SUBTRACTION; THE EQUATION
Addition
30. The Utility of Addition in Algebra.
Ex. Find the value of Zah^ + 5ab^ + 2a6^ when a = 2
and 6 = 3.
PROCESS WITHOUT ALGEBRAIC ADDITION
If we substitute directly in the given expression, we obtain
3o62+6a52+2a6«3x2x32+5x2x32+2x2x3«
= 54+90+36
= 180 Am.
PROCESS AIDED BY ALGEBRAIC ADDITION
3a6» + 5a6« + 2a62 = lOab^
= 10 X 2 X 3»
= 180 Ans.
In solving the above example, algebraic addition enables
US to save more than half the work. Algebraic addition has
other uses which will appear later.
Why do we now make definitions and rules?
31. Addition, in algebra, is the combination of several
algebraic expressions into a single equivalent expression.
Addition is sometimes described as collecting terms in an expression.
32. Similar Terms (or like terms) are terms which contain
the same literal factors and the same radical signs over the
same factors.
• * < . . 39
40 SCHOOL ALGEBRA
Thus, 7dJt^ and — 5a6' are siinilar terms. Also 5aV^and — Ga^^
are similar tenns.
Dissimilar terms (or unlike terms) are terms which are
unlike either in their literal factors or in the radical sign
over the same factor.
Thus, Mb and baJt^ are dissimilar terms. Also 3^5 and 3^5
are dissimilar terms.
The addition of dissimilar terms can only be indicated.
Thus, 6 added to a gives a f h; also a*  3o*6 added to 3a'  6»
gives a»  30*6 + 3a«  6».
33. Hethod for Addition. The most convenient general
method for addition is shown in the following examples:
Ex. 1. kM^ + Zx + 2,Z:i?^Z,27?x 5.
Arranging similar terms in the same column, and adding each
column separately, we obtain
CHECK
4x«+3a;+2= 4+3+2= 9
3x«4x3= 343 4
2a;' xb = 2l5=8
Sum 5x'2a;6= 526=3
To check the accuracy of the work, we let x = any convenient
number, as 1; find the numerical value of each row; and compare
the sum of these results with the numerical value of the algebraic
expression obtained as the sum.
Ex.2. Add2a^5a^b + 4ah^ + aW,4a^b + 2a^ab^'SaP,
a^ba^ + 2ah^.
Proceeding as in Ex. 1,
CHECK
2a»  6a'6 + 4a6' + a«6» =25+4 + 1 2
2a» + 4a'6  306*  a6* = 2 + 4  3  1  2
 o« + o'fe + 2a6' = 1+1+2 »2
Sum 3o» + 306* + a'&»  o6* = 3 + + 3 + 1  1 = 6
In the second column the algebraic sum of the coefficients is
— 5+4 + 1, which = 0; and as zero times a number is zero, the
ADDITION 41
sum of the second column is zero, which need not be set down in
the result.
The work is checked by letting a and b each  1.
Hence, the process for addition may be stated as follows:
Arratige the terms to be added in columns, placing similar
terms in the same column;
Find the algebraic sum of the numerical coefficients of each
column and prefix this result to the literal factors common to the
terms in the column.
Sometimes the algebraic sum of the coefficients of each group of
similar terms is found without arranging the terms in columns.
9
Add and check each result:
1.
2.
3.
4i
s.
 11
4
8a;
— X
7x
6
10
 6a;
3x
12a;
6.
7.
8.
9.
10.
2a
a?
7xy
a«6
7«V
5a
3a:*
lOxy
5a%
lOaV
 12o
53?
2xy
Sa'b
«y
IX. Sax, — 2ax, 5ax, ax, — Sax.
12. 5a?, 12a?,  10a?, x^,  16a?, Za?,  a?.
13. 70*6*,  12a^\?,  a^\?,  4o?62, 5a^l?, ea^ft*.
14.
3LS
16.
17.
3(o + 6)
 6(a;  y)
5Vo + a:
47rr«
5(o + 6)
4(«  y)
 eVa + X
27rr*
 4(a + V)
 5{x.  y)
2Vo + a;
i^r,^
3a;
2y
 2x + 3y
X —
■ y
i2 SCHOOL ALGEBRA
18. 19. 20.
5a? + 7 a* aa: + 4a?
a?  IQ 3a2 + 2cKC  5a?
— 7a?+l — (3? — ax — of'
21. a — 26, 3a + 46, a + 56, — 5a — 6, a — 56.
22. 3a? + 2/2 23j2 _ 72^^ _ 4aJ8 _ 5^^^ ^^^^^^ 3^^.
23. 3aa? — 56^, 2aa? + 462/*, 263/^ — 4aa?, h]^ — aa?.
Reduce each of the following to its simplest form:
24. a?a:2/ + 32/2 + 2a? + 2a:2/22/2 + a? + y^ + 3a?a:y.
*" 25. V^n —371^+ tr^'\tr?'\ 2r? — 3mn ^it^ — r^'\ mn— 2m^
26. a? + jr^  22? + 3a?  2r^ + 22? + z2  2a? + a?  2?,
27. 2a?  a:^/ + 3ar2/ .52r^ + 32/2  3a? + a? + 23/2 _ 2a:2/.
28. 7a; + y + 52 — 10a;2/ + 22/ — 32 + \Zxg — 4a» + 52
— 6x — 4as2 +2a:2/ — 32/ + 92 + 7x — ass + 21a;2 — I62 + ar — 5a:2/.
29. a? + 3a?2/ + 3an/2 ^ j^ _j_ ^^ _ 3a?2/ +3a:2/^ — j/' +2a?y
— 2a;y2 j ^  a^ _ 2^ + a?y ^ 4a? — xy^ — y^ + ^ + a? —
a?2/ + a;2/2.
Collect similar terms in the following and check each result:
30. 2a; — 3y — 5a: + 42 + 4y + 2 — 22/ — a: — 32 + 2a;— 32/.
31. 3a;y — 5aa; + 3^ — 2a;y — 3a? + 4aa; — 2]^ + 3aa; — 2xg.
32. a; — 3y + 22 + 2y — 2a; — 2 — 3a; — 42— 2a; + 2 + 2a;.
33. 2a;l + 52/2 + 3a; + 2 + 32/32a; + la;32/.
34. 3a262a2c + 3a25a26a23a2c + a26 + 6a2c2a2.
35. 5a?  3a; + 4  2a?  6a? + 4a;  7  a? + a? + 3a?
— a; + 5 + 3a?  6a;  a? + 4a;  2a? + 2a;.
36. 2a;« '5a;« + 3a?  a;»  7a; + 3a?  3 + 2a;~  5a?
+ 5 + 3a;"*.
ADDITION 43
37. Redupe Zxxxyy + Sxxon^y — 5xxxyy — 2xxxyy to its
simplest form. About how much briefer is the form you
obtain than the given form?
38. Make up and work an example similar to Ex. 37.
39. Make up and work an example showing the use of
algebraic addition (see Ex. of Art. 30, p. 39).
40. State in general language the use of algebraic addition.
(The following are mixed problems of Types I, n, m.)
41. Three partners in a retail business made $18,000 in
one year. The second partner owned the building and re
ceived twice as large a share as the first partner. The third
partner supplied most of the capital and received three times
as large a share as the first. How much did each receive?
42. Make up and work a similar example for yourself.
"^ 43. Find three consecutive numbers whose sum is 36.
44. Find four consecutive numbers whose sum is 106.
45. Make up and work an example concerning five con
secutive numbers.
■
46. The area of the United States and its outlying posses
sions is 3,742,155 sq. mi. The area of the United States
exceeds that of its outlying possessions by 2,309,045 sq. mi.
What is the area of the outlying possessions?
47. How many of the examples in this Exercise can you
work at sight?
48. Name the type to which each of the above problems
belongs (Exs. 4146).
44 SCHOOL ALGEBRA
Subtraction
34. The Xrtility of Sabtraction in Algebra.
Ex. Find the numerical value of VJaVf — Ihd^V^ when
a = 3 and 6 = 2.
PROCESS WITHOUT ALGEBRAIC SUBTRACTION
17a«6»  15a25» = 17 X 3* X 2»  15 X 3« X 2»
= 17X9 X8 15X9X8
= 1224  1080 = 144 Am.
PROCESS AIDED BT ALGEBRAIC SUBTRACTION
17o«6»  15o26» = 2a2fe»
= 2 X 3« X 2»
a 144 Ans,
In solving the above example, algebraic subtraction
enables us to save more than half of the work. Algebraic
subtraction has other advantages which will appear later.
Why do we now proceed to make definitions and rules?
35. Subtraction, in algebra, is the process of finding a
quantity which, added to a given quantity (the subtrahend),
will produce another given quantity (the minuend).
Thus, if we subtract Zab from 10a6, we obtain 7a6, for lab
added to Zab (subtrahend) gives 10a& (minuend).
36. Signs in Subtraction. From Art. 26 it follows that
Subtracting a positive quaviHy is the same as adding a nega
Hve quantity of the same absolute magnitude; and
Subtracting a negative quantity is the same as adding a
positive quantity of the same absolute magnitude.
37. Hethod for Subtraction. The most convenient general
method in subtraction is to
SUBTRACTION 45
Write the^terms of the subtrahend under the terms of the
minuendy placing similar terms in the same column;
Change the signs of the terms in the subtrahend mentally, and
proceed 09 in addition.
Ex. 1. From 5a:82ar^ + a:  3 subtract 2a:8  3ar*  a: + 2.
Check the work by letting x = 1.
CHECK
5x»2x2 4 «3 =52 + 13 = 1
2a:»3a:' a;+2 = 231+2=0
Difference 3x»+ a:«+2x5 =3 + 1+25 = 1
The coefficient of a:* is 5 — 2, or 3, of x* is —2+3, or 1, etc.
Ex. 2. Subtract 2a^  Sa^b  6a^h^  2abP + 26* from
a* + 5a^b  6a^l^  Sat^. Check the work by letting a = 1
and 6 = 1.
CHECK
0* + 5a'6  6a«6«  3a6» = 1+563 =3
2a4  3a»6  6a»y  2a6» + 26^ = 2  3  6  2 + 2 = 7
a*+8a«6 a6» 26* =1+8+012= 4
The coefficient of a'6* is — 6 + 6, or 0. The coefficient of 6* is
 2, or  2.
EXERCISE 10
Subtract and check each result:
1. 2. 3. 4. 5.
7ab 5x X 5a: — 3a?
3a6 9x 2x  Sx  At?
6.
Ixy
Zxy
7.
8.
7(a: + y)
 3(a; + y)
9.
10.
5(a + b)
3(fl + b) ■ 
 2V0 + X 
 sVa + X
 4V6  y
2V6y
u.
3r'4a:
2a? + «
12.
3a; 9
5a: + 1
13.
■ a:» + 2
14.
Sar^ + 4«  3
ar'  3a; + 5
46 SCHOOL ALGEBRA
*
15. From 3a + 26  3c  d take 2a  2b + c  2d.
16. From 7  3x + 2^2 take 15  4a;  5a?.
17. From a:^  y2  2^ + 8 tAke 2x^ + f  2s? + 10.
18. From 5xy — 3xz + 5yz + a? take 4xz — 2xy — a?.
19. From 2  a; + a? + a;* take 3 + xa?af2aJ*.
^ 20. Subtract lOxhf + SxY  13a;y2 from s?yxy^ + 23?y\
21. Subtract 3 — 2o6 + 3ac — 4cd from 5 — oc + 8cd ~ 5ad.
22. Subtract 1 + a; — a^ + x' — a?*from2 — a; — a? — a:' + a:'^.
23. Subtract a + 26 — 3c + 4d from m + 2b + d — z +a.
24. Subtract 3a;*  2a:* + 5a;  7 from 3a;» + 2a?  a;  7.
25. SubtractaJ^2a;* + a;2_.5fi^ina^a^ + a^«2a; + 5.
26. Subtract 3a;"» — 3a;* + a; — 3 from a;"» + a;*» — a;* + « — 1.
27. From the sum of 2a; and 3y subtract their difference.
28. From subtract — 3a;. From subtract a; — y.
From zero subtract 3a* — 2a6 + 6^.
29. Reduce 7aaabb + 5aaabb — 3aaa66 to its simplest
form.
30. Make up and work an example similar to Ex. 29.
If i4 = aJ»  3a? + 1 , 5 = 2a?  5a; " 3, C = 3a;» + a;* + 3a;,
find the value of
31. A + B + C 33. A + BC
32. BA + C 34. AB + C
(The following problems are variations of Types I and n.)
35. Find the value of a;, if 3a; — 2 in. = 7 in.
36. Separate $24.80 into two parts such that one part is
smaller than the other by $4.60.
USE OP THE PARENTHESIS 47
37. Separate $24.80 into two parts such that the smaller
part equals J of the larger part.
38. Separate $5000 into three parts such that the second
part shall exceed the first by $300, and the third shall exceed
the first by $800.
39. Separate $5000 into three parts such that the secouJ
part shall exceed the first by $300, and the third shall exceed
the second by $800.
4a Separate $6000 into three parts such that the second
part equals ^ of the first, and the third part equals j of the
first.
41. Separate $6000 into three parts such that the second
part is double the, first, and the third part is double the second.
42. Make up and work an example similar to Ex. 36.
To Ex. 40.
43. Name the type of which each of the above problems
(Exs. 3643) is a variation.
44. How many of the examples in this Exercise can you
work at sight?
45. How many of the examples in Exercise 1 can you now
work at sight?
Use op the Parenthesis
38. Tltility of the Parenthesis. The parenthesis is useful
in indicating an addition or a subtraction in a brief way.
Thus, 2a + 36  5c  (3a  26 + 3c) indicates that 3a  26 + 3c
is to be subtracted from 2a + 36 — 5c.
The parenthesis will also be found useful in indicating
multiplication and division in a brief manner, and other uses
of the parenthesis will become evident as we proceed.
48 SCHOOL ALGEBRA
39. Bemoval of a Parenthesis. From the processes of addi
tion and subtraction it follows that
When a parenthesis, preceded by a + sign is removed, the
signs of the terms inclosed by the parenthesis remain unchanged.
But
When a parenthesis preceded by a minus sign is remained, the
signs of the terms inclosed by the parenthesis are changed, the
+ signs to —, and the — signs to +.
Ex. Simplify 2a + 36  5c  (3a  26 + 3c).
2aH365c(3a26f3c) =2a+365c3af263c
 — a + 56 — 8c Ana,
Let the pupil check the work by letting a « 1, 6 « 1, c » 1.
40. Parenthesis within Parenthesis. Using the parenthesis
as a general name for all the signs of aggregation, it is evident
that several parentheses may occur one within another in
the same algebraic expression. The best general method of
removing several parentheses occurring thus is as follows:
Remove the parentheses one at a time, beginning with the
innermost;
Collect the terms of the result.
It is also possible to remove the parentheses in reverse order,
that is, by removing the outside parenthesis first, etc. Working an
example in this way often forms a convenient check on the first
process.
Ex. Simplify bx — y — [4a: — 6y + {  3a; 4 y + 2« —
(2a: 2)}].
5x j/[4a:% + {3x+/+22(2a: z)}]
= 5a;  y  [4a:  6y + {  3a: I y + 2«  2a: + «}]
= 5x— y— [4a:— 6y — Zx \ y + 2z — 2x + z]
= 5a:— y— 4a:f62/ +3x— y— 2z+2x— z
 6a; + 4y — Sz Ana.
USE OF THE PARENTHESIS 49
. The work may be checked by removing the parentheses in reverse
order, or by the method of substitution as follows;
Letting a; = 1, y = 2, 2 » 3, we have
= 52 [4 12 {3+2+6 (2 3)}]
= 3[8 + {5(^ 1)}] , 3  [  8 + {5 + 1}] ^•
= 3(8+6)3(2)3+2=5
Also 6iC+4y32»6+89=5
EXERCISE 11
Remove parentheses and collect similar terms. Check
each result either by substitution of numerical values, or by
reversing the order in which the parentheses are removed.
1. 3a + (2a  b). 7. x  [2x + (x  1)].
2. 2x (x 1). 8. 6a; + (1  [2  4a;]).
3. x+(l 2x). 9. 2  11  (3  a)  a}.
4. 3x  (1 + 3a;). 10. 2a;  [ a;  (a;  1)].
5. X (X 1). 11. 2y + { a;  (2y  x)}.
6. x + 2y — (2x — y). 12. a — {— a — {— a — 1)}.
13. [a;* — (ix?y — 2?) — 2?] + (xh/ — a?).
14. 1  {1  [1 + (1  a;)  1]  1} a;.
15. a; [{( a;  1)  x}  1]  1.
16. 1  12 + [ 3  ( 4  5  6) ~ 7]\.
17. a " \d + [b — (a+ b + c — a + b + d) — c]}.
18. x{2x'+ {Ss?  3a;  [a; + a?]) + [2a;  (x" + a?)]}.
19. a;*  [4a;'  [3a^  (2a; + 2)] + 3a;]  [a^ + (3a? +2a?
 3a;  1)].
20. ''[''2X { ( 2a;  1)  2a;  1]  2a;.
21. X [a; + {xy) {x+ (y x) 2y} + y]y +x.
60 SCHOOL ALGEBRA
•f»
,22. 25a;  [12 + JSx  7  ^ 12a;  6 + 16a;)  (3 +
2a:)}] + 7  (3a; + 6) + (2a;  3) + a; + 8.
23. In 3a; — (6a — 26 + c), what i^ the sign of 6a as the
example stands?
24. In Ex. 12, Exercise 10, indicate the subtraction by
use of a parenthesis. Do the same in Ex. 13.
Remove the parentheses and find the value of a; in each
of the following:
25. a; + (a; + 2) = 7. 27. 6a;  (2a;  3) = 12.
26. 3a;  (a; + 2) = 8. 28. 4x  (a;  f ) = 2i.
29. Make up and work an example similar to Ex. 16. To
Ex. 26.
30. How many of the examples in this Exercise can you
work at sight?
31. How many of the examples in Exercise 3 (p. 19) can
you work at sight?
41. Insertion of a Parenthesis. It is clear that the process
of removing a parenthesis may be reversed; that is, that
terms may be inclosed in a parenthesis.
Inverting the statements of Art. 39, we have
Terms may he inclosed in a parenthesis preceded by the plus
sign, provided the signs of the terms remain unchanged;
^ Terms may he inclosed in a parenthesis preceded hy the minus
sign, prodded the signs of the terms are changed,
Ex. a5+c+de=a6 + (c+de),
or, =a— 6 — (— c— dle) Ans.
USE OF THE PARENTHESIS 51
EXERCISE 12
In each of the following msert a parenthesis inclosing the
last three terms^ each parenthesis to be preceded by a minus
sign. Check the work either by removing the parenthesis in
the answer, or by numerical substitution.
1. a? ds^ + Sxl. 5. a:* + 4a:  ar^  4.
2. a — b + c + d. 6. a^I^ — 2cd — (? — JP.
3. 1 + 2a  a^  1. 7. 4a;*  9ar^ + I2xy  ^f.
4. la22o662. a aJ*4a:» + 4a:2 + 4a:4 a?.
It is often useful to collect the coeflScients of a letter into
a single coefficient.
Ex. Collect the coefficients of x, y, and z in the expression,
Sx " 4y + 5z — ax ^ by ^ cz — bx + ay + az.
The complete coefficient of x is (3 —a — 6) ; of y, ( — 4 — 6 + a)
or — (4 + 6 — a); of 2, (5 — c + a).
Hence, the expression may be written,
(3  o  6)x  (4 + 6  a)y + (5  c f a)z Arts,
In like manner collect the coefficients of x, y, and z:
9. mx — ny + 3z + 2x + nz — 4y.
10. X'^y''2z — ax + by — az — bx — ay + cz.
11. 7x + 12y  lOz '2ax + 3bz  cy + 2bx  Qdy .
12. 5y — Sacx — 5cdz — 4a6a: — Scdy + 2ca: — 42 — 5ax.
Collect the coefficients of a?, a?, and x:
13. 3a^ + a — 2a:^ — aa? — 5 + ax^ — 2ax — coi? — cqp^ —ex.
14. — x^ — x — aa^ + a^ — ax + bx^ — aa^ — Sbx — 26a:^+ 3a.
15. aV • ax — a — Vo? — 26^a:^ + 36a: — aV — ca? +
3cx — c.
52 SCHOOL ALGEBRA
Equations and Transposition
42. An BqnatioiL is a statement of the equality of two
algebraic expressions.
An equation, therefore, consists of the sign of equality and
an algebraic expression on each side of it; as 3a; — 1 = 2a: + 5.
The solution of an equation is the process of finding the
value of the unknown number (as of x) in the given equation.
43. Hembem of an Equation. The algebraic expression
to the left of the sign of equality is called the first member of
the equation; the expression to the right of the sign of equality
is called the second member.
Thus, in the equation Sx — 1 >« 2x +3,
the first member is 3a; — 1; the second member is 2x + 3.
The members of an equation are sometimes called sides of
the equation.
The members of an equation are similar to the pans of a set of
weighing scales which must be kept balanced. (See Art. 15, p. 18.)
44. Utility of Equations. An equation expresses the re
lation of at least one unknown quantity to certain known
quantities. By means of an equation, we are often able to
determine the value of the unknown quantity.
See the problems solved in Exercises 1, 2, etc., by the aid of
equations.
45. The Transposition of a Term is moving the term from
one member of an equation to the other member. We shall
see that when a term is transposed, the sign of the term must
be changed.
Ex. 1. Find the value of x in x—5 = ?•
EQUATIONS 53
FBOCESS WITHOUT TRANSPOSmON
We have given a;5=7
5=5
Adding 5 to each of the equals, a; = 7 + 5 ^^^' ^^' ^^
or X » 12 Arts.
PROCESS WITH TRANSPOSITION
We have given a?— 5 = 7
Transferring 5 to the righthand] x = 7 f 5
member of the equation aijid^ a? = 12 Ana.
changing its sign, , • J
Hence transposition is a short way of adding equal num
bers to the two members of an equation. The labor saved
by means of transposition is more evident when several terms
are to be transposed at the same time.
For the present, however, in order to fix firmly in mind the na
ture of the process, we shall not transpose terms, but shall add
equals to the members of an equation when we wish to transfer
terms from one member to the other.
Ex. 2. Solve 5a: (a; + 2) = 3a:  (2a;  7).
Removing parentheses, 5x — a: — 2 « 3a; —2a; + 7
Adding  3x + ^ + 2 to I 3X+2X+2 3X+2X+2
each member, ) _^
5x  a;  3a; h 2a; = 2 + 7
3a; = 9
X = 3 Arts.
46. Checking the Solution of an Equation. The result
obtained by solving an equation may be checked by substi
tuting in each member of the original equation the value of
X obtained by the solution. If the two members reduce to
the same number, the value found for x is correct.
Thus, in Ex. 2, putting 3 in the place of a;.
The left member, 5a;  (a; + 2) = 15  (3 + 2) = 15  5 = 10
Also the right member, 3a;  (2a;  7)  9  (6  7) 9 + 110
54 SCHOOL ALGEBRA
4 4 A:
LCISE 18
Solve the following equations without transposition of
terms. Verify each result obtained.
1. x + 2a: ^ 3 = 6. 6. 3a:  2 = 2a: + .74.
2. 3a: = a: + 10. 7. 5x  (2a:  3) = 6.
3. 5a:  1 = 14. a 7a:  (5a: + 4) = 2.
4. 4a:  3 = 12  a:. 9. 9a: = 10  (x + 5).
5. 5a:  1 = 3a: + 7. 10. 8x + (3x  4) = 25.
U. 10a:  (a:  5) = 4  (a: + 2).
12. 10  (3a:  5) = 8  (7a: + 2).
13. Solve Exs. 112 by aid of transposition of terms.
Solve the following problems and check each result:
14. If 5 times x equals 9 diminished by twice x, find x.
15. If f of a; equals 12 less ix, find x.
16. If 12 is added to a given niunber, the result equals
three times the given number. Find the number.
17. One number exceeds another by 5 and the sum of the
numbers is 12. Find the niunbers.
18. The difference of two niunbers is 5' and the siun of the
numbers is 13. Find the numbers.
19. Separate 12 into two parts such that one part exceeds
the other by 5.
20. One number exceeds another by 1.4 and the sum of
the numbers is 16.4. Find the niunbers.
21. The difference of two numbers is 1.4 and the sum of
the numbers is 16.4. Find the numbers.
EQUATIONS 56
22. Separate 16.4 into two parts such that one part ex
ceeds the other by 1.4.
23. Make up and work three examples similar to Exs.
1416. Also to Exs. 1720.
24. Find three consecutive odd numbers whose sum is 45.
Also five consecutive odd numbers whose sum is 45.
25. Find three consecutive even numbers whose sum is 60.
Also five consecutive even numbers whose sum is 60.
26. Make up and work an example similar to Ex. 24.
27. Make up and work an example similar to Ex. 25.
28. To what type does each of the above problems belong,
or of what type is each a variation?
EXERCISE 14
Review
1. Find the value of a + 3(6 — x), when a » 5, 6 » 2, and
«= 1.
2. Fmd the value of Sx (z — 2y + 2{x + 1) (4 x) 
V5x + 1, when x = 3.
3. Ji 8 = vt + igfi, fipdthe value of s when t; = 10, ^ = 32.16,
and < = 4.
4. If x = 3, find the value of 4x\ Also of (4x)\
Simplify:
5. 2x*  5x«  3a^ + 2a;  5 + 2x»  3a:*  2x + 2x2  2a; + 2x*
 6 + 3a;2 + x*  3a:« + 7 a; + 2 + 3x« + 2a;*  4x  2a;«.
6. 3^2" 5VT+ 8 + 5V3" 2V2" 7 + 3V3" 4V2" 2.
Subtract:
7. 3a;»  2a;* + 5a;  3 from 8a;»  a;«  1.
8. Sx*  3x^ + y» from 3a;' + 7xy*  y».
56 SCHOOL ALGEBRA
Simplify and collect:
9. ac  {  2a; + [ 4a;  (a?  2)  x]  a?}  1.
10. 9x{8a;[7a; + (6x + l) 5x]4a;} (3a; + l)2x.
Bracket coefficients of Hke powers of a;:
11. a;*a;'+23x*aa:»+aa;*cx*2aa;2+3cx'2ca;*5a;*.
12. lxa^3i^+2a2ax+ 2ax^  2aa;»  36a? + 36a;«
+ 36a;» + ex.
Solve and give the reason for each step:
13. 3a;  5 = a; + 7. 15. 4a; + (a;  1) = 3a;  (a; +2).
14. 5a;  (x  4) = 16. 16. 3  (a;  2) = 7  5a;.
17. Subtract 5a;*  3aa;  2a' from  3a;» + 2aa;*  o*.
18. Find the value of 5x'  3(a — 2a;) + 5a*, when a = 4 and
X = 1.
19. Add 5a;*  3aa; + 4a*, 5aa;  3a;* + a*, and 3aa;  a?*  2aa;.
20. Simplify a;*  [5aa; + (o*  2a;*  aa;)  3a;*]  5a*. Test the
accuracy of your work by letting a = 1 and a; = 2.
21. Solve 5  a; = 4  (7 4 3a;).
22. The land surface of the world is 51,240,000 square miles^ If
the land area of the rest of the world is seven times that of North
America, find the area of North America.
23. Add Ja;*  la;+ J, ia;* + ix  i, and fx* Ja; + f.
24. Subtract Ja;*  J a;+ { from J a;* ix  f
25. Add .5a* .15a + 2.5, 1.2a* + .3a 1.5, and .75a* f .3a .7.
26. Subtract .27a*  .12a  2.3 from 1.5a* + 2a  1.7.
27. Add 2(a; + y)  3(x+ z) + 2(y + «), 4(a; + 2)  3(x + y)
 5(2/ + 2), and 4(a; + 2/) («+«)+ 4 (2/ + «).
28. From the sum of a*  7a6 + 36* and 2a*  .66* + 7a*6*, take
the sum of 4a*6*  3a» + 2a*  b* and 3a6  26* + a*.
29. What must be added to a;* — a; + 1 that the sum may l)e x*?
That the sum may be 3a;? 15? 0?
30. What must be subtracted from 2a;* — 3a; + 1 that the re
mainder may be a;»? x* + 10? 7? a  a; + 1?
EQUATIONS 57
find the value of
31. AB+CD 33. A "{B +C) +D
32. A  [B  (Z> + O] 34. B + {A  [C  D]]
35. By a diagram show that — 7 — ( — 3) and —7+3 have
the same value.
36. In an election for two candidates, 32,544 votes were cast.
The successful candidate had a majority of 2416 votes. How many
votes did each candidate receive?
37. The Panama Canal is 49 miles long and the part of it through
the lowlands is 4 miles more than 8 times the part through the hills
(called the Culebra Cut). How long is each part?
38. How many examples in Exercise 2 (p. 13) can you now work
at sight?
CHAPTER IV
*
MULTIPLICATION
47. Mnltiplication, at the outset, may be regarded as the
process of finding the result (called the product) of taking one
quantity (the muUiplicand) as many times as there are imits
in another quantity (the mvUiplier).
The term mitttiplication has acquired a much broader
meaning than this, which is sometimes expressed as follows:
Multiplication is the process of finding a number (the
product) which is obtained from a given number (the multi
plicand) in the same way that another number (the multiplier)
is obtained from unity.
Multiplication is useful as a means of shortening addition
or subtraction. Later many other uses (often indirect) of
multiplication will become evident.
Multiplication of Monomials
48. Multiplication of Coefficients. To multiply 4a by Zb,
we evidently take the product of all the factors of the
multiplier and the multiplicand, and thus get 4 X a X 3 X 6.
Rearranging factors, we obtain as the product,
4X3XaX6or 12ab.
Hence, in multiplying two monomials,
Multiply the coefficients to produce the coefficient of the
product.
58
m
I
MULTIPLICATION OF MONOMIAI^ 59
48. Multiplication of Literal Factors or Law of Exponents.
Ex. Multiply a» by a^.
Since a* '^ a Xa Xa
and a^ = a X a
:. a^ X a* ^ a X a X a X a X a '^ aK
This may be expressed in the form
€^ Xa* '^a^^^ ^ a«,
or, in general, a*" X a** = a** "*" ",
where m and n are positive whole nmnbers.
Hence, in multiplying the literal factors of a monomial,
Add the exponents of each letter that occurs in both rrvuttiplier
and multiplicand.
Ex. 4a^b(^ X Sa^h^x = \2ofWx.
50. Law of Signs. The law of signs in multiplication
follows directly from the general law of signs as stated in
Art. 31.
(1) + SlOO taken 5 times gives + $500,
or, in general, a + quantity taken a + niunber of times gives
a + result.
(2) SlOO of debts, that is,  $100, taken 5 times gives  $500,
or, in general, a — quantity taken a 4 number of times, gives
a — quantity as a result.
(3) $100 deducted 5 times, or $100 X  5, gives as the total
amount of deduction — $500,
or, in general, a + quantity taken a — number of times, gives
a — quantity as a result.
(4) Deducting $100 of debts 5 times from a man's possessions
is the same as adding $500 to his assets; that is,
 $100 X  5 = + $500,
or, in general, a — quantity taken a — number of times gives
a + quantity as a result.
60 SCHOOL ALGEBRA
We see from (1) and (4) that
either + X +, or — X — , gives +,
and from (2) and (3), that
either — X +, or + X — , gives — .
In briefs in multiplication
Like signs give pliLs; urdihe signs give minus,
SI. Hultiplication of Honomials. Combining the results
of Arts. 48, 49, and 50, we may express the method of multi
plying one monomial by another as follows:
Multiply the coefficients together for a new coefficient;
Annex the literal factors, giving each factor an exponerd
equal to the sum of its exponents in the terms multiplied
together;
Determine the sign of the result by the rule that like signs
give +, and urdike signs give — .
Ex. 1. Multiply ba^hJ^ by  ^alfif.
The product is  30 a»6<xV.
Ex. 2. Multiply 5a*+3 by 2a^\
Since n + 3 and n — 1, added, give 2n + 2,
the product is IW^.
XZEBCI8E 16
1. 2. 3. 4. 5. 6.
Multiply
5
3a
Sab
30ry
4x
5x
by
4
7.
2
8.
5
9.
1
2x
11.
3x
10.
12.
Multiply
^ax
6xy^
7ax
5o«6
6c*d •
2aV
by J
 4ax
 7xy^
Zay
— 4aP 
3«P ■
8xyV
MULTIPLICATION OF MONOMIALS 61
13. 14. 15. 16. 17. la
Multiply 4a? iaa? .5x 2.1y» 2jx» Ja?
by .2x» faV .03x . .05i^ ix .Sx
23.
Multiply 2»i 2«i 2»i a:»i «»*
by 2^ ^ 2 ^ a?
19.
20.
21.
22.
21^1
2nl
2»i
3.111
22
2»
2
a?
Verify Exs. 1921 when n = 4. Also Exs. 22 and 23
when n = 4 and a: = 3.
24. 25. 26. 27. 2a
Multiply a*a;*"^ a^a:*^^ d^"^ — a^a;""* • a:*
by ix? — aa^ — a V aa;**'^^ g'
29. 30. 31. 32.
5(a + 6)« 3(a + 6)*  6(a + 6) 7 (a + b)^^
2(a + 6)2  (a + 6)  2(a + &)» 3(a + b^
n
n
33. Multiply 2** by 2 and verify the result when n = 4.
34. Write out all the factors of 7a«. Of (7a)».
■
35. iqby is how many times as large as 06* when a = 3
and 6 = 2?
36. How many a:'s are there in the product of 5aa? by
6aV? How many a's?
37. How much money do five empty pocketbooks contain?
5X0 = ?
38. Find the value of 7 times 0. OfSaXO. 0!OXQs?y^.
Of 3(x + y)XO.
If a = 4, 6 = f , c = 0, a? = 1, and y = 9, find the value of
39. abc. 41 5cxy^. 43. 4c^ + a.
.Q. a^c. 48. a* + 3cy. 44. (3c + x)'.
62 SCHOOL ALGEBRA
ac + y 2a + c(x + y)
45. g^. 47. ^ .
5a^ + l * _ 5(x"l)+8
46. i— . 48. ^^ ^ .
x + y 2a
49. How many of the examples in this Exercise can you
work at sight?
50. How many examples in Exercise 3 (p. 19) can you
now work at sight?
Multiplication of a Polynomial by a Monomial
52. Utility of the Distributive Law in MultiplicatioiL;
Bnle. In arithmetic we have become familiar with the fact
that, for instance, 5 X 67 = 6(60 + 7) = 5 X60 + 5 X 7; and
that this principle enables us to perform all multiplications
by committing to memory only the products up to 9 X 9.
Similarly, in algebra, a{b + c) = o6 + be. This is called
the Distribittive Law of MuMplicaiion. By use of this law,
all multiplications in algebra can be performed as a multi
plication of pairs of monomials.
Hence, to multiply any polynomial by a monomial,
Mvitiply each term of the mvUiplicani by the muUipUerf
and set down the results as a new polynomial.
Ex. Multiply 2(1?  ba^b + Sofc^ by  Soi^.
2a»  5a«6 + 3a6* = 4
Say =  12
Product  6a*6* + 15a»6»  9a«6* 48
The check is obtained by letting a *" 1, and & » 2.
EXERCISE 16
1. 2. 3. 4.
Multiply 2a + 3a; 3a; — 2y ^y — a;^ 7aa; — 4iyy
by 3ax — hxy 2xy — Zabxy i
MULTIPLICATION OP A POLYNOMIAL 63
Multiply:
5. 8ac? — 3m^ by 5an. 9. Sa:*"*"^ + Ta:** by — 4x.
6. m — m? — Sin? by — 7mhi. 10. 3x**"^ + 3a:'*"^ by a?.
7. 8a:^y — 5ar^ — y' by 3a:y. 11. 3a:^* + 5a?^ by a^.
8. 2ar* — 3aj'»"^ by a?.
12. 20""  7a«» by  2o'".
13.
Multiply 2.5a:2^3 7a.+
by .4a;
.51
14.
15.
«
16.
.25«»
17. • What is the value of 7x — 5y times zero?
Multiply:
13. 5(a + 6)2  3(a + 6)  5 by 2(a + 6).
19. 7{x  y)2 + 2(a:  y)  6 by 3(a:  y^.
20. 2(3a + 26)2 _ 5(3^ + 26) + 4 by 4(3a + 26).
21. Reduce {7aaabb — 5aaa66) X 6aa66 to its simplest
form. Compare the size of the result with that of the original
expression.
(The following problems are mixed variations of Types I,
n, and m.)
22. What number diminished by 19 equals 37?
23. What number increased by 19 equals 37?
24. What number diminished by 1.067 equals 4.5?
25. What number increased by twice itself and then by
24 equals 144?
64 SCHOOL ALGEBRA
26. What number increased by twice itself and then
diminished by 24 equals 144?
27. What number increased by  of itself and then by 20
equals 60?
28. What niunber diminished by \ of itself and then
increased by 30 equals 90?
29. What number of dollars diminished by i of itself and
then by $30 equals $160.60?
30. If a number is multiplied by 3 and then diminished
by 40, the result is 140. Find 4he number*
31. If 5 times a certain niunber is increased by 20.5, the
result is 870. Find the niunber.
32. n five times a certain number is increased by 20.5, the
result is equal to three times the number increased by 160.
Find the number.
33. A man who died left $16,000 to his son and daughter.
The share of his daughter, who had taken care of him in his
illness, was $500 less than twice the share of the son.
How much did each receive?
34. A cubic foot of iron and a cubic foot of aluminum
together weigh 618 lb. If the weight of the iron is 14 lb. less
than three times the weight of the aluminum, find the weight
of each.
35. A baseball nine has played 54 games, and the number
of games it has won is 3 less than twice the number it has
lost. How many has it lost?
36. Of which type is each of the above problems (Exs.
2235) an instance or a variation?
MULTDPLICSATION OP A POLYNOMIAL 66
Multiply each member of the following equalities by — 1
and solve:
ft
: 37. — 2a: — 5 = — a: + 4. 38. — 4x — x « — 6 — 9.
39. Make up and work an example similar to Ex. 10. To
Ex.19.
40. Make up and work an example similar to Ex. 30. To
Ex. 35.
41. How many of the examples in this Exercise can you
work at sight?
*
Multiplication op a Polynomial by a Polynomial
53. Arranging the Terms of a Polynomial. The multi
plication of polynomials is greatly facilitated by arranging
the terms in each polynomial according to the powers of some
letter, in either the ascending or descending order.
Thus, &c*+3— a;+x*— Ta:", arranged according to the as
cending powers of x, becomes
3  « + 5a;«  7x» + a:*.
Also, o* + 6* — 4a^' — 6a*6, arranged according to the descend
ing powers of a, becomes
o*  5a»6  4a«6« + h*.
54. Knltiplioation of Polynomials. By a double use of
the Distributive Law:
{a + b){c + d) ==a(c + d) + b(c + d)
^ttc + ad + bc + bd
We see that a similar result is obtained, no matter how
many terms occur in each polynomial.
Therefore, to multiply two polynomials.
66 SCHOOL ALGEBRA
Arrange the terms of the midtipUer and the multiplicand
according to the ascending or descending pmoers of the same
letter;
Multiply each term of the multiplicand by each term of the
multiplier;
Add the partial products ihus obtained.
Ex. 1. Multiply 2x  3y by 3a: + by.
The terms as given are arranged in order.
The most convenient way of adding partial products is to set
down sunilax terms in columns, thus:
2x  3y =  1.
3a; h 5 y = 8
Product 6a;' + a;y — ISy* = — 8
The check is obtained by letting a; = 1 and y = 1 (or a? = y = 1).
Note that this method checks only the signs and coefficients, not
the letters or their exponents. Mistakes in letters and exponents,
however, are rare in comparison with mistakes in signs and coeffi
cients. A convenient check for all elements in the process is ob
tained by letting a; = y = 2. A useful check on the letters and
exponents in many examples is given in Art. 56.
Ex. 2. Multiply 2x  ^ + \  Z^^hy 2x + Z  7?.
Arrange the terms in both pol3momials according to the ascending
powers of a;. (Why is the ascending order chosen rather than'the
descending?)
1 + 2a;  3a;2 
3 + 2a;  a;*
x» = 1
» 4
3 + 6x  9x2 
+ 2a; + 4a;2 
 a;« 
3a;»
63;*  2a;*
2a;« f 3a;* f a;*
Product 3 + 8a;  6x*  lla;« h x^ ■{ x^ =  4
Now multiply the two polynomials together with their terms in
the order as first given. This will show you the advantage of ar
rangmg the terms in order before multiplying.
MULTIPLICATION OF A POLYNOMI^ 67
Ex.3. Midtiply a2 + 62 + c2 + 2a6acfec by a + 6 + c.
Arranging the terms according to powers of a,
a»H2a6ac+6*6c+c» =3
g + b + c =^
a» + 2aV) a^c+ ab^  aihc { cu^
+ 0*6 + 2aJt^  abc + &•  6'c f 6c»
f g^ + 2abc  cu^ + 6»c  6c» + c»
g» + 3g»6 +Zab^ + 6« + C = 9
55. Degree of a Term; Homogeneous Expressions. The
degree of a term is determined by the number of literal fao
tors which the term contains. Hence, the degree of a term is
equal to the sum of the exponents of the literal factors in the
term.
Thus, 7g'6c* is a term of the 6th degree, since the sum of the
exponents in it is 3 + 1 + 2, or 6.
The degree of an algebraic expression is the same as the
degree of that term in the expression which has the highest
degree.
Thus, 7a^ + 3x^^ + y is of the 4th degree.
A homogeneous polynomial is a polynomial of which all the
terms are of the same degree.
Thus, 5g*& — b^ ^ ab^ 18 & homogeneous polynomial, since each
of its terms is of the 3d degree.
56. Multiplication of Homogeneous Polynomials. If two
monomials are multiplied together, the degree of the product
must equal the sum of the degrees of the multiplier and the
multiplicand.
For instance, in Ex. 3, above, the multiplicand is of the 2d
degree and the multiplier is of the 1st degree, and are both homo
geneous. Their product is seen to be homogeneous and of the 3d
degree.
68 SCHOOL ALGEBRA
The fact that the product of ttoo homogeneous expressions
miist also be homogeneous affords a partial test of the accuracy
of the work.
If, for instance, in Ex. 3, p. 67, a term of the 5th degree, such as
5a'&', had been obtained in the product, it would have been at once
evident that a mistake had been made in the work.
67; Detached Coeffloients ; Symmetrical Expressions. The
process of multiplying algebraic expressions may often be
further abbreviated by using only the signs and coefficients
of terms, omitting the letters and their exponents.
EXERCISE 17
Midtiply and check each result:
1. a:  4 by Sx + 1. 5. 7a?  5y* by 4ai« + 3j/*.
2. a: — 3 by 3a: + 2. 6. 5xy + 6 by Qxy — 7.
3. 2aj + 5 by a:  7. 7. 4a*  Vc by 8a»c + 2ak^(?.
4. 3a;4yby4a;3y. a lla^'y  7a^ by 3a? + 22/*.
9. a^ — ab + l^hy a + b.
10. ^ + s?y + xy^ + j^hyz — y.
11. 4a?  3a? + 2a:  1 by 2a; + 1.
12. 2a?3a:y + 2/by3a:~5y.
13. a?  3a? + 2a:  1 by 2a? + a:  3.
14. 3a?2/ — 4xy^ — j/* by a? — 23cy — y*.
15. a? — 3a?y + 3a:^ — fhy a? — 2xy + y^.
16. 4a?  3a? + 5a:  2 by a? + 3a:  3.
17. a? — 3a? + 5 by a? — a: — 4.
18. a? — 3xy + y* by a? — 3a:y — y*.
MULTIPLICATION OP A POLYNOMIAL 60
19. a^ab + Vhya^ + ab + V.
20. 4a^ + 9y^&xyhy4a? + 9]^ + Qxy.
21. aJ* — 7a:V + 6x}^ — y* by a*  2a^ + y*,
22. a* — 6aa? + 12a^ — 8a' by — a? — 4aa: — 4a'.
23. a* + 6* + a? + 2a6 — aa; — 6a: by a + 6 + a?.
24. ab + cd + ac + bdhy ah + cd — ac  bd.
25. Ja + ift by Ja — J6.
26. f a?  4x + i ^)y fa: + .
27. .5a — .46 by .2a — .36.
28. 1.8a?  3.2a: + .48 by 2.5a: + .5.
29. X* + 2a:»^ + 3a:*^  2 by a:  2.
30. a:»+^  3a:» + 4a;*^  Sa:*^* by a:" + 2a:**.
31. a:*^  2a:»» + 3a;**  4a:»* + 5a:» by 2a? + 3a: + 1.
32. Multiply 3a: — 5 + 4a? + a?by2x — 3 + a? without
changiiig the order of the terms. Now arrange the terms in
each expression in descending order and multiply. About
how much easier is the second process than the first?
33. .Make up and work an example similar to Ex. 32.
Arrange the terms of the following in descending order of
some letter, and multiply:
34. 4a:3a?5 + 2a?bya: + 4.
35. 3a? — 5 — a: by a; + 4a? — 2.
 "36. 2a? + y*  4tX]^ + 3a?y by y* + 3a?  2ay.
37. Which of the polynomials in Exs. 1624 are homo
geneous?
70 SCHOOL ALGEBRA
38. A number increased by 3 times itself and then by 40
equals 180. Find the number.
39. Separate 180 into two parts such that one part exceeds
three times the other by 40.
SuG. Let X = the second part.
40. A number increased by \ of itself and then by 20
equals 95. Find the number.
41. Separate 95 into two parts such that one part exceeds
\ the other part by 20.
42. A number increased by f of itself and then diminished
by 30 equals 70. Find the number.
43. Separate 70 into two parts such that one part exceeds
f of the other part by 30.
44. A number diminished by f of itself and then increased
by 30 equals 66. Find the number.
45. A number increased by .06 of itself and then by $100
equals $312. Find the number.
46. Separate 400 into two parts such that one part exceeds
3 times the other part by 60. .
47. Separate $1000 into two parts such that one part is
smaller than 4 times the other part by $100.
48. Of which type is each of the above problems (Exs.
38^7) an instance or a variation?
68. Hnltiplication Indicated by the Parenthesis; Simpli
fications. The parenthesis is useful in indicating multipli
cations or combinations of multipUcations.
Thus, (a — 6 + 2c)2 means that a  6 + 2c is to be multiplied
by itself.
(a — 6 + 2c)' means that a— 6+2cistobe taken as a factor
three times and multiplied.
MULTIPLICATION OF A POLYNOMIAL 71
To perform the multiplication expressed by a power is to
expand the power.
Again, (a — b) {a — 2b) {a \ b ■ c) means that the three factors,
a — 6, a — 26, and a {b — c, are all to be multiplied together.
Also, (a — 2zy — (a + 2x) (a — 2x) means that a + 2a; is to be
multiplied by a — 2x, and the product is to be subtracted from the
product of a — 2x by itself.
We simplify an expression in which multiplications are
indicated by parentheses and exponents by performing the
operations indicated and collecting terms.
Ex. Simplify S{x  2y) (x + 2y)  2(.t  2y)\
Z{x  2y) {x + 2y)  2(x  2yY
= 3(a;«  42/2)  2{x^  40^2/ + 4y«)
= 3x*  122/2  (2x2  8x2/ + 82/')
= 3x2  122/2 _ 2x2 + 8x2/  82/2
= x2 4 8x2/  202/2 Ans.
Check this result by letting x = 1 and 2/ = 2.
EXERCISE 18
Find the product of
1. ( a) ( a) ( a) ( a) ( a).
2. ( 1) ( 1) ( 1) ( 1) ( 1) ( 1).
3. {x  y){x  y) {x + y){x  y) (x + y) in parenthe
sis form.
4. Find the value of (— 2)*. Of a" when a = — 1 and
n = 7.
Simplify by removing parentheses and collecting terms:
5. a:*2(x + l). 8. (2a: + 3) (5x  4).
6. {x  2) (x + 1). 9. la  3(4a  8).
.7. 2x + 3(5a;  4). 10. 9a + 5(3a + 4).
72 SCHOOL ALGEBRA
11. Sx(x  2)  2xix  3).
12. (2x^Zx + iy.
13. (2a  36 + 5)2  (2a + 36  5)«.
14. (x  5)2  (« + 5)2.
15. 3a;  2(3ar^  5a; + 2).
16. a;  2(a;  1) (a; + 3).
17. (a;2)(a; l)(a; + 3).
18. 3a2  (a  26) (3a + 46).
19. (a;  y  z)2  a;(a;  2y + 2z).
20. 2ar^  3(a;  1)^ + (a;  2)2.
21. 3ar^  x(l  a;) (2 + a;) + s?.
22. 2  3(a;  2)^  2(3  2a;) (1 + x).
23. a^ — [x(a — x) — a{x — a)] — y?.
24. {x  1) (a;  2)  (a;  2) (a;  3) + («  3) (a;  4).
25. 3(a;  yY  2l(a; + yf  {x  y) {x + y)) + 2y'.
26. a;(a:  y  2)  y(z  a;  2/)  2(z  y  a;)  i^.
27. 3[(a + 26)a; + 2my]  5[(m  c)y + hx]  4[(a;  a)
a + cy\.
28. 26a6  (9a  86) (5a + 26)  (46  3a) (15a +.46).
29. Multiply the sum of (a — 2a;)* and (2a — a;)* by
3a — 2(a — x).
30. Subtract (a; — 2yf from a;* — 8^ and divide the re
mainder by X  2y.
31. Find the value of 3 X + 4. Of 8  7 X 0.
Of 6X0X6 + 7.
MULTIPLICATION OF A POLYNOMIAL 73
If o = 3> 6 = 0, x = — 2, and y ^ — .5, find the values of
32. 2ax. 36. 6y* + Zx{x — y).
33. ha^y. 37. 4a^ — a6x(4a; — y).
34. 3a:* + ofcy. 38. 3ar  5(2x + 3).
35. 6xy — as?. 39. 2(a:* + y) — ohy + aa*.
40. 2(1  2xy + (a; + y) (a* + x).
41. (a;  1)2  3(a: + 1) (a: + 2)  x{o?  2) (y  2a:).
42. 3a(a  2a:)  {a  (a  1) (a; + 1)  (a + xY) + 5aa:.
Find the value of
43. (a; + o)* — (a; — o)* when x — 2a.
44. 5(a: + p)* — (a: + 2?) (a; — 2p) when x = 3p.
45. 3a? + 4a: — 5{x — 1)^ when x ^ ah.
46. If a: = 2 and y = 1, find the value of (x + y)'. Also
of a:* + J/*.
47. From the sum of 2a + 56 and 36 — 5a, subtract three
times a — 76, and verify the result when o = 2 and 6 = 5.
Also when a = 3 and 6 = — 1.
48. If a certain number is diminished by 24 and the result
multiplied by 3, the final result will be 78. Find the number.
49. If a certain sum of money is increased by $150 and
the result multiplied by 4, the final result will be $1000.
What is the original sum of money?
50. Separate $1000 into two parts such that one part
equals four times the sum of $150 and the other part.
51. Separate .0015 into two parts such that one part equals
3 times the sum of .0001 and the other part.
74 SCHOOL ALGEBRA
52. The sum of two fractions is l,vand the larger is three
times the sum of the smaller and §.' Find the fractions.
53. Separate $100 intd two parts suck that the sum of one
part and $10 equals the other part.
54. Separate $100 into three parts such that 3 times the
sum of $5 and one of the parts equals each of the other parts.
55. Separate $100 into four parts such that twice the sum
of one part and $1 equals each of the other parts.
56. A man walked 15 miles, rode a certain distance, and
then took a boat for twice as far as he had previously trav
eled. Altogether he went 120 miles. How far did he go by
boat?
57. The sum of three numbers is 50. The first number is
twice the second, and the third is 16 less than three times the
second. Find the numbers.
58. Find five consecutive numbers whose sum is 3 less than
6 times the least of the numbers.
59. The difference between two numbers is 6, and if 3 is
added to the larger, the sum will be double the less. Find the
numbers.
60. Divide $4500 among two sons and a daughter so that
each son gets $100 less than twice the daughter's share.
61. Find two numbers, whose diiference is 14, such that
the greater exceeds twice the less by 3.
62. The difference of the squares of two consecutive num
bers is 43. Find the numbers.
63. Three boys together earned $98. If the second earned
$11 more than the first, and the third $28 less than the other
two together, how much did each earn?
MULTIPLICATION OF A POLYNOMIAL 75
64. Which of Exs. 4863 are instances or variations of
Type I? Of II? III?
65. Make up and work an example similar to Ex. 48. To
Ex. 49.
66. Make up and work an example similar to Ex. 58. To
Ex. 62,
67. How many examples in Exercise 6 (p. 29) can you
now work at sight?
CHAPTER V
DIVISION
59. Division is the process of finding one factor when the
product and the other factor are given.
The dividend is the product of the two factors, and hence
it is the quantity to be divided by the given factor.
The divisor is the given factor.
The quotient is the required factor.
Thus, to divide lOxy by 5Xy we must find a quantity which,
multiplied by Sx, will produce lOxy, The factor 5a; is the divisor,
likcy is the dividend, and the other factor, or required quotient, is
evidently 2y,
The division of a by 6 may be indicated in each of the
following ways:
b)a, a ib, ^, or a/b
60. General Prinoiple. Division being the inverse of mul
tiplication, the methods of division are obtained by inverting
the processes used in multiplication.
Division of Monomials
61. Index Law for Division. If a^ is to be divided by a^
we have
cfi aXaXaX^X^ ,, ,, ,
• = — = aXaXa '^ (T
a**
Or, in general, — = a**"*,
where m and n are positive whole numbers.
76
DIVISION OF MONOMIALS
77
62. The Law of Signs in Diviaion is obtained by inverting
the processes of multiplication.
Thus^ in multiplication, if a and b stand for any positive
quantities (see Art. 50, p. 59),
+ aX+b=+ab]
+ a X  6 =  o6
 aX +6= ab
aX6=+a6j
+ ab i +b " + a..
— ab*— b='+a..
— <iot+o»= — a. ,
,+ ofc ! — 6 = — o. .
(1)
(2)
(3)
(4)
Hence, by
definition
of divi
sion.
From (1) and (2) we see that the division of like signs
gives +• From (3) and (4) we see that the division of unlike
signs gives — • Hence, the law of signs is the same in divi
sion as in multiplication.
63. Division of Honomials. Combining the results ob
tained in Arts. 60, 61, and 62, we have the following method
for the division of one monomial by another:
Divide the coefficient of the dividend by the coefficient of the
divisor;
Obtain the eocponent of each literal factor in the quotient by
subtracting the exponent of each letter in the divisor from the
exponerd of the same letter in the dividend;
Determine the sign of the resuU by the rule that like signs give
plus J and unlike signs give minus.
Ex. 1. Divide 27a»6V by  9a«6a:».
, ,  — 3a6* Quotient
once the factor x* in the divisor cancels x* in the dividend.
Ex.2. Divide a^*^ by a^^
im— 1
o*"^ Quotient
Check the work in each of the above examples by multiplying the
quotient by the divisor. •
78 SCHOOL ALGEBRA
EXERCISE 19
Divide and check the result:
1. 15a by — 5a. lo. — rri?n by — m*.
2.  Sa:^ by X. 11.  3r^ by  1.
3. 8aV by — 4aa?. 12. — 8aa; by far.
4.  30ar^3/2 by  6a?y. 13. I662/2 by  6y.
5. — 7as2? by 7z^. 14. Sma: by .2a?.
6. 21a:^z by — ^xz. 15. .4aa:* by Ac*.
7. 186c^(? by — Oc^d. I6. .04aa; by .5aa:.
a  ZZ^i^fz' by llx!f^:f. 17. 2\;x? by far^.
9. 28aryz8 by  \ia^. is.  iar^ by .5a;.
19. 47rr2by27r. By r^. By ttt.
20. i^byflf^. Byiflr. By Jf.
21. Imi?^ by m. By .5t^. By .25t?.
22. 20(a: + yf by  4(a: + y). By  2(a: + yf.
2a  1.4(a  6)« by  7(a  h)\ By  2(a  6)*.
24. a®" by a^". By a^". — a".
25.  6a"+* by 2a»+^ By a"+^ a*.  So^^ a***.
26. a2»+5bya'»+^ By a2»+8. a"^
27. How many 2*s are multiplied together in 2^°? In 2*T
In the quotient of 2^*^ f 2*?
28. How many a;'s in a:^®? In a;*? In the quotient of
a:i«5a^?
29. Divide 2**~^ by 2 and verify your result when n = 5.
Treat 2**"^ 5 2* in the same way.
J
DIVISION OF A POLYNOMIAL 79
30. If an empty box is divided by partitions into 5 equal
•parts, will each compartment o^<he box be empty?
31. What is the value of ^ 5? Of f 7? State the
meaning of the latter in a manner similar to that used in
Ex. 30.
32. Give the value of 5 10. Of J a. Of ^ 2a:.
QjjO
7a lab 7abx 7a^b^a?'
Sax
33. What is the value of — — when a = 0? When x = 0?
7y
If a = 2, 6 = 3, c = 0, a: = 1, find the value of each of the
following:
34. ^ 36. £(26Z^)
a 4a
35. 'J^±^ 37. ^
b b + x
3a What is a polynomial? A binomial? A monomial?
Give two examples of each.
Division of a Polynomial by a Monomial
64. Utility in the Bistribntiye Law of Division; Eule. In
arithmetic we have become familiar with the fact that, fpr
instance, '
65 50 + 15 50. 15 ^.^ .^
 = — g  +  = 10 + 3 = 13,
and that this principle enables us to perform all divisions by
committing to memory only the quotients up to 81 r 9.
Similarly, in algebra, divisions can be greatly simplified by
the fact that
ac + 6c ac . be , , ^
r ^ s= j ss a + o
c c c
80 SCHOOL ALGEBRA
m
I
This is called the DutrUnUive Law of Dimsion.
Hence, to divide a polynomial by a monomial.
Divide each term of the dividend by each term of the dinaor,
and connect the reavUe by the proper signs.
Ex. 1. Divide \2cfx  lOa^ + 6aV by 2aK
2o «)12a»x  10o«y +6a^
Ex. 2. Divide 6a»»+»  4a2»^  2a«»« by 2a~^
2a'^*)6a'*+' — 4a*"+* — 2a**^
3a»*+^  2a*+»  a*»* Qiiotien/
Check the work in each of the above examples by mvJUH'ptyinii the
quotient by the divisor.
EZEBCISE 20
Divide and check:
1. 01?  3a? by — x.
2. 20q?  8xy hy 4x.
3. 4ai?  Ga^bc hy  2ab.
4. So? + 73?xhyx.
5. 15s?y — 10«y — Saiy* by 5xy.
6. — m — m^ + w' — w* by — m.
7. 14a:*3/^2 — 2\xyh? + oryz by — a:yz.
a  Sar"  2a; + 5 by  1.
9. .ear*  .\2x + 9 by  .3.
10. .02o«  .04a6  .86* by .5.
11. i«*  fa;  f by  f .
12. \a*V  Ja»f  K6« by  a6.
DIVISION OF A POLYNOMIAL 81
13. Ox'"  Go^n + 12a;" by  3a;".
14.  4ar^"+i + lOa?"^  6a;"+2 by 2a;2n^
15. a;"+'  2a;""^ + Sa:"**^ + a;" by a;"'"^
16. Sa;"''*"^ ~ 16a;*"+i  4a;»"  12a:"»"^ by  4a;'"*.
17. 9a;2n2 _ 6a;2ni + i2aJJn  Sx^n+i by 3a;"^
18. 10(a + by  8(a + 6) by  2(a + 6).
19. .5(a;  yY  .15(a;  yY by .6(a;  y)^
20. (a + 6)a;  (o + b)y by (a + 6).
21. (a — 6)a; + (a — b)y by (a — 6).
22. x{x + l) + {x + l) by (a; + 1).
23. f of a number added to twice the number gives 210.
Find the number.
24. f of a number added to 5 times the number gives 340.
Find the number.
25. f of a number added to ^ the number, gives 140. Find
the nuniber.
26. The difference between f and ^ of a certain number is
14. Find the number.
27. What number increased by .06 of itself gives 318?
28. What smn of money at simple interest for one year at
6% vnH amount to $318?
29. What number increased by .15 of itself will amount to
690?
30. What sum of money at simple interest at 5% will
amount to $690 in 3 years?
31. For every nickel which a girl put in her savings bank
her father put in a dime. If her bank contained $18.75 at
82 SCHOOL ALGEBRA
the end of one year, how many nickels did the girl save in
that time?
^ For every dime that a boy spent for books, his father
gave him a quarter to spend for the same purpose. If he
spent $52.50 in all, how much did his father give him?
33. A purse contains $10.50 in dollar bills and quarters,
but there are twice as many quarters as bills. How many
are there of each?
34. How can $2.25 be paid in 5 and 10 cent pieces so that
the same number of each is used?
35. How can $5.95 be paid in dimes and quarters using
the same niunber of each?
In the following equations divide each member by — 1
and solve, checking each result:
36. — 1 — 3a: =  a:  5 38.  a;  (2a:  1) = — 5
37. 5a:8a:= 7a: + 1 39.  7a:  5 =  3a: + 4
40. How many of Exs. 2335, pp. 8182, belong to Type
I? To Type II? III?
41. Make up and work an example similar to Ex. 31. To
Ex. 36.
42. Make up and work an example similar to Ex. 13. To
Ex. 18.
43. How many of the examples in this Exercise can you
work at sight?
Division of a Polynomial by a Polynomial
65. General Method. The method of dividing one poly
nomial by another is to arrange the polynomials according
to the ascending or descending powers of some one letter,
DIVISION OF A POLYNOMIAL 83
•
and then^ in effect, to separate the dividend into par
tial dividends, which are divided in succession by the
divisor.
Ex. 1. Divide6a?* + 7a?  3a? + 11a:  6by2a? + 3a:2.
We divide the first term of the dividend, 6a:*, by the first term
of the divisor, 2x', obtaining the quotient 3x\ Multipl3dng this
quotient, 3x', by the entire divisor, we obtain the first partial divi
dend. If we subtract this from the entire dividend and divide the
remainder by 23^, we have a process like the following:
Dividend Divisor
^ ../ ^
6aj4 ^ 7^4 _ 3a;2 + iiaj _ 6 2x«43x 2 = 15+3
6a:* + 9a:*  6g« 3a:»  a: + 3 = 5
 2a:» + 3a:* + 11a:  6 ' ^ T. '^
^2a? 3a:'+ 2a: «^^^
6x« + 9a;  6
6a:» 4 9a:  6
A quick dieck on the parts of the work in which errors are most
likely to be made is obtained by letting a: == 1, as is done in the
solution above. A more complete check is obtained by finding the
product of the divisor and quotient and noting whether the result
equals the dividend.
Now state the process of dividing a polynomial by a
polynomial as a general rule. •
Ex. 2. Divide 310^  206*  lOa^V + 6a*  0^6 by
3a2  562 + 4^^
6a<  a»6  lOa^ft* + 31a6»  206* 3a' + 4o6  56« = 6 ■^ 2
6a* + 8a»6  lOo^y 2a«  3a6 + 46« = 3
 9a»6 + 31a6»
 9o»6  12a»6g + 15a6»
. + 12a262 4 16a6»  2(M>*
+ 12a»y + 16o6»  20y
Is the dividend homogeneous? The divisor? The quotient?
84 SCHOOL ALGEBRA
Ex. 3. Divide a? + y^ + :f + 3xh/ + 3xy^ hy x + y + z.
Arranging terms according to the descending powers of x,
3^+Sx^.+ Sxy*+ y^+s^ \x hy+z =983
g* 4 a?*y H xH X* +2xy —xz + y*+z*—yz 3
+ 2x*y  xH \ 3x1/* + 1/* + «•
+ 2x'^ + 2x2/* + 2x2/«
— a^ — xs^
 2xj/2 + 2/» + «•
 xyz
XJ/* +X2*
X/«
 X2/2 4 y* + 2*
+ X2*
+ X2*
ocyz —yh +s^
— xyz — yh  /2»
 x2/»  2/*2  yz*
The process of algebraic division may often be abbreviated
by the use of detached coefficients. (See Appendix^ p. 4QQ.)
EXERCISE 21
Divide and check each result:
1. 3x2 + 7a; + 2bya; + 2.
2. 6^2 + 7a: + 2 by 3a; + 2.
3. 12a? + a:2/2(Vby3a: + 4y.
4. 3a? + a;  14 by a;  2.
5. 6a?  Slxy + S5f by 2a;  7y.
6. 12a?  Hoc  36c2 by 4a  9c.
7.  15a? + 59a;  56 by 3a;  7.
8. 44a? — xy — Zy^hy 11a; — 3y.
9. a2  462 by a  26. 12. 9a?  49 by 3a; f 7.
10. a?  ^ by a;  y. 13. 125  64a? by 5 — 4a;.
11. 27a? + 8 by 3a; + 2. 14. 8aV + »* by 2aa; + j/^.
DIVISION OF A POLYNOMIAL 85
15. 2a?  9a? + 11a;  3 by 2a;  3.
l& 35a? + 47a? + 13x + 1 by oa: + 1.
17. 6a«  17a^x .+ 14aa?  3a? by 2o  3x.
2a. Ay* 18j^ + 22f7y + 5hy 2y 5.
19. (^ {• c^x\ <?a? + (?a? + ca;* + a? by c + x.
20. llx  8a? + 5a?  20 + 2a? by a; h 4.
21. 4a; + 6a? + 3a?  11a?  4 by 3a?  4.
22. ^y  llarj^  2ar2/ H* 6ar*  6y* by 2x  3y.
23. V + 6a:«  13aj*y by 3ar^  2y..
24. ic* — 16y* by a; — 2y. 26. 7? — i^hy x + y,
25. «^ + 321/® by a; + 2y. 27. 256a:^ — y^ by 43:^ — ^.
2a 9a;  \^:x? + 8a^  13ar* + 2 by 4a:2 + x  2.
29. 10  a:* 27ar^ + 12a;*  3a; by a: + 4ar^  2.
30. 22a:2 _ 13^ + jQa:^  18a:* + 5a;  6 by a: + 5ar^  2.
31. 14a;V  lea:'^^ + 6a;S + j/^ + 5a:*2/  6ary* by 3a:2 ^ 2/2
— 2a:y.
32. 5a*63a362a263 + 3a5465bya2 + 3a6 + 262.
33. Q? — "^ + 7? — xyz — 23^z + 2y^ by a; — y — z.
34. <? + d? + 71? — 3cdn by c + d + n.
35. /2j/3 + lbyy22y + l.
36. 2a:« + 1  3a;* by 1 + 2a; + or^.
37. 6a?2/^ — 6yV — Qxh? — 13a;t/2? — 5an/% by Sxy + 22/a
+ 3a».
3a a;^  39a; + 15  2a;» by 3ar^ + 6a; + a;3 + 15.
99. 4a;«  9a;* + 25  14a;3 _ 3^8 by 2a:3 _ a;  5 + 3ar*.
86 SCHOOL ALGEBRA
40. ia2  J62 by la + §6.
41. ^x^  T^f by Ix  ly.
42. Ja'  ^V^^ by f a  6.
43. fa:^ V« + byfa;2.
44. .16x2 _ 252/^ by .4x + .5y.
45. 2.880:3 ^ 10.86a;  19.2 by 1.2a:2 _^_ 15^. _(. 6.4.
46. 6a:2n+l _'13a^» + Q^nl by 3a;n+l _ 2ajn,
47. 12a^" + 13a:3"a:'»by 3ar» + l.
4a 4a;»^^ +*5x"+2 _ ajn+i  a;« + a;"^ by x^ + 2a: + 1.
49. 6a;«+i  5a:«  Gx"^^ + ISa:*^  6a:'»^ by 2ar^  3a: + 2.
50. In Ex. 20 try to divide without arranging the terms of
the dividend either in ascending or descending order.
51. What is the value of ^^ when a: = 0? Whenv = 0?
52. Divide of + ]^ hy x — y to 5 terms and note the
remainder.
53. Divide 1 by 1 — a: to 4 terms and note the remainder.
54. Divide 1 by 1 — oa: to 3 terms.
55. If a boy walks at the rate of 3 miles an hour, how far
will he walk in 5 hours? In a hours? In x hours? In a: + 2
hours?
56. A boy starts at a given time and walks 5 hours. An
other boy then starts and rides a bicycle x hours until he over
takes the first boy. How many hours does the second boy
ride? How many if the first boy has a start of a hours? Of
y hours?
DIVISION OF A POLYNOMIAL 87
57. Two men A and B start from places 35 miles apart
and walk toward each other at the rate of 4 miles and 3
miles an horn* respectively. How many hours will it be
before they meet?
SuG. In forming an equation, it is an aid to diagram a problem
of this kind: 85^^
If the two men start 4 ( ^^ ■\ j>
at the same time and ^ • v y
walk toward each other *"^ ««^
until they meet, they must travel the same number of horn's.
Let X  the number of hours each man travels.
Then 4x = number of miles A travels.
Sx = number of miles B travels.
4c + 3x = 35 (Art. 15, 1)
7x =35
a? = 5, no. hours before they meet.
• Check. 4x — 20, distance A travels.
Zx = 15, distance B travels.
20 + 15 = 35
In working Exs. 5872, draw a diagram as an aid in each
solution:
58. Two men, A and B, start from places 42 miles apart
and walk toward each other, at the rate of 4 and 3 miles per
hour respectively. How many hours will it be before they
meet?
59. Make up and work an example similar to Ex. 58.
60. Two bicyclists, A and B, start respectively from New
York and Philadelphia, 90 miles apart, and ride toward each
other. A rides 8, and B, 12 miles per hour. How long and
how far will A ride before meeting B?
61. Boston is 234 miles from New York. If two automo
biles start from the two cities at the same time and travel
88 SCHOOL ALGEBRA
toward each other at the rate of 12 and 14 miles per hour
respectively, how far will each go before they meet?
62. Make up and work a similar example concerning trains
which travel between New York and Chicago, which are 912
miles apart.
63. One boy starts at a certain time from New York on
a bicycle and travels toward Philadelphia at the rate of 8
miles an hour. One hour later another boy starts from
Philadelphia and goes toward New York at the rate of 6
miles an hour. How long before they will meet?
64. New York and Washington are 228 miles apart. A
train starts from New York at a given time and goes at the
rate of 26 miles an hour, and two hours later a train starts
from Washington and proceeds at the rate of 34 miles an*
hom*. How long before they will meet?
65. Make up and work an example similar to Ex. 64 con
cerning trains which travel between Cincinnati and New
'Orleans, which are 830 miles apart.
66. Two boys start at the same place and travel in oppo
site directions on bicycles at the rate of 8 miles and 10 miles
an hour. How long before they will be 108 miles apart?
67. If they, travel in the same direction, how long before
they will be 16 miles apart?
68. Two boys start from New York and Philadelphia at
the same time and travel toward each other until they meet.
If one goes twice as fast as the other and they meet in 7J
hours, what is the rate per hour of each boy?
69. If in Ex. 68 one boy went 5 miles an hour faster than
the other, and they met in 6 hours, what was the rate of each?
REVIEW 89
70. Make up and work an example similar to Ex. 68
concerning automobiles traveling between New York and
Washington.
71. Make up and work an example similar to Ex. 69 con
cerning railroad trains traveling between New York and
Buffalo, which are 440 mi. apart.
72. A set out from a town, P, to walk to Q, 45 miles distant,
an hour before B started from Q toward P. A walked at the
rate of 4 miles an hour, but rested 2 hours on the way; B
walked at the rate of 3 miles an hour. How many miles did
each travel before they met?
73. How many examples in Exercise 10 (p. 45) can you
now work at sight?
EXERCISE 22
Review
1. Express the following in as few terms as possible: 3.2x^ —
2.5xy + .162/2 + 1.5x«  .82/'  ,32xy + Ay^  1.5x» h .4xy.
. 2. Subtract .15a* + .36«  2,5ab from  7a«  4a6  1.56*.
3. Add 2ip*  1.5p + 5, .75p* + p  .4, f  6p»  .5p.
4. Simplify 3.2x*  [.8x« + (3.5x  i  ,2x^)  1.5  3x].
5. Solve .3x  4 = .2a; + .5.
€• What is the root of an equation? How do you check your
solution of an equation? Check Ex. 5.
7. Shnplify 5x  3(x  2) (a; + 7) + 3(x  2)«.
If a = 0, 6 = 1, c = 4, X =» — 2, find the value of
a a{b + c)  3z.
9. (c + 2x) (b o) 3(a; + 4) (x + 5).
_ 3a + 5(2 + x)
10. •
b+c
90 SCHOOL ALGEBRA
U. Multiply a;  2 H 4x* by 2x»  1  Sx.
12. Divide a:* + 3  6x« + «« f &c  lla:» by 2x  x* + 3.
13. Find three consecutive numbers whose sum is 33.
14. In a certain kind of concrete, twice as much sand is used as
cement, dnd twice as much gravel as sand. How many pounds of
each are used in making 2800 lb. of concrete?
15. The record time for the 100 yd. swim at a certain date was
65J sec. This was 7J sec. more than 5 times that for the 100yd.
dash. What was the record time for the latter?
16. Solve Ex. 15 without using x to represent the unknown
number. How much of the labor of writing out the soluti9n is
saved by the use of x7 Is there any other advantage in using x to
solve problems?
17. What is the dividend when the quotient is x* + 2x* + 7a;
f 20, the remainder 62a; + 59, and the divisor a:* — 2x — 3?
18. What is the divisor if the quotient is «* + 3a:, the dividend
X* — 8, and the remainder 9a; — 8?
19. If a; = — § and y = — J, find the value of
(3a;  2yy (dx^ + Ay^)  Q(y  x) V^{x + 2y^ + 4).
20. Add a to h. Also add 3a  55 to 4c + 7d.
21. Subtract 3x — 2y + z from. — 7. From — o. From b.
22. Subtract 2a — 36 from 0. Also 5 from 0.
23. Can 3 + 2ab be luiited in a single term? Give a reason.
24. The product of an even number of negative factors has what
sign? Of an odd niunber of negative factors? Give an example
using not less than five factors.
25. Express the following in a simpler form: 5aaa(x — y) (a; — y)
(x y)(x  y),
26. If a boy's mark on each of three recitations is 0, what is his
average on the recitations? Give the value of h + 0. Of 3 X 0.
01 1
27. Find the value of 8 X  5 + 2
REVIEW 91
2a Form the power whose base is 5 and exponent 2. Also the
power whose base is 2 and exponent 5. Find the difference in value
between these two powers.
29. Find the value of each of the following products and verify
each result for the values x = 2, a == 3, 6 ~ 1.
(1) af.x*» (2) aj*»+*. a:—* (3) af +*. aj»*
30. From the product of Zx^ ~ 2 and 2a; — 5 subtract 7 times
the product of x and x — 2.
31. Show on squared paper that 3x4+5x4=8x4.
Also that 4X5+7X52X5=9X5.
32. Multiply %x^ ax \ §a« by Ja;* + Jox + Ja«.
33. Divide x*".— 2/*" by x* — y*.
34. Divide 2— xbyl+xto five terms in the quotient.
35. Divide [(x«  2a;  1) (a:  1) + 2x2  2a;] by [(x + 2)
(x + 1)  (a* + 2a; + 3)].
36. Multiply 3.2a;« '4.5a;y + l.Sy* by 1.5a;  3.5y.
37. Divide a;* — 15 by x' + a; — 1 to five terms.
38. Divide 36a;» + iy* + i  4a;y  6a; + iy by 6a;  Jy  J.
39. Divide 2.4x»  0.12a;V + 4.322/» by 1.5a; + 1.%.
40. Divide a» + 6» + c»  Sabc by a + 6 + c.
Solve and verify
41. (2a; + 1) (a;  3) + 7 = a;  2(x  4) (2 x).
42. 7x  2(a;  1) (2  a;)  17 = x(3x + 7)  (x + 1)«.
Simplify:
43. 6« + [42  {8x  (22 + 4x)  22x}  7x]  [7x + {142 
(42 5x)}].
44. a\b  c)  &*(o  c) + d'ia  6)  (o  6) (a  c) (6  c).
45. What is the advantage of regarding a polynomial as made up
of terms? (What is a polynomial? A term?)
46. What is the name of an expression containing two terms?
Three terms?
47. Write a homogeneous expression containing three terms, and
the letters x and y.
92 SCHOOL ALGEBRA
48. If s = ar^^^, find the value of s when a « 2, r = 3, n = 4.
Also when o = 2, r — 1, and n = 6. When a = 3, r = , and n = 5.
49. Who first used the letters x, yy and z to represent unknown
numbers in equations in algebra? (See p. 455.) Find out all you
can about this man.
50. Give some of the other symbols that were .used to represent
numbers before the use of the three last letters of the alphabet was
suggested.
51. Can you point out any advantages in the use of x, y, and z
instead of the other sjrmbols once used for the same purpose?
52. Find out, if you can, whether any other symbols (than the
last letters of the alphabet are now used to represent an unknown
number in an equation? How many different S3anbols can be used
for this purpose?
53. Who invented the parenthesis sign and when?
54. How many examples in Exercise 13. (p. 54) can you work at
sight?
1
CHAPTER VI
EQUATIONS (conHnued)
66. The Equation^ members of ah equation, and transpo*
sition have abeady been explained. (See Arts. 4245, pp.
5253.)
67. A Boot of an equation is a number which, when substi
tuted for the unknown quantity, satisfies the equation; that
is, reduces the two members of the equation to the same
number.
Ex. If in the equation, 3x — 1 = 2x + 3,
we substitute 4 in the place of x in each member,
we obtain 3a:l=12l=ll
2x43= 8+3 = II
The equation is satisfied. Hence, 4 is the root of the given equation.
68. The Degree of an Equation having One Unknown
Quantity. If an equation contains only one unknown quan
tity, the degree of the equation (after the equation has been
reduced to its simplest form) is determined by the highest
exponent of the unknown quantity in the equation.
Thus, if X is the only unknown,
2x + l =5a;— 8isan equation of the first degree.
ax ^¥ \cx ]a of the first degree,
4x* — 5x = 20 is of the second degree.
3x«— x* =6x48isof the third degree.
A simple equation is an equation of the first degree.
An equation of the first degree is also often termed a linear
equation, for reasons which will be explained later. (See Art. 148.)
03
94 SCHOOL ALGEBRA
69. Identities and Conditional Eqnations. If we take the
expression (a; — 2) (a: + 2) — 7? — 4:, and substitute a; = 1,
we obtain — 3 = 3. The two members of the expression
are found to be equal.
Similarly, they are found to be equal if we let a: = 2, 3, 4,
etc.; 0, — 1, — 2, etc.; or any munber. An expression hav
ing this characteristic is termed an identity.
An identity (or identical equation) is an equality whose
two members are equal for all values of the unknown quantity
(or quantities) contained in it.
A conditional equation is an equation which is true for
only one value (or a limited number of values) of x. For
the sake of brevity, a conditional equation is usually termed
an equation.
The equations studied in Art. 42 (p. 52) and Exercise 13 (p. 54)
are conditional equations.
Hence, the sign = is used in two senses in elementary
algebra, viz.: to indicate sometimes an equation, and some
times an identity. The context enables us to decide readily
which of these two meanings the sign = has in any given case.
Later it will be found useful to use the mark = to indicate an
identity, and = to indicate a conditional equation, or equation
proper.
70. The Aids in Solving an Equation, given in Art. 15,
p. 18, stated more precisely, are as follows:
The roots of an equation are not changed if
1 . The same quantity is added to both members of the equation.
2. The same quantity is subtracted from both members of the
equation.
3. Both members are multiplied by the same quantity or equal
quantities (provided the multiplier is not zero, or an expression
containing the unknown).
EQUATIONS 96
4. Both members are divided by the sarm quarUity (provided
the divisor is not zero, or an expression containing the
unknown).
Other principles similar to these are used later as aids in solving
equations.
Transposition (see Art. 45, p. 52) is a short way of using Prin
ciples 1 and 2 of this article.
71. The Method of Solving a Simple Equation may now
be stated as follows:
Clear the equation of parentheses by performing the operations
indicated by them;
Transpose the unknown terms to the lefthand side of the
equation, the known terms to the righthand side;
Collect terms;
Divide both members by the coefficient of the unknoum quantity.
Ex. Solve x{x  2) = x{x + 4)  3(x  3) (1)
Removing parentheses, x* — 2x = x^ +4x — 3x +9
Transposing terms (Art. 70, 1, 2), x^ x^ 2x 4x \Sx =9
Collecting terms, . —3a; =9. ... (2)
Dividmg by 3 (Art. 70, 4), a; = 3 Root
Check, xix  2) =  3( 3  2) =  3( 5) = 15
»(a; 4 4)  3(a;  3) =  3( 3 + 4)  3( 3  3)
= 3 3(6) 3 +18 = 15
Solve the following; refer to each principle in Art. 70 as
you use it, and check each answer:
1. 2a; = 15  Zx. 6. 3a:  7 = 14  4a;.
2. 15 + 3a; = 27. 7. 2a;  7 = 8 + 5a;.
3. 4a;  11 = 29. 8. 2a;  (a;  1) = 5.
4. 16a; + 3 = 15a; + 7. 9. 2 ft. + a; = 12 ft.
5. 14a;  10 = 12a;  3a;. 10. 7 in. + a; = 2 ft.
96 SCHOOL ALGEBRA
11. a?  a:(« + 6) = « + 12. i3. 7(2  3*) = 2(7  8x).
, 12. 2*  3(x  3) + 2 = 0. 14. 3  2(335 + 2) = 7.
^15. (x 8) (x + 12)  (a; + 1) («  6) = 0.
16. 5(a;  3)  7(6  «) + 3 = 24  3(8  x).
17. 3(«  1) (a; + 1) = a;(3a; + 4).
18. 4(x  3)« = (2a; + 1)*.
19. 8(a;  3)  (6  2x) = 2(« + 2)  5(5  x).
20. 5«  (3«  7) {4  2a;  (6x  3)} = 10.
21. X + 2  [x  8  2{8  3(5  x)  x}] = 0.
22. 2x(x  6)  {x* + (3x  2) (1  x)} = (2x  4)».
23. 8x* + 13x2{x*3 [(x1) (3 + x)2(x + 2)^}  3.
24. .25x  2 = .2x + 3. 26. x + 6 = §x + 8.
25. 1.6x — .7 = 1.5x — .3. 27. ^x — I = f — fa.
2a (.2x + .2) (.4x  .3) = (,4x  .4) (.2x + .3).
29. What right have we to change the equation 3x = 15 — 2x
to the form 3x + 2x = 15?
30. If 2x — 3 = 5, what right have we to transpose the
— 3, and to write the equation in the form 2x = 5 + 3?
31. What is the advantage in being able to add the same
number to both members of an equation? To transpose a
term? To divide both members of an equation by the same
number?
32. Determine which of the following are identities aitd
which conditional equations, or equations proper:
(1) (x + 3) (x  3) = X*  9.
(2) (X + 3) (X  3) = (x + 1) (x  2).
(3) (x + 2) (x  1) = x* + X  2.
(4) (X + 2) (x  1) = x».
EQUATIONS 97
33. Write an identity and an equation of which the first
members are the same. ^
34. Prove that the sum of any three consecutive numbers
equals three times the middle one of the numbers.
SuQ. Let the three numbers be indicated by n, n — 1, and n — 2.
35. Find a similar result for the sum of five consecutive
numbers. Of seven consecutive numbers.
36. Prove that the product of the sum and difference of
any two numbers is equal to the square of the first, minus
the square of the second. Illustrate by a numerical example.
Sua. Denote the two numbers by a and 6.
37. Prove that the square of the smn of any two numbers
equals the square of the first number, plus twice the product
of the two numbers, plus the square of the second number.
Illustrate by a munerical example.
za. State and prove a similar property for the square of
the difference of two niunbers.
39. Prove that if the siun of the cubes of two niunbers is
divided by the sum of the numbers, the quotient equals the
square of the first niunber, minus the product of the first by
the second, plus the square of the second.
40. State and prove a similar property of the difference of
the cubes of two niunbers.
Find the value of the letter in each of the following:
41. 3a  2 = 7. 44. 24 = 12  3p.
42. 5  26 = 1. 45. 3(y  4) = 5(2  y).
43. 5(c  1) = 12  c. 46. r  3(r  1) = 5.
47. In A — Iw, it A = 42§ and I = 8, find w.
4a If ^ = 48.36 sq. ft. and w = 6.2 ft., find I.
98 SCHOOL ALGEBRA
49. Convert each of the two precedmg examples into an
example concerning areas.
50. If F = Iwh, and V = 504, I = 12, and A = 5, find w.
51. Convert Ex. 50 into an example concerning volumes.
52. In i = prt, if i = $27, r = .05, and p = $240, find t
53. Convert Ex. 52 into an example concerning interest.
<
54. So far as we know, who first used an equation to solve
a problem? Give this first problem thus solved, and tell all
you know about the dociunent in which it was found.
(See pp. 454 and 462.)
55. Form an equation whose root is 2 and which contains
four terms.
56. Make up and work an example similar to Ex. 15. To
Ex.31. Ex.46.
57. How many of the examples in this Exercise can you
work at sight?
72. Solution of Problems. In solving problems, the stu
dent will find it necessary to study each problem carefully
by itself, as no rule or method can be found which will cover
all cases. The following general directions will, however, be
found of service:
By study of the problem, determine what are the unknown
quantities whose values are to be obtained;
Let X equxd one of these unknown quantities;
State in terms of x all the other unknown quantities which are
either to be determined or to be used in the process of the solution,;
Obtain an equation by the itse of a principle (such as, the
whole is equal to the sum of its parts, or things equal to the
same things are equal to each other);
EQUATIONS 99
Solve the eqiuxtion, and find the value of each of the unknown
quantities.
In solving problems it is especially important to note that
weletx = a definite number, not a vague quantity.
Thus, in working Ex. 1 of Exercise 24
we do not let x » A's marbles,
nor X — what A has,
but let X « number of marbles A has.
73. Checking the Solution of a Written Problem. The
best way of checking the result obtained by solving a prob
lem is to observe whether the result obtained satisfies the
conditions as originally stated in the language of the problem.
(This method is better than that used in checking the example
in Art. 46, p. 53.)
Thus, to check Ex. 18, p. 54: after the answers 9 and 4 have been
obtained, we note that the difference of 9 and 4 is 5, and that the
sum of 9 and 4 is 13. 9 and 4 thus satisfy the original conditions of
theproblem.
What is the advantage in this method of checking the solu
tion of a problem?
EXERCISE M.
Oral
1. A has a: marbles, and B has twice as many. How many has B?
How many have both?
2. There are 100 pupils in a school, of which x are boys. How
many are girls?
3. If I have x dollars, and you have three dollars more than
twice as many, how many have you? How many have we together?
4. Two bojrs together solved a examples. One did x examples.
How many did the other solve?
5. The difference between two niunbers is 15, and the leas is x.
What is the greater? What is their sum?
100 SCHOOL ALGEBRA
6. If n is a whole number, what is the next larger number? The
next less?
7. Write three consecutive numbers, the least being x. Write
them if the greatest is y.
8. John has x dollars, and James has seven dollars less than
three times as many. How many has James?
9. If I am a; years old now, how old was I ten years ago? a years
ago? How old will I be in c years?
10. A man bought a horse for x dollars, and sold it so as to gain
a dollars. What did he receive for it?
U. A man sold a horse for $200, and lost x dollars. What did
the horse cost?
■
12. If a yard of cloth cost m dollars, what wiU x yards cost?
13. A boy rides a miles an hour; how far will he ride in c hours?
14. A bicyclist rides x yards in y seconds. How far will he ride
in one second? In n seconds?
15. In how many hours can a boy walk x miles at a miles an hour?
16. A man has a dollars and 5 quarters. How many cents has he?
17. How many dimes in x dollars and y halfdollars?
18. I have x dollars in my purse and y dimes in my pocket. If
I give away fifty cents, how much have I remaining?
19. By how much does 30 exceed x?
20. What number is 40 less than x? What number is x less
than 40?
21. What number exceeds a; by a? What number exceeds a by a;?
22. By how !much does a + 6 exceed x?
23. How much did a girl have left if she had $5 and spent 15^?
If she had a dollars and spent b cents?
24. A boy had a dollars, received b cents, and then spent c cents.
How many cents did he have left?
25. What is the interest on a dollars at b per cent fore years?
26. Express algebraically the following istatement: a divided by
b gives c as a quotient and d as a remainder.
EQUATIONS IQl
27. A man having x hours at his disposal, rode a hours at the
rate of 8 miles an hour, and walked the rest of his time at the rate
of 3 miles an hour. How far did he ride? How far did he walk?
EXERCISE 26
1. Separate $84 into two parts such that on^ part is three
times as large as the other.
2. Separate $84 into two parts such that one part exceeds
the other by $12.
a Separate $84 into three parts such that the first part
is twice as large as the second, and the second part is twice as
large as the third.
4. A boy has three times as many marbles as his brother,
and together they have 48; how many has each?
5. A and B pay together $100 in taxes; if A pays $22 more
than B, what does each pay?
6. Two boys made $67.50 one smnmer by taking passengers
on a launch. The boy who owned the launch received twice
as large a share of the profits as the other boy. How much
did each receive?
7. How many grains of gold are there in a gold dollar, if
the gold dollar weighs 25.8 grains and .9 parts of the dollar
are gold and 1 part copper?
a A ball nine has played 64 games and won 12 more than
it has lost. How many games has it won?
9. A man left $21,000 to his wife and four daughters. If
the wif6 received three times as much as each daughter, how
much did each receive?
10. If he had left $21,000 so that the wife received $10,000
more than each daughter, how much would each have
received?
102 SCHOOL ALGEBRA
11. A cubic foot of water and a cubic foot of alcohol to
gether weigh 112.5 lb. The alcohol weighs j as much as the
water. What is the weight of a cubic foot of each?
12. Find three consecutive numbers whose sum is 63.
13. In a certain grade of milk the other solids equal three
times the weight of the butter fat, and the liquid part of the
milk weighs 7 times as much as the solids. How many pounds
of butter fat in 4800 lb. of milk?
14. The difference of the squares of two consecutive num
bers is 43. Find the numbers.
15. At a certain date the record time for the quartermile
run was 47 seconds, and 5 times the record time for the 100
yard dash exceeded the record time for the quartermile by
1 second. Find the record time for the 100yard dash at this
date.
16. The difference of two numbers is 13 and their sum is
35. Find the numbers.
17. John solved a certain number of examples, and William
did 12 less than twice as many. Together they solved 96.
How many did each solve?
la Three boys earned together $98. If the second earned
$11 more than the first, aijd the third $28 less than the other
two together, how many dollars did each earn?
19. The sum of two niunbers is 92, and the larger is 3 less
than four times the less. Find the numbers.
^20. The siun of three niunbers is 50. The first is twice
the second, and the third is 16 less than three times the second.
Find the numbers.
EQUATIONS 103
21. A fanner paid $94 for a horse and cow. What did
each cost, if the horse cost $13 more than twice as much as the
cow?
22. Ex. 1 (p. 95) might be stated as a problem concerning
an unknown number, thus: Twice a certain niunber equals 15
less three times the niunber. Find the number.
In like manner, convert Ex. 2 (p. 95) into a problem con
cerning an imknown number. Also Ex. 3. Ex. 8.
23. In reducing iron ore in a furnace, 7 times as many car
loads of coke as of limestone are used, and 8 times as many
carloads of iron ore as of limestone. If 800 carloads in all are
used on a certain day, how many carloads of each is this?
24. One side of a triangle is twice as long as the shortest
side. The third side exceeds the length of the shortest side
by 12 yards. If the perimeter of the triangle is 360 yards, find
each side.
25. A man spent $3.24 for coffee and sugar, buying the
same number of pounds of each. If the sugar cost 5 cents a
pound and the coffee 22 cents, how many pounds of each did
he buy?
26. The distance from New York to Chicago is 912 miles.
If this is 24 miles less than four times the distance from
New York to Boston, find the latter distance.
27. On a certain railroad in a given year the receipts per
mile were $3085. If the receipts per mile for freight exceeded
those for passengers by $265, find the receipts per mile from
each of these sources.
28. A man left $64,000 to his wife, daughter, and niece.
To his daughter he left $4000 more than to his niece, and to
his wife $8000 more than to his daughter and niece together.
How much did he leave to each?
104
SCHOOL ALGEBRA
X
a+6
29. Find the number whose double exceeds 24 by 6:
30. The perimeter of a given rectangle is 26 feet, and the
length of the rectangle exceeds the
width by 5 feet. Find the dimen
sions of the rectangle.
31. The perimeter of a given
rectangle is 18 yards, and the
length exceeds the width by 3 ft. Find the dimensions.
Make up and work a similar example for yourself.
32. The length of a rectangle exceeds a side of a given
square by 3 inches and the width of the rectangle is 2 inches
less than a side of the
square. If the area of
the rectangle equals the ^
area of the square, find
a side of the square.
Suo. Denote the sides of the square and rectangle as in the
diagram.
Since the areas of the two figures are equal,
a;2 = (a; + 3) (x  2), etc.
In working Exs. 3338, draw a diagram for each example.
33. The length of a rectangle exceeds a side of a given
square by 8 ft. and the width of the rectangle is 4 ft. less than
a side of the square. If the area of the rectangle equals the
area of the square, find a side of the square.
34. If one side of a square is increased by 4 yd., and an
adjacent side by 3 yd., a rectangle is formed, whose area ex
ceeds that of the square by 47 sq. yd. Find a side of the
square.
35. The perimeter of a rectangle is 120 ft., and the rec
tangle is twice as long as it is wide. Find its dimensions. '
048
i .
EQUATIONS 105
36. A certain rectangle is three times as long as it is wide.
If 20 ft. is added to its length and 10 ft. is deducted from
its width, the area is diminished by 400 sq. ft. Find the
dimensions of the rectangle.
37. A rectangle is 5 ft. longer than it is wide. If its length
is increased by 4 ft., and it& width by 3 ft., its area is in
creased by 76 sq. ft. Find the dimensions of the rectangle.
38. A rectangle is 4 in. longer than it is wide. If its length
is increased by 4 in., and its width diminished by 2 in., its area
remains unchanged. Find the dimen^ons of the rectangle. .
39. Make up and work an example similar to Ex. 38. j
40. A tennis court is 42 ft. longer than it is wide. If a
margin of 15 ft. on each end and of 10 ft. on each side is
added, the area of the court is increased by 3240 sq. ft.
Find the dimensions of the court.
41. The length of a football field exceeds its width by 140
ft. If a margin of 20 ft. is added on each side and end of the
field, the area is increased by 20,000 sq. ft. Find the dimen
sions of the field.
. 42. A boy is three times as old as his brother. Five years
hence he will be only twice as old. Find the present age of
each.
43. A man is twice as old as his brother. Five years ago
he was three times as old. Find the age of each at the present
time.
44. How many pounds of coflFee at 30jf a pound must be
mixed with 12 pounds of coffee at 20jf a pound to make a
mixtiu^ worth 24ff a pound?
45. How many pounds of tea at OOff a pound must be
mixed with 25 lb. of tea at 40jf a pound, to make a mixture
worth 45ff a pound?
106 SCHOOL ALGEBRA
46. Make up and work an example similar to Ex. 45.
47. Find five consecutive numbers whose sum shall be 3
less than six times the least.
4a Find three consecutive odd niunbers whose sum is 63.
49. A telegram at a 252 rate cost 47 cents. How many
words were in the telegram?
Sua. A 252 rate means a cost of 25 cents for the first 10 words
and 2 ce;ats for each additional word.
50. Make up and work an example, similar to Ex. 49, con
cerning a telegram sent at a 403 rate.
51. A talk over a long distance telephone at a 507 rate
cost 85{5. How many minutes did*the talk last?
*
SuG. A 507 rate over a long distance telephone means a cost
of 50 cents for the first 3 minutes and 7 cents for each additional
minute.
52. A rectangle is 8 ft. longer than it is wide and the pe
rimeter is 120 ft. Find the dimensions of the rectangle.
53. If 5 is subtracted from a certain number and the differ
ence is subtracted from 115, the result equals three times the
given number. Find the number.
54. If J is added to double a certain fraction, the result is
the same as if f had been subtracted from three times the
fraction. Find the fraction.
55. What number subtracted from 100 gives a result fequal
to the sl^n of 14 and the number?
56. Find the number which exceeds 12 by as much as
three times the niunber exceeds 24.
57. Find five consecutive nxunbers such that the last is
twice the first.
EQtATlOMS 107
58. Find two consecutive integers such that the first plus
5 times the second equals 5^
59. A man is 48 years old and his son is IS. How many
years ago was the father four times as old as the son? Also
how many years hence will the father be twice as old as the
son?
60. Find two numbers such that their difference is 20^ and
one is four times as large as the other.
61. The length of a single tennis court exceeds the width
by 51 ft. If the width is increased by 9 ft., we have a double
court, the area of which exceeds that of the single court by
702 sq. ft. Find the dimensions of each court.
62. A boy sold a certain number of newspapers on Monday,
twice as many on Tuesday, on Wednesday 5 more than on
Monday, and on Thursday 7 less than oii Tuesday. If he
sold 310 newspapers on the four days, how many did he sell
on each of the days?
63. Twentyfive men agreed to pay equal amounts in
raising a certain siun of money. Five of them failed to pay
their subscriptions, and as a result each of the other twenty
had to pay one dollar more. How much did each man sub
scribe originally?
64. A boy starts from a certain place and walks at the
rate of 3 miles an hour. Three hours later another boy starts
after the first boy and travels on a bicycle at the rate of 6
miles an hour. How many hoiu^ will it be before the second
boy overtakes the first? (Draw a diagram.)
65. If the boys had traveled in opposite directions, how
many hours after the second boy started would it have been
before they were 81 miles apart?
108 SCHOOL ALGEBRA
ee. A boy was engaged to work 50 days at 75j5 per day for
the days he worked, and to forfeit 25jf every day he was idle.
Oh settlement he received $25.50; how many days did he
work?
67. Which of the above problems belong to, or are varia
tions of, Type I? Of Type II? III?
68. How many examples in Exercise 15 (p. 60) can you
now work at sight?
CHAPTER VII
ABBREVUTED MULTIPLICATION AND DIVISION
Abbreviated Multiplication
74. Utility of Abbreviated Hnltiplioation. In certain
cases of multiplication, by observing the character of the
expressions to be multiplied, it is possible to write out the
product at once, without the labor of the actual multiplica
tion. This is true of almost all the multiplication of binomials,
and that of many trinomials, and by the use of the abbre
viated methods at least three fourths of the labor of multi
plication in such cases may be saved. The student should
therefore master these short methods as thoroughly as the
multiplication table in arithmetic.
75. L Square of the Sum of Two Quantities.
Let a + 6 be the sum of any two algebraic quantities.
By actual multiplication, a + b
, a + b
a^ + ab
+ ab +V
a^ + 2ab + b^ Product
Or, in brief, (a + by = a^ + 2ab + ly^,
which, stated in general language, is the rule:
The square of the sum of two quardiiies equcds the square of
the first, plus twice the product of the first by the second, plus the
square of the second.
109
110 SCHOOL ALGEBRA
Ex. 1. (2a: + SyY = 43?+ 12xy + V Product
Ex. 2, 1042 = (100 + 4)2 = 1002 + 8 X 100 + 42
= 10,000 + 800 + 16 = 10816 Ans.
76. n. Square of the Difference of Two Quantities.
By actual mukiplication, a — b
a — b
a^ — ab
ab +
a^ 2ab + V Product
Or, in brief, (a  by = a^ 2ab + V,
which, stated in general language, is the rule:
The square of the difference of two quavtities equals the square
of the first, minus twice the product of the first by the second,
plus the square of the second,
Ex. 1. (2a;  ^yf = ^  12xy + 9f Product
Ex. 2. [{x + 2y) 5]2 = (a + 2y)2 10(a: + 2y) + 25
= a? + 4a:j/ + 4y2  lOx  20y + 25
Product
To check the work of Ex. 2, let a; = 2, y = 1.
Then [(x^ + 2y)  5p = (4 + 2  5)« = (1)« = 1 ,
Also
«* +4x1/ +42/» 10a: 2O2/ +25 =4+8+4 2020 +25 = 1.
EXERCISE 26
Write by inspection the value of each of the following and
check each result:
1. (n + yY 5. (5a; + 1)2
2. (ca;)2 6. (a: 2+ 1)2
3. (2a;y)2 7. (a; 2/2)2
4. (3a:  2y)2 8. (1  Tj/^)*
ABBREVIATED MULTIPLICATION
111
9.
(3a?* + 5a?y
18.
(1.5m  .02)2
10.
{&a?y  llyV)2
19.
[(a + b)+ 4p
11.
(5a:*  3y"2"»)2
20.
[(a + 6)  3p
12.
(4a?y^z^^ + V»)*
21.
[(a + b) + cp
13.
(i^ + hy
22.
[(2a  a:) + 3y]^
14.
aoh  f o:^)*
23.
[3 + (a + fc)]2
15.
{.2x + .3yy
24.
[5a  (a: + y)]^
16.
(.3a + .04fc2)2
25.
[2a2  (6  2c)p
17.
(.02a;  myy
26.
[(a: + y) _ (a + 6)p
27. Find the value of 998^ by multiplying 998 by itself.
'This product might also have been obtained in the following
way:
998*  (1000 2)« = [1000*  2 x 2 X 1000 + 2«]
« 1,000,000  4900 + 4
= 996,004
After practice the part of the work in the brackets may be omitted.
Compare the amount of work in the two processes of finding the
value of 998*.
By the short method obtain the value of:
1. 9992 31. 5P 34. 9962
29. 9972
30. 99982
32. 10032
33. 972
35. 99972
36. (99.2)2
37. Make up and work an example similar to Ex. 19. To
Ex.29. Ex.36.
3a How many of the examples in this Exercise can you
answer orally?
112 SCHOOL ALGEBRA
77. m. Product of the Stun and Difference of Two
Quantities.
By actual multiplication, a + b
a — b
a^ + ab
c? — y Product
Or, in brief, (a + 6) (a  fc) = a^  b",
which, stated in general language, is the rule:
The proctud of the sum and difference of two quardiiies equals
the square of the first minus the square of the second.
Ex. 1. {2x + Sy) {2x  3y) = ia?  9f Product
Ex. 2. Multiply x + (a + b)hy x  {a + b).
We have
lx + (a+ b)] [x  (o + b)] = x«  (a + b)\ by III.
= x»  (o« + 2a6 + 6^), by I.
= a?*  a*  2a6  6» Product
Let the pupil check the work.
It is frequently necessary to regroup the terms of trino
mials in order that the multiplication may be performed by
the above method.
Ex. 3. Multiply x + y — zhyx — y + z.
(« + y  2) (X  y + 2) = [X + (y  «)] [X  (y  Z)]
= a;2 _ (y _ 3)2^ by III.
= a;2 _ (2,2 _ 2yz + ««), by II.
 x^ — y^ + 2yz — z^ Product
Let the pupil check the work.
EXERCISE 27
Write by inspection the value of each of the following
products, and check the work for each result:
1. (x + z){x z) 3. (3x  y) (3a: + y)
2. (y  3) (y + 3) 4. {7x + 4y) {7x  Ay)
ABBREVIATED MULTIPLICATION 113
5. (a?  2) (a^ + 2) 9. (Ja + §6) (Ja  §6)
6. {cue' Vy){a^^ 6*y) lo. (2ix  \y) {2\x + \y)
7. (1  11a?) (1 + 11«») u. (.2x + .3y) (.2x  .3y)
a (2x« + 6y) (2a;»  5y") li (.05o«.3y)(.05o*+.36»)
M. (fa; + .7y) (fa;  .7j/)
14. (a"+» + J6"i) (o»+i  6»i)
15. [(o + 6) + 3][(a + 6)3]
16. [(« + y) + o] [(x + y)  o]
17. [(2a;  1) + y] [(2a:  1)  y]
la [4 + (a; + 1)] [4  (« + 1)]
ift [2a; + (3y  5)] [2a;  {Zy  5)]
20. (o + 5 + 3) (o + 6  3)
21. (a; + 2/ + o) (x + y  a)
22. (4 + a: + 1) (4  a:  1)
23. (2a: + 3y  6) (2a:  3y + 5)
24. (4 + a: + y) (4  a:  2/)
25. (ar« + 3a: + 2) (ar^ + 3a:  2)
26. (a + 6 + 3a:) (a + 5  3a:)
27. (a + 6  3a:) (a  6 + 3a:)
2a (a?  a:y + y^) (ar^ + a:2/ + y^)
29. (a2 + a + l)(a2a + l)
30. (2ar^  3x  5) (2a:2 + 3^. _ 5)
31. (2a:2 + 53.y _ ^^2) ^^2^ ^ ^^ ^ p
32. {^ + xy  y^) {7?  xy  y^)
33. [(a + 6)  (c  1)] [(a + 6) + (c  1)]
34. [(a:^ + J/2) + (a:V + 1)] [(a:^ + 2/') ~ ix'f + D]
35. (a: + y + 2 + 1) (a: + 2/  2  1)
114 SCHOOL ALGEBRA
36. Work Ex. 16 in full (see Art. 54, p. 65). How much
of this labor is saved by the short method of multiplication?
37. Make up and work an example similar to Ex. 36.
3a Multiply 93 by 87. This product may also be obtained
thus:
93 X 87  (90 + 3) (90  3)
 8100  9 » 8091
Compare the amount of work in the two processes.
39. Make up and work an example similar to Ex. 38.
Find the value of each of the following in the short way:
40. 92 X 88 43. 1005 X 995
41. 103X97 44. 1032972
42. 105 X 95 45. (17.31)2  (2.69)2
Find in the shortest way:
46. The area of a rectangle 102 ft. long and 98 ft. wide.
47. The cost of 32 doz. eggs at 28>5 per dozen.
4a The cost of 67 yd. of cloth at 73jf a yard.
49. Make up and work two examples similar to Exs. 4748.
5a Work Ex. 40 in full. How much of this labor is saved
by Using the short method of multiplication?
Write at sight the product for each of the following miscel
laneous examples:
61. {x + 2a)2 58. [{x + 2y) + 5]2
52. (a + 2a) (a:  2a) 59. (a; + 2y + 5) (x + 2y5)
53. (x — 2o)* 60. (.3a; + .5y) (.Sk — .5y)
54. (3a;  1) (3a; + 1) 61. 998«
55. (3a;  1)* 62. 998 X 1002
56. (3o*26»)2 63. 97*
57. (3o«  26») (3o« + 2i») 64. 97 X 103
ABBREVIATED MULTIPLICATION 115
65. Make up and work an example in each principal fonn
of abbreviated multiplication studied thus far.
66. How many of the examples in this Exercise can you
answer orally?
78. IV. Square of any Polynomial
By actual multiplication^
a + b + c
a + b + c
a^ + ab + ac
+ ab +b^+ be
+ ac + be + c?
a^ + 2ab + 2ac + b^ + 2bc + c^ Product
Or,in brief, {a + b + cf ^ a^ \V + (? + 2ah \2ac + 2bc.
In like manner we obtain
{a + b + c + d)^^a^\V + (? + d? + 2ab + 2M + 2ad
+ 26c + 26d + 2ci
Or, in general,
The square of any polynomial equals the sum of the squares
of the terms plus tvdce the product of each term by each term
which follows it.
It is often useful to indicate the order in which the products of
the terms are taken as shown in the following diagram. (If the
curved lines joining the terms are drawn as each product is taken,
the nmnbers on these lines may be omitted.)
:. (a —
Ex. (a  26 + c  Sxy = o« f 46* + d» + 9x»  4a6 + 2ac  6aaj
I N 4 ^
5
46c+ 126a;  6cx.
Let the pupil check the work.
116 SCHOOL ALGEBRA
Find in the shortest way the value of the following and
check each result:
1. {2x + y + iy 8. (2a« + 5a3)2
2. {x2y + 2zy 9. {xy + ziy
3. (3a: 22^ 5)* lo. (2a; + 3y  4z  5)*
4. (2a6 + 3c)2 u. (3Q?'4a^ + x2y
5. (a;2y32)« 12. Hx"  ^x + 5^
6. i4x + Sy ly 13. (It? i^ + ix + 6)«
7. (a?a; + l)* 14. (.2a + .36  .Sc)^
15. Expand (2a — 36 + c — 4d)* by multiplying in full.
Now obtain the same result by the method of Art. 78, p. 115.
About how much of the work of multiplication is saved by
the latter method?
16. Make up and work an example similar to Ex. 15.
Write at sight the product for each of the following mis
cellaneous examples:
17. (2o36)« 21. (dx' + f)*
la (2a + 36) (2a  36) 22. (Sa?  ^y'y
19. (2 + 36)« 23. (^ + 2y 3) (« + 2y + 3)
20. (3a?  y») (3*2 + y») 24. (x + 2y3)*
25. (4x + iaiy
26. (4a; + ia  f ) (4a: + §0 + I)
27. (a;»+* — 3a;»'y)*
28. (x»+*  3a;»^) (a;"+' + Sx^^y)
29. (.02**  .3x + .5)*
>J
x + a "'^
x + b
01? + ax
+ bx + ab
ABBREVIATED MULTIPLICATION 117
ao. Make up and work an example in each principal form
of abbreviated multiplication studied thus far.
79. V. Product of Two Binomials of the Form x + a,
x + b.
By actual multiplication^
x + 5 X — by
g +3 x + Zy
a? + 5a; ^ — bxy
+ 3g + 15 + Sary  15y^
a? + 8a; + 15 7?  2xy  Ibj^ a? + (a +'h)x + ab
By comparing each pair of binomials with their product,
we observe the following relation:
The Tproduct of two binomials of the form x + a and x + b
consists of three terms:
The first term is the square of the first term of the binomials;
The last term is the product of the second terms of the
binomials;
The middle term consists of the first term of the binomials
with a coefficient equal to the algebraic sum of the second terms
qf the birurmials,
Ex. 1. Multiply a:  8 by a; + 7.
8 + 71. 8X7 56.
.'. (a?  8) (x + 7) = x*  a?  56 Product
Ex. 2. Multiply (x  6a) (x  5a).
( 6a) + ( 5a)   11a. ( 6a) X ( 5a)  + 30a«.
.*. (x  6a) (a?  5a)  x*  llaa; + 30a« Product
Ex.3. MuItiplya:4y + 6bya: + y2.
(» + y + 6) (» + y  2) = [(a; +y) +6] [(x + y)  2]
= (a? + yy + 4(a; +y)  12 Product
118 SCHOOL ALGEBRA
KXBBCISB 29
Write the product for each of the following and check each
result:
1. (x + 2) (« + 5) 10. (a; + .2) {x + .5)
2. (a;5)(x3) ll. (« + J) (x + i)
3. («  7) (x + 4) 12. (x + .02) (x + 5)
4. (x  4) (x + 8) 13. (o + .02) (a + .5)
5. (x + l)(x7) 14. (xi)(x + i)
6. (x^  2) (x»3) 15. (a + f) (a  h)
7. (x* + 3)(x*+l) 16. (oft + x) (a6 + 3x)
a (o + 3x) (o  lOx) 17. {ah + x) {ah  3x)
9. {x7y){x + y) la (xy  7a*) (xy + 38*)
19. (a:* + 5) (a;«  5)
20. [{x + y) + 3] [{x + y) + 5\
21. [(a: + y)3][(a: + y) + 5]
22. (a; + y3)(a; + y + 5)
23. (a + 26 + 5) (a + 26 + 3)
24. (2a: + 3y + 3a) (2a: + 3y  5a)
25. (a: + a + 6) (a: — a — 6)
26. (2a: + a + 36) (2x  a  36)
27. (2a: + a  36) (2x  a + 36)
28. Find the product of a: + y + 6 and x + y — 3 by
multiplying in full. Then find the same product by the
method of Art. 79. About how much of the work of multi
plication is saved by use of the latter method?
29. Make up and work ah example similar to Ex. 27.
ABBREVIATED MULTIPLICATION 119
30. A building lot is 167 ft. wide and 213 ft. deep. If the
width and depth of the lot are each increased by 1 foot, find
the increase in area without multiplying 167 by 213.
Write at sight the product for each of the following mis
cellaneous examples: ^
31. (or + 5) (a:  5) . 35. {x + 5) («  3a)
32. {x + 5)* 36. {x  3a) {x + 3a)
33. {x — 5)^ 37. {x  3a) {x + 5a) .
34. (a; + 5) (a; 3) 3a {x  baf
39. (a; + y + 5a) (a; + y — 5a)
40. {x + y + baf
41. (a; + y + 5a) (a: + y + 3a)
42. (x + y + 5a) (a: + y  3a)
43. (a:2 + §a; + 3)2
44. (a? + ia: + 3)(x2 + j^_3)
45. {x'\\x + Z){^ + hxS)
46. {o? + ax + ^) {a^ " ax '\ 7?)>^
47. Make up and work an example in each principal form
of abbreviated multiplication studied thus far.
48. How many of Exs. 3145 can you answer orally?
80. VI. Product of Two Binomials whose Corresponding
Terms are Similar.
By actual multiplication^
2a 36
4a + 56
8a2  12a6
+ 10a6  156^
8a2  2a6  156^ Product
120 SCHOOL ALGEBRA
We see that the. middle tenn of this product may be ob
tained directly from the two binomials by taking the alge
braic smn of the cross products of their terms. Thus,
(.+ 2a) ( + 56) + ( 36) ( + 4a) = lOab  12a6 = 2a6.
Hence, in general, •
The product of any two binomiala of the given form consists
of three terms:
The first term is the product of the first terms' of the binomials;
The third term is the product of the second terms of the
binomials;
The middle term is formed by taking the algebraic sum of
the cross products of the terms of the binomials,
Ex. Multiply 10a: + 7y by 8a;  Uy.
To show the method of obtaining the middle term of the
product, we write the given expression in the form
(idx + 7y){Sx  ily)
Hence,
(lOc) ( Ily) + (7y) (&r) =  llOxy + mxy =  5ixy
.'.. (lOc + 7y) (8x  Ily) = SOc*  5ixy  77y« Produd
EXERCISE 30
Write at sight the product of each of the following and
check each result:
1. (2x + 3) (a: + 4) 7. (5a;  1) (a; + 7)
2. (2a;  3) (a;  4) 8. (x + 3y) (3a;  8y)
3. (2a; + 3) (a; 4) 9. {Sa^ + b) {4a^  5b)
4. (2a; 3) (a; + 4) lo. (a; + i) (fa; + J)
5. (3a + 5) (2a + 3) ll. (a + .26) (2a  .36)
6. (3a 5) (2a + 3) 12. (ia; + f a) (fa;  Ja)
ABBREVIATED MULTIPLICATION
121
13. How many examples in Exercise 9 (p. 41) can you
now work at sight?
EXERCISE 31
Review
Write at sight the value of each of the following and check each
result:
1. (2a;+3)»
2. (2x  3)«
3. (2x + 3) (2a;  3)
4. (x + 3) (x  6)
5. (2x + 3) (3x  6)
e. (z+ 3y) (x + 2y)
7. (2x + 3y)«
a (2x + Sy) (3x  4y)
9. (2x  3y)»
10. {2x + 3y) (2x  Sy)
11. (Sa  3x) (4a + 5x)
12. dTx + 3y«)«
13. (5a« + 3y») (6a*  Sy»)
14. (a» + 3x) (a«  5x)
15. (2a + 3x + 5)«
31. Why is it that the result of expanding {— 2x — 3y)' is the
same as that of expanding {2x + SyY ?
32. Give two expressions sunilar to those in Ex. 31 for which
the product is the same.,
33. Why is (a — 6)* equal to (6 — a)* ? Make up two expressions
similar to these.
34. Make up and work an example in each principal form of
abbreviated multiplication studied thus far.
16. (
;2a + 3x + 5) (2a + 3x  5)
•
17. (
;2o + 3x + 5) (2o + 3x  3)
la (
:i  2x  3x« + x»)»
19. (
[a +b+x+y){a+b xy)
20. 1
[a* +ax + x*) {a* ax + x»)
21. 1
;«•+»  x^»)*
22. (
:4o» + 2o + 1) (4a«  2o + 1)
23. {
[Sax"  2o">x)*
24. 1
[x» + 2xj/»i)*
25. 1
[1  a)»
26. 1
[oD* y,
27. 1
:2x+3j/)*
2a 1
;2x3i/)«
29. I
[ab)ia+b}
30. (
:x+3)(x3)
122 SCHOOL ALGEBRA
Simplify, using the methods of abbreviated multiplication as far
as possible:
35. (a + 26)» + (a  26)*
36. (a + 26)*  (a  26)*
37. (2x  1)* + (1  2zy
38. {2x  1)»  (2a: + 1)*
39. (3a  1)* + (2  3a) (2 + 3a)
40. (2a;  7y) (2x + 7y)  4(a;  2y)* f 13j/ (5y  x)
41. (3a;* + 5)* + x* (10  3a;) (10 + 3a;)  (5 h 13a;*)*
42. (a  c + 1) (a + c . 1)  (a  1)* + 2 (c  1)*
43. (x +y xy) {x y xy) \x^  (x  y*) (a; 4 y*)
44. Show that a* ^ {a+ 6) (a  6) 4 6*.
45. By use of the relation proved in Ex. 44, obtain the value of
(7J)* in a short way.
SuG. We have (7J)* = (7J + J) (7J  i) + (i)»
= 8x7+i=56i Ana.
Using the method of Ex. 45 find the value of :
46. (8J)* 49. (15i)* 52. (7.5)* 55. (75)*
47. (19})* 50. (49i)* 53. (19.5)* 56. (195)*
48. (199})* 51. (99})* 54. (99.5)* 57. (995)*
58. (9.7)* (Use (9.7)* = 10 X 9.4 + .3*)
59. (9.8)* 60. (9.6)* 61. (4.8)* 62. 98*
63. Find the value of (a + 6)' by multiplication. Examine the
result obtained. Make a rule for obtaining similar products in a
short way. Treat (a — 6)' in the same way.
64. By use of the rule obtained in Ex. 63, write out by inspection
the value of {x + y)'.
65. Also of (a  x)\ 66. Of (6 + j/)».
Solve the following equations, using methods of abbreviated
multiplication wherever possible:
67. (2a;  1)*  4x* = 19
I (2a; + 1)*  (2a;  1)*  16
ABBREVIATED DIVISION 123
Compute in the shortest way:
69. The area of a field 103 rd. long and 97 rd. wide.
70. The area of a square field each side of which is 98 rd.
71. The cost of 62 yd. of cloth at 58^ per yard.
72. The cost of 85 A. of , land at $95 per A.
73. How many of the examples in this exercise can you work at
sight?
Abbreviated Division
81. Utility of Abbreviated Division. In certain cases
much of the labor of division may be saved by the use of
meclianical rules. We discover these rules by performing
the division operation in a typical case, notiitg the relation
between the quantities divided and the quotient, and for
mulating this relation into a rule.
82. I. Division of the Difference of Two Squares.
Either by actual division, or by inverting the relation of
Art. 77, p. 112, we obtain
_— = a — and = a + 6.
a + b a — b
Hence, in general language.
The difference of the squares of two quantities is divisible by
the sum of the quantities, and also by the difference of the quarts
tities, the quotients in the respective cases being the difference of
the quantities and the sum of the quantities.
Ex. 1. y ^y = 2x{:Sy Quotient
2x — oy
Ex. 2. "^ 7 i^ t ?r = a:  (a + 6) Quotient
X + {a + b)
Let the pupil check the work in these examples.
124 SCHOOL ALGEBRA
KZEBGISE SS
Write at sight the quotient for each of the following, and
check each result:
, o''a ^ , a%*  36c«d»
W + 6c»d*
.250"  .16^
.5o— .45
.04y' .Oy
.2a; + .3y
a?  .256*
3.
a — X
94a;*
3 2a;
a?8l3/*
25a?  36y*
5x  ,6j/*
16a;* 4V
25aJ«  y"
8.
10.
11.
12.
13. Divide a^ + 2a6 + P  4a? by a + 5  2ar by long
division. Write the result of dividing (a + 6)^ — 4a:^ by
a+6 — 2x by the method of Art. 82. Estimate how much less
the labor of the second process is than that of the first.
14. Make up and solve an example similar to Ex! 13.
Obtain in the shortest way the quotient for each of the
following:
^j, ix + D'  g' la («  ^>'  ('^  D'
X
+ 1 +
d?
 (6  2c)«
a 
(b2c)
43*
(f + iy
16. iz 4 19.
17. . . .... /■ ■ 20.
(a  6) + (c  1)
1  (o + 6  c)^
1 + (a + 6  c)
(2a + Sby  (5a;  4y)*
2a;* + (j/* + 1) (2a + 36)  (5a;  4y)
ABBREVIATED DIVISION 125
Write a divisor and quotient for each of the following:
22. = 25. ^^ ■ ^^ — —^^
23. — = 26. ^^ ' —
Find two factors for each of the following:
27. 2500 16 29. 2491 31. 99.19
28. 2484 30. 9919 32. 6319
33. Divide a^ — Vhy a — b. Divide (a — by by (a — 6).
34. Find the difference in value between {x + yy and
a? + 2/2, when x = 2 and y = 3.
83. n and IH Division of Sum or DifEbrence of Two Cubes.
By actual division we can obtain,
a^ + V €? V
—^ = a2  oj + 62 and r ^a^ + ab + V.
Hence, in general language.
The sum of the cubes of two quavJtities is divisible by the sum
of the quantities y and the quotient is the square of the first quavr
tity, minus the product of the two quantities, plus the square of
the second quantity; also
The difference of the cubes of two quantities is divisible by
the difference of the qiuzntities, and the quotient is the square of
the first quantity, plus the product of the two quantities, phis the
square of the second.
126 SCHOOL ALGEBRA
8a^  27y» _ (2a;)»  (3y)'
2a;  3y 2a;  3y
= (2x)* + (2«) (3y) + (Zy)*
= 4a^ + Qxy + Qy* Quotient
Ex.2. ^^±f = (a_6)«_3(a6) + 9
=a22a6+6*3a+36+9 Qmiieni
Let the pupil check the work in these examples.
EXIBCI8E 88
Write at sight the quotient for each of the following and
check each result:
1. — r^r 6. ^
3.
+ 2
a»l
x1
27a? 64
3x4
1 + ^
l + 2sf
125  «»
7.
9
10.
ofi + 1^
a? + f
.008ai»y»
.2a;  y
5 a? ia + lb"
11. Divide 8a« + 276^ hy 2a + 36 by the method of long
division. Now write out the same quotient by the method
of Art. 83. Estimate how much of the labor of division is
saved by using the second method of obtaining the quotient.
12. Make up and work an example similar to Ex. 11.
13. Treat (a + 6)' — &c* divided by (a + 6) — 2a; as in
Ex. 11.
ABBREVIATED DIVISION 127
Obtun in the shortest way the quotient for each of the
following:
c* + (1  «)» (g  1)»  afr
c + (lx) (a  1)  a?
, 8  (a; + y)« Sa^ + («»  D'
2  (x + y) 2a: + a^  1
27i« + 12V 8(x  w)»  a»
16. ., . . . / / 19 '^
3a? + 5y» *•• 2(« y)z ,
Write the binomial divisor and the quotient for
So^a? 8o» + 1
aa = 24. =
Scfi27a? a* + j^
21. = 25. — 
«» + 86» ^ + y^
22. = 26. — =
23. = 27. ^ ^ =
^ (g + 6)» + (a; + y)' _
find a factor of each of the following:
29. 20« + 3» 31. 8027 33. 125027
30. 8000 + 27 32. 7973 34. 124973
35. Divide a' — 6* by a — 6. Also divide (a — 6)' by
a — 6.
36. Find the difference in value between a? + ^ and
(x + y)' when x = 2 and y = 3.
Write a binomial divisor and the corresponding quotient
for each of the following miscellaneous examples:
37. = 38. =
128 SCHOOL ALGEBRA
39. ^ = 46. = =^ =
42. lQ«*9 ^ ^, 8(» ^yf^if ^
^ 8a'27y» ^ • ^ a*  9(x  y)« _^'
^ «« + l_ _, 27a»  (X  y)» _
44. = 91. ^
45. ^1 :^ 52. (;g + y)^  (a:  y)^
53.
(g; + 1)» + (a: ~ 1)» _
54. How many of the examples in this Exercise can you
work at sight?
55. How many examples in Exercise 1 (p. 8) can you
now work at sight?
84. 17, 7, and 71. Division of Sum or Difference of any
Two Like Powers.
By actual division we can obtain.
a + 6
= a' — a^fc + oft^ — 6* QuotieTd
=^ a^ + a^b + abl^ + V Quotimt
a — b
a* + 6* is not divisible by either a + 6 or a — 6. But
^' + ^ = a^  a% + a'b'  al/+ ¥ Qiu)tieTd
a + b
a — b
= a^ + a?b + a^b^ + ab^ + ¥ Quotierd
ABBREVIATED DIVISION 129
\ Hence,
The difference of two like even powers of two quantities
is divisible by the sum of the quantities, and also by their
difference;
The snp of two like odd powers of two quantities is divisible
by the snm cf the quantities;
The difference of two like odd powers of two quantities is
divisible by the difference of the qmrdities.
For the quotient in all these cases —
"^ (1) The number of terms in a quotient equals the degree
of the powers whose sum or difference is divicjed ;
(2) The terms of each quotient are homogeneous (since
the exponent of a decreases by 1 in each term, and that
of h increases by 1 in each term).
^ (3) If the divisor is a difference, the signs of the quotient are
all plus; if the divisor is a sum, the signs of the quoOent are
aUemately plus and minu>s.
In the above statements the parts in italics should be
committed to memory.
The last statement forms a general rule for signs of a
quotient in all the cases of abbreviated division, including
IVI.
^ J ^2^ + f _ {2xf + f
■ " 2a;+y 2ac + y
= {2xY  {2x)^y + (2a:) V  {2x)ji' + y*
= 16a^&r«y + 4«*j/2«2a;2/» + 2^ Quotient.
a^ + 7P _ (g^)^ + {7?f
= (a2)4(a2)8(a^) + (a2)2(x2)2(a2) {^x^f + ^x^f
^a^a^^+d^a^+x^
130 SCHOOL ALGEBRA
EXERCISE 84
Write at sight the quotient for each of the following and
check each result:
^ a« + 32
5.
6.'
7.
1
a« + «»
A«
o + a;
9
asx»
m*
a — X
«
b' + f
O.
b + y
A
Vy'
+ 2
o^128
o 2
a:»l
x 1
3:^ + 1
9.
Z23i?t
2xy
lO
o" + «"
xv.
o + a;
n
^ yH
AX«
ar'y
TO
243 o».
6 — y x + 1 3 — a
13. Divide 32a^ + x^ by 2a + a: by the method of long
division. Now write the same quotient by the method of
Art. 84, p. 128. Estimate how much of the labor of division is
saved by using the second method.
14. Make up and work an example similar to Ex. 13.
Write a binomial divisor and the corresponding quotient
for each of the following:
15. ^ = 19. =
16. ^ = 20. ■ =
•
Obtain a factor of each of the following:
23. 100,001 25. 100,032
. 100,243 26. 99,757
ABBREVIATED DIVISION 131
27. Divide a^ + ¥ by a + b. Also divide (a + b)^ by
a + b.
28. Find the difference in value between o^ — j/* and
(z — y^, when a: = 3 and y = 2.
EXERCISE 36
Review
Write at sight the quotient for each of the following:
 62^2 ^ 62_4a;2 a:88(a+6)8
±. r O. ; — —
6 7* X 6 + 2x
« 5»  x3 \ 63 _ 8x3
0— X — 2x
^ 6«  x« o ^'  32x8
b X 622
'6+2 ' b +2x
5 ^+a^ 10 x'  4(0 + by
' b +x ' X 2(0 +b) ' jfi + y*
16. In Ex. 11 remove the parenthesis in the dividend and divisor,
and divide by long division. The work required is about how
many times that required in the abbreviated process?
Write a binomial divisor and the corresponding quotient for each
of the following:
17. ^8y* ^ 24 o'  Mx + yy ^
^ y4g« .^ ^g g* + 8(2 + yy ^
^ y+Sy* _ 2g a»  8(2 + yy ^
20. «^^ . 27. §^±i^ =
22. ^'» = 29. 32«°  y° ,
23. 21^  30. «^y' 
mmIi*
X  2(a + b)
12.
8x3 a'
2x a •
13
27cfi  8(x + t/)3
X<9.
3a2  2(x + 2/)
14
a.12 _^. y9
x*+2/»
15.
xi2  2/8
132 SCHOOL ALGEBRA
Divide each of the following by x — a in a short way:
31. a:»  a» + x»  o* 34. 3(x»  c^) + 4(x  o)
32. a^  a» + 5(x«  o«) 35. (x  a)» + 5(x»  a«)
33. x»  o» + 6(x  o) 36. 7(aj  a)» + 6(aj  a)
Find the value of each of the following in the shortest way:
37. (a + 6) (a + h) (a  6) (a  b)
38. (a + 26) (a + 26) (a  26) (a  26)
39. (3aj  ?y) (ar  2y) (3x + 2y) (3x +2y)
Simplify:
40. 5x  3(aj  2)«  3(3  2x) (1 +x) .
41. 7  6(x  2)«  3(3  2x) ( x)
Solve:
42. (x  8) (» + 12)  (x + 1) (x  6) 
43. (2x  1) (x + 3)  (x  3) (2x  3)  72
46. Four times a certain number diminished by 12.07, equals
twice the number increased by 1.13. Find the number.
. 47. Separate 1000 into three parts such that the second part is
three times as large as the first part, and the third part exceeds the
first part by 100.
^ . 4a The Suez Canal is 100 miles long. This is 2 miles more than
8 times the length of the Simplon Timnel. Find the length of the
tunnel.
49. The temperature of the electric arc is 5400® F. This is 464°
more than 8 times the temperature at which lead fuses. Find the
temperature at which lead fuses.
50. The velocity of sound in the air is 1090 ft. per second. This
rate is 10 ft. more than 9 times the rate at which sensation travels
along a nerve. Find the rate at which sensation travels. How
does this compare with the velocity of an express train going 60
miles per hour?
51. Who fiist used the aigo + to denote addition, and when?
Oee p. 467.)
abbrewiaTed division 133
52. Give some other S3n3ibols used to represent addition before
the sign + was invented. Discuss as far as you can the relative
advantages of these signs.
5a Answer the questions in Ex. 52 for the subtraction sign.
54. Answer the questions in Ex. 52 for the sign X.
59. Answer the questions in Ex. 52 for the sign f*.
55. Answer the questions in Ex. 52 for the sign «•
CHAPTER VIII
FACTORING
86. The Factors of an expression (see Art. 11) are the
quantities which, multiplied together, produce the given
expression.
Factoring is the process of separating an algebraic expres
sion into its factors.
86. Utility of Factoring. If it is known that
7? 8a; + 15 = ( a:  3) ( x  5)
and 2ar^  13a: + 21 = (2a;  7) ( a;  3)
Then ^^ 8a; + 15 ^ ( a;  3) ( a;  5) ^ a;  5
2a^  13x + 21 (a;  3) (2a;  7) 2a;  7
The above reduction of a fraction to a simpler form illus
trates the usefulness of a knowledge of factoring in enabling
us to simplify work and save labor.
Why do we now proceed to make definitions and rules and
to divide the topic. Factoring, into cases?
87. A Prime Quantity in algebra is one which cannot be
divided by any quantity except itself and unity; as a, 6,
a" + h\ 17.
In all work in factoring, prime factors are sought, imless
the contrary is stated.
88. Perfect Square and Perfect Cube. When an expres
sion is separable into two equal factors, the expression is
134
TfACTOKlISG 135
called a perfect square, and each of the factors is the square
root of the expression.
Thus, 9aV = Zax^ • Sox*.
/. 3ax* is the square root of 9aV.
Also, x* — 4a? H 4 = (x — 2) (x — 2), and is therefore a perfect
square, with a;  2 for its square root.
When an expression is separable into three equal factors^
the expression is called a perfect cube, and each of the factors
is its cube root
Thus, 27a^7^^ = SoxV • 3arV * SoxV
.'. 3axV is the cube root of 27a^x^^.
89. The Factors of a Monoinial are recognized by direct
inspection.
Thus, the factors of 7aV are 7, a, a, x, x, x,
90. Factors of Polynomials. Multiply 4a:^ + 2xy + j/^ by
4a;^ — 2xy + y^. What terms are canceled in adding the
partial products? Because these terms have been thus can
celed and have disappeared, it is diflScult to take the final
product 16a^ + ^a^ + y* and from it discover the original
factors which were multiplied together to produce it.
Hence, in factoring polynomials various methods must be
devised to meet different cases, and the cases must be care
fully discriminated.
Case I
91. A Polynomial having a Common Factor in all its Terms.
Ex. Factor 3ar^ + 6a;.
3a;« + 6x = 3a;(x + 2) FoxiUyrs
At first, in working examples of this kind, it is well to put the
work in the following form:
3x )3a;' + to = 3x(x + 2) FadUyrs
. X +2
136 SCHOOL ALGEBRA
Check by Substitution If we let x » 2
dx(x+2) =6(2+2) 24
Also 3x* + to  12 + 12  24
Check by Multiplication
X +2
Sx
3x* +&C
Hence, in general,
Divide all the terms of the polynomial by their largest common
factor;
The factors vriU be the divisor arid qtiotient.
86
Factor the following and check the work for each example
either by substitution or by multiplication, or by both as
the teacher may direct:
1. 22? + 53? 6. 3aV  15aV ll. f a^fc  f oi^
2. a:"  2a; 7. 18«^  27a?y 12. ^aa?  2a;*
3. a;* + q? a, a? — a? — a? 13. •2a:' + .4aa?
4. 3a*  a 9. a*x  2aV 14. .02aa?  .4aV
6. 7a + 14a* 10. ^2? + ia? 15. 1.2mn — .6m*
16. 3a*  6aa; + 9a:* 19. a*6»c  a« W + 2a*6V
17. 2a: + 4a:*  6a:» 20. 2xy  8a:*y + Ga?"^
la 10a»6*  35o»6» a. o"6»c*» + llo"6»c*"+i
22. 7(a + 6)x + 5(a + 6)y
23. 7(o + 6)a:*y + 5(a + 6) V
24. 21 (x  yY  14(x  yy
29. 9(2x  a) » 12(2a;  o)»
FACTORING 137
In the shortest way find the value of
26. 847 X 915  847 X 913
27. 312.75 X 87  312.75 X 84
28. 8 X 11 X 232 + 7 X 11 X 232  5 X 11 X 23?
29. Trip __ ^^ when tt = ^, 2J = 8, and r = 6.
3a Trip  7rr2 when tt = ^^, /J = 410, and r = 60.
Find the value of a; in the following equations:
31. ax + hx — 10. (What does the value of x become
when a = 5 and 6 = 15?)
32. ox = 10 — 6a:
33. oo: + 6a; + car = 12
34. 2aa: — 6aj + 3ca: = 15
Factor the numerator and denominator of each of the fol
lowing fractions and then simplify the fraction by canceling
factors:
,. 3a*66a62 ^„ aV
35. — — 1^ 37.
30^6 + 606* 4ar*  6a;«
3^ a:» + 2a;^ 3^ Zyq  ^(f
Z^  6a? \2f(i^  6pg
39. From an examination of Exs. 2638, state the uses or
advantages of being able to factor by the method of Case I.
40. How many of the examples in this Exercise can you
work at sight?
Case II
/
92. A Trinomial that is a Perfect Sqnare. By Arts. 75
and 76 a trinomial is a perfect square when its first and last
terms are perfect squares and positive, and the middle term is
twice the product of the square roots of the end terms. The
138 SCHOOL ALGEBRA
sign of the middle term detennines whether the square root
of the trinomial is a sum or a difference.
. Ex. 1. Factor 16r^  24a:y + V.
16x2  2ixy + V = (4a?  3y) (4a;  3y) Factors
Ex. 2., Factor (a + by + 4(a + b)x + 4r^.
(a + by + 4(a + b)x + 4x* = [(a + 6) + 2a;p
= (a + 6 + 2a;)* Ans.
Hence, in gen€!ral,.to factor a trinomial that is a perfect
square,
Take the square roots of the first arid last terms , and conned
these by the sign of the middle terrri^
Take the resvU as a factor tvnce.
EXERCISE 37
Factor and check:
1. 4x^ + 4xy + y^ 9. a^ + 2a^ + c?
2. 16a2  24ay + 9y^ lo. 4a? + Ux^f + 121xi/^
3. 25r^  lOx + 1 11. Sla^b + 126a^b^ + 49a6»
4. a;2  20xy + lOOy^ 12. 8a^y  4:0axy + 50xh/
5. A9c+28b(^ + 4b^€? 13. 2a;*  Sa:^ + 8ar^
6. a^b^ + 4a^b + 4(j^ 14. SOa;^^ ^ 3^^ + 75^2
7. xy^ + 2xy + x 15. a^x + aa? — 2a^a?
8. 2m^7i — 4mw + 27i I6. a?"^ + 2a;'*y + y^
17. (a  6)2  2c{a  6) + c^
18. 9(:c + yf + 122(a; + t/) + 42*
19. 16(2a  3)2  16a6 + 246 + 6^
20. 25(x  y)2  120a;y(a;  y) + 144a;V
21. a2 + 62__c2 4.2a6 + 2ac + 26c
* FACTORING 139
22. i'j2 + 4xy + 9f 24. .04a2 _ i2ah + .OOft^
23. i«2 + ij^ + ^j^ 25. 25a2  30aa: + 9a?
Solve the following equations for x:
26. ax3bx = a^ Qab + 96^
27. oo: + Sfea; = a^ + 6a6 + 96^
28. a; — 2ax = 1 — 4ax — 4a^
«
29. 2ax + dbx = 4a^+ 12a6,+ 96^
Factor the following examples in Cases I and II, and check
each result:
30. a? 43? + 4x 36. lOa^  20a + 10
31. a? 3x^ + 4x 37. 20a2  10a + 10
32. m«  2m% + mW 38. 16ay  24a2p + 9a?
33. m^ — m% + mV 39. ioc^ + 4^2/ + a;^^
34. aV  8a«a: + IGa^ 40. 8a? + 16a?y + xy^
35. aV — 6a?x + 4a? 41. 16mV — 9mn^ + v?
42. How many of the examples in this Exercise can you
work at sight?
Case III
93. The Difference of Two Perfect Squares.
From Art. 77, p. 112, (a + 6) (a  6) = a^  6^
Hence, a^ — 6^ = (a + 6) (a — 6)
But any algebraic quantities may be used instead of a and
6. Hence, in general, to factor the difference of two squares,
Tdke the square root of each square;
The factors vrill be the sum of these roots and their difference.
14D SCHOOL ALGEBRA
Ex. 1. Factor a?  16y*.
x«  16j/*  (x + 4y) {x  4y) Padora
Ex. 2. Factor «*  y*.
» (a^ + y*) (x f y) {x  y) Foctora
Ex. 3. Factor (3« + 4^)^  (2x + Zy^.
(3x+4y)«(2x+3y)«[(3x+42/) + (2x + 3y)] [(3x + 4y) 
(2a; + 3y)]
 (3x tI 4y f 2a; + 3y) (3z +4y2a;3y)
 (&c + 7y) (x + y) FacUir%
Let the pupil check the above examples.
Factor and check:
1. a?  9 10. a?* — 9aV 19. aa? — aa?
2. 25  16a^ 11! m  64a^ ^ ao. a'x  «
3. 4a*  496« 12. 242  2a:» 21. 225a?«  y^
4. a;* — 4y* 13. a^ — a:* 22. 2ia? — ^
' 5. 100  81m2 14. Zijf  75ay» 23. ^^a*  Oft*
6. 9a* 7? 15. a*  aj* 24. .09a:*  .16y*
7. 1  64m* 16. a*  816* 25. .Ola*  .046*
a 3a:*  12y* 17. a:«  y» 26. .253/*  ^6*
9. a?  9a*a: la a:^  a: 27. .81a:*  .00256*
28. a:*  2/« 35. (a: + 2y)*  (3a: + 1)* .
29. a^«  y*»2« 36. 25(2a  6)*  (a  36)*
30. (a: + yf  1 37. ^^  yz^*
31. a?  (y + 1)* 38. 81x"  16y*
32. (x — y)*  9 39. a:^ — 144a:y*z*
33. 4(a:  y)*  25 40. (a  6)*  4(c + 1)5N
34. 1  36(a: + 2y)* 41. 1  100(a?  a:  1)1
y
\
FACTORING 141
Solve the following equations for x:
42. aa; + 26a: = a^  4fc2 44. 3a:  oar = 9  o^
43. ax — 2bx — a^ — 4i^ 45. a: — 6a: = 1 — 6*
Factor the following miscellaneous examples:
46. a* — 4a «3. a:* — 9a:
47. a* — 4 54. a* + 9a^a: + 6aa?
4a a*  4a + 4 55. a' +j*a*a: + 9aa?
49. a? — 4a 56. a* — aa:*
50. a* — 4a2 + 4a 57. a^ + aa^
51. a^  4a« + 4a* 58. (a + a:)*  9
52. a:* — 6a: 59. (a + 6)* — (a: — y — of
60. Make up and work an example in factoring in each of
the cases treated thus far.
61. How nmny of the examples in this Exercise can you
work at sight?
62. How many of the examples in Exercise 2 (p. 13) can
you now work at sight?
Case IV
94. A Trinominal of the Form ac;>+to + c.
It was found in Art. 79 (p. 117) that on multiplying two
binomials like a: + 3 and a: — 5, the product, a:* — 2a; — 15,
was formed by taking the algebraic smn of + 3 and — 5 for
the coefficient of x (viz. — 2), and taking their product (— 15)
for the last term of the result. Hence, in undoing this work
\ ; to find the factors of a? — 2a: — 15, the essential part of the
., process is to find two niunbers which, added together, will
^ give — 2 and, multiplied together, will give ^ 15.
1
y^
142 SCHOOL ALGEBRA
Ex. 1. Factor ar^ + llx + 30.
The pairs of numbers whose product is 30 are 30 and 1, 15 and 2,
10 and 3, 6 and 5. Of these, that pair whose sum is 11 is 6 and 5.
Hence, x» + llx + 30 = (x + 6) (x + 5) FadmB
Ex. 2. Factor x^ _ a;  30.
It is necessary to find two numbers whose product is — 30, and
whose smn is — 1.
Since the sign of the last term is minus, the two numbers must
be one positive and the other negative; and since their smn is — 1
the greater number must be negative.
x«  a;  30 = (x  6) (re + 5) FadUnz
Ex. 3. Factor ar^ + 3ary  lOy^.
Since % and — 2y^ added give Zy^ and multiplied give — IO9*,
x» + 3a:y  lOy* = (a: + %) (a;  2y) Factors
Hence, in general, to factor a trinomial of the form
a? + bx + c,
Find two numbers which, muMplied together, produce the
third term of the trirwmial and, added together, give the coefficient
of the second term;
X (or whatever takes the place of x) phis the one number, and
X plus the other number, are the factors required.
EXERCISE 39
Factor and check:
1. a? + §a; + 6 6. a? + a;  30
2. 7?x& 1, x^ + Qxy  16/
3. a? + a:  6 B, tx?  &xy  \&f
4. ar^ + 7a;  44 9. ar^ + 8a; + 16
5. x^  llx> 30 10. ar^ + 6a:  36
FACTORING 143
11. ar^  5a;  36 19. o^f  23ary + 132
12. «*  5a:2 _ 3g ^ 3^2 _ 5a^ _ 240^
13. ar^ + 3a:  28 21. a:*  Ox^ ^ g
14. x2 _ 2a;  48 22. 2a  14aa;  maa?
15. a?  8a;  48 23. 2a;3 _ 22ar^  120a;
16. a;2 + 13a;  48 24. a;^  25a;» + 144a;
17. ar^  22a;  48 25. o?""  x^  56
18. ar^  4a;  96 26. a^fc^  liaftc^  26c*
27. a;2 + (a + 6)a; + a6
28. a;2 + (2a  36)a;  606
29. a;2 + (a + 26 + c)a; + (a + 5) (5 + c)
30. a;^ + (a + 6)a; + (a — c) (6 + c)
31. (a;  y)2  3(a;  y)  18
Factor and check each of the following miscellaneous
examples:
32. a;2  4a; + 4 3a a*  4^^
33. a;^ — 4 39. a* — 4a^y + a^
34. a;^ — 4a; + .3 40. a^ — 1
35. a;* — a;* — 6a; 41. a;^ + 5aa; + ^
36. a? — 4a; 42. a; — a;^
37. a;3 + 6a;2 ^ 93. 43. a*  Ta^ + 12
44. Make up and work an example in factoring to illus
trate each case treated thus far.
45. How many of the examples in this Exercise can you
work at sight?
144 SCHOOL ALGEBRA
Case V
95. A Trinomial of the Form aac^ + bx + e*
•
From Art. 80 (p. 119) it is evident that the essential part
of the process of factoring a trinomial of the form aa? + bx + c
Ues in determining two factors of the first term and two factors
of the last term, such that the algebraic smn of the cross
products of these factors equals the middle term of the
trinomial.
Ex. Factor lOa? + 13a:  3.
The possible factors of the first term are lOx and z, 5x and 2x.
The possible factors of the third term are — 3 and 1, 3 and — 1. In
order to determine which of these pairs will give + 13x as the sum
of their cross products, it is convenient to arrange the pairs thus:
lOx,  3 5x,  1
\ •• ••. .••
• • • •
•
•• • • .' •
• * •
z, 1 2x, 3
Variations may be made mentally by transferring the minus sign
from 3 to 1; and also by interchanging the 3 and the 1.
It is found that the sum of the cross products of
6x,  1 .
2z, 3
Hence, 10x« + 13a;  3 = (5x  1) (2x + 3) Fadora
Let the pupil check the work.
Hence, in general, to factor a trinomial of the form
aa? + bx + c,
Separate the first term into two siich factors, and the third
term into two siich factors, that the sum of their cross prodticts
equals the middle term of the trinomial;
As arranged for cross muitiplicaiion, the upper pair taken
together and the lower pair taken together form the two factors.
FACTOBINQ 145
IZIRCI8S 40
Factor and check:
1. 2a? + 3a; + 1 15. 6*^  2xy  4y
2. 3a?  14a; + 8 * 16. 16a?  6xy  27y*
3. 2a? + 5a; + 2 17. 12a? + a;y  63y^
4. 3a? + 10a; + 3 la 320^ + 4a6  456«
5. 6a?  7a;  5 19. 4a?  13a? + 9
6. 2a? + 5a;  3 20. 9a?  148a? + 64
7. 6«» + 20x»  16« 21. 12a^  7aa  12a»
a. Sa*4a*^ 22, 24a? + 104aY  ISa^
9. 8o* + 2o  15 23. 25«« + 9o*6*  166*
10. 2«* + a;  10 24. 16a;*  10«V  V
u. 12a:* 5x  2 25. 3«*»  8a:y  3y*
* 12. 4a? + 11a:  3 26. 25a*  41a^ + 166*
13. 5a:* + 24a;  5 27. 20  9*  20a?
^14. 9a?  15a?  6a: 2a 5 + 32a5^  21a:y
29. (o + 6)* + 5(a + 6)  24
^ "^ 30. 3(«  »y + 7(«  y)3  6s?
^ 31. 3(a? + 2a;)«  5(a? + 2a:)  12
•^ ^ 32. 4a;(a? + 3a:)2 « g^^^ ^ 3^.) _ 323.
^ ' 33. 2(a; + ly  5(a?  1)  3(a;  1)«
Factor and check each of the following miscellaneous
examples:
34. 4a? — 1 3aa?l
35. 4a? + 4a; + 1 39. a? — a? — 6a;
36.3a? + 4a; + l 4a5a2 + 9a2
37, a? + 4a; + 3 4i. a?  9a; + 18
146 SCHOOL ALGEBRA
42. a:* 6a? + 9a; 46. ««  a»
43. a^ — 4(a: + yY 47. x* — x
44. Sa? + 7x  6 48. a?  ox  2a'
45. (a + 6)2 + 2(fl + 6)x + a? 49. 2a?  5a?  3a:
50. Make up and work an example in factoring in each
case treated thus far.
51. How many of the examples in this Exercise can you
work at sight?
52. How many examples in Exercise 25 (p. 101) can you
work at sight?
Case VI
96. Sum or Difference of Two Cubes.
From Art. 83 (p. 125), ^^4r = a^  a6 + 6*.
Hence, a» + 6» = (o + 6) (a^  ofc + &«) . . . (1)
In like manner, a* — 6* = (a — 6) \(j? + ofc + &*) . •• • (2)
But any algebraic expressions may be used instead of a and
6 in formulas (1) and (2).
Ex. 1. Factor 27a?  Sj/*.
27x»  8y» = (3x)«  (2y)«
Use 3x for a and 2y for h in (2) above. \
27x»  82/» = (3x  2y) (9x« +6x2/+ 4y«) Faxiofr%
In working examples of this type, it is often convenient to call
3x — 22/ the "divisor factor" and 9x* + ^xy + 42/* the "quotient
factor." Why are these names appropriate in this case?
Ex.2. Factor a« + 86®.
o« + 86» = (a«)« + (268)»
= (a« + 26») (a*  2a*6» + 46«) FQxim%
FACTORING 147
Ex. 3. Factor (a + by  a?.
(a + 6)'  a^ = [(a +6)  x] [(a + by + (a + b)x + z^]
Let the pupil check the above examples.
Hence, in general, to factor the sum or difference of two
cubes,
Obtain the values of a and b in the given example, and substi
tute these values in formula (1) or (2).
^ 97. Sum or Difference of Two Like Odd Powers.
Since the difference of two like odd powers is always divis
ible by the difference of their roots (see Art. 84, p. 128), the
factors of a** — 6**, when n is odd, are the divisor, a — 6, and
the quotient.
Ex. 1. Factor o^  6^
a6  6« = (a  6) (a* + a»6 + a«6« + a6» + b^)
Since the sum of two like odd powers is divisible by the
sum of the roots (see Art. 84, p. 128), the factors of a** + 6%
when n is odd, are the divisor, a + b, and the quotient.
Ex.2. Factor a:^ + 322/^.
x^ + 32y^ = X* + (2yy
= (x + 2y) [x*  x»(22/) + xK2yy  x{2yy + (2yy]
= (x + 2y) (x*  2x^ + 4xV  8x2/« + 1%*) Factors
98. Sam or Difference of Two Like Even Powers.
The difference of two like even powers is factored to best
advantage by Case III (p. 139).
Ex. 1. x^  y\
= i^ + y'){x' + y^) (x + y){x y) Factors
The *MW,of two like even powers cannot in general be fac
tored by elementary methods unless the expression may be
148 SCHOOL ALGEBRA
regarded as the sum or difference of two cubes (Art. 96), or
other like odd powers.
Ex. 2. a« + 6« « (a^y + (6*)»
= (a* + 6*) (o*  aV + 6*) Factors
But a' + V,a^ + b\ and a' + 6* cannot be factored by any
elementary method, and are therefore prime expressions.
Let the pupil check the examples of Art. 97 and 98.
ESERGISS 41
Factor and check:
1. m?  n* 14. a*  64n^ 27. a" + oj"
2. c» + &P 15. 25ac 2x' 2a a* + 6*
3. 27  a? 16. &• + y» 29. 32a*  1
4. cf + 86»c» 17. (a + 6)» + 1 30. a"  6"
5. a?  125 la 125 + (26  a)» 31. 243  a?
6. 64y»  27 19. 8  (c + (i)» 32. 64  (a  by
7. aV + 1 20. (a;  yY  27a:» 33. 8(x  2y)» + 1
8. 1  1000a:» 21. 16aV  54a»» 34. o^®  6^®
9. 27a^.+ cfx 22.of + j^ 35. a^® + ¥^
10. 612x»  j/» 23. a:^  y^ 36. 32a:5  a^^
n. a + 343a* 24. a^ + m^ ' 37. a* + y*
12. a*  «• 25. a:" + y" *3a 8a:^* + y^
13. x"  y« ^26. a'  1285^ 39. 51^a:»  (a + 6)»
40. Make up a binomial expression whose terms contain
unlike exponents and which can be factored as the sum of two
cubes. Also one that can be factored as the sum of two 5th j
powers. *
41. Make up a binomial the exponents in T^hose terms'
are even numbersi and which can be factored as the sum of '
1^
FACTORINa 149
two cubes. Also one that can be factored as the sum of two
5th powers.
42. State which of the following expressions can be fac
tored:
s? + y^ ^ + 1^ a?yi® 7? + y^^
x» + y* ^y^ ^ + y^ ^ + y^
3? + ]^ ofiy^ sfi + y^ a? y^^
Factor and check each of the following miscellaneous
examples:
43. a* _ 4^2 51 ofi + a^^
44. a*  8a' 5X ofi  cfi
45. X* — 4aa; + 4a* 53. a" + y*
46. a?  4ar + 3a* 5^, x^  y^
47. ai^a^ 55. 6a*  13a + 6
4a a:^ + a* 56. 16a:* — 8xy + y^
49. a?  4(a + 6)* 57. a:* + 27a»a;
50. a:»  8(a + 6)» 58. a:" + y»
59. Make up and work an example in factoring to illus
trate each case treated thus far.
60. How many of the examples in this Exercise can you
work at sight?
Case VII
99. A Polynomial whose Terms may be Grouped so as to
be Divisible hj a Binomial Divisor.
Ex. 1. ax ^ ay bx ^by ^ (ax  ay)  (bx. by)
» a(x  y)  b{x  y)
« (a — 6) (x — y) Factors
The last step in the preceding process is sometimes clearer to the
pupil when written Id the following form:
(X V) )a(x  y)  6(x  y) _ (, _ y) („ _ j) p„aara
a —0 '
15a SCHOOL ALGEBRA
Ex. 2. 1+ 15a*  5a  3a» = 1  3a»  5a + 15a*
= (1  3a»)  5a(l  3a»)
= (1  3a») (1  5a) Fadors
Ex. 3. a» + 3a«  4 = a* + 2a« + a«  4
= a\a + 2) 4 (a 4 2) (a  2)
= (a + 2) (a« + a  2)
= (a + 2) (« + 2) (a  1) Factors
Let the pupil check the above examples.
EXERCISE 42
Factor and check:
1. ax + ay + bx + by 14. a:* — o^ — 4x + 4
2. x^ — ax + ex — ac 15. oV— 6V— aV+6V
3. 5xy lOy 3x + 6 16. x(x + 4)^ + 4(a: + 4)
4. 3am— 4mw— 6ay+8ny 17. a^Ca + 3) — 3(a + 3)
5. a^x + Sax + OCX + 3cx 18. 2(a? — rf) — {x — y)
6. 3a^ + 3a6 — 5an — 56n 19. Ax{x — 1)^ + x — 1
7. ar* + a? + 2a:2 __ 2a; 20. a:^  1 + 2(ar^ 1)
a 2a^2aJ»2aV+2a2x 21. ^f + ^f
9. 2^ + 2r^ + y + l 22. a:y + a?j^
10. a;3i? — 2a^x — x + 2a 23. a? — ]^ — a? + y^
11. s? + Sy — Sx — xy ' 24. o^ — 2^ — (a; — y)*
12. 2? — 2? — z + 1 25. 4a^ — aV + a? — 4
13. ab — by — a + y 26. ^—j^+^—j^+x—y
27. 4aa:^ + 8aa; — 8a — 4aa?
28. a(3a  xf  6aa:2 ^ 2a:»
29. a?  8  7(a:  2)
30. 4(a? + 27)  31x  93
31. (2ar + l)'(2a: + l)(3ar + 4)
32. (2x3)« + 2a?9a: + 9
FACTORING 151
33. a? 7x  Q
34. a? Sx^lOx + 24:
35. a? 8ix?+17x10
Factor and check each of the following miscellaneous
examples:
36. a^ — Sa:' 44. a:^ — a^^ + a: — 1
37. a^  16ar^ 45. a?  9x^ + 18x
38. a^6ax + 9a? 46, s^ + {a + bf
39. dx — bx + ay — by 47. a^ — y^
40. a2a2 48. (a + 6)22(a + 6)p + p2
^41. a?  a» + 2(a:  a) 49. (a + 6)^  (o + 6)  2
42. 3a^ — 4a — 4 50. a? + a? — a^ — a'
43. a:* + 2/^ 51. a: + o' — a  x'
52. Make up and work an example to illustrate each case
in factoring treated thus far.
53. How many of the examples in this Exercise can you
work at sight?
SpifciAL Cases Under Case III
100. By the Oronping of Terms we may often reduce an
expression to the difference of two perfect squares.
Ex. 1. Factor a? — 4:xy + 42/^ — 9z^.
X*  4xy + 42/*  9z2 = (a;2  ixy + ^y^)  92?
ss (x 2i/)* 92*
= [(X  2y) + 32] [{x  2y)  Sz]
= (x  22/ + 32) (X  22/  32) Factors
Ex. 2. Factor a^  a?  y^ + b^ + 2ab + 2xy,
a»x«2/*+6*+2a6+2x2/ = (a22a5 +6*)(x* 2x2/ +2/')
= (a + by  (x  yy
= (a+b+X'y)(a+bx+y)
Let the pupil check the above examples. r actors
152 SCHCX)L ALGEBRA
EXERCISE 48
Factor and check:
1. a2 + 2a6 + 6»a? u. 2ab + a? a* l^
2. a*2a6 + 624a:2 12. x^ + a^ y  2flKC
3. a*a?2ayj/* 13. a^ + f  a? + 2ay
4. 9a2a?^4a:y4y* 14. a*  ar*  2r^  y*
5. l&a^ix? + 2xyf 15. a?  2^  1  2y
6. m* — a? — y^ — 2xy 16. 1 + 2a^ — o? — '^
7. a2 + 62 + 20^4x2 17. c2a262 + 2a6
8. a2 + 624x2 + 2a6 la 0^ + 6*  c*  2a6
9. a^  4^2 __ 52 + 2a6 19. 2a6 + a^ft^ + i  ai^
10. a:2_2a6a2fe2 20. 22?  42  22* + 2
21. 2O2/2; + ar^  4y2  25z2
22. a2 + 2a6 + 62_c22cd<P
23. x^ + 4y2 _ 9^2 _ 1 ^ 4^^ _ gg
24. 9a2  25a:2 + 4J2 _ 1 _ lOa;  I2a6
25. c?  962a:2  1 + 66a:  10a6 + 256^
Factor and check each of the following miscellaneous
examples:
26. ax — bx + ay — by 34. a:* — 6* + a? — 6*
27. a^a?2xyy^ 35! 02 + 6^^2 + 206
28. o2 + oa: — 06 — 6a: 36. o' — 27^
29. a22o6 + 62a;2 37. a^Qay + 9f
30. 2o + 26  3o  36 38. a'^x — 16x2^
31. 4a2 + 4o + 1  62 39. 3o2  4a + 1
32. 9x2 _ 4^2 _ 4^ _ 52 4Q a^_8(a + 6)»
33. .9a:2 _ 9y2 + J. _ y ^ . o* + y*
FACTORING 153
42. Make up and work an example in each case in factor
ing treated thus far.
43. How many of the eicamples' in this Exercise can you
work at sight?
101. The Addition and Subtraction of a Square will some
times transform a given expression into a difference of two
perfect squares.
Ex. 1. Factor a* + aV + 6*.
Add and subtract a^.
»
o* + a«6« + 6* = o< + 2a«6« + 6*  a»6«
 (a« 4 6* + a6) (a« + 6«  ab) Factors
Ex.2. Factor aj*  7ry + y*.
Add and subtract 9xV«
 (x* + »* + 3xy) (.3^+y'  3xy) Fadon
Let the pupil check the above examples.
EXERCISE 44
Factor and check:
1. <j* + cV + «* ' ' 6. 49c*llc*<P + 25d*
a. a;« + «* + l 7. 16a* 9x^ + 1
3. 4a^  13** + 1 8. 100a;*  61a? + 9
4. 4a*  21a«6» + 96* 9. 225o*6*  4o*6» + 4
5. 9«* + 3«V + V 10. 32a* + 26*  56a»6»
U.a* + 46« i2.H64a;* W. aY + 324
154 SCHOOL ALGEBRA
Factor and check each of the following miscellaneous
examples:
14. a^ + 2aV + a^ 20. 4a*4aV + aj*
15. a'^ + ah^ + ai* 21. a^  ax  6x^
16. a^ + 4oV^3aj4 ^2. b^ + a^b^ + a^
17. a^  4a^ 23. a* + 20^62 + 6*
18. o* + 4a^ 24. a* — x^
19. 4a*13aV + a:* ^5. a8 + 64a;8
26. Make up and work an example in each case in factoring
treated thus far.
102.. Other Methods of Factoring algebraic expressions will
be treated later. Thus it will be found that ^
a2 + 62 = (a + 6 + V2ab) (a + 6  ^206)
Also, 0" + ^= {a + V^b) {a  V^b)
Factoring by use of the Factor Theorem is treated in the
Appendix (pp. 467^9).
103. General Principles in Factoring. It is important in
factoring to reduce each expression to its prime factors.
Therefore it is important to use the diffferent methods of
factoring in such a way as to obtain prime factors most
readily.
Hence, in factoring any given expression, it is useful to
1. Observe, first of all, whether all the terms of the expression
have a common factor (Casel); if so, remove it,
2. Determine which other case in factoring can be used next
to the best advantage.
3. If the expression comes under no case directly, try to dis
cover its factors by rearranging its terms; or by adding and sub
trading the same quantity; or by separating one term into two
terms, •
FACTORING
155
4. Contmue the process of factoring until each fador can he
resolved no further.
EXERCISE 46
Factor:
1. 3x»  3a:
2. 2x8  8xy + Sxy*
3. x»  llx* + 30x
4. 4x5 + 5a:*y  Qxy*
5. 12a*  2a6  306«
6. X*  1  j/« + 22/
7. 40a?  5
8. 16x*  4ac»2/ + 25xV
9. ^>kdaa; — 3a — x
10. 3x^^^
U. 4a*.\^^l
12. 2a;«  32
13. X* + 4x  45
14. 4x* + 2a  a»  1
15. Sax* — 5a
16. 18x8  3x»  36x
17. X* + 3xV + 42i
18. a*x«  9x«  a« + 9
19. 110  X  x»
20. 3x« + 13xy  302/»
^21. 7a  7a»6*
22. 6x* + 14x + 8
23. X*  (x  2)«
^24. 3a +3a«
25. a'  a« + 2a  2
Review
26.
27.
28.
29.
30.
31.
/32.
33.
34.
35.
36.
37.
3a
39.
40.
41.
42.
43.
44.
45.
46.
47.
N' 4a
49.
5a
6x«  2x  4x«
1  232* + «*
128  2y»
1  a«  6»  2a6
21a«  17a  30
x" + 2/"
8x« + 7292»
405xV  45x*
a^  4a' + 5a2  20
(c +dy 1
(x  2/)« + 2(x  y)
24x» + 5x2/  362/'
x»  2x*2/  4x2/* + 82/»
(a*  6)*  a*
2* + 0* + 1
(a*  6«  c2)«  46«c«
21x«  40x2/  2l2/*
32 Hn"^
5x^ + 5x2/'
2ax' + ia2/'
1 + X — X* — x'
x*  9  7(x  3)«
4a^  37a2 + 9
x«64
156 SCHOOL ALGEBRA
51. a:»  27  7(x  3) ^ 56. 4(a«  6«)  3(a + 6)«
52. 32x^  y^^® 57. (a  6)» + (x  y)»
53. («« + y*)*  16aV 58. (a«6  a5*)*
54. X* + x V  j/V  2* 59. x" + a»
55. oo:*  ax  xV + y 60. x* + 2/* + (x + y)»
61. (a  6)« (x + y) + (a  6) (x + y)*
62. (a  6)* + 4(a  6) (x f y) + 4(x + y)«
63. a"  1 65. 4a*  96«  1  66
64. 4a«  96« + 4a  66 66. (x*  y«)«
67. (x»  1)* + (2x + 3) (x  1)«
68. a*  6*  a*x^ + 6*x«
69. 3x»  27 + ox*  9a
70. a» + 3a^ + 3a6« + 6«
71. o^  3a«x + 3ax*  x»
72. a^6cx — amnpx + m*npy — a6cmy
73. 4x + 4on + X* — 4a* — n* 4 4
74. 2(x»  8) + 7x*  17x + 6
75. a*  46< + o* I 26*
76. (3x*y  3xy»)*
77. 18x* + 62xy  6y* 81. x«  79x* + 1
78. (x + 1)'  «• 82. a*  9 + 96*  6a6
79. (1  2x)*  X* 83. X*  4x*  16x* + 64
*80. ax*  ex + ax  c 84. (x* + 3)»  64x*
85. X*  49y* + 9  6x*
86. xV — 4x* + 4 — y* — 4x*y* + 4xy
87. ahix — bcm*yz + acmxe — abmny
8a How many of the examples in this Exercise can you
work at sight?
89. Work again the oddnmnbered examples on p. 88.
FACTORING 157
104. Factorial Hefhod of Solving an Equation.
Ex. i; Solve a? + 5x  24 = 0.
Factoring the lefthand member, we obtain
(» + 8) (a;  3) 
If any factor of a product equals zero, the entire product equals
zero. Hence to obtain the roots for the above equation, we .may
let each factor in the lefthand member equal zero and obtain the
value of X from the two resulting simple equations.
Hence we have for the above equation
a; + 8 0
a? « — 8 Root
Check for a?   8
«« + 6x  24
«64 40 24
0
Also a;  3 «
x  3 Rooi
Check for x « 3
a^ +5x 24
9 + 1524
 24  24
0
Ex. 2. Solve x(x  2) (3a: + 4) (a: + 1) = 0.
Using the above method, we obtain
X  0, 2,  J,  1 Roots
Check for a?  0. x(x  2) (3x + 4) (x + 1) Let the pupil
= 0(0  2) (0 + 4) (0 + 1) apply the checks
= 0(2)4 X 1 0 for the other
values of x.
Ex. 3. Solve a*  a? = 4a;  4.
Transposing all terms to the lefthand member, we have
x*— »*— 4a?+4=sO
Hence, x*{x  1)  4(x  1) =
{x  1) (a?  4) 
(a:  1) (x 4 2) (a?  2) 
a; « 1, 2,  2 Roots
Let the pupil check the work.
158 SCHOOL ALGEBRA
EXEBCISE 46
Solve and check each of the following;
1. a?  5x + 6 = 14. 0^2 _. 2x =
2. o^x2 = 15. a? + ax==0
3. x^ _ 7a. = _ 12 16. a?  a^x =
4. a^ — a: = 6 ^ 17. a:' + a:^ = 4a: + 4
5. ix? = x + 12 IB. a? + a?9x9 =
6. ar*  16 = 19. a^5a? + 4: =
7. ar* = 9 20. Q^x^ix + ^^O
a a:(ar»4) = 21. 3(3:^ _ j) _ 2(a: + 1) =0
9. a:^  25a; = 22. l+a^ = 2a?
10. x' = 9a; 23. 2/* — 9y2 =
11. 2ar^  3a; + 1 = 24. p^  3p + 2 =
12. 3a;^ — 4a; = 4 25. 3m^ — 4m + 1 =
la a? — s? — 6a; = 26. 2? — 42 + 4 =
27. 2^  132/2 + 36 = '
Form the equation whose roots are
28. 3 and 4 30.3,7 32. 0, 2
29.  5, 2 31. 1, 2,  2 33. 2, 3,
34. The square of a certain number diminished by 4 times
the number equals 45. Find the nmnber.
35. The square of a certain number increased by 6 times
the number equals 40. Find the number.
36. What number plus its square equals 12?
37. The square of a certain number diminished by 9 times
the number equals zero. Find the number.
FACTORING 159
38. The square of what number equals 25 times the
number?
39. The cube of what number equals 25 times the number?
40. Find two consecutive numbers whose product is 72.
tt. If to 3 times the square of a certain number we add 4
times the number, the result equals 4. Find the number.
42. The depth of a certain lot equals three times the front,
and the area of the lot is 7500 sq. ft. Find the dimensions
of the lot.
43. The temperature at which iron fuses is 2800° F.,
which is 332® more than 4 times the temperature at which
lead fuses. Find the temperature at which lead fuses.
44. The area of Texas is 265 J80 sq. mi. This is 29,240
sq. mi. less than 6 times the area of New York. Find the
area of New York.
45. How many of the examples in this Exercise can you.
work at sight?
46. How many examples in Exercise 30 (p. 120) can you
work at sight?
CHAPTER IX
fflGHEST COMMON FACTOR AND LOWEST
■ COMMON MULTIPLE
105. Utility in the Highest Common Factor and Lowest
Common Multiple. The advantages in the knowledge and
use of the largest factor common to two or more expressions
and of their lowest conmion multiple are similar to those
found in arithmetic for the same principles. They aid in
reducing fractions to a simple form, in adding and subtract
ing fractions, and in multiplying and dividing fractions.
Other advantages will appear later.
Why do we now proceed to make definition^ and rules?
Highest Common Factor
106. A common factor of two or more algebraic expres
sions is an expression which divides each of the given expres
sions without a remainder.
The highest common factor of two or more algebraic
expressions is the product of all their prime common factors.
Thus, the highest common factor, or H. C. F., of 4a^, 12a;', and
16x*i/ is 4a!*.
107. The Method of Finding the H. C. F. is to
Factor the given expressions, if necessary;
Take the H. C. F. of the numerical coefficients;
Annex the literal factors comrrum to all of the expressions,
giving to each factor the lowest exponeni which it has in any
expression.
160
HIGHEST COMMON FACTOR 161
Ex.1. Find the H. C. F. of 6ar^  12aJ3/2 + 6»« and SicV
f 9a:^  122^.
6xV  12xy* + 62/»  ey(x  y)'
3xy + 9itj/»  122/*  32/«(a;» + 3x2/  V)  32/*(x + 42/) (a;  y)
.. H. C. F. = Syix  2/)
^Ex. 2. Find the H. C. F. of 3a(a26  o62)2 and a^ib  a)^
3o(a«6  db^y = 3a[o6(o  6)]« = 3a»6*(a  6)*
a\b  a)« = a«[  (a  6)]« = a«(a  6)«
.. H. C. F. = a\a  6)*
EXERCISE 47
Fmd the H. C. F. of
1. 4a26, 6a6« 6. ar^  3x, ar*  9
z 5a^y, 15aV 7. ix' + Ox, Gx' + 9x
3. 24a2x3, 56aV s. c^  a?, a^  a?
4: 24a:j/, 48a«*, 36a: 9. ocy — y, a? — x
5. 34aV, 51aa:^ 10. 4a' + 2a2, 4a«  a
12. 4a»a:  Aas?, Sa^oi?  Scuc*, 4a2ar*(a  x)^
13. 2a:3  2x, Bar*  3a;, 4a:(a;  1)^
14. 6a? + 5xy — 4^/^, 4a? + 4xy — Sj/^
15. 3a?  5a?  2ar, 4a?  5a?  6a:, a?  4a:
16. b a% 36  a^b  2a^6, 6^  a^J^
17. 1  o^ 1  a«, 3a + 3a2 + Sa', 1 + a* + a*
la Find the H. C. F. of the numerator ond denominator
of the fraction in Ex. 5 (p. 170).
J.9. Beginning with Ex. 17 (p. 170), treat each example
through Ex. 22 in the same way.
162 SCHOOL ALGEBRA
*FindtheH. C.F. of
20. (a%  ai^)\  anS'ia  6)*
21. 9(a?  xy)\ \23^{q?  2/»)*
22. (a + h){x y), (a  6) (y  a:)
23. (a + 6) (a:  y)2, (a  6) (y  a:)*
^24. 4a:2^ar^_3._2, (2a:)2
25. 3a*  10a + 3, 9a  a», (3  a)»
2& a:*(x — a)^ ar(a* — a^
27. In Exs. 1 and 6, name some common factor of the two
given expressions which is not their H. C. F.
28. Write two expressions whose H. C. F. is aV.
29. Write also two expressions whose H. C. F. is 3x(x — 1).
30. Write three expressions whose H. C. F. is a{x — 6).
31. How many of the examples in this Exercise can you
work at sight?
Lowest Common Multiple
«
108. A common multiple of two or more algebraic expres
sions is an expression which will contain each of them with
out a remainder.
The lowest common moltiple of two or more algebraic
expressions is the expression of lowest degree which will
contain them all without a remainder.
Thus, the lowest common multiple, or L. C. M., of So*, 6a^, and
iaa:* is 12a'x'.
109. The Method of Finding the L.C.H. is to
Factor the given expressions y if necessary;
Take the L. C. M, of the numerical coefficierUs;
Annex each literal factor that occurs in any of the given ex
LOWEST COMMON MULTIPLE 163
preasions, giving the factor the highest eocponerU which U has in
any one expreesion.
Ex. 1. Find the L. C. M. of 3a^  9a?, a?  9x, and
a?  6a: + 9.
ar*  9a:» = 3x»(a;  3)
0^  9a; ^x{x + 3) (x  3)
»«  6x + 9 = (a;  3)*
. • . L. C. M. = 3a:»(a; + 3) (a;  3)»
Ex. 2. Find the L. C. M. of (a^b  06^)8^ 206(6  a)^ and
(a«6  a62)» = [a6(a  6)]» = a»6»(a  6)»
2a6(6  o)« = 2a6[  (a  6)]« = 2a6(o  6)«
a^a^  62)2 ^ fl2(^ + 5)j (^ _ 5)1
. • . L. C. M. « 2a«6»(a + 6)» (a  6)» Arw.
EXERCISE 48
Find the L. C. M. of
1. 30*6, 2a62 6. 120^6, 1606^, 2ia^b^
2. 12aV, day 7. 2a:(a: + l),a?l
3. 2ac, 36c, 4a6 8. Sa^ + 3(i6, 2a6 + 26^
4. 3a% 4ac^, eir^c 9. Ta:^^ 20*  6x
5. 42a?2/^ 282^2* 10. a:^  1, ar^  1
11. Qi?y^,a?3xy + 2f
12. Sqi?  3ar, 6x2 _ 12a: + 6
13. 5aar^(ar  y)2, 362^(a:2 _ y2)
14. a:»3a:2_40a;, a?9a: + 8
15. a*  62, a'  6«, a' + 6»
16. 6a:2 + 63.^ 2a:3 _ 2x2 3^ _ 3
17. 4a26 + 4a62, 6o26  6062, 3a2  362
la 2ar^ + X  1, 4a:2 _ 1^ 2ar^ + 3a. + 1
19. 3x»  3, 6x2  i2aj + 6, 2x» + 2x2 + 2x
164 SCHOOL ALGEBRA
20. 12a^  2a?  140a:, l&x? + Qx 180, 6a*  39a? + 63a:
21. la; + a?a?, l+a: + a? + a?, 2a;2a?
22. (x  1)\ 7xf{x^  l)^ 14sfy{x + 1)»
23. 18a?  12a? + 2a;, 27a?  3a?, 18a?  24a? + 6a;
24. (x  1) (a; + 3)2, (a; + 1)^ {x  3), (a?  1)^, a?  9
25. Find the L. C. M. of the denominators of the fractions
in Ex. 18 (p. 181).
26. Find the L. C. M. also of the denominators of the
fractions in each e3cample from Ex. 21 to Ex. 28 (inclusive),
p. 181.
FindtheL. C. M. of
27. (a«6  oft^)*, a»i?(a + 6)*
2a (ofcc  bcd)\ {Sa^c  Saed)\ 6aV  6a*(P
29. {a^b  ab^y, {a"  ab)\ (a» + aV
30. 9(a? xyf, 12(a?  f)\ 18(a? + a?y)^
31. a — 6, 6 — a ,
32. 9(a  h)\ 12(6  of
33. (a + 6) (a;  y), (a  6) (y  a;)
34. (a + 6) (a;  y)^ (a  6) (y  a;)»
35. 4  a?, a?  a;  2, (2  a;)^
36. a?(a; — a)*, a;(a2 — a?).
37. Find two consecutive niunbers the diflFerence of whose
squares is 5.
3a Make up and work an example similar to Ex. 37.
39. The reclaimable swamp land in the United States and
the land that is capable of irrigation equal 178,000,000 acres
all together. If the irrigable land exceeds the swamp land
by 22,000,000 acres, how many acres of each of these kinds
of land are there?
LOWEST COMMON MULTIPLE 165
40. The distance from New York to Havana is 1410 mi.
If a steamer leaving New York travels at the average rate
of 260 mi. per day, and one leaving Havana at the same time
travels at the average rate of 280 mi. per day, how many
days and hours will elapse before the two steamers meet?
41. The distance of the sun from the earth is 92,800,000
mi. This distance exceeds 107 times the diameter of the sun
by 95,200 mi. Find the diameter of the sun.
42. A man bequeathed $20,000 to his wife, daughter, and
son. To his daughter he left $2000 more than to his son, and
to his wife three times as much as to his son. How much
did he leave to each?
43. The distance of the moon from the earth is 238,850
mi. This exceeds 110 times the moon's diameter by 1030 mi.
Find the diameter of the moon.
44. If 10 m. exceeds 10 yd. by 33.7 in., how many inches
are there in a meter?
45. Write a conunoh multiple of the expressions in Ex. 1,
which is not their L. C. M.
46. Write a common multiple of the expressions in Ex. 10
which is not their L. C. M.
47. Write two expressions whose L. C. M. is 24a^6V.
4a Write two expressions whose L. C. M. is 12a?(x — 2)*
(X  1).
49. Make up and work an example similar to Ex. 27. To
Ex.31. To Ex. 47.
' 50. How many of the examples in this Exercise can you •
work at sight?
51. How many examples in Exercise 35 (p. 131) can you
work at sight? ' .
CHAPTER X
FRACTIONS
110. "Utility of Fractions. In algebra, as in arithmetic,
fractions are useful in indicating new units, and in indicating
quotients and thus opening the way to save labor by cancel
lation.
In algebra fractions also have other uses besides those
which appear in arithmetic. Thus, in algebra, a fraction is
often useful in expressing a general formula.
Ex. If an automobile goes a miles in h hours, how far
would it go in c hours at the same rate?
 » no. of miles the automobile travels in 1 hour
rr = no. of miles the automobile travels in c hours
Why do we now proceed to make definitions and rules?
111. A Fraction is the indicated quotient of two alge
braic expressions. This quotient is usually indicated in the
form —
b
The fraction  is read "a divided by 6," or, for brevity,
b
"a over 6."
Note that the dividing line of a fraction takes the place of
a parenthesis and is in effect a vincidum.
166
FRACTIONS 167
Another method of writing the preceding fraction is a/6. This is
called the solidus notation. It is convenient in printing mathematical
expressions, and is much used in European mathematical hterature.
^ ^ written in the solidus notation would be (x + l)/{Sx — 5)
3x — 5
The numerator of a fraction is the dividend and the <fo
rumtmator is the divisor of the indicated quotient.
Terms of a fraction is a general name for both numerator
and denominator.
KXEBCISE 49
1. If three boys weigh a, 6, c pounds respectively, what is
their average weight?
2. If four boys can run the quarter mile in p, q, r, s sec
onds respectively, what is their average time?
3. How many acres are there in a field a feet long and b
feet wide?
4. How many acres are there in a field c rd. X d rd.? In
one/ yd. X e ft.? p ft. X g rd.?
5. If sugar is worth a cents a pound, how many pounds
can be obtained in exchange for b pounds of butter worth
c cents a pound?
6. If coal is worth c dollars a ton, how many tons of coal
can be obtained in exchange for p tons of hay worth b dollars
a ton?
7. Make up and work a similar example concerning c
calves, worth a dollars each, exchanged for chairs worth d
dollars each.
8. If coal is worth c dollars a ton, how many tons can be
obtained in exchange for / bushels of wheat worth h cents a
busbd and for w bushels of com worth y cents a bushel?
168 SCHOOL ALGEBRA
9. Who first used the letters a, b, c to represent known
numbers? (See p. 456.) Tell all you can about this man.
10. Before the use of a, b, c, what other symbols were used
to represent known numbers? Discuss the relative advan
tages in these different sets of symbols.
U. As a notation, in what respects is a/b superior to
a 5 6? To 7? In what respects is it inferior to each of these?
6
12. How many examples in Exercise 45 (p. 155) can you
now work at sight?
112. An integral ezpression is one which does not contain
a fraction; as 3a:* — 2y.
An expression like Sx* + $2 + i in which fractions occur only
in the numerical coefficients is sometimes regarded as an integral
expression.
A mixed expression is one which is part integral, part
fractional.
Thus,3x«+a:5+^±^
113. Sign of a Fraction. A fraction has its own sign, which
is distinct from the sign of both numerator and denomina^
tor. It is written to the left of the dividing line of the fraction.
The sign of — y i^ "> <^d the sign of ^^ is + understood.
General Principles
114. A. // the numerator and the denominator of a fraction
are both multiplied or divided by the same quantity , the value of
the fraction is not changed.
For if a dividend is denoted by D, its divisor by d, and the quo
tient by Q T)
^Q,andZ)=dxO 1
a
TRANSFORMATIONS OF FRACTIONS 169
If m denotes any multiplier, DXTn^dXmxQ,OT
^^Q (Art. 16, p. 18)
Also if m denotes any divi9or except zero,
Dirm  d 8 w X 0, or ^^^ = Q (Art. 15)
a i m
116. B.' Law of Signs. By the laws of signs for multipli
cation and division (see Arts. 50, 62, pp. 59, 77),
a__ — a a ^ — a _ a ^ _ o, ^ a
b " ^' "6 6~ " ^' Vc bXc "  6 X c
«
x + y ^ x + y ^ _ x + y
y  X  {x y) X y
Or, in general,
The signs of any even number of factors of the numerator
and denomiruxtor of a frojction may he changed withovt changing
the sign of the fraction.
But if the signs of an odd number of factors are changed^ the
sign of the fraction must be changed.
Transformations of Fraction?
I. To Reduce a Fraction to rrs Lowest Terms
116. A Fraction in its Lowest Terms is a fraction whose
numerator and denominator have no common factor.
To reduce a fraction to its lowest terms, as in arithmetic,
Resolve the numerator and the demmiinator into their prime
factors y and cancel the factors common to both.
Ex. 1. Reduce ^ , ^ to its lowest terms. .
Divide both numerator and denominator by 12aV (see Art. 114).
36a»x« 3a
48aVy' 4xy'
Ans.
170 SCHOOL ALGEBRA
„ „ 9o6  12^ _ 36(3o  46) _ 36 .
^'' • 12a« 16ab~ 4a(3a  46) ~ 4o ^"*
Notice particularly that in reducing a fraction to its lowest terms
it is allowable to cancel a, factor which is common to both denomina
tor and numerator, but it is not allowable to cancel a term which
is common imless this term is a factor.
Thus, — reduces to  :
ac c
but in , a of the numerator will not cancel a of the denominator.
a+y
This is a principle very frequently violated by beginners.
EZEEGI8E 60
Reduce each of the following to its simplest form:
27 > 3a^  6a^b ,^ ^5{x  yY
36 4a262  Saff^ I8(a:  y^
108 ^ 2a ,^ a^b + ab^
16.
' 144
9.
3 ^2
'• 150
la
4 So***
■ 12aV
11.
12ar*j^
12
■ 153?y^
A^.
t. 3a^ 13.
6o«  9a»a;
^ 72xyz*
14
96a^z» '
3 (a;y)M
a; + y)«
^'' («* 
y2)8
2a:*
82^
4a*  2a 2a«6  2a6*
Zx — Qy ^^ Qxy
6ax — 12oy 93^ — 12a;y*
4a: + 4y „ 6a*6* + 1206*
18.
x^  f ^^ 2a?  3xy
(x + yy ' 4a:»  9a:^
12oV  Sa^xy 49ar^  64^^
ISaa:^  12ax^y ' lis?  IGxh/
8(a^  1) ^, a:5  27
21.
12x  12 ar^  6x + 9
6a? — ajy — 23/*
24.
25.
6a?  7aJ3/ + 2y2
(a + 6)2  c2
4a?  2a:y  12j^ ' a2 ^ {b + c^
TRANSFORMATIONS OP FRACTIONS 171
26. V"(^^)! 29. ^^^
a?  (a  1)2 • a^  ^  6x2
27. —Tn r^r 30. ^
4a?  2aa:  6a2 a^ + a^y + y4
23^ a?  8 axbxay + by
xh/^ + 2xf + ^f ' a^  62
a? — 2? _ 4 _ 2ajy — 4z + j/^
32.
2? — a? — 4 — 2^2 — 4a; + y2
1+4
33. What is the correct value of the fraction ■z——j ? If
6 + 4
the 4's are struck out, what does the value of the above
fraction become? Is it allowable, therefore, to strike out the
4's in the above fraction?
34. Make up and work an example similar to Ex. 33.
35. Which of the following fractions can be simplified by
striking out the 4's?
I + 4 a; + 4 4a; 3X4 4a 4a
II + 4 y + 4 4(1 + y) 4 + 11 X + 4 4a;
36. Make up and work an example similar to Ex. 35,
involving 3's.
37. Which of the following can be simplified by striking
out the 62's?
b' + x ^ 62,4 ggfcz a262
b^ + y Wy 362 + 4 a2 + 62 62a;*
38. Which of the following can be simplified by striking
out a + 6 in both numerator and denominator?
a + 6 5(a + 6) 3a;(a + 6) 3(a + 6)
o + 6 + c 3(a + 6) + c 4y{a + 6) 5(a + 6) + c
39. Make up and work an example similar to Ex. 38 con
cerning the striking out of a;. Of (p + qf.
172 SCHOOL ALGEBRA
40. Why is it allowable to subtract 4 from each member
of the equation x + 4 = a + 4 and not from each term of
the fraction — — ?
a + 4
41. How many of the examples in this Exercise can you
work at sight?
EXEBCI8E 61
1. Reduce ~ f to its lowest terms.
4 — ar
(x  2)« ^ (x  2) (a?  2) ^ (2  x) (2  x) 2  x
4  X* i2+x)(2 x) (2 + x) (2  x) " 2 + X
Check. Let x » 1, then, ^ " I = ^ ~ ;: = 
4 — X* 4 — 1 S
A, 2X211
^' 2Ti=2TT"s
Reduce to simplest form and check the work:
^ (a — 6)^ a + b — c
3. ^ ^ 9.
10.
6»
a*
(2x
y)*
»*
4**
9  m*
m*
 7m + 12
9
a*
(X
3)»
2
4
c 
b
5. r:^ :: 11.
12.
c*  (o + A)*
%3x
12ay — 600;
4a«6»8aft»
a? 27
96a; + a?
4  (o + 6)»
(o  2)*  6*
a« — 6a; — dy f 6y
60 "■ i*  o«
TRANSFORMATIONS OF FRACTIONS 173
Without changing the value of the fraction
14. Change each of the following so that the denominator
of the fraction shall be a — 6.
3 3 X xy  3a; 2a  36
h — a h —• a b — a b — a 6 — a b — a
15. Change each of the following so that the denominator
shall be (x — y) (x — y).
3 4 a6
(y  x)(y  x) (y  x) (xy) {y  xY
16. Show that ■■ — r equals
(y  ar) (a  y) ^ {x  y) {y  z)
17. By changing signs of f actors, write each of the follow
ing in three different ways:
5 ' a — b __ a — b a — b (b — a) /c — d)
a b X  y c  d {x  y) (y  z) (y  x) (y  z)
Solve the following equations, after reducing the fraction
in each equation to its simplest form:
la ^Zl+2a? = 5 20. ^i^ = 7 + a?
a; — 1 a; + 3
19.  a:* = 7 21. r — = 5  a:
a: — 1 7?
22 ^ "" 3o + fea;  36 ^ ^^
a + 6
aa; — 5a — 6a; + 56
23. ^ — ^ ^ = 15  3a;
25. How many of the examples in this Exercise can you
work at sight?
174
SCHOOL ALGEBRA
II. To Reduce an Improper Fraction to an Integral ob
Mixed QuANTrrr.
117. An Improper Fraction is one in which the degree of
the numerator equals or exceeds the degree of the denomi
nator.
Since a fraction is an indicated division, to reduce an im
proper fraction to an integral or mixed expression,
Divide the numerator by the derum/inator;
If there is a remainder, write it over the denomiruUor, and
annex the resvU to the quotient with the prosper sign.
Ex. Reduce
a? + x + 2
5
X +3
3x»  2x  5
ar' +3a; +6
5x 11
• • x^+x+2 ""^^^ x^+x^2^'^'
When the remainder is made the numerator of a fraction with
the minus sign before it, as in this example, the signs of terms of
the remainder must be changed, since the vinculum is in effect a
parenthesis (see Art. 41, p. 50).
EZEBCISB 62
Reduce each of the following to a mixed quantity:
1.
32
5
4.
a?
■2a; + 3
X
n
4a? + 6x5
2x
2.
121
9
3.
181
17
7.
lOa'a:' + 5ax — 7 — a
5ax
a?3a^ + xl
x + 1
TRANSFORMATIONS OF FRACTIONS 175
„ ^■+3xy2fl „ 9a»
O. ; 15,
x + y 3a*  2fc
„3aJ«13a;28 ,^3? + a?4x + 7
* ?^^3 "•
10. ^^x + 2a ^^
X — 1
X3. 2^1+7
x + 3
2a^
a + b
35^ — «* + a:* 
2x
a» + l
«
—  — (To three tenns.)
a? + x + l * l + aai^
14. ittt: 21.
a? + 2 ' 2 + xa?
22. Make up an improper fraction with a monomial de
nominator and reduce it to a mixed number.
23. Make up an improper fraction with a binomial de
nominator and reduce it to a mixed number. '
24. How many examples in Exercise 1 (p. 8) can you
now work at sight?
IIL'To Reduce a Mixed Expression to a Fraction
118. To Beduce a Mixed Expression to a fraction^ it is nec
essary simply to reverse the process of Art. 117. Hence,
Mvitiply the integral eocpression by the denominator of the
fraction, and add the numerator to thd residt, changing the signs
of the terms of the numerator if the fraction is preceded by the
minus sign;
Write the denomirudor under the result.
176 SCHOOL ALGEBRA
" a;y ^ X y
2y«
iln«.
y  aj
EXERCISE 68
Reduce to a fraction:
1. 3^ 2. 12f 3. 13i«^
4. al +  icoa + l ?^
a + a;
a
5. x + l+^r 11. ^ ^^ 4 a  1
X — 1 a — 2
1 av — c? .
6. a? + a:  1 7 12. x a ^ — + y
X  1 x + a
7. 4a;  2  ^ , , 13. 1   — ^
2a: + 1 26c
9.a:l^\ 15. (^i).^.a6
4
"■'("'•+rT^)
TRANSFORMATIONS OP FRACTIONS 177
20. ^e distance from New York to Chicago is 912 mi.,
which is 100 mi. more than one fourth of the distance from
New York to San Francisco. Find the latter distance.
21. A running horse with a rider has gone 1 mi. in 1 min.
35 sec., which is 13^ sec. more than three times the time in
which an automobile has gone one mile. Find the latter
time.
22. Make up two mixed numbers of your own and reduce
them to improper fractions.
23. How many of the examples in this Exercise can you
work at sight?
rv. To Reduce Fractions to EgunrALENT Fbactions op
THE Lowest Common Denominator
119. To Beduoe Fractioiui to their lowest common denomi
nator, as in arithmetic, we
Find the lowest common mvUiple of the denominators of the
given fractions;
Divide this common multiple by the denominator of each
fraction;
Midtiply each quotient by the corresponding numerator; the
results will form the new numerators;
Write the lowest common denominator under ea^h new
numerator.
2 3 5 .
Ex. Reduce  — , r^, and :; — = to equivalent fractions
dax 4a^x oaar
having the lowest common denominator.
The L. C. D. is 12aV.
Dividing this by each of the denominators, we get the quotients
4ax, 3x, and 2a.
178 SCHOOL ALGEBRA
Multiplying each of these quotients by the correspondii^ numer
ator and Bettiog the results over the common denominator, we obtain
Sax 9x 10a .
Ana.
12a*x«' 12aV* 12o*x*
EXERCISE 64
Reduce the following to equivalent fractions having the
lowest common denominator:
57^ „ ^ 2^ ic^ od
8*12 ' M'^'^'hc
2 31 ® a i2 3
5' 15' 20 a*  a a  1
2x bx X  1 1
9'6 1 + x ' xx + x"
^ 12a 7 a ^^ x 1
4. — r. — :. T 10.
• 56 '10' 6 x'Vafl
« 2 3  1 „ 1 + g  1 «
6. — ;,  — , 2a, 12 j; — r—, 7,
3o»'4aa;' 'a; 2  2a;' ' 3 + 3a;
14
a;»  1' a;* + « + 1'
3 4
IS.
16.
15 3
3x  6' 2a; f 4' a?  4
2 a: a;
«a;»' 3 + 3a;' 2 2aj
ADDITION AND SUBTRACTION OF FRACTIONS 179
Processes with Fractions
I. Addition and Subtraction of Fractions
120. The Method of Adding or Subtracting Fractions, as in
arithmetic, is to
Reduce the fractions to their lowest common denominator;
Add their numerators, changing the signs of the numerator of
any fraction preceded by the minu^ sign;
Set the sum over the common denominator;
Reduce the restdt to its lowest terms.
a — 1 a* — a a
o — 1 1 a* — a a
^ o»  a» + o» 4 1 + g  1
a{a  1)
■ ^  (i> + 2o» f o _  o' + 2o h 1
a{a — 1) a — 1
Ex. 2. Simplify ^^ + ^ ^
Arts.
7? — \ X + 1 1 — X
The factors of a;* — 1 are a; + 1 and a; — 1. Hence, if the sign of
the denominator, 1 — x, is changed, it will become a; — 1, and be a
factor of x' — 1. But by Art. 115 (p. 169), if the sign of 1 — a; is
changed, the sign of the fraction in which it occurs must also be
changed. Hence, we have
, «* , X , X x^ {x^ —x '\x^ {x 3a;* .
H rT H ;r = ;; :; = : r Arw.
x'lx + lxl x*l a;*l
Where the differences of three letters occur as factors in the va
rious denominators, it is useful to have some standard order for
the letters in the factors. It is customary to reduce the factors so
that the alphabetical order of the letters is preserved in each factor,
except that the last letter is followed by the first. This is called
the cydic order.
Thus, a — 6, 6 — c, c — a are written in the cychc order.
180 SCHOOL ALGEBRA
Ex. 3. Simplify
1
+ ^ r~^ TT +
(a  6) (c  a) (o  b) (c  b) (c  b) (a  c)'
Changing e — 6 to 6 — c, and a — c to c — a where they occur,
we obtain
(a  6) (c  a) (a  6) (6  c) ' (6  c) (c  o)
(a — 6) (6 — c) (c — a)
2a2c 2
r (a  6) (6  c) (c  o) (o  6) (6  c)
Ana,
Find the value of
3.21 a
1, ~ +   ;= 6.
2x X Sx a — b a + b
o 2 3,1 „ 3a: + 1 . 1  3x
2, J  7. X 1
3.
3a 4ax x 8
J Li a <^ + l _ <^~1
2ac Sab be 2 2
^ a + 26 6o 1 ^ a; 1 a; + l
' 2a6 6a2 ' « + 1 a:  1
^ 2^_+3, 3a + g ,^ a: + l o,73«
^ 4^^+^ — fe ^^ ^r"^+3^
3a ~ 46 2a — 6 — c ^ 15a — 4c
"• ~2~ 3 ^ 12
2a?2/ — 3z _ xs? — y^z . y — 3x2^ _ 2
* ~~3^ 2xy^ 6a?2 3
,, a; + 1 , 1  a; m + 1 , 2m
ar  2 a; + 2 (m  1)* ' m*  1
ADDITION AND SUBTJ^CTION OF FRACTIONS 18
„ (a + by 26 ,^ 3a: 2z , 10a:
1
4(ofc)* 06 a; + 2 a2!B*4
1 o . 20?
3a;3 2a; + 2 '6a?6
2a:  1 4x + 2 4ar^  1
X .^ a: — 1 z — 2
20. —  — + 2 —  —  —  — 
a:* — 1 a: + l a: — 1
a: + l a: + 2 x + 3
x + 2 a:3 , 2a: + 5
22. ^ ■ . 7 — 7~5 T +
2a
' 2ar^ + a:  1 40:^ _ j ' 2ar^ + 3a: + 1
6 ttb' __ akl^
M^ (a + fc)^ (a + hf
24 2ay ^ 3y ^ 3a: 3a:^  3y^
' a? — 2/^ 2x 2y 2ay
^ 3a:  y , 14ay 3a: + y
' x + 2y a? — 4y^ a: — 2y
^ 2 , 3a:  1 2a:  5
26. 1  + X — —
^ X — 1 x + 1 2
_ 2 a: 3 x^
27.
a: + 4 a:2_4a. + 16 a:* + 64
5a: 7 V 26
" 2(a:3)2 3a: + 9 4a:2.36
Reduce each of the f oUowmg fractions to its lowest terms
and collect:
a:29 x4 ' x^l^ (x + 1)^
\
\
182 SCHOOL ALGEBRA
31.
46* (o*)* , o 3a + 36
: — rr + ^ — t,
o»6* o*6» 3o36
32 _3?_ + _^ + l
33.' 2i,.+ 1 + 2
a^ — 6* a + 6 6 — a
a?^— 4^ 2y + x 2y — x
1 1 2a:
« 1 1 + a: ' 1 a?
36. t+t^^^+ y
^y^ x + y y X
37. +
88a4a + 4 Sa^S
3 2 5:c 1
39.
X . X—\ \ — 7? X + \ X + ^
1
{x  2) (3  a;) 10  7a; + a:^ (5 _ ^.j (3. « 3)
2 3,4
+
(a 3) (6 2) (a 2) (2 6) (a 2) (3 a)
5
(a  3) (2  h)
26 + a 26  a 46aj  2a*
_ a; + 1 2a;  1 , 2
*2. :r — — — — — +
41
2
43.
' 6a;6 12a; + 12 3  3ar* 12a;
a:*a;6 g* + 4a; + 3 15a;
a? + 5a; + 6 ar^4a; + 3 9a?
g* + 2a6 + 6^ 4a*  6* o*  2a6 + 36*
a*6* 2a*3a626*"^ a*3a6 + 26*
MULTIPLICATION OF FRACTIONS 183
45. — ;: 7
■ Qa? 3{x + 3y 5(x3)* 5a?  45
Sx + 2 , X ■ 4 — X
46. r +
■«*5a: + 6 8a!ar'15 7a; a? 10
47. ^^#, , + ,^^, .+
{a — V} (a — c) (6 — c) (6 — a) (c  a) (c — b)
48. , /, . + .. .^. .+ '^
(a — 6) (o — c) (6 — c) (6 — a) (c — a){c f b)
{x  y) (a:  z) (yz)(y' x) (z x){z y)
50. 2^ + ^ .+ "^
(xy) (xz) (y  z) (y  x) (z  x) (z  y)
(/ — m) (Z — n) (m — n) (m — (n I) {n — m)
52 1 ( a; r la; 1 1 ^? 1_
'a; (a;+l Lar*a: + l a; + lj ) a? + 1
53. Make up and add three fractions with monomial
denominators.
54. Also three with binomial denominators.
55. How many examples in Exercise 2 (p. 13) can you
now work at sight? * '
II. Multiplication op Fractions
121. The Hethod of Finding the Product of two or more
fractions, as in arithiiietic, is to
Multiply the numerators together for a new numerator, and
mvMply the demmiinators together for a new denominator , canr
celing factors that are common to the two products.
This method reduces the multiplication of fractions to
the multiplication of integral expressions, and enables us to
use agam our knowledge of the latter process. «
184 SCHOOL ALGEBRA
Ex. ^±lx^^X ^
g +y y (3; 4 y) (x  y) 4x«
a; ^ «(«» + y«) '^ (^ + y) (x + y)
4(x  y)
»«+y»
AnSi
II. Division op Fractions
122. The Method of Dividing one fraction by another is
the same as in arithmetic. For
a c aXd bX c
b ' d bXd ' bXd
aXd
(see Art. 114, p. 168)
 (?) (f )
bXc
Hence, to divide one fraction by another.
Invert the divisor and proceed as in midtiplication.^
x{x + l) z' + x + l ' (x + ly
_ (xl)(x«+x + l) (x«  1) (a*  1) (x + 1)* ^
a;(z + 1) ^ X* +X+1 ^ (z  1)»
 <^±il' An..
The reciprocal of a number is the result obtained by divid
ing unity by the given number.
Thus, the re iprocal of 2 is 1 5 2 or 5; of x is .
Hence, the reciprocal of a a fraction is the fraction inverted.
2 2 . 3 3
Thus, the reciprocal of g is 1 i «; that is, 1 X o* or ^,
Similarly, the reciprocal of t is ; of —7— is 1.
•" ■ ^ a z +y a —
DIVISION OF FRACTIONS 185
EZIBGI8E S6
Simplify:
1 50!^ 28a'y am^ + Zab . ab + 3
' 14o»c 15a^ ' 4a*  1 '20 + 1
21xy^ . 28a:» a:* 9 ^ a3
• 13z» ■ 39z* ■ ar« + X ■ «»  1
^ 9o*6 28ar' , 21a?'a: ^ (o  1)» ^ » + 1
ZL
8c*a; 15fc*c 106c» a(,x + 1)« (a  1)*
*• 49y ^ "^'^^ ^ 40x» 9**  1 ^ 12ar  18
5 15» 2a;(a; + 1) 2a:*  a;  1 4x'  1
■ 2x(2a;  1) 5a? ' 2a? + «  1 a:*  1
a»y  aa?y , oV  2aay + a^
c?7? + o^x^ ' o* + oy
a» 1 _ a»o / 1 \ 2x  2
3a? + x2 >. 4a?l /x \ . fa? . y^\
"•4x*4x3>^9?^ "V^ + V^W + ^J.
5xy x + y {xyr 2xy
17 3(aj&)^ ^ 7(a»  6*) . 14a6
4(o + hf 9(o  6)» ■ 8(o + 6)
x' + 2x3 x* + 2x15 . x' + Sx'
x* + x12x' + 2x3 ■x» + 4x'
^ 6x*y  4xy' ^ aOx + 20y xy
45a? — 20i/* 4a?y* x + y
^ 6x*5x4^6x* + x2^ 2x*+ 5x  12
^" „ . . _ 7 X r^ : X X
2x» + 7x4 4x*4x3 9x*6x8
27
186 SCHOOL ALGEBRA
fy* »' + y' V /'i L y ^ • ' a!*  ay + y*
„ :^{alY (a + xY\ . g + gl
■ o*  (a; + ly 1  (o  «)« * a  a;  1
26ao''6^464 yo»46 + 4
■ 62a o2 + fc2 + 2oi4 6«o* + 4o4
\a6 6c ac) \6 c o/ o*6*c*
„ a+uiiUp+"('»)x('i+?+«')i
\a^ ar ac / L ^^ \ ^ /J
tm + 2n , m_— _2nn _^ r m + 2n _ m — 2n "
m — 2n m + 2n J * L^ — 2n m + 2n J
30. Write the reciprocal of each of the following: 3, a, 2x,
4 1 a^^J_ a + 2x 1
5' 5' 2a:' 2x a  26' a^  2V
31. Make up and work two examples involving both mul
tiplication and division of fractions.
32. How many of the examples in Exercise 15 (p. 60) can
you now work at sight?
IV. Rbduction of CoBiPLEx Fractiokb
123. A Complex Fraction is one having a fraction in its
numerator, or in its denominator, or in both.
* ^ In simplifying any complex fraction, it is important to
write the entire fraction a± ^h step of the ppcess.
REDUCTION OF FRACTIONS 187
Ex.1 ^L. '—^xX^ ^^fl*.
y y
When the numerator and denominator of a complex frac
tion each contain fractions, the expression is often simplified
most readily if we
MvUiply both numerator and denominator by the lowest
cominon denominator of the fractions contained in them.
1+1+1
^Ex.2. Simplify  — ?^ — .
 +  + 
y z X
Multiplying both numerator and denominator by xyz, obtain
124. A Continued Fraction is a fraction whose denomina
tor is a mixed expression, having another mixed expression
in its denominator, and so on until the fraction ends.
1
;
Ex. Simplify
<:
x+ ^ ' ^
1 1
X\^ :i —  x+ ^f xH
a: — 5
['  ^] m
Ana.
X — 5
Hence, in general, to simplify a continued fraction,
Reduce the last mixed expression in the fraction to an im
proper fraction (see the brackets in the examples);
/
188 SCHOOL ALGEBRA
Then invert the last fraction and mtdtiply H into the numer
aior under which it is placed (see the brace) ;
Thus aUemately reduce a mixed number and irvoert a dimor
fraction until the simplificaticn is completed.
Simplify:
4
X
X
1.
21
X
2.
1
4
3?
1
2x
4**
1
1 
'2i
1
3
2
X
a?
«»
(i)'
67
1
X
3 *
1 1
, a + 1
11
a — 1
"*2d
2
a? — 
4. '^ ^
2cd
6. 3
6
2a:
+ 1
_£
7. ^^=1 11.
y «
y a;
I
a; 1 — X
8. =^ r r 12.
1 + a: a:
9  X 1 ~ a;
a? 1 la; a:
(I+IY «+4ti
11,1 "' c*(o + fe)«  a*t*
a^ ax 31? a^lf(?
a .x^
10. 14.
r
REDUCTION' OF FRACTIONS 189
H 2 21. 2ol
.ax ax n a
15. ■■'■■ " 1 ■ Z —
X a 2 . 1 a
H a
a X a ax i 1 + a
"• 2a;l * M ^
1 2a;
3a;
(at  1)»  c'd* "* x + 1
(a6 + «i)«l 23. 1^
x1
la 3^^^ + 6
2x • u
3 qo
1 1 24. 1^,
1+O + T
20. 3 ^—Y 25. §_ f. (o  2)
^rra ' + 2+4
1 1_ '^y^ 2be )
a 6 + c
4
a;
27. : — X
a: + 2 « — 2 ar
1 + a? 1a*
1 ? 1+ ^
1 +.^^ 1  *
1 — a; 1 + as
190 SCHOOL ALGEBRA
29. ^y»*
^f^ + ^
a^ + aj^y + ^y .
{^fy ' /i +
l./^Uiy l./'iiY /"UlIY'
a' \b cj ^b" \a cj ^\a^ c h)
30. >l zJ X ^ ^ X
1^ _ /I 1 Y 1^ _ /I _ 1 Y / 1 _ 1 Y  i
62 Va "*" c; c^ U 6/ \a ^7 ^^
31 1 + 2a; 4(i  ^x + a:^)  f
' 1 + oi + 2? I(J + ^ + i^)  *■
"^ 1  2x
Find the value
32. Of ^ . ^ when t) = =•
1 + 2i? 7
r^ 22 , 1
33. Of ; when « = — «•
34. Of JF when F =^—,m = 10.2, d = 5, and r  ^•
r 2
35. Make up and simplify a continued fraction.
36. How many examples in Exercise 19 (p. 78) can you
now work at sight?
EXERCISE 58
Oral Review
1. Give the value of each of the following:
(1) 11. (2).l  1. (3) ?^t^ + 5^.
^ ^ 2a 4a ^ ^ 3a 6a '^ ^ 2 ^ 2
•<4) i±i  iz_». „, 1 + J. <e, 1  >.
.E^a,^+iy (.,g)' (3,(iD
PROCESSES WITH FRACTIONS 191
3. On the foot rule show the meaning of i in. 4 2. Of } in. 4 2.
4. Divide each of the following fractions by 2: , }, {, , f,
2a £ a a f 6 6 7 3^ 5£ 46^ 3^
6 ' 26' 6' 2 'a' a' 2b' 4b' a' a'
5. Divide by 2. By 2a. 36. 46.
6. Divide 1 by each of the fractions jr, ^r, % t, f^r^
Ji Z Z Ub >
7. Give the reciprocal of 3, =, 4, r, ti ri  •
3 4 4 6a
1 1 111232
8. Give the value of  when x  s When ir  , , ~, , , ,
X 2 3 4 8 3 8 6
2 _1 6
3' a' c
9. State the value of? when a; 1 Whenx i , ^, 1 i,
 3' 2' 5
15 1
10. Give the value of when x  — •=■.
2 f X 5
n Tm,o+;al«#^9 rv ^ 3 4 a  1 a 4a 1 2a 1«
12. If 4 is subtracted from both nimierator and denominator of
■f^y is the value of the fraction changed? By how much?
13. If a = — j: and 6 = — ;r, state the value of ^^. Of — pr,
4 2 46
14. YThat is the value of 1 ^ 2/3? Of 1 4 a/6? Of 2 ^ x/2yl
15. Simplify those of the following fractions which can be reduced
to lower terms:
4x Ax 4a 4 ^'a — 6 a + 6 a* + 6* a*6
4a + 6' 4a + 46' 46' a + 4' 6  a' a« + 6^' a* + x«' aV
Give the value of
 31 2 8 %_ x^  1
8 '  4  6'  2/3' a/6' 1  « '
17. Of 4a« 4 ^« ia» 4 Ja« x* 4 Jx* Jx* + x ^ 1 s Jx»
1 + x* 2a» + }a 2 + S/Ox
192 SCHOOL ALGEBRA
18. Make up an illustration to show the value of 5 X (for
instance, in connection with a pupil's mark for ^ examples wluch
he failed to work correctly).
19. Give in the briefest form the product of (a — 6) (6 — o). Of
(x  2y) (2y  x). Of (a  6) (a + 6) (o« + 6«) (a* +6*) (a* +68).
1 1 o
2a Does
(y  ^)* («  yY
EXERCISE 69
Written Review
1. Indicate by a parenthesis that 2a — 36 + c is to be subtracted
from 5a + 26 — 3c. Then remove the parenthesis and simplify.
2. Subtract the sum of^+2y—z and a — ic — 3^/ from — 5.
Also from 0.
3. Write by inspection the value of [(3a — 6) — c + 2dp.
4. Factor (a + 6  c)»  (x + y  x)' • ^  i . 1  ^
z* y* 8
1 1 ' 1
5. Chajige — h : so that it shall be a perfect square.
x^ xy y*
6. What is the difference between an exponent and a power?
Give an illustration.
7. Subtract (5a + 1) (2a  3) from (a + 2) (a + 1) + (a + 2)*.
8. Find the value of Z{x  1)«  3(a; + 1) (x f 2)  x{x  2)
(y — 2x) when a? = — 2 and y = — 5.
2A* — B^
9. Find the value of — «^ — when A = 5a and B = 2a.
10. By factoring find the roots of x' — 5a; + 6 = 0. Prove your
answer.
11. Show that J is equal to ^ *
c — a ^ a — c
12. If a « 12^, 6 » 37^, c = 33^, and d  10, find in the shortest
4cP 4cP 4d^
wfty the numerical value of each of the following: — , — , r
13. From 7.08a2 take  ^aK
(at ^ 5»)a
14. Reduce — 1 H — tm — to an improper fraction.
PROCESSES WITH FRACTIONS 193
15. When we change x— 3=5toa;=5f3, what is the change
called? What right have we to make this change? Why do we
transpose — 3 instead of adding 3 to each member of the given
equation?
16. What is the use or advantage in being able to jfind the H. C. F.
of two given expressions? In being able to find their L. C. M.?
Illustrate.
17. Show that the sum of two numbers (as of a and 6), divided by
the sum of their reciprocals, equals the product of the given numbers.
18. If s = z , find the value of s when a =» 2 and r » — «.
I — T Z
2 3
Also when a = — ^ and »" = — 7*
19. If 8 = r, find the value of s when r = h, ^ = "TTf and
r — 1 J 46
1
a =  2
20. If a = 3, which is greater, ttt or —t^ — ?
' o ' 10 — a 3a
21. Divide 2a» + 10  16a  39a2 + 15a* by 2  5a»  4a.
22. Give an illustration to show why 3x0 gives zero. Also
why « gives zero.
23. Show that a common factor of any two algebraic expressions
is also a common factor of their %um and difference. Of the sum
and difference of any multiples of the given expressions.
24. Prove that if half of the smn of any two numbers (as of a
and 6) is added to half their difference, the result will equal the greater
of the two munbers. Illustrate by two numerical examples.
25. Prove that if half the difference of any two numbers is sub
tracted from half their siun, the result will be the smaller of the two
numbers.
26. Write an example of a continued fraction and reduce it.
27. Why is it allowable to change both minus signs to plus in
— aj = — 3, and not in — a; — 3?
2a CoUect in a short wfty ^^1^2 + r:r2 + JTa + ^qn
SuG. Collect the first two fractions first.
104
SCHOOL ALGEBRA
29. Collect in a short way
X X
90. Also
' + *
X +2 X 3 x2^a;+3'
2xy 4xy*
X " y X +y x*+2/* x*+y*
Simplify:
i(f x* ■hix2)
31
32.
+ xy .2i/»)
«* + icy — 22/*
1.2 3 .
34. T^ — 7 H r — s  z — 5 H
2 X
1 2x 2x + l ^4a^ 1
« +2 ftp
4x 3
'• X  1 "^ a?  2 X  3 ^ (x»  x) (x  2)
^ +^
2 +_A
o — 0+ o — :3 00+
a a <r a
3 2 1
^ x«  3x + 2 + (x  1) (3  x) "^ (2  x) (x  3)*
fl+x _l+x«1 flha:' l+x»l
'^'U+x* l+a;»J Urhx' 1+W
39.
i
X
1+'
ax
x»
."oJ
X
6  1)'
X — 4 —
40.
X — 4
X —
x4 
1
X
X  2
2
X — 4
X —
X — 5
2
41.
1 
9
. a;«l
x»
+ ^
9
X*
X*
9 J
X
(ii)e>)
PROCESSES WITH FRACTIONS 196
1 +8g« 1  27x»
2x 3x
2x 3x
1 +r^^ 1 
1 2x 1 +3a:
g* 5g» +4 g a; 2 ^g
*® g» + l ^ 1 ■*■, l'
g g*
44. Given o + 6 4 c  2a,
show that a 4 & — c = 2(« — c) and that o — 6 + c «» 2(« ^ 6).
45. Also show that
a^+d^ b' 2(s  a) («  c)
1 
2ac ac
a* +1^ (^ 28(8  c)
46. Also show that 1 + 2fjh ~ ah '
47. Show that
(2o  3g)» [8g^(o + 2g)» + 5g^ (o + 2g)^l t 27g» (a h 2g)^ (2a  3g)«
(2a3g)^»
reduces to
2ag* (24g 4 5a) (g + 2g)»
(2a  3g)i«
48. The distance from New York to San Francisco by way of
Cape Horn is 13,800 mi. This is 1920 mi. less than three times the
distance from New York to San Francisco by way of Panama.
Find the latter distance.
49. Make up and work an example similar to Ex. 48, using the
fact that the distance from London to Bombay by way of the Cape
of Cxood Hope is 11,220 mi., but by way of the Suez Canal is 6332
mi.
CHAPTER XI
FRACTIONAL AND LITERAL EQUATIONS
125. A fraotional equation is an equation that contains
an unknown number in a denominator.
Ex.  + 5 = 3a;.
X
Equations containing binomial numerators and numerical de
nominators are frequently termed fractional equations, since they
are solved in the same manner as fractional equations proper. See
Ex. 1. of Art. 126.
An integral equation is an equation which does not con
tain an unknown number in a denominator.
126. The Method of Solving a Fractional Equation. If an
equation contains fractions, it is necessary first to midtiply
the members of the equation by such a number as will remove
the fractions.
^  o 1 a; + 1 '2x 5 11a: + 5 a;  13
Ex.l. Solve ^ j^ 3—
The L. C. D. of the denominators is 30.
Multiplying both members of the equation by 30 (see Art. 70,
3), we have
15(a; + 1)  6(2x  5) = 3(1 Ix + 5)  10(a;  13)
Hence, ' 15a; + 15  12x + 30 = 33a: + 15 lOx + 130
15a:  12a:  33a: + lOc 15  30 + 15 + 130
 20x = 100
a: = — 5 Root
Ghkck. 541 ^ =:^^l ^14^ = 2+3  1
2 5 2 5
11a: + 5 a:  13 ^  55 4 5  5  13 ^ , g 
10 3 10 3 oipx
196
FRACTIONAL AND LITERAL EQUATIONS 197
Ex.2. Solve ,^ + ?J^  ^^ = 0.
1 + a: 1— arl— ar
Multipljdng by the L. C. D., 1 — x*,
4(1  x) H (x + 1)2  x2 + 3 =
44x+a;2H2x + laj*+3=0
2x ^  8
a; = 4 Root
Let the pupil check the work.
Hence, in general,
Reduce each fraction in the equation to its lowest terms;
Clear the equation of fractions by mvitiplying each member
by the L. C. D. of all the fractions;
Complete the solution by the methods of Chapter VL
EXERCISE 60
Solve and check each result:
1.
X Sx 7x 34
3 5 5 15
2.
2a;3 z + l_5x + 2
4 6 12
3.
1 3 5 _ 3 _ 19
2x X 3x 4a; 24
4
2x 2x + 1 1
4.
3 5 3
«
3a; + 5 _ . x + 4
4 6
6. 7 = f(a;2).
7. 2a;  8  (24  2x) = 0.
8. (a;  1) = i(x  2).
9. 3(§x  I) (^x + f ) = x«.
198 SCHOOL ALGEBRA
,^ Z2x x3 . « + 4,l
8 6 3 ^24
3xl a + 14a: + 1 ^ 3(gl)
7 6 21 4 •
2a; + 5 x + lj _ 5g  10^ 1
5 10 20 5
^ 13. 3^f(5 + *)+?^i(2:c + 5)=^^
6x + 5 , x + 5
i4.2(. + i) + .(ll)=«^ +
4
a! + 5 _ g + 7 g + l _ 2a;5 ^ x + 22
7 5 2 10 70 *
16. f (5x + 2)  i(7a:  2) + f (Sx  2) = X  1
17. .5x — .4x = .3. 5 g
23.
18. 1.6x — 5 = x. 2x— 1 3x + l
19. .6x — 1.5 = .2 — .15x. 6x — 5 8x — 7
1.5X1.6 _ 3.5X2.4 ^ si^Ts = :^+T
1.2 .8
^, 3.2X3.4 .6x + 4 25.   ^:i^ = .
21. —^^ 2:5 3 3X7 3
X1 3 26 _5_4._6 L.
"^ ;M^"5 l_a;^l + x l;c«
3 , 4 8x + 3
27 I =s •
3x^3 + x 9x»
2x + 1 10 _ 2x  1
■2xl 4r'l~2x + l*
1,1 2
29, — rT +
X+1 X1 X+2
FRACTIONAL AND LITERAL EQUATIONS 199
30.
x\ _ 3 lSx
a? "8 a;  2 a^ + 2x + 4'
31.
X — 1 x+l
32.
x3lx^ + l_x + d
2x + 6 3? 9 3x9
33.
3 4 5 _ 11 4
xlx + 1 2x2 3x + 31x»
34.
x + l x* + 7 2 x1
2x3 4x^9 2x + 3 6  4x
35.
3a?5 7 o ^_ 7
3X.6 6X + 12 ' ~ 2x^8
36.
6x + 6 2x + 1 2x
2a? + 5a: + 3 2ar^  a:  1 a:^ ^ 2a:
Reduce eacji fraction in the following to its lowest terms
and then solve:
37. ? = ^:ii. 39. ^ ~ ^ ~ ^ ,= 4  6(x3).
2 x+l x2 ^
38. lE±i  ? = 0. 40. ^'^"^ ^ = 83(x+4).
x* + x4 x*+2x + 4 ^ ^
Find the value of the letter in each of the following:
«. _J_ + 1 = ?. 43. _i^ ? 14 = 0.
t) + 2 3(j) + 2) 3 32a 64*
42.^_+i.^ ?_.44. ^_+ 31 1
3(p7) 6 2p14 2< + 2 3< + 3 6
„ r + 6 2r18, 2r + 3 ., ,3r + 4
4S. = 54 ^ .
11 3 4 '12
46. li A = ho, A = 600 and w = 20, find the value of /.
Do you know the meaning of this process in arithmetic in con
nection with the rectangle?
200 SCHOOL ALGEBRA
47. In like manner, find / when ^ = 80 and t^; = llf .
48. If F = Iwh, V = 720, / = 10, and w = 6, find h.
Do you know the meaning of this process in arithmetic in con
nection with the study of volmnes?
49. In like manner if F = .36, w = .8, and h = .9, find /.
50. If p = 6r, p = 9 and 6 = 45, find r.
Do you know the. meaning of this process in arithmetic in con
nection with the subject of percentage?
51. If p = 6r, p = 760 and r = .05, find 6.
52. If i ^pH,i=^ $66, p = $440 and t = 3, find r.
Do you know the meaning of this process in arithmetic in con
nection with the subject of interest?
53. J{i=^pH,i = $66, p = $360, and t = 3f , find r.
54. I{i = pH,i = $15.75, p = $75, r = .06, find t
55. If C = f(F32) find F when C = Sjp. When
C = 100.
Do you know the meaning of this process?
56. If LW = Iw Bind L = 8, W = 100, and «? = 40, find I.
Can you find out the meaning of this formula and process?
57. I{R = ^^, find s when iJ = 10 and or = 32.
9 + s 
58. JiV = (l + J^v, find V when F*= 20 and t = 13.
59. It K^ ^h{b + 6'), K = 280, h = 12, and b = 10,
find b\
60. If F = ttR'H, V = 1540, TT = ^, andiJ = 7, find H.
61. If r = 7riJ(/i + L), r = 1144, TT = ^S R = 14,
find Z«
FRACTIONAL AND LITERAL EQUATIONS 201
62. Make up and work an equation containing fractions
with the denominators 4, 6, and 12. Can you form the equa
tion so that the root shall be 1? 2? 4?
63. Make up and work an equation containing fractions
with the denominators a; + 2, x — 2, and o? — 4.'
64. Work again Exercise 24 (p. 99).
127. Special Methods. The work of solving an equation
may often be lessened by using some special method or device
adapted to the peculiarities of the given equation.
First Special Method. If in a given equation ih/e denomir
naiors of some fractions are monomials, and of others are poly
nomials, it is best to make two steps of the process of clearing
the equation of fractions: (1) remove the monomial denom
inators and simplify as far as possible; (2) remove the re
maining polynomial denominators.
^ , c, , 2a: + 8 13a;  2 , a; 7x x + 16
Ex. 1. Solve —  — — r + 7: = 77: ^r;;
9 17a: 32 3 12 36
Multipljdng by 36, the L. C. D. of the monomial denominators,
8a; +34 ?^^f^ + 12x = 21a:  a:  16
Transposing all terms except the fraction to righthand side,
36(13a:  2)
17x  32
= 50
Dividmgby2, i^g!:=^=25
234a:  36 = 425a:  800
191x = 764
a: = 4 Root
Let the pupil check the work.
Second Special Method. Before clearing an equation of
fractions, it is often best to combine some of the fractions
into a single fraction.
202 SCHOOL ^ALGEBRA
X — I x — 2 x — S X — 4
Ex. 2. Solve
x — 2 x — 3 X — 4: a; — 5
In this equation it is best to combine the fractions in the left
hand member into a single fraction, and those in the righthand
member also into a single fraction, before clearing of fractions. We
obtain
1 1
{x  2J (a?  3) " (x  4) {x  5)
7
Clearing and solving, ^^o ^^
Let the pupil check the work.
EXERCISE 61
Solve the following and check each solution:
1 3a; — 1 , 4x _ x + 5
6 3a; + 2 2
2 3 2a; a; lGx ^ 2 3a;
4 '*",6 157a; 9 *
3 2a;  1 x
3. 2i.
2a; + 4 4 2
^ 5a;H13 _ 2a; + 5 23 5  ja;
12 6 4a; ^36 3
5
2ix3 a; + ll lla; + 5 _Q
7 a; 4 8 16
g 3a;  1 4a;  7 __ a; _ 2a;  3 7a;  15
30 15 4 12a; 11 60
J 6a;  7 a; + 1 . 2a;  1 _ 199
' lla; + 5 15 30 10 '
8 oi a; + 4 ^ 43a; 4a; + 9 _ 4a ; 5
■^7a; + ll 8 12 24 "^4*
3 o^2^  — + ^""^ = 2a; 1^ _ 2a; + 3^
3a;
'9 12 ' fa; +11
J
FRACTIONAL AND LITERAL EQUATIONS 203
, 2r2a; 1 fSa:  1 , 2a;  5 , IT , 19* + 3 ^
•3L92l6 + ^T8 + ^J+^4 ?•
f O^ 1 C A^ I 1 A^ I t
^ 1.2a;  1.5 .4a; + 1 ^ .4a; + 1
1.0 »^X "^ m^ »0
12.
13.
14.
1
a; — 2 a; — 3 a; — 4 x — 5
X — 1 _ X — 3 _ X — 5 _ X — 7
a; — 2 a: — 4 a; — 6 a; — 8
■
a; — 7 a; — 8 a; — 4a; — 5
15.
a; — 8 a; — 9 a; — 5 a; — 6
3 2 ^ 2 3
3a; 2 2a;  3 2a; + 3 3a; + 2*
^^^ 2a; + l 2a; + 9 ^ 2a; + 3 2a; + 7
a; + l a; + 5 a; + 2 a; + 4 '
17. 4a;  17 10a;  13 8a;  30 5a;  4 ^
a;42a;3.2a;7 a;l
18. Work Ex. 2 by clearing all denominators at once. Then
work the same example by the method of Art. 126. About
what fraction of the work is saved by thesecond process?
19. Treat Ex. 13 in the same way as you treated Ex. 2.
20. On the average, the distance one must go below the
surface of the earth to get an increase of 1° in temperature,
is 62 ft. This is 1 ft. more than one third the distance one
must go above the earth's surface to get a decrease of 1° in
the temperature. *Find the latter distance.
21. Who, so far as we know, first invented transposition
in solving equations, and when? Who first brought the use
of transposition into prominence?
, * Transpose the second and third fractions.
204 SCaoOL ALGEBRA
22. From what language does the word (dgehra come?
What does the word algebra mean?
23. Work the oddnumbered examples on p. 101. How
many examples on that page can you work at sight?
128. Two Equivalent Equations are equations which have
identical roots; that is, each equation has all the roots of the
other equation and no other roots.
Thus, x» — 4a; = 0, and x(x f 2) (x — 2) =0 are equivalent,
since each is satisfied by the values a; = 0, 2, — 2, and by no other
values of x.
If we multiply the two members of an equation by the
same expression, the resulting members are equal, but the
resulting equation may not be equivalent to the original
equation.
Thus, if we take the equation a; = 3 and multiply each mem
ber by a; — 2, we obtain x{x ~ 2) = 3(aj — 2) or
{x  3) (x  2) = 0,
which is not eqmvalent to the original equation, since it has the
root X s» 2, which the original equation does not have (Art. 103).
In general, if the two members of an integral equation are
mvUiplied by x — a, the root a is introduced and the resulting
equation is not equivalent to the original equation.
129. An Extraneous Boot is a root introduced into an
equation (usually iminlentionally) in the process of solving
the equation.
The simplest way in which an extrai)^ous root may be
introduced is by multiplying both members of an integral
equation by an expression containing the imknown number.
See the example of Art. 128.
A more conmion way in which extraneous roots are intro
duced during a solution — and one more difficult to detect —
FRACTIONAL AND LITERAL EQUATIONS 205
is by a failure to reduce to its lowest terms a fraction con
tained in the original equation.
2x— 4
Thus, in solving / _ iw ^o^ ~ ^' *^® ^* ®^P should be to
2a;— 4
reduce the fraction 7 777 ^ to its lowest terms. If this is done,
(a;— l)(x— 2) '
2
we obtain the equation r « 1, whence a; = 3.
If, however, we should fail to reduce the fraction to its lowest
terms and should multiply both members by {x — 1) (a? — 2), we
obtain 2x— 4=x'— 3x42, whence x* — 5x + 6 = 0,
or (z  3) (x 2) = 0, andx = 2, 3.
On testing both of these results, we find that the extraneous root
2 has been mtroduced.
Often the fraction which can be reduced to simpler terms
occurs in a disguised and scattered form. In this case it is
best to solve the equation without attempting to collect the
parts of the fraction. An extraneous root may then be
detected by checking the results obtained.
Thus, the fraction in the above equation might be changed
in the following way so as to make it difficult to detect its
presence in the equation:
, 2a 2 2
^'^'''^ (x l)(x 2) " ix  1) (x  2) " ^'
whence • ^^2 " (z  1) (x  2) = ^^
There is nothing in the appearance of this last equation to indi
cate that it implicitly contains a fraction which should be simplified
before proceeding with the solution proper.
Hence it is important constantly to remember that a root of an
equation is not such because it is the result of a series of operations,
as clearing an equation of fractions, transposition, etc., but because
it satigfies the original equation.
206 SCHOOL ALGEBRA
130. Losing Boots in the Process of Solving an Equation.
If both members of the equation (x — 2) (x — 3) =0 are divided
by X — 2, we obtain a; — 3 « 0.
The resulting equation is not equivalent to the original equation
since it does not contain the root x ^ 2, which the original equation
contains.
Hence, in general,
// both members of an equation are divided by an expression
containing the unknoion quantity, write the divisor expression
equal to zero, and obtain the roots of the equation thus formed
as part of the answer for the original equation.
EZEBCISE 62
1. Multiply each member of the equation a: — 2 = 1 by
X — 2. Is the resulting equation equivalent to the original
equation? Why?
2. Make up and work an example similar to Ex. 1.
3. Multiply each member of the equation a: = 2 by a: —5.
Is the resulting equation equivalent to the original equation?
Why?
4. Divide each member of the equation x^ — 9 = a; — 3
by a; — 3. Is the resulting equation equivalent to the orig
inal equation? Why?
5. Make up and work an example similar to Ex. 4.
ic — 3
6. Solve the equation — —  = 1 after first reducing the
fraction to its lowest terms. Now solve the equation without
reducing the fraction to its lowest terms. Do the two meth
ods of solution give the same result? Which result is correct?
Why?
7. Make up and work an example similar to Ex. 6.
Solve each of the following, check each result, and point
FRACTIONAL AND LITERAL EQUATIONS 207
out each extraneous root, giving the probable reason for the
occurence of such a root:
8. 5+ 1 ^
2 x+1 x+1
9. ^ = 1 9
x + 1 (x + l)(a;2)
10. ^ + 1 = ^
(x + 2)(a; + 3) x + 2
a? — 1 7
11. ? — t = a;  .
a^l 6
I
12. ^^4^ — = Z.
a?l a;l x + l
13. Form an equation in which 3 is the extraneous root.
14. How many exami)les in Exercise 31 (p. 121) can you
now work at sight?
131. A nnxnerioal equation is an equation in which the
known quantities are expressed by figures. Thus, all the
equations on p. 199 are numerical equations.
A literal equation is an equation in which some or all
of the known quantities are denoted by letters; as by a, b,
c . , ,, ot 7n, n, p . , ,
The methods used in solving literal equations are the same
as those used in solving numerical equations.
Ex. 'Solve a{x — a) = b(x — b).
ax — a^ ^bsc "l^
ax —bx — a^ —b^
(a  6)x  a«  &«
X = a +b Root
Chxck. a(x — a) = a(a \b — a) ^ ab
bix 6) =6(a+66) = a6
208 SCHOOL ALGEBRA
EXERCISE «8
Solve for x and check:
1. 3a; + 2o = X + 80. u. o^ = (o  6)* + 6*it.
2. 9ax  3b = 2ax + ^. * 12. (o 6)x = o«(o+6)a!.
3. 5axc = ax5c. ^^ 9^,bx^a_b
*. ax + b = bx^2b. b a b a
5'3cx = ai2ba + cx). ^^ a + x^ ^ ajx_
6. 5a; — 2ax = 3 — 6. a — 2x o + 2a;'
7. 2ax  Sb = ex + 2d. ,^ 4xa ,_x + a
8. (x + a){xb) 3^. ' 2xa xa
9. ab{x + 1) = a* + Vhc. XXX
10. (a!l)(x2) = (xo)«. ' 06 c"
,Q ax^b.bx — ccx — a ^
X». 1 1 = y,
dO DC ttC
' 3a + 63a6" Oa^ ^ 6^ *
21. ? — ^ la;
22.
a a — 6 a + 6 6
a^ — X V — X (? — X _(j? V
c a ' b c a
„ 5a*7x , ofc^ + lOa; 10cr' + 3x , 5(a  c) , 6«
Sob 5ac 6&C . 36 5c
FRACTIONAL AND LITERAL EQUATIONS 209
a + x ^ ^x x — 1 X
24. = . 25. w
X — a 1 a+1 a
X
a X a + x
26. Make up and solve two literal equations.
27. How many examples in Exercise 35 (p. 131) can you
now work at sight?
EXERCISE 64
Oral
Solve the following orally, without transposing any term contain
ingx:
1. 4a; = — 12 4. ox » 6 7. 6a; + c = d
2. 3a; = a t 5. 2x — 4 = 6 8. ax +bx = c
3. ax = 5 6. ox— 5 = 7 9. ax=c+5x
10. 6 = 3x 11. 10 =  5x
2 V 1 X 1
12. ^ = 3X ^21. =2 30. g = j
13. a = 5x 22. =3 31. 4=^
X z
a 2x
14. c = dx 23. = 1 32.  3 = =
X o
15. 3x = ^ 24. ^=9 33. 4 = y
4 111 bx
16. 2x= ? 25. ^!^=37 34. a= —
5 X c •
17. 5X = 4 26. j=p 35. =
18. ax = T 27. ttx = 2 36. xx = ■=
h 3 2 7
4 M "i; M MM ^ A MM 3? ^
"•3=2 28.^=4 3^1= 3
20. «6 29. 7X=6 38. =i
a 4 X 2
210 SCHOOL ALGEBRA
39  =  41  = 2 43 25_£
x" 5 " 3~ X ' 6 " d
^71 ^o 2 1 ^ 4 12
45. p+Q= — '— 47. = m
^ ^ X X — a *
46. ——^p+q *»• ""7"="—^"
a; X o •
49. How many examples in Exercise 45 (p. 155) can you now
work at sight?
EXERCISE 66
Review
Solve for x and check:
2 5 2 X a;' 1
^' a;2 x+2"x24* ^' 2 a:l~^
6x4 1 2x 1 2a;~4 _^
^' 15 5 7X16""'
a;&_ X + 6 4a^ ~ ^^ _ ^
' a; — 2a x+2a . x^—4a^
SpxjSqx p2q Vq _^
x^— if x{q q— X
X— 2 x— 3 X— 5 X— 6
6.
xZ x4 x6 xl*
X— 5 a;
7. 6+l(x9i^)+3+ g 2
Find the value of x in the shortest way, when
8. yX^Xl9 + ^X41.'
9. 3.1416X = 3.U16(723)  3.1416(476).
X X (. _ 2x X 
10. =2. IX. ^=5.
3 2 1
12. If = 4, find the value of x when y = t; Also when
X y 3
= ~ • When y = ^'
4 5
FRACTIONAL AND LITERAL EQUATIONS 211
13, Solve for Z: TQg= _ , •
.14. Solve for a: ^= «(w)
15. In adding — j — h ^ we retain the L. C. D. 24. In solving
the equation — j— = ^ and clearing of fractions, the L. C. D.
24 disappears. What is the reason for this difference?
16. Make up an example similar to Ex. 15.
17. Make up and solve an equation which contains fractions
with the denominators 8, 2(x — 1), and 4.
18. Make up and solve an equation which contains fractions
with the denominators a +b, a — b, and 6* — a\
EXERCISE 66
1. Find the number the sum of whose third, fourth, and
fifth parts is 94.
2. Make up and work a problem concerning one fourth and
one sixth of some number.
3. State  — 2 ~ 28 as a problem concerning a number
and find the niunber.
4. A certain number exceeds the sum of its third, fourth,
and tenth parts by 38. Find the number.
5. A piece of bronze weighs 415 pounds. It contains twice
as much zinc as tin, and 8 times as much copper as tin. How
many pounds of each material are in the bronze?
6. Find two consecutive niunbers such that one seventh
of the greater exceeds one ninth of the less by 1.
7. Express in symbols 15% of x. 5% of x. 115% of 6.
212 SCHOOL ALGEBRA
8. Two men kept a store for a year and made $4800.
The man who owned the store building received 40% more
of the profits than the other. How much did each receive?
9. In building a macadam road the county pays twice
as much as the state, and the township pays 50% more than
the state. How much does each pay if the road costs $18,000?
10. Separate $770 into two parts so that one shall exceed
the other by 20%. By 33%.
11. The difference of two numbers is 9. 3 increased by
jl of the less of the two numbers equals f of the greater.
Find the numbers.
12. The iron ore in the United States is J of the iron ore
in the rest of the world. If there are 75,000,000,000 tons of
iron ore in the entire world, how many tons of iron ore are
there in the United States?
13. The population of India is r that of China, and the
population of the rest of the world is 3f times that of India.
What is the population of India and China, if that of the
entire world is 1,500,000,000?
14. A man bequeathed $60,000 to his wife and three chil
dren. In a first will he bequeathed his wife three times as
large a share as one child received. Later he changed his
will and bequeathed his wife $10,000 more than the share of
a child. By which of the two wills would she have received
the larger amount?
15. In one kind of concrete, the parts of cement, sand,
and gravel are 1, 2, and 4. In another kind of concrete these
parts are 1, 3, 5. How many more cubic feet of cement are
needed to make 5600 cu. ft. of concrete of the first kind than
«
of the second?
FRACTIONAL AND LITERAL EQUATIONS 213
16. A girl's grades are, arithmetic 87, reading 92, and
geography 85. What grade must she have in spelling to
make her general average 90?
17. The average wheat crop of the United States for
four years was 660 millions of bushels. What would the
crop for the fifth year need to be in order to make the
average for the five years 700 million bushels?
18. A pupil has worked 15 problems. If he should work
9 more problems and get 8 of them right, his average would be
.75. How many problems has he worked correctly thus far?
19. A baseball nine has played 36 games of which it has
won 25. How many games must it win in succession to
bring its average of games won up to .75?
20. Make up and work an example similar to Ex. 19.
21. A baseball nine has won 19 games out of 36 games
played. K after this it should win f of the games played,
how many games would it need to play to bring its average
of games won up to .66f ?
22. A baseball nine has won 25 games out of 36 played.
It still has 12 games to play. How many of these will it need
to win in order to bring its average of games won up to .75?
23. How much water must be added to 50 gallons of
milk containing 8% of butter fat to make a mixture contain
ing 5% of butter fat?
Sua. The 50 gal. of milk contain 50 X .08 or 4 gal. butter fat.
4 5
If X denotes the number of gallons of water, — —  — = —, etc.
50 +x 100
24. A certain kind of cream is f butter fat, and a certain
kind of milk is 3% butter fat. How many gallons of the
cream must be added to 40 gallons of milk to make a mixture
which is 6% butter fat?
214 SCHOOL ALGEBRA
25. Of what type is each of the above problems an example
or variation? ^^
26. A mass of copper and silver alloy weighs 120 lb. and
contains 8 lb. of copper. How much copper must be added
to the mass in order that 100 lb. of the resulting alloy shall
contain 10 lb. of the copper?
27. A mass of copp)er and silver alloy weighs 120 lb. and con
tains 8 lb. of silver. How much silver must be added to the
mass in order that 1 lb. of the resulting alloy shall contain
2^ oz. of silver?
' 28. If 100 lb. of sea water contains 2\ lb. of salt, how much
^ fresh water must be added to it in order that 100 lb. of the
mixture shall contain 1 lb. of salt?
29. How much fresh water must be added to 100 lb. of sea
water in order that 20 lb. of the mixture shall contain 4 oz.
'"^ of salt?
30. How much water must be evaporated from 100 lb.
of salt water in order that 8 lb. of the water left shall con
tain 1 lb. of salt?
31. How much water must be added to a gallon of alcohol
which is 90% pure, in order to make a mixture which is 80%
pure?
32. If it takes a man 9 days to do a piece of work, what
part of it will he do in one day? If it takes him x days to
do the work, what part of it will he do in one day?
33. If a boy can do a piece of work in 15 days which a
. man can do in 9 days, how long would it take both working
together to do the piece of work?
SuG. What fractional part of the work will the boy do in 1 day?
The man? If together the boy and man can do the piece of work
in X days, what part of the work can they do together in 1 day?
FRACTIONAL AND LITERAL EQUATIONS 215
34. A can spade a garden in 3 days, B in 4 days, and C in
6 days. How many days will they require working together?
35. A and B together can mow a field in 4 days, but A
alone could do it in 12 days. In how many days can B mow it?
36. A and B in 5f days accomplish a piece of work which
A and C can do in 6 days or B and C, in 7^ days. If they all
work together, how many days will they require to do the
same work?
37. One pip)e can fill h given tank in 48 min. and another
can fill it in 1 h. and .12 min. How long will it take the pipes
together to fill the tank?
38. Two inflowing pipes can fill a cistern in 27 and 54
min. respectively, and an outflowing pipe can empty it in
36 min. All pipes are open and the (^stern is empty; in how
many minutes will it be full? ' ^ ff
Sua. Since emptjdng is the opposite of filling, we may consider
that a pipe which empties ^ of a cistern in a minute will fill — nV
of it each minute.
39. A tank has four* pipes attached, two filling and two.
emptying. The first two can fill it in 40 and 64 min. respect
ively, and the other two can empty it in 48 and 72 min.
respectively. If the tank is empty and the pipes all open, in
how many minutes will it be full? , '2^ j  x:
40. At what time between 3 and 4 o'clock are the hands
of a watch pointing in opposite directions?
Solution. At 3 o'clock the minutehand is 15 minutespaces
behind the hourhand, and finally is 30 spaces in advance: therefore
the minutehand moves over 45 spaces more than the hourhand.
Let X = the number of spaces the minutehand move*
Then x  45 = " " " " « hourhand
But the minutehand moves 12 times as fast as the hourhand;
hence, x = 12(a: — 45). Solving, x = 49xi.
Thus the required time is 49xt ^J^ past 3.*
216 SCHOOL ALGEBRA
41. When are the hands of a clock pointing in opposite
directions between 4 and 6? Between 1 and 2?
42. What is the time when the hands of a clock are to
gether between 6 and 7? Between 10 and 11?
43. At what instants are the hands of a watch at right
angles between 4 and 5 o'clock? Between 7 and 8?
44. The planet Mars is in the most favorable position to
be observed from the earth when it is in
line with the earth and on the opposite
side of the earth from the sun (Mars
is then said to be in opposition). If
the year is taken as. 365 days, and it
takes Mars 687 days to make one revo
lution about the sun, how long is the
interval between two successive opposi
tions of Mars?
Sua. If it takes the earth x days to overtake Mars and thus put
Mars again in opposition, how many revolutions about the sun does
the earth make in x days? How many revolutions does Mars make
in X days? In the interval from one opposition to the next, how many
more revolutions about the sun does the earth make than Mars?
■
45. It takes the planet Jupiter 12 yr. to make one revolu
tion about the sun. How long is it from one opposition of
Jupiter to the next?
46. The interval between two successive oppositions of
Mars is 780 days. Determine the time it takes Mars to make
one revolution about the sun (i. e. the length of the year on
Mars).
47. A courier travels 5 mi. an hour for 6 hours, when an
otlier courier starts at the same place and follows him at the
rate of 7 mi. an hour. In how many hours will the second
overtake the first? *
FRACTIONAL AND LITERAL EQUATIONS 217
SuG. If a? = the number of hours the second courier travels,
how many hours does the first courier travel? How many miles (in
terms of x) does the first courier travel? The second? Do the two
couriers travel equal distances?
48. A eourier who travels 5j mi. an hour was followed
after 8 hours by another, who went 7^ mi. an hour. In how
many hours will the second overtake the first?
49. A woman can write 15 words per minute with a pen,
and a ^rl can write 40 words per minute on the typewriter.
The woman has a start of 3 hours in copying a certain manu«
script. How long before the girl using the typewriter will
overtake the woman?
50. A train running 40 mi. an hour left a station 45 min.
before a second train running 45 mi. an hour. In how many
hours will the second train overtake the first?
51. A gentleman has 10 hours at his disposal. He walks
out into the country at the rate of 3§ mi. an hour and rides
back at the rate of 4^ mi. an hour. How far may he go?
52. A and B start out at the same time from P and Q, re
spectively, 82 mi. apart. A walked 7 mi. in 2 hours, and B
10 mi. in 3 hours. How far and how long did each walk
before coming together, if they walked toward each other?
If A walked toward Q, and B in the same direction from Q?
53. A certain room is 20 ft. long and 12 ft. wide. The
walls and ceiling of the room together have an area of 752
sq. ft. How high is the ceiling?
54. A rifle ball is fired at a target 1100 yd. distant and
4^ sec. after firing the shot the marksman heard the impact
of the bullet on the target. If the bullet traveled
at the rate of 2200 ft. per second, what was the rate
at which the sound of the impact traveled back to the
marksman?
218 SCHOOL ALGEBRA
55. A rifle ball is fired at a target 1000 yd. distant and 4
sec. after firing the shot, the marksman heard the impact of
the bullet on the target. If sound traveled at the rate of
1100 ft. per second, at what rate did the bullet travel?
56. A 21 lb. mass of gold and silver alloy when immersed
in water weighed only 19 lb. If the gold lost yV of its weight
when weighed under water, and the silver yV of i^s weight,
how many pounds of each metal were in the alloy?
Sua. If X denotes the number of pounds of gold, how many
pounds of silver were there in the mass?
The law involved in the above example is that when any object
is weighed in water, it loses in weight an amount equal to the weight
of the water which it displaces. Hence, if the specific gravity of
gold is approximately 19, the weight of the water displaced by the
gold ^ ^Y of the weight of the gold.
Find out if you can who first used this method of determining
the relative amounts of metal in an alloy and what use he first
made of the method. .
57. An alloy of aluminum and iron weighs 80 lb., but
when immersed in water it weighs only 60 lb. If the spe
cific gravity of aluminum is 2\ while that of iron is 7^, how
many pounds of each metal are in the alloy?
58. A mass of copper and tin weighing 100 lb. when im
mersed in water weighed 87.5 lb. If the specific gravity of
copper is 8.8 and that of tin is 7.3, how much of each metal
was in the mass?
59. If a bushel of oats is worth 40ji and a bushel of
com is worth 55f5, how many bushels of each grain must
a miller use to produce a mixture of 100 bu. worth 48j4 a
bushel?
60. A man has $5050 invested, some at 4%, and some at
5%. Hqw much has he at each rate if the annual income is
$220?
FRACTIONAL AND LITERAL EQUATIONS 219
61. Divide the number 54 into 4 parts, such that the first
increased by 2, the second diminished by 2, the third multi
plied by 2, and the fourth divided by 2, will all produce equal
results.
62. Find three consecutive numbers such that if they be
divided by 2, 3, and 4 respectively, the sum of the quotients
will equal the next higher consecutive number.
63. In the United States the gold dollar is 90% gold and
10% copper. If a mass of gold and copper weighing 24 lb. is
75% gold, how many pounds of gold must be added to it to
make it ready for coinage into gold dollars?
64. My annual income is $990. If J of my property is in
vested at 5%, f at 4%, and the rest at 6%, find the amount
of my property.
65. If one pipe can fill a swimming tank in 1 hour and an
other can fill it in 36 minutes, how long will it take the two
pipes together to fill the tank?
66. At what time are th6 hands of a watch at right angles
between 10 and 11 o'clock?
67. If one baseball nine has won 16 games out of 42
played, and another has won 18 out of 40 played, how many
straight games must the first team win in order at least to
equal the average of games won by the second team?
68. If the interval between two successive oppositions of
the planet Saturn is 378 days, how long is the year on
Saturn?
69. If A, B, and C together can do in 5^ days a certain
amount of work, which B alone could do in 24 days, or C
alone in 16 days, how long would A require?
70. How much water must be added to 1 gal. of a 5%
solution of a certain chemical to reduce it to a 2% solution?
220 SCHOOL ALGEBRA
71. A baseball player who has been at the bat 150 times
has a batting average of .240. How many more times
must he bat in order to bring his average up to .250, pro
vided that in the future his base hits equal half the number
of times he bats?
72. A girl has worked a certain number of problems and
has f of them right. If she should work 9 more problems and
get 8 of them right, her average would be .75. How many
problems has she worked?
73. If the sum of two consecutive integers is 4p + 5, find
the integers.
74. A man has a hours at his disposal.' He wishes to ride
out into the country and walk back. How far may he nde
in a coach which travels b miles an hour, and return home in
time, walking c miles an hour?
75. Generalize Ex. 33; that is, make up and work a similar
example where letters are used instead of figures for the
known numbers.
76. If E denotes the number of days it takes the earth to
revolve once around the sun, P denotes the number of days
it takes a planet (as Mars) to complete a revolution about
the sun, and S the number of days between two successive
oppositions of the planet, show that ^ — ^ = o'
77. The fore wheel of a carriage is a feet in circumference
and the hind wheel is h feet. What distance has been passed
over when the fore wheel has made c revolutions more than
the hind wheel?
78. Make up and work three^ examples similar to such of
the examples in this Exercise as the teacher may point out*
FRACTIONAL AND LITERAL EQUATIONS 221
EXERCISE 67
1. Given V = Iwh, find h in tenns of the other letters.
Also solve for /. For w.
2. Given i = prt, find each letter in terms of the others.
Find each letter in terms of the others in the following
formulas used in geometry:
3. K = ^bh 6. S = ttRL
4. ii:  ih(b + 6') 7. T = 7rR{R + L)
5. C = 27rR a. T = 27rR{R + H)
Also find each letter in terms of the others in the following
fonnulas used in mechanics and physics:
9. S = rf
11. c = 3(^32)
13. R= ^'
g + s
10. LW = Iw
"• ^=1
14.1 = 1+1
f P f
15. By use of the formula in Ex. 2 determine in how many
years $325 will produce $84.50 interest at 5 per cent.
16. Also find the rate at which $176 will yield $43.56 in
terest in 5 yr. 6 mo.
17. Change the following temperatures on the Centigrade
scale to Fahrenheit readings:
(1) 50^ (2) 0° (3) 2700°
18. Metals fuse at the following temperatures on the Cen
tigrade scale. What are the temperatures at which they fuse
on the Fahrenheit scale?
Tm 228° Lead 325° ^ Copper 1091° Iron 1540°
bc + d
19. Solve the following equation for 6: d _ 2d^
Also solve for c. For d. be a
1
<
222 SCHOOL ALGEBRA
20. A boy who weighs 80 lb. is on a teeter board at 4,
6 ft. from the fulcrum F. He just balances a boy who
is at B on the same boards 8 ft.
from F. What does the second
boy weigh? (Use the foiinula of
Ex. 10.)
21. Make up and work an example similar to Ex. 19.
22. How many examples in Exercise 48 (p. 163) can you
now work at sight?
CHAPTER XII
SIMULTANEOUS EQUATIONS
132. Heed and Utility of Simultaneous Equations.
Ex. A farmer one year. made a profit of $2221 on 27
acres of corn and 40 acres of potatoes. The next year with
equally good crops, he made a profit of $2028 on 36 acres of
com and 30 acres of potatoes. How much did he make per
acre on his com and on his potatoes?
Let X = no. of dollars made on 1 acre of corn
y = " " " " " 1 " " potatoes
Then 27a; +40y = 2221
36x + 302^ = 2028
From these equations the value of x may be found by combining
the equations in some way which will get rid of, or eliminate, y,
(See Arts. 136138.)
Try to solve the above problem by the use of only one unknown,
Bsx, If you succeed at all, you will find the method awkward and
inconvenient.
Why do we now proceed to make definitions and rules?
133. Simultaneous Equations are a set or system of equa
tions in which more than one unknown quantitr is used, and
the same symbol stands for the same unknown number.
«
Thus, in the group of three simultaneous equations,
x h 2/ f 22 = 13
X 2y \z =0
2a; f 2/  2 = 3
X stands for the same unknown nmnber in all of the three equations,
y for another unknown nmnber, and z for still another.
223
224 "^ SCHOOL ALGEBRA
134. Independent Equations are those which cannot be
derived one from the other.
Thus, z+y = 10,
and 2a; = 20  2y,
are not independent equations, since by transposing 2y in the sec
ond equation and dividing it by 2, we may convert the second equa
tion into the first.
But 3x —2y = 5\ are independent equations, since neither one
5aj +y =6.
of them can be converted into the other.
135. Elimination is the process of combining two equa
tion^ containing two unknown quantities so as to form a
single equation with only one unknown quantity. Or, in
general, elimination is the process of combining several sim
ultaneous equations so as to form equations one less in
number and containing one less unknown quantity.
There are three principal methods of elimination: I, adr
diiion and subtraction; II, svbstitiUion; and III, comparison.
These methods are presented to best' advantage in connec
tion with illustrative examples.
Ex. Solve
136. I. Elimination by Addition and Snbtractioii.
12a; + 5y = 75 *. (1)
9a:  4y = 33 (2)
In order to make the coefficients of y in the two equations alike,
we multiply e(]^ation (1) by 4, and (2) by 5,
48x + 20y = 300 (3)
• 45a;  202/ = 165 W
Add equations (3) and (4), 93a; = 465
Divide by 93, a; = 5 Root
Substitute for x its value 5, in equation (1),
60 4 51/ = 75
.' .y =3 Root
Check. 12x + 5i/ = 12 x 5 + 5 X 3 = 75
9a;42/=9x54x3=33
SIMULTANEOUS EQUATIONS 225
Since y was eliminated by adding equations (3) and (4)
the above process is called elimination by addition.
The same example might have been solved by the method
of subtraction.
Thus, multiply equation (1) by 3, and (2) by 4,
3ea; + 15y =225 :(6)
36a;  1% = 132 (6)
Subtract (6) from (5), 312/ = 93
2/=3
and a; = 5
It is miportant to select, in every case, the smallest multipliers
that will cause one of the unknown quantities to have the same
coefficient in both equations.
Thus, in the last solution given above, instead of multiplying
equation (i) by 9, and (2) by 12, we divide these multipliers
by their conunon factor, 3, and get the smaller multipliers, 3
and 4.
Hence, in general,
MuUvply the given equations by the smallest numbers that
will cause one of the unknown quantities to have the sams co
efficient in both equcdions;
If the equal coefficients have the same sign, subtract the corre
sponding members of the two equations; if the equal coefficients
have unlike signs, add.
EXERCISE 68
Solve by addition and subtraction:
1. Sx 2y = I 4. 5x  3y = 1
X + y = 2 3a; + 5y = 21
2. 2x — 7y = 9 5. x + 5y = —S
5x + Sy = 2 7ar + 8y = 6
3. 4x + 3y = 1 6. 3x  2t/ = 4
2x  6y = 3 5x  4y = 7
226 SCHOOL ALGEBRA
7. 2y + x =
13.
52 = 3
4x + Qy= 3
3 5
8. 9x  8y = 5
5 + 2^
15* + 12y = 2
9. 4x6y + l =>
14.
2^ + 2 = 1
3 ^4
5a:  7y + 1 =
^ + ^ = 2
10. 8* + 5y = 6
2 8
Qy'+ 2a; = 11
IS.
4«. 3y_ 7
5 2
11. 5a;3y = 36
^^ mm
3ar . 2i/ 7
7a;  5y = 56
4 + 5 ~2
"•11 = 1
16.
5a; 8y „
2 3
6 9
« y_i
3a; 5y »
4 9
4 6
17. Find two numbers whose sum is 12 and whose differ
ence is 2.
18. The half of one nimiber plus the third of another
nmnber equals 13^ while the sum of the numbers is 33.
Find the nmnbers.
19. State Ex. 1 as a problem concerning two nmnb«g^
20. State Ex. 2 as a problem concerning two nmnbers.
21. .7 lb. of sugar and 3 lb. of rice together cost 57)6; also
5 lb. of sugar and 6 lb. of rice cost 6Qff. Find the cost of a
pound of each.
22. Make up and work an example similar to Ex. 18. To
Ex. 21.
23. How many examples in Exercise 50 (p. 170) can you
now work at sight?
/
SIMULTANEOUS EQUATIONS 227
137. n. Elimination by Substitution.
Ex. Solve 5 a:+ 2y = 36  . . (1)
2a; + 32/ = 43 (2)
From (1) &r =» 36  2y
• 36  2t/
.. ^ 5^ /3)
In equation (2) substitute for x its value given in (3),
72 43/+ 15^215
112/ =143
2/ = 13 i2oo<
Substitute for y in (3), x = — = — = 2 Root
o
Let the pupil check the work.
Hence, in general,
In one of the given equations obtain the value of one of the
unknown qtumtities in terms of the other unknown quantity;
Substitute this value in the other equation and solve,
EXERCISE 60
1. Work the examples of Exercise 68 (p. 225) by the
method of substitution.
Find out which of the following sets of equations are worked
more readily by the method of addition and subtraction, and
which by the method of substitution, and work each example
accordingly :
2. a; = 3y5; 4. a:3=0
2a; + 52/ = 12 2j/ +* 3a; = 5
3. 3a;  4y = 1 5. 2a; + 3y = 1
4a;5/ = l 3aJ + 4y = 2
228 SCHOOL ALGEBRA
6. 7a; + 8y = 19 a y = 3
5x + 6y = 13^ 2x = 3y  17
*7. z2y — 3 9. y = Zx
y = 5x21 4x + 5y = 38
10. Make up and solve an example in simultaneous equa
tions which is solved more readily by the method of addition
and subtraction than by the method of substitution.
11. Make up and solve an example of which the reverse
of Ex. 10 is true.
12. How many examples in Exercise 51 (p. 172) can you
now work at sight?
138. m. Elimination by Comparison.
Ex. Solve 2x  3y = 23 . : (1)
5a; + 22/ = 29 (2)
From(l) 2x= 23+32/ . . . ^^ (3)
From (2) 5a;= 2922/ (4)
From(3) a;=?^ • • (5)
From (4) ^^ 29 2y ''{%)
Equate the two values of x in (5) and (6),
23+32/ 29 2y
25
Hence, 115 + ISr/ = 58  42/
192/= 57
2/ = — 3 Root ,
23—9
Substitute for y in (5), x = — — = 7 Root
* *
Let the pupil check the solution.
Hence, in general.
Select one unknown quantity, and find its value in terms of
the other unknown quantity in each of the given equations;
Eqmte these two values, and solve the resulting equation.
SIMULTANEOUS EQUATIONS 229
EXERCISE 70
1. Work the examples of Exercise 68 (p. 225) by the
method of comparison.
Ascertain by which of the three methods of elimination
each of the following examples can be worked most readily,
and solve accordingly:
2. a; = 3y + 9 7. 9ar + 12y =  6
x = 5y + 13 6a;  52/ =  17
3. a: = 3y + 9 8. a: = 5
3a;  5y = 10 3x  2y = 13
4. 6a; + 5y  8 = 9. 5a; + 3y = 8
4a;  3y  18 = 5x  4y ^ 7
5. y = 2a; 10. y = f (a;  3)
3a; + 2y = 21 y = f x + 1
6. y = 6a;3 U. y = 2x + 1
85a; = y 3a; + ^ = 8
12. Make up and solve an example in simultaneous
equations whiclr is solved more readily by the method of
comparison than by either of the other two methods of elim
ination.
13. Make up and solve an example in simultaneous equa
tions which is solved more readily by the method of substi
tution than by either of Jhe other two methods.
14. Make up and solve an example solved more readily
by the method of addition and subtraction than by the other
two methods.
^
230 SCHOOL ALGEBRA
EZEBCI8E 71
Solve and check each result:
3 4 ^ 40 ^
3 4 • 3 "^ '
2. X ^  4. jg ^j  1
4y^±i0 = 3 3yx = 2
2a;y 3a; + 2y _„
'• ~5~+~ll ^
_2x '4« + y^^
'3 4
^'e. L_ 3 =0
y+3 «+4
y(a;  2)  a:(y  5) + 13 =
'• f(x + 3y)K« + 2y)=^
3y(x + 4y + f)=0
8. .4a:  .Zy = .7 10. .5a; + 4.5y = 2.6
.7« + .2y = .^ 1.3a; + 3.1y = 1.6
9. 2a: + 1.5y= 10 U .8a;  .7y = .005
.3x  .05j/ = .4 2a; = 3y
a; H
^ 3
10» + 1 2a; + 3
S—l = y . _—
5 x + 3i±2^
SIMULTANEOUS EQUATIONS ' 231
L/ x — y 1
x + y 5
y _ 3x % _ ?5
I 2 12 3^ „
Hi li
1*. (a:5)(y + 3) = (a;l)(y + 2)
ay + 2a; = a;(y + 10) + 72y 
a;  2 x + 10 10y _
15. __ + ^ + _^_13
2y + 6 4a! + y + 6 ,_q
3 8
6y + 5 " 3a; + 5 _ 9y4
8 5a; 2y 12
2yTf3 x + y _ iy + 7
4 3x2y 8
3x2 ^ 6a;5 _ a; + y + 6
5 10 6x + y
3y  2 _ 2y  5 ■ 3 + 7a;
12 8 10y3a;
18. Practice oral work with small fractions as in Exercise
58 (p. 190).
139. Literal Eqaations.
Ex. Solve ax + by •= c (1)
ax + b'y = c (2)
Multiply (1) by a', and (2) by a,
adx + a'6y = (/c .. . . , (3)
<w!x + ch'y = €ui (4)
Subtract (4) from (3), {clb  ab')y "olcai
dc  ad „ .
• •
232 SCHOOL ALGEBRA
Again, multiply (1) by 6', (2) by 6,
ab'x+Wy ^Vc (6)
a'hx^lib'y ^W (6)
Subtract (6) from (5), (a6'  o'6)x = 6'c  6c'
Let the pupil check the work.
In solving simultaneous literal equations, observe that if the
value obtained for the first unknown is a fraction containing a
binomial term (or the value is complex in other wajrs), it is better
not to find the value of the other unknown as in numerical equations,
i. e. by substituting the value found in one of the original equations
and reducing. A better method is to take both of the original equa
tions and eliminate anew. See the solution of the preceding example.
EXERCISE 72
Solve and check each result:
1. 3a; + 4y = 2a 7. aar + 6y=sc
5a; + 6y = 4a mx\ny = d
2. 2aa; + 36y = 4a5 s. hx + ay = a^\h
bax + 4i)y = 3a6 ab{x  y) = a^  &«
3. ox + hy^l 3 c^a:rfV = cd
ax^byl cd(2(fe  cy) = 2(Pc*
4. x — y — 2n
mx — ny  m^ + n^ 10. ^"^^ = Vl
5. 26a: + ay = 46 + a V "^ ^
ahx''2(jiby = 46 + a x + y==2n
6. axby = 0^ + 1^ U. {a + l)xby = a + 2
bx + ay= 2(a2 + 6^) (a  l)x + 3by = 9a
12. (a  b)x  (a + 6)3/ = a* + 62
bx + ay =
13. ^ + l = 2 15. (± l>)x+Ja + b)y ^^
a + b a — b a^ + b^
xy =2b aa:26y = a226*
14. ax — bx = ay — dy > 16. (a + b)x + cy = 1
a? — y~l CX+ (a + b)y ^ 1
SIMULTANEOUS EQUATIONS
233
17. (a + h)x  (a  h)y = 3a6
(a — h)x — (a + b)y =» ab
X — b
a + a — b
x + 2a y2b_ a^ + b^
19.
X — 1 y — a _
61
x + 1
+
+
b — a
y1 __!
b
a— 6 a+b o^— &^ b ' 1— a
20. (a;  1) (a + 6) = ii(2/ + a + 1)
(y+l){ab)=:b{xbl)
21. Make up and work an example similar to Ex. 7.
Ex. 11.
140. Three or More Simultaneous Equations.
3x + 4y  52 = 32
To
Ex. Solve
(1)
4x 5y + Sz = 18 (2)
5x3y 4^ = 2 ...... (3)
If we choose to eliminate z first, multiply (1) by 3, and (2) by 5,
9x + l2y  152 = 96
20a;  25t/ + 152 = 90
Add (4) and (5), 29x  13y = 186 . . .
Also multiply (2) by 4, (3) by 3,
16x  202/ + 122 =72 ...
15a;  9y  122 = 6 . . .
Add (7) and (8), 31a;  292/ = 78 . . .
We now have the pair of simultaneous equations,
29a;  13y = 186
31a; 29y =78
Solving these, obtain a; = 10
2/=8J
Substitute for x and y in equation (1),
30 + 32  52 = 32,
2=6 Root
Check. 3a;+ 4^/ 52 =3x 10 +4x85x6 =32
4a;5i/+ 32 =4X 10 5X8+3X6 = 18
5x 32/ 42=5X 10 3X84X6=2
(4)
(5)
(6)
(7)
(8)
(9)
Rods
234 SCHOOL ALGEBRA
In like manner, if we have n simultaneous equations con
taining n unknown quantities, by taking different pairs of
the n equations, we may eliminate one of the unknown quan
tities, leaving n — 1 equations, with n — 1 unknown quanti
ties; and so on.
EZSBCISE 78
Solve and check:
1. x + y + z^6 ^ 9. ix + iy + iz^2
3x + 2y + z = 10 ix + iy + z = 9
Sx + y+^z^U lx + ^y + lz=3
2. 3a;  y  23 = 11 10. 2x + 2yz = 2a
4a: — 2j^ + z=— 2 3a; — y — 2 = 45
6a;:2/ + 3a= 3 6a; + 3y3z = 2(a + 6)
3. 5a;  62/ + 23 = 5 ^ ^ ??_l??«^ = iq
8a; + 4y52==5 * 3 4 5
9a; + 5y — 63 = 5 \^_^j??— — f;
4. 3a;j2/ + 2 = 7j "g" S" T ~
2a;i(y32) = 5i §« __ 7y 3z^^ _ .. ,
2a;iy + 4z = ll 2 5 10
5. 2x + Sy = 7 12. x + y + 2z = 2{a + b)
3y + 42 = 9 a; + 2 + 22/ = 2(a + c)
5a; + 62 = 15 ' y + 2 + 2a; = 2(6 + c)
6. 2a; + 4y + 32 = 6 13. a; + 1/2 = 3a6
6y  3a; + 22 = 7 x + zy = 3a6l
3a; 82/ 72 = 6 , 2^ + 2a; « 36a— 1
7. a; + 32/ + 32 * 1 14. 3x + 22/ = ^/a
3a;  52 = 1 62  2a; = f 6
91/ + IO2 + 3a; =1 5j/  132 + a; =
8. w + ij — 1^ = 4 15. — a; + y + 2 + iJ=«a
w+'D a; = l . a; — y + 2 + i) = 6
t) + t(? + a; = 8 a; + 2/ "~ 2 + tj = c
u^ W'\x — 5 X'\y + Z — V = d
SIMULTANEOi^S EQUATIONS
235
16. Practice the oral solut ^n of simple equations as in
Exercise 64 (p. 209).
141. The Uce of ^ and j; as TTnknown Qnantitief enables us
to solve certain equations which would otherwise be diffi
cult of solution.
^ + ^ = 49 (1)
X y
^ + ^ = 23 ......... . .(2)
X y
Multiply (1) by 7, and (2) by 5, j
Ex. 1. Solve
X * y
X y
(3)
(4)
76 1
Subtract (4) from (3), ■« 228, /. = 3, or y = J Root .^ .
Substitute the value of y in (2), hence, a: = i Root
Xet the pupil check the work.
Ex. 2. Solve
^3 + A
2a: 3y
11
(1)
(2)
?.Jl = 29
X 4y 4:
Whenx and y in the denominators have coefficients, as in this
example, it is usually best first to remove these coefficients by mul
tiplying each equation by the L. C. M. of the coefficients of x and y
in the denominators of that equation. Hence,
Multiply (1) by 6, and (2) by 4,
'5+1266
X y
?i29 .......
[X y
Solving (3) and (4) by the method used in Ex. 1,
■ * ^ " il Roots
Let the pupil check the work. .^
i
236 SCHOOL A.LGEBRA
EXF JiCISE 74
Solve and check;
X y X y a
3 + 5 = 2 1 + ^ = 1^11
X y X y a
2. = 7 8. ~H — = mr + n
X y X y
3,4 li_^«._i_*,2
 — = 1 — I — = mtrr
X y X y
3.2 + 1=9 9 .f + A = 2
3x 2y ox 'ay
A + ±=13 ^ + « = ^
&x 5y X y ab
2x 3y X y
2 I 3  a . b , ,
1 =—5 — I — = a\ b
Sx 2y ' X y
5 _3 A = 1 ii 11. 5y — 3a: = Ixy
' 4x dy ^^ 15x + QOy = IQxy
10i 1=,. 1 + 1 + 1. 2
OX Zy X y z
6.1 + 1 = 1 l_i + l = 7,
X y n X y z
1 1 3,2,5 ..
X y X y z
3 1 ,1
13. = 3i
a: y
21 =
X Z
SIMULTANEOUS EQUATIONS 237
14.
3x~2y'^5z ^^
IS.
1_
X
_1_
1_
z
1
a
 + .'=12
X y 2z
1_
1
z
_1_
X
1
5 3 1 _i6
2a; 4y as "
1_
z
1
X
1.
_1
c
16. £+^£ =
X y z
/
a .c h _
X z y
TO
6 , c o _
y z X
«
17. 5yz + 6a» — 3a:y = ^xyz
Ayz — 9xz + Qcy = 19xyz
yz — 12ocz — 2xy = 9xyz
18. Make up and work an example similar to Ex. 1.
To Ex. 4. Ex. 13. Ex. 15.
19. Work again such examples on pp. 212 and 213 as the
teacher may point out.
142. In the Solntion of Problems Involving Two or More
Unkiiown Quantities, it is necessary to obtain as many inde
pendent equations a>s there are unknown quantities invohed
in the equations and to eliminate. (See Art. 134, p. 224.)
Ex. Find a fraction such that if 2 is added to both nu
merator and denominator, the fraction becomes ; but if 7 is
added to both numerator and denominator, the fraction be
comes f. ; '
Two unknown numbers occur in this problem, viz, : the numera
ator and denominator of the required fraction. Hence two
equations must be formed in order to obtain a solution of the
problem.
238 SCHOOL ALGEBRA
Let  represent the fraction.
y
Then, ^^^112 '2 "^^ iT+T^S
Clearing these equations, and collecting like terms,
2x y = 2
3a;  2y =  7
The solution gives a; = 3 and y = 8.
Therefore  is the required fraction.
Let the pupil check the work.
EXXBCISE 75
1. Find two numbers whose sum is 23 and whose difference
is 5.
2. Twice the difference of two numbers is 6, and  of their
sum is 3. What are the numbers?
3. Find two numbers such that twice the greater number
exceeds 5 times the less by 6; but the sum of the greater num
ber and twice the less is 12.
4. 2 lb. of flour and 5 lb. of sugar cost 31 cents, and 5
lb. of floiu* and 3 lb. of sugar cost 30 cents. Find the value
of a pound of each.
5. A man hired 4 men and 3 boys for a day for $18; and
for another day, at the same rate, 3 men and 4 boys for $17.
How much did he pay each man and each boy per day?
6. In an orchard of lOQ trees, the apple trees are 5 more
than f of the nmnber of pear trees. How many trees are
there of each kind?
7. One woman buys 4 yd. of silk and 7 yd. of satin, and
another woman at the same rate buys 5 yd. of silk and 5f
yd. of satin. Each woman pays $17.70. What is the price
of a yard of each material?
SIMULTANEOUS EQUATIONS
239
8. Solve Ex. 7 without using x and y to represent unknown
numbers (see Art. 1). About how much of the labor of writ
ing out the solution is saved by the use of x and y1
9. 1 cu. ft. of iron and 1 cu. ft. of lead together weigh
1180 lb.; also the weight of 3 cu. ft. of iron exceeds the weight
of 2 cu. ft. of lead by 40 lb. What is the weight of 1 cu. ft.
of each of these materials?
10. In an athletic meet, the winning team had a score of
26 points and the second team had a score of 2J points. If
the winning team took first place in 7. events and second
place in 5 events, while the second team took 6 firsts and 3
seconds, how many points does a first place count? A second
place?
U.. In an athletic meet, the three winning teams made
scares as follows:
Team
Ists
2ds
3ds
Total Score
A
B
C
5
3
1
2
3
4
2
1
6
33
25
23
What did each of the first three places in an event count in
this meet?
12. Make up and work an example similar to Ex. 10.
13. Two partners agree to divide their profits each year in
such a way that one partner receives $1000 more than f of
what the other receives. If the profits f(ir a giVen year are
J10,000, what does each partner receive?
!*• Separate 240 into two parts such that twice the larger
part exceeds five times the smaller by 10.
240 SCHOOL ALGEBRA
15. If the cost of a telegram of 14 words between two
cities is 62^, and one of 17 words is 71 jf, what is the charge
for the first 10 words in a message and for each word after
that?
16. Make up and work an example amilar to Ex. 15
concerning telegraph rates between two cities near your
home.
17. A farmer one year made a profit of $1640 on 20 acres
planted with wheat and 30 acres planted with potatoes.
The next year, with equally good crops, he made a profit
of $1210 on 30 acres planted with wheat and 20 acres
planted with potatoes. How much per acre on the average
did he make on each crop?
18. In three successive years, th^ farmer raised crops with
profits as follows:
(1) 20 A. wheat, 30 A. com, 40 A. potatoes; profits $1720
(2) 30 A. wheat, 40 A. com, 20 A. potatoes; profits $1520
(3) 40 A. wheat, 20 A. corn, 30 A. potatoes; profits $1440
What were his average profits per acre for each kind of
crop?
19. The freight charges between two cities on 400 lb. of
first<dass freight and 600 lb. of secondclass freight were
$14.24, while the charges on 500 lb. of firstclass freight and
800 lb. of secondclass were $18.48. What was the rate per
100 lb. on each class?
20. The freight charges on shipments between two places
were as follows: 800 lb. of 4th class + 500 lb. of 5th dass +
700 lb. of 6th class, $17.11; 1000 lb. of 4th class + 600 lb. of
5th class + 800 lb. of 6th class, $20.66; 600 lb. of 4th dass +
1000 lb. of 5th class + 900 lb. of 6th class, $20.52. Find the
rate per 100 lb. for each of the dasses named.
(o
SIMULTANEOUS EQUATIONS 241
21. The com and wheat crops of the United States in the
year 1909 were together 3,509,000,000 bu. ; the com and oat
crops 3,779,000,000 bu.; and the wheat and oat crops,
1,744,000,000 bu. How many bushels were in each
crop?
22. One cubic fpot of iron and one cubic foot of aluminum
weigh 636 lb.; a cubic foot of iron and one of copper weigh
1030 lb.; a cubic foot of copper and one of aluminiun weigh
706 lb. How much does one cubic foot of each of these ma
terials weigh?
23. In boring holes in a metal plate, three circles touching
each other are to be drawn, the distance^/
between their centers being .865 in.,
•650 in., and .790 in., respectively.
Find the radius of each of the three ^
circles.
24. The Eiffel Tower is taller than
the Metropolitan Life Building of New York, and the latter
building is taller than the Washington Monument. If the
difference between the heights of the first two is 284 ft.;
between the first and last is 429 ft.; and the sum of the
first and last is 1539 ft., find the height of each.
25. A ton of fertilizer which contains 60 lb. of nitrogen,
100 lb. of potash, and 150 lb. of phosphate is worth $21.50;
a ton containing 70, 80, and 90 lb. of these constituents in
order is worth $19; and one containing 80, 120, 150 lb. of
each in order is worth $25.50; what is the value of one pound
of each of the constituents named?
26. If a bushel of oats is worth 40f!f and a bushel of com is
worth 55)lf, how many bushels of each must a miller use to
produce a mixture of 100 bu. worth 48fi( a bushel?
242 SCHOOL ALGEBRA
27. How many pounds of 20)!f coffee and how many jpomids
of 32jf coffee must be mixed together to make 60 lb. worth
28^ a pound?
28. Make up and work an example similar to Ex. 27.
29. If two grades of tea worth 50jf and 75jf a poimd are to
be mixed together to make 100 lb. which can be sold for 72j!f
at a profit of 20%, how many pounds of each must be used?
30. A farmer wishes to combine milk containing 5% of^
butter fat with cream containing 40% of butter fat in order
to produce 20 gal. of cream which shall contain 25% of
butter fat. How many gallons of milk and how many of
cream must he use?
31. A man has $5050 invested, part at 4%, and the rest
at 5%. How much has he invested at each rate if his annual
income is $220?
Can you work this example by use of one unknown quan
tity?
32. A man wishes to invest part of $12,000 at 5% and the
rest at 4% so that he may obtain an income of $500. How
much must he invest at each of the rates named?
' 33. Make up and work an example similar to Ex. 32.
34. If a rectangle were 3 in. longer and 1 in. narrower it
would contain 5 sq. in. more than it does now; but if it were
2 in. shorter and 2 in. wider its area would remain unchanged.
What are its dimensions?
SuG. Draw a diagram for each rectangle considered in the prob
lem. See Ex. 30, p. 104.
35. If a rectangle were made 3 ft. shorter and 1 ft. wider,
or if it were 7 ft. shorter and 5\ ft. wider, its area would
remain unchanged. What are its dimensions?
/
SIMULTANEOUS EQUATIONS 243
36. A party of boys purchased a boat and upon payment
for the same discovered that if they had numbered 3 more,
they would have paid a dollar apiece less; but if they had
niunbered 2 less, they, would have paid a dollar apiece more.
How many boys were there, and what did the boat cost?
SuG. Let X = the number of boys, and y — the number of dollars
each paid. Then xy represents the number of dollars the boat cost.
37. After going a certain distance in an automobile, a
driver found that if he had gone 3 mi. an hour faster, he would
have traveled the distance in 1 hr. less time; and that if he
haS gone 5 mi. faster, he would have gone the distance in
Ij hr. less. What was the distance?
38. Make up and work an example similar to Ex. 37.
39. If a baseball nine should play two games more and
win both, it will have won f of the games played. If, however,
it should play 7 more, and win 4 of them, it will then have
won I of the games played. How many games has it so far
played and how many has it won?
40. If a physician should have 12 more cases of diphtheria
and treat 10 of them successfully, he will have treated f of his
cases successfully. But if he should have 32 more cases and
succeed with 30 of them, he will have succeeded with  of his
cases. How many cases has he had so far and how many has
he treg,ted successfully ?
41. If 1 be added to the numerator of a certain fraction,
the value of the fraction becomes \\ but if 1 be subtracted
from its denominator, the value of the l^:action becomes J.
Find the fraction.
42. There is a fraction such that if 4 be added to its numer
ator the fraction will equal ^\ but if 3 be subtracted from its
denominator the fraction will equal f . What is tbi? fraction?
244 SCHOOL ALGEBRA
43. Make up and work an example similar to Ex. 42.
44. A certain fraction becomes equal to ^ if if is added to
both numerator and denominator. It becomes ^ if 2j is
subtracted from both numerator and denominator. What is
the fraction?
45. Find two fractions, with numerators 11 and 7, respec
tively, such that their sum is 3y J, but when their denominar
tors are interchanged, their sum becomes 3^.
46. If f is added to the numerator of a certain fraction, its
value is increased by ^ ; but if 2f is taken from its denomi
nator, the fraction becomes f . Find the fraction.
47. The sum of two numbers is 97, and if the greater is
divided by the less, the quotient is 5 and the remainder 1. *
Find the numbers.
Sua. The divisor multiplied by the quotient is equal to the divi
dend diminished by the remainder.
48. Divide the number 100 into two such parts that the
greater part will contain the less 3 times with a remainder
of 16. >^ ,. '
49. The di^erence between two numbers is 40, and the
less is contained in the greater 3 times with a remiainder of
12. Find the numbers.
60. Separate 50 into two such parts that J of the larger
shall exceed f of the smaller by 2.
51. A tank can be filled by two pipes one of which runs 4
hr. and the other 5; or by the same two pipes if the first runs
3 hr. and the other 8. How long will it take each pipe running
separately to fill the tank?
52. Two persons, A and B, can perform a piece of work in
16 days. They work together for 4 days, when B is left
SIMULTANEOUS EQUATIONS 246
alone, and completes the task in 36 days. In what time could
each do the work separately?
53. A and B can do a piece of work in 8 da.; A and C can
do the same in 10 da.; and B and C can do it in 12 da. How
long will it take each to do it alone?
54. 37 means 10 X 3 + 7. Does xy mean 10a; + y? Why
this difference?
55. How would you write a niunber whose digits in order
from left to right are I, m, and n? Why may not such a
number be expressed as ImrCt
56. Express in symbols a number whose digits in order are
a, 6, c, and d. Whose digits are x, y, and z. x and y.
57. A number consists of two digits whose sum is 13^ and
if 4 is subtracted from double the number, the order of the
digits is reversed. Find the number.
58. The sum of the digits of a certain number of two
figures is 5, and if 3 times the units' digit is added to the
number, the order of the digits will be reversed^ What is
the number?
59. Twice the units' digit of a certain number is 2 greater
than the tens' digit; and the number is 4 more than 6 times
the smn of its digits. Find the number.
60. In a number of 3 figures, the first and last of which are
alike, the tens' digit is one more than twice the sum of the
other two, and if the nmnber ib divided by the sum of its digits,
the quotient is 21 and the remainder 4. Find the number.
61. An oarsman can row 12 mL down stream in 2 hr., but
it takes him 6 hr. to return against the current. What is his
rate in still water and what is the rate of the stream?
Make up and work a similar example.
246 SCHOOL ALGEBflA
62. A boatman rows 20 mi. down a river and back in 8
hr.; he can row 5 mi. down the river while he rows 3 mi. up
the river. Find the rate of the man and of the stream.
63. A man rows down a stream 20 mi. in 2f hr., and rows
back only f as fast. Find the rate of the man and of the
stream.
64. 3 cu. ft. of cast iron and 5 cu. ft. of wrought iron to
gether weigh 3750 lb.; also 7 cu. ft. of the former and 4 cu.
ft. of the latter weigh 5070 lb. What is the weight of 1 cu. ft.
of each?
65. Regarding the orbits of the earth and of the planet
Mars as circles whose center is the sun, the greatest distance
between the earth and Mars at any time is 234,000,000 mi.,
and the least distance between them is 48,000,000 mi. How
far is each of them from the sun?
66. 2 lb. of tea and 5 lb. of coffee cost $2.50. If the price
of tea should increase 10% and that of coffee should diminish
10%, the cost of the above amounts of each would be $2.45.
Find the cost of a pound of each.
67. Two bins contain a mixture of com and oats, the one
twice as much corn as oats, and the other three times as
much oats as corn. How much must be taken from each
bin to fill a third bin holding 40 bu., to be half oats and half
corn?
68. If A gives B$10, A will have half as much as B; but if
B gives A $30, B will have ^ as much as A. How much has
each?
69. Two grades of spices worth 25)i and 50ff a pound are
to be mixed together to make 200 lb. which can be sold at
52ff per lb. at a profit of 30%. How many pounds of each
grade must be used?
SIMULTANEOUS EQUATIONS 247
70. A train maintained a uniform rate for a certain dis
tance. If this rate had been 8 mi. more each hour, the time
occupied would have been 2 hr. less; but if the rate had been
10 mi. an hour less, the time would have been 4 hr. more.
Find the distance.
71. If the greater of two numbers is divided by the less,
the quotient is 3 and the remainder 3, but if 3 times the
greater be divided by 4 times the less, the quotient is 2 and
the remainder 20. Find the numbers.
72. Why are we able to solve problems like Exs. 70 and
71 by algebra and not by arithmetic?
73. Find two numbers whose sum is a and whose difference
is 6. ^
74. If. a pounds of sugar and b pounds of coffee together
cost c cents, while d pounds of sugar and e pounds of coflFee
together cost / cents, what is the price of one pound of each?
75. If a bushel of oats is worth p cents, and a bushel of
com is worth q cents, how manjr bushels of each mi^st be
mixed to make r bushels worth s cents per bushel?
76. Find a fraction such that if a be added to both nu
merator and denominator the value of the fraction is p/q;
but if b is added to both numerator and denominator, the
value of the fraction is r/s.
»
77. Generalize Ex. 34 (p. 242), by using a letter for each
number in the example.
78. Generalize Ex. 53 (p. 245), by using a letter for each
number in the example.
79. Make up and work three examples similar to such of
the examples in this Exercise as you think are most interesting
or instructive.
248 SCHOOL ALGEBRA
143. TTtilities in Algebra.
1. Brevity of expressions which represent numbers. Brevity
means a saving of time and energy.
Thus, for instance, for " number of feet in the length of a rect
angle,'' we may use a single letter as x,
2. The saving of space also opens the way for the use of
auxiliary quardily of various kinds.
See, for instance, the process of factoring o* + a'6* + 6*, p. 153.
3. By using a letter to represent any number (of a given
class), we are able to discover and prove general laws of numbers.
Thus, (a f 6)* = a* + 2ab + 6* is true for any numbers whatever.
As an example of the discovery of new and useful laws of niunber,
we may take the case where we know half the sum and half the dif
ference of two niunbers and desire to find the numbers themselves.
In the above description of the known facts, there is nothing to
suggest a method of obtaining the desired end. But if we express
the given facts in the algebraic form, thus, —^ and ^ , it is
at once suggested that half the difference added to half the sum wiU
give a, the greater of the two numbers, and subtracted will give b, the
smaller.
It may be well to notice that one soiuce of this discovery is that
in the algebraic expression we used separate symbols, a and b, of
nearly equal size for the two niunbers considered.
4. Combination of several rules into one formula.
Thus, the single formula p ^ br combines three cases (and rules)
used in arithmetic in treating percentage. Similarly, the formula
i = prt covers all cases used in treating interest in arithmetic.
This advantage comes (1) from the fact that a letter may be
used to represent any nmnber. See 3 above.
(2) From the fact that an equation can be solved for any letter
in the equation. ^
(3) From the approximately uniform size of the letters employed,
which suggests that we treat all the letters alike and give each the
leadership in turn.
5. The use of letters to represent unknown numbers often
enables us to begin in the middle of a complex problem and work
SIMULTANEOUS EQUATIONS . 249
in several directions and thus solve problems which otherwise
we could not analyze. See the examples on pp. 242243.
6. We should also remember constantly that the symbols
used in algebra (and the advantages coming from their use)
are but a part, or detail, of the more general subject of sym
bolism as a whole and of its utilities; and that a training in
algebra should give a better grasf of the whole subject of symbols
and their uses,
EXERCISE 76
1. Abbreviate the following as much as you can by use of
the letter x:
J a certain number + J the nmnber = 25. How much
shorter is your expression than the given expression?
2. Make up and work an example similar to Ex. 1.
3. Why does a knowledge of algebra suggest to us that a
number like 27001 can be factored and also the method of
doing this, while a knowledge of arithmetic does not do the
same? (See Ex. 29, p. 127.)
4. Is a railroad ticket a symbol or representative of the
money paid for it? What are the advantages in the use of
the ticket? The disadvantages?
5. Discuss in the same way a check drawn on a bank and
used in paying a bill.
6. In canceling a railroad ticket, what are the advantages
in punching the ticket as compared with crossing it with a
pencil mark? With burning it?
7. What is a newspaper (or a book) a symbol or represen
tative of? What are the advantages in its use? The disad
vantages?
250 . SCHOOL ALGEBRA
8. A certain firm occupied a building running from 10 to
20 Barclay St. in a certain city as their place of business. In
advertising in one magazine they gave their address as 10
Barclay St.; in another they gave their address as 12 Barclay
St.; in another as 14 Barclay St. What was the advantage
in doing this? By this means what double use was made of
the symbols 10, 12, 14, etc.
9. If a teacher has a set of papers from each of several
classes, what is the advantage in arranging them at different
angles when piling one set upon another?
10. Can you give another instance where difference of
position is utilized as a symbol?
11. What are the advantages in using a flag as a symbol
or representative of a nation?
12. What are the advantages and disadvantages of read
ing a book of travels as compared with traveling?
13. Why does a policeman in a large dty have a number
as well as a name? Name other classes of* men which have
numbers as well as names.
14. What are the advantages to a person in having a name?
15. Let each pupil make up (or collect) and work as many
examples as possible similar to the examples in this Exercise.
Sua. This work is of such a nature that it may readily be ex*
tended in various directions at the option of the teacher.
CHAPTER XIII
GRAPHS
144. A variable is a quantity which has an indefinite
number of different values.
A function is a variable which depends on another variable
for its value.
Thus, the area of a circle is a function of the radius of the circle;
the wages which a laborer receives is a function of the time that the
man works.
A g^aph is a diagram representing the relation between a
function and the variable on which the fimction depends for
its value.
A fimction may depend for its value on more than one variable;
thus, the area of a rectangle depends on two quantities — the length
of the rectangle and the breadth. The present treatment of graphs,
however, is limited to functions which depend on a single variable.
In algebra we study only those functions which have a definite
value for each definite value of the variable.
145. Uses of Graphs. A graph is useful in showing at a
glance the place where the function represented has the
greatest or least value and where it is changing its value most
rapidly, and in making clear similar properties of the function.
Graphs of algebraic equations are useful in making clear
certain properties of such equations which are otherwise
difficult to understand. A graph also often furnishes a rapid
method of determining the root (or roots) of an equation.
251
252
SCHOOL ALGEBRA
146. Framework of Reference. Axes are two straight
lines perpendicular to each other which are used as an auxil
iary framework in constructing graphs; as XX' and YY' .
The Xaxis, or ^s of abscissas, is the horizontal axis; as
XX' . The yaxis, or axis of "brdinates, is the vertical axis;
as YY'.
Y
m^
■'^^
T
t
Y'
The origin is the point
in which the axes inter
sect; as the point 0.
The ordinate of a point
is the hne. drawn from
the point parallel to the
2/axis and terminated by
the a;axis. The abscissa
of a point is the part of
the araxis intercepted between the origin and the foot of the
ordinate. Thus, the ordinate of the point P is APy and the
abscissa is OA.
The ordinate is
sometimes termed
the " 2/ " of a
point, and the ab
scissa, the " X " of
a point.
P
+ — j(a, 4)
(8,2)
I
XI — 'h^ — I — I — h
o
Ordinate s
above the araxis
aretakenasplus;
those below, as
minus. Abscissas
to the right of
the origin are plus; those to the left are minus*
The coordinates of a point are the abscissa and the ordi
nate taken together. They are usually written together
H 1 1 1 1 — \X
Y'
n
GRAPHS 253
in parenthesis with the abscissa first and a comma
between.
Thus, the point (2, 4) is the point whose abscissa is 2 and ordi
nate 4, or the point P of the figure. Similarly, the point ( — 3, 2) is
Q; (2, 2) is R; and (1, 4) is S. "v
The quadrants are the four parts into which the axes di
vide a plane. Thus, the points P, Q, R, and S lie in the first,
second, third and fourth quadrants, respectively.
EXERCISE 77
Draw axes and locate each of the following points:
1. (3, 2), (1, 3), (2, 4), (4,1).
2. (2,'§), (3, li), (5, f), (2,i).
3. (2, 0), (3, 0), (0, 4), (0, ^), (0, 0).
4. (1,a/2). (l,V2),(v^,0),(V5,3), (iVE, 2V^).
5. Construct the triangle ^hose vertices are (1,1), (2, —2)
(3.2).
6. Construct the quadrilateral whose vertices are (2, — 1),
(4, 3), (3, 5), (3, 4).,
7. Plot the points (0, 0), (1, 0), (2, 0), (5, 0), (1, 0),
(3.0). I .
8. Also (0, 0), (0, 1), (0, 2), (0, 3), (0, 5), (0,  1), (0, 3).
9. All points on the a;axis have what ordiqate?
10. All points oir the 2/axis have what abscissa?
11. Plot the following pairs of points and find the distance
between each pair of points:
(1) (6, 5), (2, 8) (S) (3, 6), (2, 6)
\ (2) (3, 0), (0, 6) (4) (0. 0), (3, 5)
12. Construct the rectangle whose vertices are (1, 3),
(6, 3), (1, 2), (6, 2), and find its area.
254 SCHOOL ALGEBRA
ft
^ 13. Construct the rectangle whose vertices are (—3, 4),
(4, 4), (—3, —2), (4, —2), and find its area.
14. Construct the triangle whose vertices are (3, A)j
(1, 3), (2, 4), and find its area.
15. In which quadrant are the abscissa and ordinate both
plus? Both tninus? In which quadrant is the abscissa minus
and the ordinate plus? In which is the abscissa plus and the
ordinate minus?
16. Pritctice oral work with small fractions as in Exercise
58 (p. 190).
Graphs of Equations of the First Degree
147. To ConstrHct the Graph of an Equation of the First
Degree Containing Two Unknown Qnantities, as x and y,
Let X have a series of convenient values, as 0, 1, 2, 3, etc.,
— 1, —2, —3, etc.;
Y
GRAPHS
255
X
y
1
1
1
2
3
3
5
etc.
etc.
1
3
2
5
etc.
etc.
Find the corresponding wlues of y;
Locate the points thus determined, and draw a line through
these points.
Ex. Construct the graph of the equation y = 2a; — 1.
Construct the points (0, 1), (1, 1), (2, 3), (3, 5),
(— 1, —3), (—2, —5), etc., and draw a line through
them. The straight line AB is thus found to be the
graph of 2/ = 2a; — 1.
148. Linear EqnationB. It will always be
found that the graph of an equation of the
first degree containing not more than two
unknown quantities is a straight Une. Hence,
A linear equation is an equation of the first degree.
149. Abbreviated Method of Constmcting the Graph of
a Linear Equation. Since a straight line is determined by
two points, in order to construct the graph of an equation
of the first degree it is suflScient to construct any two points of
the graph and draw a straight line through them.
Ex. 1. Graph 3^^  2a; = 6.
When a;=0, y—2;
when y^OfX— — 3.
Hence, the graph
passes through the
points (0, 2) and
(3,0), or CD is the
requhed graph. '
The greater the x^ i ( ( j^ i i
distance between
the points chosen, the
more accurate the
construction will be.
It is usually advis
able to test the _
result obtained by
locating a third point and observing whether it falls upon the
graph as constructed.
256 SCHOOL ALGEBRA
If the given line does not pass through the origin, or near the
origin on both axes, it is often convenient to construct the line
by determining the points where the line crosses the axes.
Ex. 2. Graph 4x + 7y = 1.
When X = 0, 2/ = I; when t/ = 0, x = J. Hence, the graph
passes close to the origin on both axes. Hence, find two points on
the required graph at some distance from each other, as by letting
a: = 0, 9. and finding y = ^, —5. Let the pupil construct the figure.
EXERCISE 78
Graph the following. (It is an advantage, if possible, to
draw the graph line in red, the rest of the figure in black ink.)
1. y = X + 2 7. 4a; — 5y = l
3. 3a; + 2y = 6 ^ i
4. 3a; 22/ = 6 ^' « = 3(j/ .1)
5. 3a;  51/ + 15 = 10. y = x
e, y = 2x IX. y = i
12. If a; = 2, show that whatever value y has, x always
= 2. Hence the graph of a; =2 is a fine parallel to the yaxis.
13. Graph a; = 0; also y = 0.
14. Show how to determine from an inspection of a linear
equation whether its graph passes through the origin; near
the origin on one axis; near the origin on both axes.
15. Graph 5a; + Gj/ = 1 ; also 6a; — y = 12.
16. Obtain and state a short method of graphing a linear
equation in which the term which does not contain a; or ^ is
missing, as 2y — 3a; = 0.
GRAPHS
257
Before graphing the following, determine the best method
of constructing each graph, and then graph:
17. a: + 2y = 4 20. if^x + Jy = § 23. a; — y = 5
18. 2y = a: 21. x =  3 24. y + 2 =
19. 5x 6y = l 22. 5a: + 42/ = 25. 3x2y + ^^0
26. Construct the triangle whose sides are the graphs of
the equations, y—2z + 1 = 0, 3y— a:— 7==0, y + 3a: + 11=0.
27. An equation of the form y = b represents a line in
what position? One of the form x = a^
28. Make up and work an example similar to Ex. 4. To
Ex. 26.
150. Oraphic Solution of Simiiltaneoas Linear EquationB.
If we construct the graph of the equation a: — y = 3 (the
line AB) and the graph of 3a: + 2y = 4 (the line CD), and
258 SCHOOL ALGEBRA
measure the coordinates of their points of intersection^ we
find this point to be (2, —1).
{/p # s 3
by the ordinary algebraic method, we find that a: = 2 and
V = 1.
In general, the roots of two simvitaneoris linear equaiions cor
respond to the coordinates of the point of intersection of their
graphs; for these coordinates are the only ones which sat
isfy both graphs, and their values are also the only values of
X and y which satisfy both equations.
Hence, to obtain the graphic solution of two simultaneous
equations.
Draw the graphs of the given eqvaiions, and measure the co
ordinates c/ the point {or points) of intersection.
Graphing two simultaneous equations forms a convenient
method of testing or checking their algebraic solution.
151. Simnltaneons Linear Equations whose Graphs are
Parallel Lines. Construct the graph of a; + 2y = 2 and also
of 3x + 6y = 12.
You will find that the graphs obtained are parallel straight lines.
Now try to solve the same equations algebraically. You wilLfind
that when either x or y is eliminated, the other unknown quantity
is eliminated also, and that it is therefore impossible to obtain a
solution. The reason why an algebraic solution is impossible is
made clear by the fact that the graphs, being parallel lines, cannot
intersect; that is *to say, there are no values of x and y which
will satisfy both of these lines, or both equations, at the same
time.
152. Graphic Solution of an Equation of the First Degree
of One Unknown Quantity. By substituting for y in the
[y = a; — 3
first equation of the pair \ _ ^ the two equations
GRAPHS
259
reduce to a: — 3 = 0 Accordingly, the graphic solution of
an equation like a: — 3 = can be obtained by combining
the graphs of y = a; — 3 and t/ = 0. In other words, the root
of a; — 3 = is represented graphically by the abscissa of
the point where the graph of y = a; — 3 crosses the xaxis.
' EXERCISE 79
Solve each pair of the following equations both graphically
and algebraically, and compare the results in each example:
a; + 7y+ll =
aj  3y + 1 =
9x  6y = 3
2/ = 2x + 3
8. Solve graphically 2a:+3=0
9. Solve graphically 3a:— 5=0
1.
2.
3.
i2x + 3y = 7
\x — y = l
fy = 3x  4
ly =  2a: + 1
2y = a:
la; + y + 6 =
fy = 2a:
5.
6.
7.
10. Discover and state the relation between the coeflScients
of two hnear simultaneous equations whose graphs are par
aUd lines. (8x + 5y = 7.
11. Solve graphicaUy 6x + 2t/ = ll. ,^ , _,
12. Solve both algebraically find graphically 1 gx — Qw = 5
13. Construct the quadrilateral whose sides are the
graphs of the equations, a: — 2y — 4 = 0, x + y = 1,
3y — 5a:— 15 = 0, a: + 2y — 4 = 0, and find the co
ordinates of the vertices of the quadrilateral.
14. Make up and work an example similar to Ex. 1. To
Ex. 6.
15. How many examples in Exercise 26 (p. 110) can you
now work at sight?
260
SCHOOL ALGEBRA
163. OrapMo Solution of Written Problems.
I. Aailway Problems.
Ex. The distance between New York and Philadelphia
is 90 mi. At a given time, a train leaves each city, bound
for the other city, the train from New York going at 40 mi.
an hour and the one from Philadelphia at 30 mi. In how
many hours will they meet, and at what distance from New
York?
The train dispatcher represents the distance between the stations
by the line AB, each space denoting 10 mi. Each space on AI rep
resents 1 hour. He lo
cates E three units to the
right of A and one imit
above AB^ and F four
units to the left of B and
one unit above AB. He
produces AE and BF to
meet at C, and draws
CD perpendicular to AB.
He obtains the distance from A at which the trains meet, by
measuring AD to scale (and hence determines the siding at which
one train must wait for the other). He obtains the time that elapses
before the trains meet, by measuring CD to scale.
The problem may also be solved algebraically in the same way
as Exs. 5761, p. 87.
The advantage of the graphical method is that in this solution
it is easy to make allowance for any waits which trains may make
at stations. Hence, railroad timetables are often constructed
entirely by graphical methods.
ihr.
Shr.
Ihr.
•

""^
^^
^
:^
^
^
<i
ft
1
M
'^^
^
^
1
i
i
i
•
1
:i
1
i
8
8
9
s
s
s
8
II. Problems in the Miztnre of Materials.
Ex. In order to obtain a mixture containing 20% of butter
fat, in what proportion should cream containing 30% of fat
be mixed with milk qpntaining 4%?
30
20
«
16
4
10
GRAPHS 261
Graphical Solvtiori
We construct a rectangle, and write in two adjacent cotnetiS
(here the lefthand corners) the per cents
of fat (30 and 4) in the two given fluids;
and in the middle of the rectangle we
write the per cent (20) desired in the mix
ture. The differences between the num
ber in the middle and the numbers in the
corners (16 and 10) are then found and placed as in the diagram.
The diflFerences thus found show the relative amounts of the given
fluids to be used, viz. : 10 parts of milk, and 16 of cream.
Now solve this problem algebraically by the method used for
Exs. 2630, pp. 241242; and by an examination of this solution,
discover for yourself the reason for the above graphical solution.
EXERCISE 80
Solve the following problems graphically:
1. The distance between New York and Philadelphia is
90 mi. If a train leaves New York at noon and goes 40 mi.
an hour, and another train leaves Philadelphia at the same
time and travels 20 mi. an hour, at what time and how far
from New York will they meet?
2. Make up and work an example similar to Ex. 1.
3. The distance between New York and Buffalo is 440
mi.' If a train leaves New York at 11 a. m. and travels at
the rate of 40 mi. an hour, and a train traveling 30 mi. an
hour leaves BuflFalo at the same time, at what time and how
far from New York will the trains meet?
4. Make up and work an example similar to Ex. 3.
5. In order to obtain a mixture containing 22% butter
fat, in what proportion must cream containing 32% of fat
be mixed with milk containing 5%?
262 SCHOOL ALGEBRA
6. In order to obtain a mixture containing 28% of butter
fat, in what proportion must cream containing 35% of fat
be mixed with cream containing 25%?
7. Make up and work an example similar to Ex. 6.
8. In what proportion must oats worth 50j!f a bushel be
mixed Vith corn worth SOjf a bushel in order to make a mix
ture worth 60jlf a bushel?
9. Make up and work a similar example concerning mixing
different grades of coffee.
10. The distance PQ is 48 mi. At 8 a. m. one boy starts
from P and walks toward Q at the uniform rate of 4 mi. an
hour. At the same time another boy starts from Q on a
bicycle and rides toward P at the rate of 8 mi. an hoiu' but
at the end of each hour of riding rests ^ an hour. By means
of a graph determine where and when the two boys will meet.
11. Make up and work an example similar to Ex. 10.
12. Umw many examples in Exercise 27 (p. 112) can you
now work at sight?
EXERCISE 81
Review
1. Tell the degree of each term of
5x«  4a;V  Ux  xy + xy^  x*y h ^'^  7y + 11.
2. Factor (1) x* + 4.
(2) m^ — 2mn + n^ + 5w — 5w.
(3) a*  n^  w*  2a6 + 2mn + 6«.
3. Factor 2(x»  1) + 7(a;2  1).
Simplify:
^ 4a;  5 4 +x ,2 x 5
*• — 7^ — — tjt; r « —
45 30 • 3 18
1 ~ 5x 3a; 4 5 2a;  3
6a;*6'*"4a;+4'^33x'
GRAPHS 263
6 2 , 1 2 I
(xl)»^(li)« 1
X X
' .i(x»x2) •
9. 1 ^
a —
1 «
Solve:
Airr* 2 3 5
6 7x+15 4 ^'
^ 13 ^ __ a;+ 1 _ X— 8 _ a;— 9
a;2 X— l" x^ xl'
14 ??_?=^ 16 a;2 10a; _* y 10
■ 2 3 9' * 5 3 4 *
^_L.32/__„, 2y44 2a; + y a;+13
4 "^2 "^* "1 8 4
^^' ?"3*^"^' ^^ (a6)a: + (a + 6)y=a46.
y = 9 (x 1/) (a^  62) « a^ + h\
18. .3a: +.2?/= 1.3.
.31/ +.22 =.8.
.32 4.2x=.9.
19. =. 20. Solve +=A
y 6 a ' 6 . a6
2 a; 2/ 1
^ 15 a' ^6' a'6'
21. Is it allowable to divide each term in 16a; = 96 by 16?
Is it allowable to divide each term of 16a; — 96 by 16? Give
reasons.
264 SCHOOL ALGEBRA
22. Obtain the value of
_ ^a(x+l) J, when a: —
23. Solve — T+— — = =+
a+h X a—b z
24. Why is it proper to change — a; = — 3 into aj = 3, and not
proper to change —x —3 into a; + 3?
ct c
25. What number added to the denominators of r and j, respec
tively, will make the results equal?
c c c CL "4" h d h
26. Does r equal — r? Does equal — \ ? Does
a—b ^ a b c ^ c c
c c c
— — r equal ""+r? Verify your statements for the special case
when a  4, 6 = 8, and c = 2.
27. The sums of three numbers taken two and two are 20, 29,
and 27. What are the numbers?
28. Factor 8(x + vY  (2a;  y)\
29. What is the advantage of being able to add the same number
to both members of an equation? In being able to transpose a term?
To divideboth members of an equation by the same mmiber? (See
Art. 70.)
30. Solve (a 4 c)x — (a — c)y = 2ab,
(a+ b)x — (a — b)y = 2ac.
31. Does (o* 4 &*) (o + b) equal a' + 6'? Verify your state
ment by letting a and b have convenient numerical values. Can
you prove your statement without the use of numbers?
32. UK^nR^ and C= 27rR, find K when C= 10 and fr= V.
33. If s = igl^ and v = gtj find s when ^ = 32 and v — 64.
34. l{C^2nR and 7= j7r/iJ», find V when C= 33 and fr= V
(use cancellation wherever possible).
Solve:
35. 4(x + y) + ^^=13. 36. 3x+^ = 6.
jBug. Let p = (a; + 2/), g =
a; y
37. Show that
reduces to
GRAPHS 265
12<(<» f 2Y (2<»  3)«  6<» (2^  3)» ffl + 2)
(<'+2)»
38. Graph y—2x\h when 6 « 1. On the same diagram
graph y = 2x+ 6 when 6=2. When 6 « — 1. When 6=0.
39. Graph y—ax{2 when a» 1. On the same diagram
graph 2/= ax+ 2 when a= 2, 3, 1, 3.
40. Graph y= 3x + 2, and j/ = — Ja; + 2 on the same diagram.
41. Make up and work an example similar to Ex. 38. To Ex. 39.
42. The Fahrenheit reading at the boiling point of alcohol is
95° higher than the Centigrade reading. Find each of the readings.
43. Make up an example similar to Ex. 42, using the fact that
ether boils at 96° Fahrenheit.
44. Give the value of  4 a, a & , — x* ^ x*.
a a o o
45. What is the reciprocal, of  + r ?
a
46. Show that elimination by comparison is a special form of
elimination by substitution.
47. Show that elimination by addition and subtraction may also
be regarded as a form of elimination by substitution.
48. Eliminate a between the equations F = Ma and 8 » iafi.
49. Given I ^ a\ {n — \)d and 8= o(^+ 0> fi^d * ^^ terms of
dy n, and L
SuQ. What letter must be eliminated?
rl—a
50. Eliminate I between I = ar'^'^ and a =
r1
CHAPTER XIV
INEQUALITIES
154. The Signs of Inequality are >, which is read " is
greater than," and <, which is read " is less than."
Thus, a>b means that a is greater than b,
c<b means that c is less than 6.
Observe that both signs of inequality are written with the
opening toward the greater quantity.
155. An Inequality is a statement in symbols that one
algebraic expression represents a greater or less number than
another; asx + y <a^ + 6^.
Remember that any positive number is greater than any negative
number, and that of two negative niunbers the smaller is the
greater. Thus, 2 > — 5, and — 2 > — 3.
The first member of an inequality is the expression on the
left of fte sign of inequality; the second member is the ex
pression on the right of this sign.
156. Inequalities of the Same Kind. Two inequalities are
said to be of the same kind, or to subsist in the same sense,
when the greater member occupies the same relative position
in each inequality; that is, is the lefthand member in each,
or the righthand member. Hence, in inequalities of the same
kind the signs of inequality point in the same direction.
266
INEQUALITIES 267
Thus,
but
x>2x
3 ^^
•3
•4
a<b
■
2a>b'
a
"2
<
are of the same kind;
are of opposite kinds.
157. Properties of Inequalities. The following primary
properties of inequalities are recognized as true:
(1) Adding and subtracting quantities. An ineqiuility vrUl
be unchanged in kind if the same quantity is addM to oi
svbtracted from each member. Hence,
(2) Terms transposed. A term may be transposed from one
member of an ineqicality to the other, provided its sign is changed.
(3) Signs changed. The signs of all the terms of an in
equality may be changed^ promded the sign of the inequality is
reversed.
(4) Positive multiplier. An inequcdity wUl be unchanged in
kind if all its terms are multiplied or divided by ifie same posi
tive number.
(5) Saised to a power. An inequality will be unchanged in
hind if both members are positive and both are raised to the same
power.
(6) Equalities combined with inequalities. If the members
of an inequality are subtracted from equals y the result will be an
inequality of the opposite hind. If the members of an inequality
are divided into equals, the result will be an inequality of the
opposite hind.
(7) InequaUties combined. If the correspcmding members
of two inequxdities of the same hind are added, or multiplied,
the resulting inequality will be of the same hind. But if the
members of an inequality are subtracted from, or divided by.
268 SCHOOL ALGEBRA
the corresponding members of another irisqucdity of the same
hind, the resulting inequality wHl not always be of the same
kind.
The following are numerical illustrations of the above
principles:
(1)
COM
II V
coot
(5)
7> 5
.'. 72> 52
10>8
«
or
49>25
(2)
•
•
12>73
.12+3>7
(6)
20=20
7> 5
13<15
(3)
•
•
.7>10
.10> 7
(7)
17>15
8> 2
or
7< 10
Adding,
25>17
(4)
7> 5
3= 3
17>15
8> 2
31 > 15 Subtracting, 9 < 13
158. A conditional inequality is an inequality which is
true only for certain special values of the letter or letters
involved.
Thus, (3 — xy > (a; — 4)* is a conditional inequality, since it
may be proved that it is true only when x >3J.
An nnconditional or absolute inequality is an inequaUty
which is true for all possible values of the letter or letters
involved.
Thus, a* + 6* > 2ab is an unconditional inequality, since it may
be proved to be true for all possible values of a and b (the value
zero not being considered in this case).
Hence, the conditional inequality corresponds to the conditional
equation, and the unconditional inequahty to the identity (see
Art. 69, p. 94). As with the equality sign, so with the signs > and
<, the particular sense in which each is used is, for the present, to
be determined by the context.
INEQUALITIES
269
159. Solntion of Conditional Ineqnalities.
Ex. 1. For what value of a; is 3a; — 4 > 1 — 2x?
TranspofiiDg terms, 3a; + 2a; > 5
5a; > 5 . * . a; > 1 Arts,
This process may be illustrated graphically as in the diagram.
Graphing p = 3a; — 4, we
obtain line il^. Also graph
ing 5 = 1 — 2Xf we obtain
line CD. These lines inter
sect at the point F, whose
abscissa is 1. To the right
of F (where x>l),p>q, i. e.
3a;  4 > 1  2a;.
Ex. 2. Given that x is
an integer, determine its
value from the following
inequalities:
'4a; 7<2a;+3
.3a; + 1>13 a;
Transposing terms,
(
2a;<10
4a;>12
/
B
TT
/
'
i
1
{
y
^
1
j/
\
t/
HI
>
\
1
\
*
"5
\
^
1
Y
X\
1
y
/
'D
i
/
a'
i
Dividing by coefficient of x in each inequaUty,
x<5
x>3
.', x =4 Ana.
Let the pupil illustrate this solution graphically.
160. Proof of ITnconditional Inequalities.
Ex. 1. Prove that the sum of the squares of any two un
equal quantities is greater than twice their product,
Xiet a be the greater of the two quantities, and b the less.
Then, ^ ab >0
.•.(a6)2>0
f\a^2ab+b^>0
^ i
270 SCHOOL ALGEBRA
Ex. 2. Prove (a + 6) (6 + c) (a + c) >8a6c.
The lefthand member when expanded becomes
a(6> f c2) + 6(a2 + c2) + c(a» + 6«) + 2afcc
But from Ex. 1, a(6*+c*)>a(26c) (1)
6(o2Hc*)>6(2ac) (2)
c(a*+6«)>c(2a6) (3)
Also, 2a6c=2a6c (4)
Adding (1), (2), (3), (4),
(af 6) (6 + c) (a+ c) > 8a6c
EXERCISE 82
Reduce:
1. (a; + l)»<a? + 3 5. *l°Z^<xJ5L7
2. (3x)^>(x4)« ^ ^
^+1 v^^ + O
3. 7aa; + 6>3aa; + 56 ^' a:  2 a;  4
^ 4a:3^a: , 3ar + 8 . a^x^h^x
3 221 0X26X
4(x + 3) 8a; + 37 7x  29
9 18 5«12
Find the limits of x:
9. 3a:+l>2a: + 7 ^ 60>^:^>50
2a:l<a; + 6 5
10. 3(a:4)+2>4(a:3) ^ ioo>a: + ^>90
2(a:+l)<4(a:l)+3 2
Solve the following:
13. xy>h 14. 3a:42/>6
a: + y = 12 4a: + 5y = 80
15. What number is that whose fifth plus its sixth is
greater than 6, while its third minus its eighth is less than 4?
16. A certain integer decreased by \ di itself is greater
than \ of the number increased by 5f ; but if J of itself i3
INEQUALITIES 271
added to the number, the sum is less than 20. Find the
number.
If the letters employed in each are positive and unequal,
prove:
17. Za^ + V^>2a(a + b) ^^ a , 6. ^
18. o«63>3a263a62 ^ ^
21. a + 6>2Va6
19. a« + 6«>a26 + a62 ^ ei^ + j2+e2>a6 + ac + 6c
23. 6a6c<a(62 + a6 + c2) + c(62 + 6c + d2)
24. oh{a + 6) + ac(a + c) + 6c(fe + c) < 2(a» + fc» + c*)
25. a' + 63 + c3>3a6c
26. A baseball team has won 22 games out of 35 gaines
played. What is the least niunber of games which the team
must win in succession in order that its average of games
won may exceed .75?
27. Make up and work a similar example concerning games
won by a basketball team.
28. A boy has worked correctly 13 examples out of 18.
What is the least number of examples which he must work
correctly in succession in order to bring his average above
90%? Above 80%?
29. Who invented the signs > and < to represent inequal
ity? What other signs were invented for this purpose?
Which set of signs do you consider superior and why?
CHAPTER XV
INVOLUTION AND EVOLUTION
Involution
16L InvolTition is the operation of raising an expression
to any required power.
Since a power is the product of equal factors, involution is a
species of multiplication. In this multiplication, the fact that the
quantities multiplied are equal leads to important abbreviatioDS of
the work.
Powers of Monomials *
162. Law of Exponents or Index Law.
Since, a^ = aXaXa,
(a^y ^ (aXaXa)(aXaXa){aXaXa)(aXaXa)
In general, in raising o** to the m* power, we have the factor
a taken m X n times, or
(a**)*" = a"»" I.
Also, (06)" = abX ah X ah to n factors
= (aXaXa . ... to w factors) {bXbXb . , ton factors)
.*. (aby = o'^ft" IL
This law enables us to reduce the process of finding the
power of a product to the simpler process of finding the power
of each factor of the given product.
163. Law of Signs. It is evident from the law of signs
in multiplication that
272
INVOLUTION 278
1
(1) An even power of a quantity {whether plus or minvs) is
always positive.
Thus, (  3)* = 9, and (  a6)* = a*6*.
(2) An odd power of a quantity has the same sign cw the oHq'
inal quantity.
Thus, {ay =  a\ and ( +a)« = a^
164. Involiition of Honpmiah in Oeneral. Hence, to raise
a mononiial to a required power.
Raise the coefficient to the required power;
Multiply the exponent of ea^h literal factor by the index of the
required power;
Prefix the proper sign to the result.
Ex.1. Find the cube of Sa^y.
{3z^y = 27xV Ana.
Ex. 2. {2ab^y = 32a^b'^ Ans.
165. Powers of Fractions. By a method similar to that
used in Art. 162, it can be shown that
6p«
Hence, to raise a fraction to a required power.
Raise both numerator and denominator to the required power,
and prefix the proper sign to the resulting fraction.
^ ( 2a^x\ _ 16a^V .
EXERCISE 83
Write the square of
1. 7a26 2. bxif^ 3. fa:y
^ 3a: ^ 6a6
Ssr* 11
274 SCHOOL ALGEBRA
6.  13a:»y
8. — 4y»+^
^nym
lOz
Sod
Write the cube of
•
10. 3xy 11.2*2
12. Ja^sy^ 13. 5a;"y
3a;
14. .
4f
Write the value of
16. (7afc*c')3 ^g
17. (lta^b*y
c
2xVY ^^ (2«')'
xz y . 22. ( §m*)«
18. (a;^)* 20.
(
Vi»)« 23. (3^a^»)?
24. (.03)2
25. (.(X)3)«
28. f^"V
\ 3aV
26. (.03)»
27. (.(X)3)«
29 y^^'
■ 2V3y 30. (2^)'
•
31. Commit to memory
the values of the various powers of
(1) 2 up to 2^0
(3) 4 up to 4«
(2) 3 up to 3« (4) 5 up to 5*
(5) 6 up to 6^
(6) the squares of the numbers up to 29
32. Give the value of 2 X 3^ (2 X 3)^.
33. Give the value of ( ^ J ' T' k2*
34. Give the value of each of the following: [(—2)']*,
[(2)2]3, [(3)2]2. Does [(a)^2 equal [(a)2]3?
35. Does 2X3^ equal 6^? Does ^(4^) equal 2^?
36. On squared paper show the meaning of (.3)^ =* .09.
Also of (1.5)2. Of (1.5)2, (2.5)2, (.5)2.
INVOLUTION 275
37. Given 2* = 32, find in the shortest way the value of
2^\ Also of 2^^ Of 2^.
38. Make up and work an example similar to Ex. 37 con
cerning powers of 3. Of 4.,
39. Give the value of (2 y\ Of (2)^
40. In x^ • 7^y how many ar's are multiplied together? How
many in (a:^)^? Write out in full each of these sets of factors.
41. Treat in the same way a^  a^ and (a*)°.
42. Make up and work an example similar to Ex. 40.
43. Treat [(a^yY and (oi^f  jf" in like manner.
44. In obtaining the value of mTT^* is it allowable to
cancel the 4's? •
(off
45. In reducing 3— g to its simplest form, is it allowable
to cancel the 5's? Why?
46. Is it allowable to cancel the 4's in obtaining the value
°UX2»^ Of 3,?
47. Does 2^ X 2* equal 4^? Give a reason for your answer.
The value of the second expression is how many times as
great as that of the first?
48. Express as a power of 2 the number of greatgrand
parents a person has. Also the number of greatgreat
greatgrandparents.
49. If a decimal fraction contains four places, how many
places will its square contain? Its cube? Its fourth power?
Give a numerical illustration of your answer to the first of
these questions.
50. To ^rs add the square of ~ of ^r*
2a? ^ 2 2a
276 SCHOOL ALGEBRA X/ ^
W
12
51. 'Find the value of ar""'^ when a = ,r = , and n = 6.
In each of the following let a = 0; ; 1; 2; 3; — ; —1; —2;
—3, in succession, and tabulate the results obtained as in
Ex. 54, p. 24.
52. 0^2 _ 3^. ^ 2 5^. Q!? + X 56. a? 01? + I
53. Qi? — 2xl 55. a:^  a; 57. a:^ _ 2^^  2
58. Show that 2^ X 5* = 10®. Is there any advantage in
knowing that 2® X 5« = 10®?
59. Work again such examples on p. 105 as the teacher
may indicate.
60. How many of the examples in this Exercise can you
work at sight?
Powers of Binomials
166. Oeneral Method. In obtaining a required power of
a binomial, it is possible to abbreviate the work even more
than in the involution of a monomial.
It is suflScient, in taking up the subject here for the first
time, to obtain several powers of a binomial by actual mul
tiplication, and, by comparing them, to obtain a general
method for writing out the power of any binomial. A formal
proof of the method is given later.
(a + by = a^ + 2ab + U".
(a + by = a^ + Sa% + SaP + ¥
(a + by = a^ + ia?b + Ga'V' + Aab^ + ¥
(a + by = a^ + 5a% + lOa^V" + lOaW + 5ab^ + V
If b is negative, the terms containing odd powers of 6 will
be negative; that is, the second, fourth, sixth, and all even
terms, will be negative.
Comparing the results obtained, it is perceived that
INVOLUTION 277
I. The number of tenns equals the exponent of the power
of the binomial; plus one.
II. Exponents. The exponent of a in the first term equals
the index of the required power, and diminishes by 1 in each
succeeding term. The exponent of 6 in the second term is 1,
and increases by 1 in each succeeding term.
III. Coefficients. The coefficient of the first term is
1; of the second term it is the index of the required
power.
In each succeeding term the coefficient is found by multi^
plying the coefficient of the preceding term by the exponent of a
in that term, and dividing by the number of the preceding
term,
IV. Sig^s of terms. If the binomial is a difference, the
signs of the even terms are minus; otherwise the signs of all
the terms are plus.
Ex. (a + by = a^ + 7a% + 21a^}y' + SSa^ft^ + SSa^if
+ 21a^b^ + 7a6« + 6^
7x6
The coefl5cient of the third term = — — = 21.
The other coefficients are detennined similarly.
Observe that the coefficients of the latter half of the expansion are
the same as those of the first half in reoerse order,
167. Binomials with Complex Terms. If the terms of
the given binomial have coefficients or exponents other than
unity, it is usually best to separate the process of writing out
the required power into two steps.
Ex. (2a:'  \fy = (2r^)^  ^2:x?y {ly") + Q{2x?f {\ff
4(20?) {\ff + {\fY
Let the pupil check the work by letting a; = 2, i^ = 2.
278 SCHOOL ALGEBRA
168. Application to Polynomials.
Ex. (.x + 2y + 3zy = [(x + 2y) + 3z]»
= (a; + 2yy + 3(a; + 2yf{Zz) +
3(a: + 2y) {Zzf + (3z)'
= a^ + 6a:^ + 123^* + Sj/^ + 9A +
^xyz + 36y% + 27«z* + 54^^ + 272» 4n».
Let the pupil check the work.
ESERCISE 84
Expand and check:
1. (o + xf 13. {23?  1)«
2. ' (a  «)s 14. (2a2  36)*
3. (6 + 2/)^ "• (a^' + 2o)«
4. (6y)« 16. (3a« + )''
5. (ay)» "• (2^ + 5)«.
6. (a + j/)» 18. (2)^
'• (^a + t)"* 19. (a^ + a;l)s
8. (2a + 6)» 20 {x'Zxlf
9 (l«)* 21. {jo? + ac + <?)*
10. (2c(P)5 ^ 22. {xy>tzf
11. (73a:*)» 23. (2a^  a; + 3)»
12. {Z\<?f 24. (l + xar')<
25. How many terms are there in the expansion of (o+6)*?
Of (a + 6)'? (a + 6)"?
26. Write out the last three terms in the expan^on of
(a + 6)«.
27. Who first suggested the writing of an exponent in its
present position (that is, a little above and to the right of the
base)? (See p. 456.) Tell all you can about this man.
EVOLUTION 279
28. Give some of the ways in which powers were previously
indicated and exponents written. Discuss the relative merits
of these different ways of writing exponents.
29. How many examples in Exercise 35 (p. 131) can you
now work at sight?
Evolution
169. A Boot of a quantity is a quantity which, taken as
a factor a given number of times, wiU produce the given
quantity.
170. Evolution is the process of finding a required root
of a quantity.
What is the radical or root sign? What is. the meaning of
V9? Of^? v^?
171. iramber of Roots. Taking a particular example, we
find that V4 has two values, viz. : + 2 and — 2, for (+2)^ =4,
and (2)2 = 4.
A number containing a square root of a negative quantity
is tenned an imaginary number.
A real nnmber is a nmnber which does not contain an
imaginary number.
The nature of the square of an in^iginary number, as of V— 4,
is explained in Chapter XVIII (p. 334).
If we include imaginary roots, it may be shown that when any
root of a given number is extracted, the number of possible roots
equals the index of the root to he extracted.
Thus, in taking the cube root of 8, we find three possible roots,
viz.: 2, 1 + V^, and 1 V^.
172. The Principal Root of a number is that real root of
the number which has the same sign as the number itself.
Thus, the principal root for V4 is 2; for ^^27 is 3; for ^'^
is 3.
In this chapter only the principal roots of numbers are considered.
280 SCHOOL ALGEBRA
Evolution op Monomials
173. Index Law. Since (a"*) = a"»» (Art. 162, p. 272), it
follows that
V a""* = a"* I.
where m and n are positive integers.
Hence, the process of finding the root of a quantity affected
by an exponent becomes simply a division of exponents.
Also, v^ = ^/a^/h . _ II.
For let ^a = ar, a/6 = y;
. • . a" = a . . . (1) y* = 6 • • • (2)
But x»2/* = {xyY (by Art. 162)
Substitute for x* and y" from (1) and (2),
ab = {y/a^lY (3)
Extract the nth root of each member of (3),
^ab = \/ay/h
This reduces the process of finding the nth root of a product
to the simpler process of finding the nth root of each factor.
174. Hethod. Hence, to extract a required root of any
monomial.
Extract the required root of the coefficient;
Divide the exponent of each letter by the index of the required
root;
Prefix the proper sign to the result.
How may the work be checked?
Write the square root of
1. 9xY 4. ma?f 6. aV*
2. 25a« ^ S6a^ ^ 121aV^
3. 1442/2 ' 49x^^ ' 81y2« + 2
EVOLUTION 281
Write the cube root of
9. 1252/»2» ^ _216^ ^^ 1000
10. iaW 3432/^ ' a:«V
Write the value of
15. v^512a:« 18. v''64^^ ^^ / /625^
16. 4^16y^^ 19. V^^^xV^ ^**^
17. v^32a^V 20. V^^ 22.. v^^a»V^
23. Express 32 as a power of 2. 81 as a power of 3.
24. Express each of the following as a power of some niun
. . ^ ^i 27 32 121
ber (a prime nimiber, if possible) : 32, 243, 256, ^> glS' o^'
and 729.
25. Find the value of a; in each of the following equations:
:r3 = 64, x^ = ^^a? = S^x' = 128.
26. If 2* = 16, what is the value of x?
SuG. Write 2* = 2^
27. Solve each of the following for x: 2* = 32, 3* = 81,
^ 3 = 243. and ()'= 
28. Find the value of r or w in each of the following:
a' = 16, r» = 32, r* = g, ^3 = ()". and 2 = 32.
29. Find the largest square factor in each of the following
numbers: 45, 128, 192, 112, and 147.
30. Find the largest cube that is a factor of 54. Of 81,
,^^
282 SCHOOL ALGEBRA
81
128, r^' Also the largest 4tli power that is a factor of 32.
Of 162, 256.
Extract the square root of each of the following by taking
out factors which are perfect squares:
31. 5625 33. 46,656 35. 48X16X18X24
32. 1296 34. 63 X 28' 36. 27 X 12 X 98 X 50
37. Who first used the sign V to. denote a root? Who
first suggested the use of the vinculum instead of the paren
thesis (as in the latter part of V .)? How were roots like
V, V, first indicated after the invention of the radical
sign?
38. Make up and work an example similar to Ex. 25. To
Ex.29. Ex.33.
39. How many of the examples in this Exercise can you
work at sight?
Squarb Root
175. Square Soot of Pol]rnomial8. In order to determine
a general method for finding the square root of any poly
nomial which is a square, we consider the relation between the
terms of a binomial and the terms of its square; as between
a + b and its square, a^ + 2ab + 6^. This relation stated in
inverse form gives us the required method. The essence of
the method consists in writing a^ + 2ab + 6^ in the form
o2 + 6(2a + 6).
Ex. Extract the square root of 16a:^ — 24a^ + 9j^.
16x*  24xy h V 4g  3y Root
16x»
8x 3y
2ixy+9y^
'V
EVOLUTION 283
TaJdng the square root of the first term, 16a;*, we obtain 4x, which
is placed to the right of the given expression as the first term of the
root. Subtract the square of 4x from the given polynomial.
Taking twice the first term of the rOot, 8x, as a trial divisor, and
dividing it into the first term of the remainder, we obtain the sec
ond term of the root, — 3j/. This is annexed to the first term of the
root and also to the trial divisor to make the complete .divisor,
Sx Sy. . ' .
176. Square Boot to Three or More Terms. In squaring
a trinomial, a + b + c, we may regard a + 6 as a single
quantity, and denote it by a symbol, as p, and obtain the
square in the form p^ + 2pc + (?.
Evidently we may reverse this process, and extract a square
root to three terms by regarding two terms of the root, when
found, as a single quantity. So a fourth term of a root, or
any number of terms, may be obtained by regarding the root
already found as a single quantity.
Ex. Extract the square root of a^Gaj^ + lOx^aOa: + 25.
X* 6x« + 19x2  30x + 25 1 x'  3x + 5 BxhA.
X* 
2x« 3x
3x
 6x» + 19x2
6x«+ 9x2
2x*6x +5
+ 10x2  30x + 25
+ 10x2  30x + 25
The first two terms of the root, x* — 3x, are foimd as in the ex
ample in Art. 175.
To continue the process, we consider the root already found,
aj2 — 3x, as a single quantity, and multiply it by 2 to make it a trial
divisor.
Dividing the first term of the remainder, 10x2, by the first term
of the trial divisor, + 2x2, we obtain the next term of the root, f 5.
The process is then continued as before.
The work may be checked by squaring the result obtained, or by
numerical substitution.
Let the pupil state the above process as a rule.
7
284 SCHOOL A3LGEBRA
EXERCISE 86
Find the square root and check:
1. a^4a? + 6x^4x + l.
2. 1  2a  a2 + 2a^ + a\
3. 9a^  12a:8 + IOt?  4a: + 1.
4. 25 + 30a: + 19ar^ + &7? \r^.
5. w«  4n5 + 4n^ + Bn^  1271^ + 9.
6. 4a:« + 12a:5 + a:*  24a:8 _ 143^ j^ 12a: + 9.
7. l + 16m«40# + 10w~8m« + 25m^
8. 4Cn2 + 2cW + 4n« + 25  44/1^  407i  12w^
9. 9a:« + V + 24a:^y + 24a:y^  8a:y  8ar^y*  50a:'2^.
10. m2 + 9 + ar^ + 6m + 6a: + 2ma:.
11. l + b7?\2Q^ + 7?'h? + 23^ +2x.
12. 2V  47a:* + 49a:« ^a:^  V + 16a; + 4.
13. Jar^5a: + 25. 16. ;xA^2:x?  z^\.
14. l^bxy^^. 17. ia*ia3 + ^a24a+36.
15. —^ 14. ^® :^l hll + i + —
4^^ y or X or a
19. ia:*ia? + W^3a:+^'.
20. ^V  la:' + !V  f ^ + tI.
21. 1 + a  iV  Ja3  V + a^ + a^.
22. £. + ca; + 5 + i5 +  + i.
4 cr 4ar 2a: c
a^ . a:^ «> . a^ . a*
23. ^aa: + 2 +  + .
Find to three terms the square root of
24. 1 + 4a:. 26. a^ + 46. 28. a? + 3.
25. a? 6. 27. 4a26a6. 29. a^ + 3ab2l^.
EVOLUTION 285
30. Solve a?  6a: + 9 = 25.
Extracting the square root of each number we have x — 3 — * 6
whence x = 3 =*= 5, and x = 8, or — 2.
Let the pupil check these results.
Solve the following equations:
31. «2 _ 6x + 9 = 36 35. 4x* + 12x + 9 = 16
32. ar* + 4x + 4 = 25 36. 16a:2 _ g^. + j ^ 4
33. a? + 8x + 16l 37. 2/2 _ 2y + 1 = 9
34. ar« + 8a: =  15 38. r^ + lOr + 25 = 16
39. Make up and work an example similar to Ex. 3. To
Ex.26. Ex.34.
40. How many examples in Exercise 45 (p. 155) can you
now work at sight?
177. Square Soot of Arithmetical Numbers. The same
general method as that used in Art. 175 may be used to
extract the square root of arithmetical numbers.
The details of the method of extracting the square root of num
bers are explained in arithmetic (see Diu'ell's Advanced Arithmetic),
As illustrations of the process, we give the following examples:
Ex. 1. Extract the square root of 1849.
1849 140+3 , 1849 Ii3 Root
40* = 1600 16
2x40 =80
3
249 or more briefly, 83
249
249
249
83
Let the pupil check the work.
Ex. 2. Extract the square root of 18.550249.
18.55^49 1 4.307 Root
16
83
255
249
8607
60249
60249
286 SCHOOL ALGEBRA
Ex. 3. Extract the square root of f .
f = .66666666+
Extracting the square root of .66666666+, we obtain .8164+
as the desired result. •
178. For extraction of Cube and Higher Boots, see Ap
pendix, p. 472.
EXERCISE 87
Find the square root and check where possible:
1. 7225 5. 337561 9. 199.204996
2. 2601 6. 567009 ^lo. 10.30731025
3. 105625 7. 36144144 il. .0291419041
4. 182329 a 8114.4064 12. 1513689.763041
Find to four decinaal places the square root of
13. 7 16. 3f 19. 6f 22. .049
14. 11 17. 2i 20. If 23. 1.0064
15. 12.5 18. .9 21. ^V 24. 36ri
r
Compute to three decimal places the value of
25. V2 + V3 29. / ^ V^
26. V^V3 ^^ ^ 5( V^  V2)
27. V3 V3 + Vl _ ^
_ 31 / 7V3 + 2^/5
28. V3V6  2V7 ' 4
32. Find the diagonal of a square whose side is 5 in.
33. Find the side of a square whose diagonal is 5 in.
34. If a city park is § mi; long and 300 yd. wide, how much
EVOLUTION 287
is saved by walking from a comer to the opposite comer
along the diagonal instead of along the sides?
35. In JSl = ^/8{sa) (sb) (sc), if a = 25, 6 = 63,
c = 74, and s = J(a + b + c), find K.
Do you know what figure the above formula gives the area of?
36. Show the meaning of V^i09 = .3 on squared paper.
Also of V.009 as well as you can.
37. If 47riP = 10 and tt = V^, find R.
38. On squared paper construct the triangle whose ver
tices are the points (3, 0), (1, 5), and (—2, —3), and find
the length of its sides.
39. The area of Texas is 265,780 sq. mi. Knd the side of
a square having an equivalent are& Think of some distance
familiar to you which is approximately equal to a side of this
square, then picture to yourself the whole square and thus
visualize the area of Texas.
40. By the method of Ex. 39 visualize the area of the state
in which you live.
41. In like manner visuaUze the area of Germany, which
is 208,830 sq. mi. '
42. The population of New York City in the year 1910
was 4,766,883. What is the side of a square which would be
covered by this number of people if each person occupied a
space 30 X 15 ? Hence visualize the population of New
York City as it was at this date.
43. By the method of Ex. 42 visualize the population of
a large city in your neighborhood.
44. In like manner visualize 30,000 sheep herded
together.
288 SCHOOL ALGEBRA
45. In a recent year the record yield of com for an acre
was 255 bu. If a bushel is taken as Ij cu. ft., find the side
of a square bin 3 ft. deep which would just hold this com.
46. In a recent year the record yield of milk for one cow
was 27,432 lb. If 7^ lb. make a gallon and a gallon is 231
cu. in., find the side of a square tank 2 ft. deep which would
just hold this milk.
47. A baseblall dropped from the top of the Washington
Monument has been caught by a player standing on the
ground at the foot of the monument. The velocity attained
by a falling body is given by the foraiula v = V2g8, where v
denotes the velocity attained, s the distance through which
the body falls, and g = 32.16 ft. The height of the monu
ment is 555 ftfc Pind the velocity of the ball when caught.
48. A stone is dropped from a balloon a mile above the
earth. Find the velocity with which the stone strikes the
earth (resistance of the air being neglected).
At sight give the value of
49. 2 + V9 52. >v/25 X vie 55. 9 ^ V^27
«
50. Vi6V9 53. Voi^Vi . 56. 2v^i253v^
51. 3'v/l62\/9 54. IOa/16 57. 3V92V'4
2 + 4 V^ 3 V64 V9
"• ~7r~ ''• — 3 —
60. Make up and work an example similar to Ex. 21. To
Ex.39. Ex. 53*
CHAPTER XVI
EXPONENTS
179. Positive Integral Exponents. Using o^ as a brief
symbol for a X a X a, and o*" as a brief symbol for a X a
X a X a to m factors, we have already found the
following laws to govern the use of positive integral ex
ponents:
I. a'^Xa* =^ a~+" III. (a'»)'» = a*"
o« IV. v^a*"* = a"»
11. —=a"»", if m>n tt / i v ^
o" V. (at)** = a*6»
180. Practional and Negative Exponents. We have seen
that by usiAg fractions as well as integers, and negative as
well as positive quantity, the field of quantity and operation
in algebra is greatly extended and some processes are made
simpler, others more powerful. These same advantages are
secured by the use of fractional and negative exponents.
Let us suppose that the first and fundamental Index Law,
a** X a" = 0"*+**, holds for fractional and negative exponents,
and then inquire what meaning must be assigned to these
exponents.
We limit the fractional and negative exponents here treated to
those whose terms are either positive or negative integers, and com
mensurable; that is, expressible in terms of the unit of quantity
used in the given problem.
Exponents like V2y as in a ^, are not included in the discussion,
though the student will find later that the same laws hold for these
exponents.
289
290 SCHOOL ALGEBRA '
181. I. Heanin^ of a Fractional Exponent.
Since by Index Law I,
it follows that (t is one of the three equal factors which may
be considered as composing a^; that is, a' is the cube root of a\
So^ in general,
P V V V
aFX(^ Xa^Xcf tog fafctors,
^.^1^1 totftemis
^«
Hence, in general, in a fractional exponent the numerator
denotes the power of the hose that is to be taken, and the derumirwr
tor denotes the root that is to be extracted.
Ex. 1. 8* = '^ = V^64 = 4 Ans.
Ex. 2. o* Xc^ Xc^ ^ a***^ = a** Ans.
Ex. 3. Vae^' • 2*«^» = 2*a:«+» • 2*a?^» = 2V« = 4a:*' Ara.
Ex. 4. 32* = '^^ = 2« = 64 Ans.
Note that in Ex. 4 it is best to extract the required root first.
In the examples which involve letters, the work may often be
checked by nimierical substitutions.
• ■
t
EZEBGISS 88
Express with radical signs: r
1. a*. 3. 2a*. 5. 5a: V* . ^ <^'
2. c*. 4. 2a%^ 6. 2c*d*. . 8. «S^,
EXPONENTS 291
Express with fractional exponents:
9. A?'^. 12. aVi. 15. Vi^. "is. ^^
10. V?. 13. b^y. 16. 2'^Vy.
11. 2V^. 14. 2a;Vj;i. 17. v^^^. i« 3^23^^
2V8IV3*
5
Find the value of
20. 27*. 23. v^. 26. (27)*. 29. (243)*.
21. 25*. 24. vW. 27. (32)*. 30. (^V)*
.22. 16*. 25. (8)*. ' 28. (216)*. 31. (f ^)*.
Simplify the following by performing the indicated opera
tions:
32. a* X a*. 35. 2M X 2M. 38. 'V2V^.
33. 2a X a*. 36. aW'^. ^9  a*'^^ . a* Va.
34. a^xiXahK 37. 7Vo'^. 40. xi\^a^'2x^^.
*
^^ 41. 2* 2* 2*. ^44. a2«iva2«+i.
42. iC*"'"* • X*^*. 45. ic2(a+&) . /j2(a&)^
43. ^p+*« . y2ji8«^ 4g x''"'"* • a:^^ • x^"**.
V
^» o*V^ ^ 2a;*V^ ,. V2^ ^ 2*7*
^8. — ^^ — X 77= — • 49. 3 — ^ — X
V^ ^c V^3v^5 v^V^
50.' Find the value of 5' to three decimal places. Also
of 5* or Vl25. Multiply the two results. Compare the
amount of work in this process with that of finding the
value of 5^. Which process gives the more accurate result?
51. What two parts are there to every power? What is
the difference between an exponent and a power?
292 SCHOOL ALGEBRA
52. Make up and work an example similar to Ex. 34. To
Ex.49. Ex.50.
53. How many of the examples in this Exercise can you
now work at sight?
54. How many examples in Exercise 1 (p. 8) can you
now work at sight?
182. n. Meaning of the Exponent Zero, or of ao.
By direct division, — =1
By subtraction of exponents, — = a® .*. a® = 1
Thus, (fi may be regarded as the result of dividing some power '
of a by itself.
An expression like px^\qx^r is sometimes written px^^qx^n^^
the advantage being that in the latter form every term contains an x.
183. in. Heaning of a ITegative Exponent.
By subtraction of exponents, — — = a
— IX
t+n
By cancellation, —  = — = —
0*+" a* X a"" a"
.•.a« = —
Ex. 1. 4"^ = 7T=77; 47W.
4i Id
Ex.2. 8l=L=T ^^•
8t 4
Negative exponents are useful in enabling us to write
certain decimal fractions in an abbreviated form.
EXPONENTS 293
Ex. 3. Express .0000002 in a briefer form by the use of
negative exponents.
.0000002 = =A= 2 X 107
^^^ 10,000,000 107 ^
184. Transference of Factors in Terms of a Fraction. It
follows from the meaning of a negative exponent that any
factor may be transferred from the numerator to the denominator
of a fraction, or vice versa, provided the sign of the exponent of
the factor is changed,
Ex. 1. Transfer to the numerator the factors of the de
ab^
nominator of
xy~i
1= ab'^xr^yi Ans.
xy~^
Ex. 2. Express with positive exponents — zrrr
xy~^z ^
2a^b _2byh
xy~^z~^ a^x
Ans.
EXERCISE 89
Transfer to the numerator all factors of the denominator:
1 « 2. ^ 3. 3^  1
(? cdr^ 2(? 2^x^y^
Express with positive exponents:
294 SCHOOL ALGEBRA
Find the numerical value of
"•*'L 17.11? 22. fiy
12. 27J 3» V 27/ '
./T" 18. 3''X4i 23. (125)*
46 19. 2»i8~* (8)*
^„. 23. 32. 4*
27.
16. VSP
91. 8*
28. Express .000,000,003 in a briefer form by the use of
negative exponents. How many more figures and symbols are
there in the first form than in the second?
29. Treat .000,000,001 in a briefer form by the use of
negative exponents.
30. Show that 1 millimeter equals 10"® meter.
31. The micron is a small unit of measure equal to one
millionth of a meter. Express it as a part of a meter by use
of a negative exponent.
32. The length of a wave of violet light is .000,016 in. ,
Express this number as 16 in. multiplied by a power of 10.
Give the value of
^ «"' «^' QT' iiJ' ^'^'' (4T' h' r
34. 3X20, 3o», 3(i)«, 4*5«, 4* f 5«, ^.
35. 1*, r*, $1», 1"*, r* X 4"*, 20 X 1  n.
36. 8* X 8"^ X 8*, 37. 16* X 16"* X 8* X 8*.
EXPONENTS 295
38. Express 4° as some power of 4 divided by itself.
39. Express x^ as some power of x divided by itself.
40. Express 4"^ as the quotient of two powers of 4.
Express xr^ as the quotient of two powers of x. Express
ar"^ in like manner.
41. State the value of $4®. Of $42. $4~*.
42. Which is greater, {{)^ or (i)^?
Simplify ^e following by performing the indicated opera
tions, and reducing the results:
44. 4a:2 ^ 2a;». 3Va^
45. 6a*a;"*a*a:*. 53. ^^Y' 
46. a~*2a*. Ta^'^i"^
47. 8a;"*y 4 4a:V So"^^^
48. To'x"* 5 5a;*y. a:**Vy^
50. . ^^* n^ •
51. . 57. —=
58. State which of the following has the greatest value:
Give the value of
59. 4"* + 8* + 8'. 61. 3*5X4« + 8"* + l*.
60. 5" + 42  (I)"*. 62. 4" + «"  5 X 4*.
296 SCHOOL ALGEBRA
63. Make up and work an example similar to Ex. 21. To
Ex.28. Ex.41. Ex.59.
64. How many of the examples in this Exercise can you
work at sight?
65. How many examples in Exercise 51 (p. 172) can you
now work at sight?
185. Meaning of (a"")** for Fractional and Negative Expo
nents. We now extend the law (a"*)** «= a'"'* to fractional
and negative exponents.
Ex. 1. Find the value of (4"*)"*.
(4"*)* = 4^ = 32 Am.
Ex. 2. \{sa^y = (8a«)* = sM = 1 Am.
''^''^' \8W) *81i6* 16*
= ^^Am.
Hence, in general,, to simplify a complex expression in
exponents.
Convert each radical sign into a fractional exponent;
Convert each power of a power into a power ivith a single
exponent;
Convert each negative exponent into a positive exponerU;\
Simplify by cancellations and coUectibns.
EXERCISE 90
Reduce to the simplest form:
1. (a*)». 3. (o»)i 5. (c*)*.
a. («»)!. 4. {2?)\ 6. (o^)"*.
i
f
o.
— \
>
>?
EXPONENTS
I)
J 11. (9^)"*.
7. aj'^^'ai'^
8. (a:»+')"^ y.^ 12. (3o^)«.
9. (a;"*2/*)'. * ' "« ^"l*"*^^
10. (641)*.
19. (9ar«2r'0"*
•13. (6a:"*)^
14. («°+* ir"*)".
297
15. (4x<)"*.
16. (9a2a;V)"*
17. (2o2a;*)«.
18. (Sa;^)^.
20.
O
"(»l)
29.
21. (o'^Jo^)"*,
30.
22. VCai)"*.
23. (oV3a»6»)^.
24. {/(V^)*}"*.
25. (c^ar^) ".
26. f 25Vi^"*.
27a;V^l~*
c
*/
31. Va:iv^^ ^ '^yVic^.
33.
27.
82/* v^
28.
{y*^yVa;y)
34. ^^Ji^i.
35. ^^ —.•
36. [\/rHFi"*.
/^"+'\'' . / a; Y\
\x^) ' \x^~'J
/
~"fx
37
38. 8* + 94  2^ + H  7°.
V^ a* c
061 7 ■
298
SCHOOL ALGEBRA
J a^x^ I aVx
»
2* . y^
ri ' .i
(x^)
n+2
' ^«!!ii) (a:«^)
X'
46. By a numerical substitution show that —  does not
X'
x^
equ^l a?. Also that — does not equal a?*.
47. Write the square root of each of the following: 9a;"*,
9x^, h'^, 9x^, 16a"*, 16a"*.
48. Solve a:"* = 27.
Sua. Raise both sides to the power ( — f);
Then (x"*)"*  (27)"* = "^ = 
•'•^"9
Find the value of a; in each of the following:
55. a;"" = 2.
1
49.
a;* = 2.
52. a;"* = 4.
50.
a;* = 27.
53. x^ = — i
51.
a"* = 3.
54. a:"* = 1.
56. a:"^ = —3.
57. a;"* = — ^.
58. Make up and work an example similar to Ex. 11. To
Ex.28. Ex.55.
59. Practice oral work with small fractions, as in Exercise
58 (p. 190).
EXPONENTS 299
186. Polynomials whose Terms contain Fractional or
Vegative Exponents.
Ex. 1. Multiply x + 3x^  2a;* by 3  2a;"* + ix^.
a;+3a;* 2a;*
3  2x^ + 4x"*
at+Qx* 6x*
 2x*  6a;* + 4
+ 4a;* + 12  8a;"*
3a; + 7a;*  8x* + 16  8a;"^ Product
Ex. 2. Extract the square root of
9a»  ^ + ^ + 20V^ + 4a^
Writing the expression by use of exponents only
9a« 3Qo"* H13a'420o"^ +4o^ )3o"*5o^ 20"^ i2oo<
9a»
6a~*5ai
30a"^hl3a«
~30a"* + 25a«
6a"*10oi  2o*
12a«+20a"*f4ai
12a«+20a~*h4ai
EXERCISE 91
1. Arrange a? + bx"^ + x'^ + x in ascending order of 
powers of x.
Arrange x^^ + 1 + a; * in descending order.
Multiply:
2. a*  a* + 1 by a* + 1.
3. 3a;*  2a;*2/* + 3y* by 3a;* + ,2yK
4." 2a;*  3a;* + 4 by 2 + 3a;"*.
5. o^a"*6* + 6byai + a"*6* + 6.
300 SCHOOL ALGEBRA
6. Multiply ^xy +2y^hy 2x^ + x^hf^ + jT*.
7. Multiply 2x*  3yi + x'^y^ by 2a;"*2^ + 3x"*.
First try to multiply the following expressions as they
stand; afterward rearrange the terms and multiply:
8. 2a:*  4  3a:* + a:"* by 3a:*  2a:* + x.
9. 2 + c^x'^ + a^x^ by 2a"*a:*  4a"'*a:* + 2a2a^.
Va: va: va:y^ V^
Divide:
11. 5a: + 2a:*  2a:* + 1 by a:* + 1.
12. 8a:2 + — + 32/^  18a:y«  8ar*2/^ by 2a:i + 3»^
xy
+ Axy'^^.
13. a:"*a:"* + 52a:*by l+2v^.
14. Vx — Vy by V^ — Vy .
15. Va + ^ + V6 by '^ + '^ + ^.
16. 27a2  30a2/i + 3y2 by 3a  2a*y"*  y^
17. x~^ + a:"*2^"2 + 2/"^ by a:"^ + a:~*2r^ +.a:'^y"*.
18. ?a:*y*4v^^by^ + 2y*.
y Va:
,^ 9 zVx , 10a: . V^ «, 3 , ^
a va^ Va va va
20. 4^2_8V^5 + 4?= + =by2a^' V^a^^
Va Va^ va
Extract the square root of
21. a:*  4a:V + 4a:y. 22. %xy^ + I2y"^ + ^'\
23. ai  4a"*6* + 106  12a*6* + Qafc^.
EXPONENTS 301
24. a:"* + 8a;*  2a!~* + 16x~*  8a;"* + 1.
25. 9z*  30a;"*y + 13a;y + 20a;"V + ^~Y •
26. 25o*6»  10o*6"*  49 + 10o"*6* + 25a"*6».
a;* ar ar X
_ . 24V«^ , ^ , ^aJx , 4
29. ^J^+28
4« Sx* 9Vx
30. Make up and work an example similar to Ex. 5. To
Ex. 14. Ex. 24.
31. Practice the oral solution of simple equations as in
Exercise 64 (p. 209).
EXERCISE 92
Review
1. Simplify (t)* +4'* 8*. Also ()^ • 4" • 8*.
2. Which is greater, (2*)* or 2' • 2*? How many tunes greater?
3. Which is greater, (4"^)' or 4~* • 4*? How many times greater?
4. Does {jf^y equal sf^? Illustrate numerically.
5. Divide 9o»  2la*^/x  ^ox + 12a~ix* by fa'  4ax^.
Simplify:
^ a.n(a.n.l)n 9. (x«)«+* • (x*)*"* ^ (x«+^)«.
a.n+i.a.n1' j^Q Solve 3x* = 32  6x1
7. (x^)«+* 5 xr^.
11
12. Arrange and extract the square root of
iir^+y^+ 2x~^y^  2z^y  xr^^.
3n _3n n n
13. Kndthevalueof (x2 y 2)^ (a;2_ ^ a).
■^"^"^f(3)'*:4^
302 SCHOOL ALGEBRA
14. Simplify
s/ 2,
047 ^
Factor:
6' „jr* in ^■'"—•.i
ab 17. a; y*.
15. Simplify ^ ^^^. "• ^* f + «•
16. Simplify [(o»)""»J»=i. 20. Divide o«6» by ^Vft.
21. Find the difference between the value of (J)' and that of
( 2)2. Also between ( 2)* and ~2«.
22. Does 2o* equal ;r;? Why? Give a numerical illustration.
2o'
23. Does ^'"tf"^ equal ~^? Why? Dlustrate by giving
o, 6, :c, and y convenient numerical values.
a 1
24. Does o"* + IH equal "T ? Explain as in Ex. 23.
23. Make up an example similar to each of the three preceding
examples.
26. Show that ^<^ °*)*3^<^«)* reduces to ^'(tr°')*
1 5^ n(x»+ 1)
27. Reduce (n— l)a;"(x*»H n)"» 4 (ic''+n) « to "i'
28. Show that *(^+^^*(^"^i"* ^ ^^+^)*^^%^)* .
{z + o)*  (»  o)* 3(a;*  o«)*
29. Express 8* as a power of 2. Also 4»' 8* • 8^, 4» • 4»+» • 8*"*.
30. Simplify ^^(^^>^ ^ Expand (xt4a:»)».
2«+i.4« 32. Expand (Vi2>^)*.
33. In the year 1910 the record time for 1 mile traveled on a
bicycle was 1 min. 7 sec., which was 12 sec. more than twice the
record for 1 mile traveled by an automobile. Find the latter
record.
34. How many pounds of 18^ coffee must be mixed with how
many pounds of 30^ coffee to make a mixture of 100 lb. worth 22ff
^ pound?
J
EXPONENTS 303
35. Who first suggested the use of a fractional exponent and
when? Who first showed that such exp>onents could be used accord
ing to mathematical laws? Who first used zero as an exponent and
when?
EXERCISE 93
Obal
Give the value of each of the fc^owing:
1. 4«, 4*, 4«, 40, (4«)l, (4«)», ,.
2. (I)*, (♦)•, (t)"*, 4* + (I)*
3. af+* . a^* . (a^)*. lo. (a* + 6*) (a*  6»).
4. a*+*a**(a:«^)**. 14*4
^i._, H. (a* 6*)+ (a* 6*).
12. (x** + 1/) (x"  y).
5. x~^x *
6. 2»+i2"i.
7. 2* • 2*, 3* • 3~*.
8. 4*4*, 4* 40, 7a:« + 6ao.
13. (a*+6V.
14. (a*H6*)«.
1111 15 ^r* !
9. (a* + 6*) (a*  6*). * arijr*'
Factor:
16. a* 6*. "• a*^"^.
17. a* 6*. !»• ^"^9.
3^ 3 30
20. Give the value of each of the following: — , ^, 7;^ 3® X 6,
3 X 50, 3« X 50, 30 + 60, 305.
21. Give the value of 16* • 16*. Of (16*)*, 16"* • 16"*, (16"*)"*.
22. Give the value of ^. Of^. Of 7f^^  s^^  x^^.
a;*'* 2n
Give the value of x when
23. X* « 3. 25. x"* = 2. 27. a:» » 3.
24. x*  4. 26. a:"* = 4. 28. x"*  8.
304 SCHOOL ALGEBRA
29. Give the square root of each of the following: 4s , 4x %
30. State the value of $80. Of $8"*. Of $8"*.
31. Express 5^ as the quotient of two powers of 5, and henoe find
its value.
32. Express 4"? as the quotient of two powers of 4, and hence
find its value. Treat 4"* in the same way. Also x~*.
33. Give the value of each of the following : (.3)"*, l"* X P X ^
(.01)*, 1"* ^ 40.
34. Give the reciprocal of 2. Of f , f , 42, 8"*, 1*^.
CHAPTER XVn
RADICALS
187. Indicated Koots. The root of a quantity may be
indicated in two ways:
(1) By the use of a fractional exponent; as a*.
(2) By the use of a radical sign; as Va.
For some purposes, one of these methods is better; for
some, the other method.
Thus, when we have a^ Xar X a~*, where the quantities are alike
except in their exponents, it is usually better to use fractional
exponents to indicate roots; but if we have 5 v3 — 7 V^ + 8 Vl2,
where exponents are alike, but coefficients and bases unlike, it is
usually better to use the radical sign to indicate roots.
In the preceding chapter we considered exponents; we
have now to investigate the properties of radicals.
188. A radical is a root of a quantity indicated by the
use of the radical sign; as "yx, "^27.
The radicand is the quantity under the radical sign.
In treating radicals, we deal only with principal roots (see
Art. 172, p. 279), unless the contrary is stated.
189. Surds. An indicated root which may be exactly ex
tracted is said to be rational; as V 27, since the cube root of
27 is 3.
A snrd is an indicated root which cannot be exactly ex
tracted; as V3, \^.
305
306 SCHOOL ALGEBRA
190. The Coefficient of a radical is the number prefixed to
the radical proper, to show how many times the radical is
taken.
Thus, the coefficient of 5 V3 is 5; of 6av^ is 6a.
191. Entire Surds. If a surd has unity for its coefficient,
it is said to be entire.
192. The Degree of a radical is the niunber of the indicated
root.
Thus, V^ is a radical of the third degree.
193. Similar Badicals are those which have .the same
quantity under the radical sign and the sam^ index. (The
coefficients and signs of the radicals may be unlike. Hence,
similar radicals must be alike in two respects, and may be
imlike in two other respects.)
Thus, 6 Vs, — 4a/3 are similar radicals.
194. Fundamental Principle. Since a radical and a
quantity affected by a fractional exponent differ only in
form, in investigating the properties of radicals we may use
the properties obtained for fractional exponents.
Thus, since
(a6)» = a»6»
Transformations of Radicals
195. Simplification of a ftuantity under the Eadical Sign.
Ex. 1. SimpUfy V^56.
V^66  ^8x7  2^7 Ana. (Art. 194)
RADICAL 307
Ex. 2. Simplify 5V18aW.
Hence, in general,
Separate the quantity under the radical sign into ttoo factors,
one of which is the greatest perfect power of the same degree as
the radical;
Extract the required root of thisfa^^tor, and mvMply the coeffir
dent of the radical by the resuU;
The other factor remains under the, radical sign.
196. A Fraction under the Kadical Sign. To simplify when
the quantity under the radical sign is a fraction,
Multiply both numerator and denominator of the fraction by
such a quantity as will make the denominator a perfect power
of the same degree as the radical;
Proceed as in Art, 195.
Ex. 1. Simplify ^^.
Ex. 2. Simplify f^.
186
^186 ^ 186 ^ 26 ''^ 366*
= y oHp X 10a6 = gT'v/lO«6 Ans.
In studying radicals in examples involving letters, the work may
often be checked by the substitution of numerical values.
197. Meaning of Simplification. By simplification radicals
are reduced to their prime form, so that it is made easier to
determine, for instance, whether a number of given radicals
are similar or not.
308
SCHOOL ALGEBRA
Thus, it is difficult to determine whether T'v/lS and — 5\/72 are
dmilar, but when the given radicals are put in the form 21 V^ and
» 30a/2, it is easy to see that they are similar.
Again, the radicals (a  1) 4/l±i and (a + 1) i/^Zi,
^ a  1 ^ a + 1
although unlike in their present form, may be reduce d not only to
similar radicals, but to the same expression, ^a^ — 1.
The pupil should show this reduction for himself.
Hence, a radical in its simplest form is one whose radicand
is integral and contains no factor which is a power of the
same degree with the radical.
£xpress in the
1. Vi2.
2.. V27.
3. V50.
4. 3V28.
5. V45.
6. V50.
7. ^^48.
8. ^.
Simplify:
26. V.
27. 2V.
28. 3V.
29. V33J.
EXERCISE M
simplest form:
10. Jv^.
11. fvT08.
12. v^48.
13. '^128a^.
14. V250a2fe«.
15. V99a.
16. 2V4a^.
17. aVSaV.
18. V200a^
19. VT47^.
20. 2\/63a:Y"
21. v^81aV.
22. A/a2(a;y)».
23. V49a:3(a + 1)5.
24. 10
n/
12a'c*n
25
25a;«
9o*
^^ tvg
32a*«
3a
'34. 2«J^.
6^ 27a
35. 3vf.
RADICALS 309
36. 5a^. 39. ^n__ ^ I^^Z^
37. i/A. 40. ^d^^. ^ 5{x\y)'
38
44. (a + WIEZI. 45. Wl47a^fc^^
^ (a + 6)5 ^V 320ay
46. Given V3 = 1.73205 +, find in the shortest way the
value of Vl2. Of V27. a/75, a/243.
47. Usmg a/5 = 2.23607 +, make up and work an exam
ple similar to Ex. 46.
48. Make up and work an example similar to Ex. 5. To
Ex.23. Ex.31.
EXERCISE 96
Oral
9. a/*. 13. V7^
10. 3a/§. 1* 2A/4i.
11. 2A/f. ' 15. y ^
" \fi "■ v/f
17. How many examples in Exercise 94 (p. 308) can ^ou work
at sight?
198. I. Making Entire Surds. It is sometimes desirable
to introduce the coefficient of a radical under the radical
sign. This may b^ done by reversing the process of Art. 195.
Ex. 1. Express 3a^ as an entire surd.
3^ = V¥xb = v^135 Am.
Ex.2. 2v^=  v^='^^^"^?w.
Reduce by inspection:
1.
a/8.
5. VT.
2.
a/o^.
6. V4.
3.
^€^7^,
'V^^
4.
Vf.
8. n.
310 SCHOOL ALGfiBllA
EXEBCISE96
Express as entire surds:
1. 2VS. 1. 2v^3. 3m.y2»
«• 3V5. 8. 2mV^. "■ ^ ' ^*
_ / 9. mW4n\ 13. 2a;V — ^•
6. 21^:^. 11. fVio. "• ^^Q^'
15. (X  1) V2i. 17. i^y^lT^^
16. (a + 2,)v/:^^ 18.(1.)V/^. 
19. Make up and work an example similar to Ex. 5.
20. To Ex. 7. Ex. 16.
21. Work again Exercise 76 (p. 249), or similar examples
suggested by the teacher or pupils.
22. How many of the examples in this Exercise can you
work at sight?
199. n. Simplification of Indices. If the exponent of
the quantity under the radical sign and the index of the
radical sign have a conmion factor, this factor may be canceled
and the radical thereby simplified.
Ex. 1. V^ = a* = a* = ^ Ans.
Ex. 2. v^ = v^3 ^ ^5 ^^^
EXERCISE 97
Simplify the indices of the following:
1. V^«. 3. ^^, 5. v ^27a».
2. ^. 4. v<i9. 6. ^lOOaV.
Ans,
RADICALS 311
8. vSlaV. 10. ^9^^. . 12. q4^.
13. v^a24o6+462. 14. 4^8{a  26)».
15. Make up and work an example similar to Ex. 4.
16. To Ex. 7. Ex. 13.
200. m. Bednoing Badicals to the Same Index. Radicals
of different degrees may be reduced to equivalent radicals
of the same degree.
Ex. 1. Reduce a/2 and VE to equivalent radicals having
the same index.
.^5  5* = 5*  ^« = ^S^j
Ex. 2. Arrange in ascending order of magnitude VS, Vs,
and V2.
We obtahx , ^^"125, ^^, Voi
hence, the ascending order of magnitude is, \/2, v3, V5 Ans.
EXERCISE 98
Reduce to equivalent radicals of the same (lowest) degree:
1. V7 and vlT. 7. v^, 'i/g, \^.
2. v^ and v^. 8. v^, ^, ^^.
3. v'S and v^5. 9. \^Sa, ^26, 1?^.
4. VJ and v'f. 10. '^i + y and \^x — y.
5. 't^m and '^25. U. ^!^ and v^x^. ;
6. Ve and ^200. 12. '^?, ^?, <^.
Show which is greater:
13. VS or v^. 15. 2V5 or Sv'll.
14. v^orVe, 16. v^or2V2.
10 or 2<^. W
312 SCHOOL ALGEBRA
—  ~ «
»
17. v^ or 2^. V i 19. JV7 or <^. '
, 20. ^4 or v^.
Show which is the greatest:
21. V3, V^,oril^. 22. 3^51, 2 Ve, or 2 v^.
23. Make up and work an example similar to Ex. 1.
24. To Ex. 7. Ex. 16.
25. How many examples in Exercise 83 (p. 273) can you
now work at sight?
Operations with Radicai^
I. Addition and Subtraction op Radicals
201. The Addition of Similar Radicals is performed like
the addition of similar terms, by taking the algebraic sum qf
the coefficients of the terms.
The addition of dissimilar radicals can only be indicated,
Ex. 1. Add Vl28  2V50 + V72  VTs.
Vl28 2\/56+ V72 Vl8= 8a/2 10^/2+ 6\/2 3>v/2
= V2 Sum^
Ex.2. 2\/ + V60 + Vi5 + V.
= *\/i5 + i\/i5 + Vis + i Vis
= ^Vl5 Sum
Ex.3. Vi28 + 2 Vi  3 VsT.
= 4^2 + v^  9^3
= 5^  9^^ Sum
EXERCISE 99
Collect: ;
1. Vis + Vs. 3. 2V27 + V75.
2. V5OV32. 4. V5 + V20 + V^.
ADDITION AND SUBTRACTION OF RADICALS 313
5. 4v^2v^54. 10. 2Vi + V^.
6. 3^625  4v^i35. IZ fZ
7. 2^189  V448. * o ^ X
8. 'S^ + v^ ^^375. 12. \^Js + \y/m.
14. 2V256 + 3V462V366.
15. 3v^2c + 3^?^54c w^OOOc.
16. Vi2oP + fcViSo  6 A/3oi^.
17. 2a/^6?  3aVl66<r' + BcVOc^.
18. 6v^2o + iS''250a^  26^432^.
19. v^ + Vis  V&O + Vl62.
20. a/75  4 V243 + 2^108.
21. evf  5a/24 + 12A/f .
22. 5V12Vf + 6\/6030VS.
23. 3V5  10\4 + 2 V45  5V^.
24. ■v/27  •v/iS + VSOO  l/i62 + 6\/2  7V3.
25. 2 V63  3^^  V + iV^  f a/?.
26. a/2^ + Vis  V768 + 9VF+ v/75  3V33.
27. 2lVf5V + 6V410V3i + ^Vrii.
28. 5oVi2aP  36 V27^ + 2V300^  40afeVo.
29. 3 Vie  3V12 + 2V54  5V^.
^^0. 3V54  2 vis + 5 VJ + 5VJ.
31. 3Vo*x + 2abx + 6^ + 2V(a  6)^3;.
/ 32. 8Vo(a; + 2yy  ZVaa? + 4axy + 4a f.
314 SCHOOL ALGEBRA
33. How many more symbols are used in Ex. 25 as it is
given than in your answer?
34. Compute the value of V1282V50 + a/72 by ex
tracting the square root for each radical and combining
results. Now obtain the value of the whole expression by
first reducing each radical to its simplest form and then
collecting. Compare the labor in the two processes.
35. Make up and work an example similar to Ex. 34.
36. Name some of the advantages in being able to reduce
radicals to their simplest form.
37. Compute to three decimal places the value of
^20  V + 2\/45  Vi in the shortest way.
38. Make up and work an example similar to Ex. 13.
39. To Ex. 17. Ex. 29.
40. How many of the examples in this Exercise can you
work at sight?
II. Multiplication of Radicals
202. Mnltiplioation of Honomials.
Since a\^b X cA^d = ac^/hVd
= ttcX^bd
we have the general rule,
Reduce the radicals if necessary to the same index;
Multiply the coefficienis together for a new coefficient;
Mtdtiply the quantities under the radical sign together for a
new quantity under the radical sign;
Simplify the resuU.
Ex. 1. Multiply sVe by 2^3.
5 V6 X 2 V3  lOylS  30v^ Prodvd
MULTIPLICATION OP RADICALS 3l6
Ex.2. 5V2X2v^ = 5V^X2^
= 10iJ^72 Produa
203. Multiplication of Polynomials.
Ex. Multiply 3 V2 + 5 a/3 by 3 V2  V3.
3\/2f5V3
3V^ V3
18 + 15V6
 3^/6  15
3 f 12\/6 Procitid
100
Multiply:
1. 3V5by Vi5. 7. V2byV^.
2. 2v^ by 3^^. 8. \^2^ by Vte.
3. 2VT5 by 3V36. 9. '^ by a/6.
4. a/28 by a/35. 10. v^6 by 'V^.
5. i Vf by A/i. 11. Vf, VI, and V2J.
.. m by }^i.  v^ "y ^
>) 13. V3  v/e + 2VIO by 2 V2.
14. 4^/6  3 V3 + 3\/2 by 2 Ve.
15. ioa/  5V + UvU by V^.
^6. 3 + \/2 by 2  2V2.
17. 2V3  3 V2 by 4V3 + 5V^.
18. V? + V5 by V?  V5.
19. Vo + V« by V^o  V±.
316 SCHOOL ALGEBRA
Multiply:
20. 4V2  3Vf by 3 V2 + 4^^.
21. V2 + V3 V5by V2 V3 + V5.
i22. 2 V3  3 V6  4a/15 by 2 V3 + 3\/6 + 4Vl5.
/23. 3V30 + 2V'53V6by2\/5 + 3V6.
24. JV8 + V32  V48 by 3 V8  iV32 + 2 Vi2.
25. 12 vf  4a4 + 1^216 by 6 V  2 vf + 3 V6.
26. V2x + V«l by V3«.
27. VSx + Vx + l by Vx + 1.
28. Vajl  3Va; + l by 2^* + 1.
29. P . Vp'  4g . p Vp^  4g
2 2 "2 2
30. a — Va — ir+ Va by Vo — a;+ Va.
31. 3\/2x  5 Vx  1 by ZV2x + 5 Va:  1.
32. Va + x — Va — X by Va + ic + Va — x.
33. (2V2 + V3)(3V2 V3)(3V3 V2).
34. (2 Vx + 2 + 3 V^ (6«  5 V2a: + 4) (3a/x+2 2 V^).
35. In the year 1910.the greatest mountain height climbed
by man was 24,853 ft., which waa 3441 ft. less than twice
the height of Pike's Peak. Find the height of Pike's Peak.
36. If 5 mi. exceed 8 kilometers by 153 ft. 4 in., how many
yards in a kilometer?
37. Make up and work an example similar to Ex. 4.
38. To Ex. 9.
39. How many examples in Exercise 85 (p. 280) can you
now work at sight?
DIVISION OF RADICALS 317
III. Division of Radicals
iiM. Division of Radicals. Reversing the process for
multiplication, we have the rule,
// necessary, reduce the radicals to the same index;
Find the quotient of the coefficients for a new coefficient, and
the quotient of the quantities under the radical signs for a new
gmntity under the radical;
Simplify the resvU.
Ex. 1. Divide 6a/8 by 3^6.
J:^ = 2A/i = 2VV = V3 Qmtieni
3V6
Ex. 2. Divide 6^ by 2\/2.
EXESCISE 101
Divide:
1. V27 by Vs. ■ 8. 4Vf by aVp.
2. 4^12 by 2 V6. 9. 5Vfby2V'^.
3. 8Vi25 by IOa/IO. lo. 2v^by3V2.
4. ZVm by 9V45. il. V54 by <^.
5. a»Vo^ by 2a V^. 12. ^^by V6. \
6. 4A/i8 by 5 V32. ^13. V^ by ^^^^
'• ^by V. / 14. V6 by Vsj.
15. 5 V35  Tv^ by Vs.
16. 12\/7  60V5 by 4^3.
17. 6 Vi05 + 18 ViO  45 Vl2 by SVTS,
318 SCHOOL ALGEBRA
18. Divide 12^f^+ 30^20 + 42^^ by 2^J^.
19. Divide 10i5l8  4v^60 + 5^5^00 by 3v^. '
20. Divide Va:^ — y^ by Va: — y.
21. Make up and work an example similar to Ex. 3.
22. To Ex. 12. Ex. 19.
23. Work again such examples on pp. 239240 as the teacher
may indicate.
205. Eattonalizing a Monomial Denominator. If the de
nominator of a fraction is a surd^ in order to make the
denominator rational^
Multiply both numerator and deru/minator by such a number
as will make the denominator ratUmal,
«
One object in rationalizing the denominator of a fraction
is to diminish the labor of finding the approximate value of
the fraction.
5
Thus, if we find the approximate numerical value of ^
directly, we must find the square root of 2, and divide 5 by
thfe decimal which we obtain. On the other hajid, if we find
the value of the equivalent expression, f V2, we extract the
square root of 2, multiply by 5, and divide by 2. In the latter
process we therefore avoid the* tedious long division, and di
minish the labor of the process by nearly one half.
206. Eationalizing a Binomial or Trinomial Denominator.
If the denominator of a fraction is a binomial containing
radicals of the second degree only, since
(Va + V6) (Va  \/6) = a  5
DIVISION OF RADICAI£I
Muliiply both numercUor and denominator by the denomina
tor, with one of Us signs changed;
For a trinomial denominator repeat the process.
jj^ J 2V5 + 4V3 _ 2V5 + 4V3" 3V5 + Vz
zVEVs 3V5V3 zVl+Vz
_ 42 + 14V15 _ 42 + I4V15 ^ 3 + Vl5 ^^
453 "42 3 '
l + Vf  V2
Ex.2.
X
1 + A/5 + V2 1 + V3 + a/2 1 + a/S t a/2
^ '^'+f/~'^ x\^^2+V2V6Ans.
I + a/3 Ia/3
EXERCISE lOa
Reduce to equivalent fractions having rational denomi
nators:
1.
2.
1
V2'
a/2
2a/3
3. — Ti 6.
3a/5
10. 1+^.
2 a/3
^ 3a/3  2a/2
2\^ + 3a/5*
jj 3a/6  2a/3
' 4a/63a/5
4. 1^
. 2+A/5
\
2a/7
2a/6
7 ^1
. ' 3v^ '
„ 2^3^4
sv'e
9. 3V2
3 + A/2' .
2a/15 + 3^/10
4V3 + 3a/2
3a/o  4a/6
2a/o3v^
X9. •
a/«T1 + 2
13.
320 SCHOOL ALGEBRA.
^g 2V2o  1 + 3Va ^ 2V6  3v^  V^
3^20  1 + 2V0 ' 2V'6 + 3V2 + V42
„2+'v/6V2i/ ^ Va?1  Va? + l
17. , rr r= 80.
2  V6 + a/2 Va^l + Va^+l
_ V5V6 + I „ Va+VbVa + b
V6+V5 + I >^^ Va Vb + Va + b
Find the approximate numerical value of
22 3 .V 25. L.. 28. ^A
V2 V300 ,^ V2 + V3
oa 2V5 .. 3V7 ^^ 5V7I
23. . 26. =• 29. — =
3V2 5V5 V7 + 2
«. 12 ^^ I + V2 ^^ 3\/34
24. =. 27. — ! — . 30. —
V7 2 a/3 4V35
31. Make up and work an example similar to Ex. 2.
32. To Ex. 14. Ex. 23. Ex. 29.
33. How many examples in Exercise 45 (p. 155) can you
now work at sight?
IV. Involution and Evolution op Radicals
207. The process of raising a radical to a power, or of
extracting a required root of a radical, is usually performed
most readily by the use of fractional exponents.
Ex. 1. Find the square of sVx.
(3V^)« = (3x*)2 = 9a;* = 9 V^x* Ans.
Ex. 2. Extract the square root of 4a Va^6^.
(AaVcfib^)^ = (4o • o*6*)* = 4*a*aM
2a*6* = 2v'a666 = 2a6v^ Ans.
INVOLUTION AND EVOLUTION OF RADICALS 321
Ex. 3. Extract the cube root of Va^b\
( Vo^ * = (a»6*)*  06* = aVb* = abVb An8.
This process might have been performed by extracting the
cube root of a%^ as it stands under t^e radical sign; thus,
^ "' y/Vo^^VcW'abVb Ana.
EZEBCI8E lOS
Perform the operations indicated:
1. (Vm)*. 5. (\^)«. 9. (^J^«.
13. Show how the computation of (VS)" may be short
ened by the use of (V'2)^ = 2, and also of the fact that
2» = 32.
14. Compute the value of each of the following in the
shortest way: (v^)«, (V3)», (V3)», (V2)», (V2)",
(V3)", (a/3)', (^3)".
Expand:
15. (V3+V2)*. 16. (VsV^)". 17. (2V3iA/2)«.
18. Make up and work an example similar to Ex. 14.
19. To Ex. 15.
20. Practice oiial work with small fractions as in Exercise
58 (p, 190).
322 SCHOOL ALGEBRA
v. Square Root of a Binomial Surd
208. A quadratic surd is a surd of the second degree; as
VZ and Vofe.
A binomial surd is t binomial expression, at least one
term of which contains a surd; as V2 + 5 VS, or a + Vt.
200. A. The prodvct'of two dissimilar quadratic surds is a
quadratic surd.
Thus, V2 X V6 = Vl2 = 2V3
or Va6 X Vabc = abVc
Proof. If the surds are dissimilar, one of them must have
under the radical sign a factor which the other does not
contain. This factor must remain under the radical sign in
the product.
210. B. The surn or the differeru^e of two dissirnikir quudraHc
surds cannot equal a rational quantity.
We use z ^yas A short way of writing x +y and z— y.
Proof. If Va ^ Vb can equal a rational quantity, <*.,
squaring, a =*= 2Vab + 6 = c*
=t 2Va6 = c^ab
But Vab is a surd by Art. 209 ; hence we have a surd equal
to a rational quantity,. which is impossible.
211. C If a+ ^/b = 05 + V^ ^^ a = 05, 6 = y.
Proof. If a+ Vb.= x+ Vy
transposing, Vb — Vy — x — a
If 6 does not equal y, we have the difference of two surds
equal to a rational quantity, which is impossible; hence,
6 = y a = ar
SQUARE ROOT OF A BINOMIAL SURD 323
In like manner, show that if
a — Vh = a: — Vy then a = ar 6 =y
212. D. If V^ + Vif = ^a+ VS; thm »+ y = a.
Squaring the given equals, x+y\ 2'\/xy = a+ ^/h
Hence, x+y^a (Art. 21 1)
In like manner, if ^/x— Vy = Va— Vfr
it may be shown that a;+ y^ a
Also, since 2 \/^ = V6 (Art. 211)
aj+ y — 2^^^ = a  Vft
and V«— Vy=VaV&
213. Extraction of the Square Boot of a Binomial Surd.
Ex. Extract the square root of 5 + 2^6.
Let Vx+Vy= V5+2V6.'.a;+2/=5 (Art. 212)
Then, yS Vy= V 52V6 (Art. 212)
Multiplying, a;y= V25 24 .•.xy= 1
J^=2
.'. V^+ V^= V3+ V2
.. ^5+2^6= \/3+ \/2 ilrw.
214. Finding the Square Boot of a Binomial Surd by In
spection.
By actual multiplication we may find,
(V2 + V5)2 = 2 + 2VIO + 5 = 7 + 2VlO
In the square, 7 + 2 VlO, 7 is the sum of 2 and 5, 10 is the
product of 2 and 5. Hence, in extracting the square root of
a binomial surd,
Transfofm the surd term so that its coefficient shall be 2;
Find two numbers such that their sum shall equal the rational
term, md their product equal the quantity under the radicai;
324
SCHOOL ALGEBRA
Extract the square root of ecich of these numbers, and oomied
the results by the proper sign.
Ex. Find the square root of 18 + SVE.
18+8^/5=. 18+ 2 VSO
The two numbers whose siun is 18 and product is 80 are 8 and 10.
. • . Vl8+8>v/5= ^8+ \/lO
 2^/2+ VlO Root
8. 7724A/iO,
9. 8736V5.
10. 14 + 3 Vs.
13 4fV ^.
14. 2m + 2Vm*  n*.
EXERCISE 104
Find the square root of
1. I7I2V2.
2. 23 + 4V15.
3. 35I2V6.
4. 96A/2.
5. 42 + 28V2.
6. 7312\/35.
7. 26 + 4V30.
s\ 15. 10a« + 9 + 6a Vfl?+T.
Find the fourth root of
/16. 28  16\/3.  c r. .18. 193  I32V2.
^^7. 97  56^3. 'I: j^:^ 19. ^^ + ^V6.
Find by inspection the square root of
20. 3 + 2V2. Jf'^j 23. 23 6\^.
21. 92Vi4. 24. I8I2V2.
.88. 21 + 12V5. 25. 7 + 4A/3.
9€, Prove that Va * Vft cannot equal Vc.
i7. Prove that Va cannot equal b + Vc,
as. Make up and work an example ^iipllMr to Bbc. 2#
EQUATIONS CONTAINING RADICALS 325
29. To Ex. 11.
30. Practice the oral solution of simple equations as in
Exercise 64 (p. 209).
VI. SoLtmoN OF Equations Containinq Radicals
215. Simple Equations containing Radicals.
Ex. 1. Solve Vx^ + 7  1 = x.
Transpose terms so that the radical shall be alone on one side of
the equation.
Squaring, x'^+7 = x^ +2x +1
.'. 2x=6
z=3 Root
Check. \/^a^+7 = V9 +7  4
x + 1 •= 3 + 1 = 4
Observe that only the principal value of a radical is used in
checking a result, ms in the other processes m this chapter.
Ex. 2. Solve Va: + 3 + V^ = 5.
Transpose terms so that one radical shall be alone on one side
of the equation.
\^x +3=5 — "s/x __
Squaring, a; + 3 = 25 — lOVaJ + »
. • . 10\/i_ = 22
5Vx = 11
Squaring, 25a; = 121
25 =• W Root
Let the pupil check the work.
In general,
Transpose the terms of the given equation so that a single
radical shall form one member of the equation;
Raise both members of the equation to the power indicated by
the index of this radical;
Repeat the process if necessary.
326 SCHOOL ALGEBRA
216. Fraotioaal Equations containing Badioals.
Ex. 1. V^  Va:8 = — =L=
Vx8
Multiply by Vx 8, Vx'8g a;+8 = 2
.. ^x* Sx^x 6
a;« 8x =a;« 12x+36
4a =36
X =9 Root
Ex.2. ^ + 3 _ 3\g5
Vi  2 3 Va  13
Clearing of fractions, 3x  4\/x  39 = 3a; — ll\/a? + 10
Va:«7
a; » 49 /^oo<
EXERCISE 106
Solve the following equations: /
1. 3  Va; + 1 = 0. 7. 2\/3a;5^vi+T=0.
2. 5V^ = 3. a 3Vxl = V« + l.
3. 1  V3a;  5 = 0. 9. Va;+168+V«=0.
4. l = v^l. 10. Va:1515+Vi=0.
5. V^2a:1 + 1=4. 11.  + V« = V^T^.
6. iclVar^ + 3 = 0. 12. f V^ V2a:+=0.
13. Va+ Vx + 88 = 0.
14. V4a; + 3 = 2Vx ~ 1 + 1.
15. 2 Vx  V4a;  22 = V2.
16. V9a; + 35 = 7V5  3V«.
17. y 13 + v/7 + v/3 + Vi = 4.
18. VVi + 3  V^Vi  3  V^2^ « 0.
EQUATIONS CONTAINING RADICALS 327
^^19. V25a:29V4a;ll3Vi=0.
20. Vx + V4a + x 2Vb + a; == 0.
..) a. V3 + a:+ V5 =
V3 +
a;
1/22. V9 + 2a; = , + V^.
Vd + 2x
J 23. 3V2a;+l3V2a;3 =
V2a:3
Va; + 3 V^  2
^^ '6V«7 , 7Vx26
25. = — 5 = —7= •
Va:  1 7Vx  21
Va: + a — Va:
Voa; + 6 + Voi ^ .1
27. r n v^ = 1 + ^•
Voa: + 6  Voa: &
a: — a V« — Va . ^ /
+ 2Va.
; ' Vx+Va
y «« Vg^T23Vac .
X 29. , ;= = 4.
V9a: + 2 + 3Va:
/' ®^ l^2^ + ^li + V^ + VS=V2,
31. Vi + Va  ^Qx •\Q^^ Va.
a + 3 a3 2a
32.
Vi + 2 Vic2 «4
33. /«7T/1 = 1.
a: a:
X h 25 aV ar
328 SCHOOL ALGEBRA
'^^35. Va? + 4a: + 12 + Va?  12a;  20 » 8.
36. V9a? + X + 5  V9qi? + 7x + 6 1 = 0.
37. The square root of a certain number plus the square
root of the sum of the number and 7 equals 7. Find the
number.
38. State Ex. 1. above as a problem concerning one un
known number. Similarly state Ex. 4. Ex. 19.
39. Practice oral work with exponents as in Exercise 93
(p. 303).
217. Extraneous Bootft in Badical Equations. It may
readily be shown that squaring each of the two members of an
eqiuition does not necessarily produce an equivalerU equaHon.
Ex. 5 + Vx = 2
Hence, Vx = — 3
a; 9
Checking, 5 + V^ » 2, or 8 » 2. But this is impossible; hence,
9 is not a root of the given equation.
Note that if the sign of the radical vx is changed in the origiDal
equation, by solving the equation thus formed the result x » 9 is
obtained; this answer can be proved.
EXERCISE 106
Solve the following equations and check each result. In
each case where the root is impossible, change the origmal
equation so as to make the result obtained a root.
1. 1  V^ = 3. 3. 3 + V2x = 5.
2. 4  Vx\\ = 5. 4. \/a: + 7 = 2  VxS.
5. V7ar+18= V7a;+1 + 1.
6. Vix + 9 + V^ = Va:45.
7. Vx + 9  Vx + 2  V4a;27  0^
RADICALS 329
8. Vi  Vx + 8 = 8.
9. Va; + 8  8 + Vi = 0.
10. Practice the oral simplification of radicals as in Exeiv
cise 95 (p. 309).
EXERCISE 107
Review of Radicals
Collect:
^1. §\/i5  zoVi + JV^20  6V^ + Vsoo.
2. 4^147  3a/75  Wi + 18 V^  24 Vf.
v:>3 2a./456^ 1Gb fc^ 14x /5^ 5x HT
' 36 V 16a« a» V 562 "^ a^ V 49^. 2aV 201?'
Multiply:
4. 2 + V3V5by2fV3 + V5.
5. 'v/o 4 Vfl + X — V^ by Va — \/o Hx — Vx.
6. iVlbySv^. Also2>^f byi^;^. ,.
Divide:
^7. 6 \/l2 4 3 VS  6 V30 + 4 VI5 by 2 V6.
8. x' — a; + 1 by a; + Vx — 1.
Rationalize the denominator of
0^ 2\/33V2 j^Q 2Vl5+8 , 8^/3  6\/5
3_2V6 ' 5 + V15 5^/3 3\/5'
Find the nmnerical value of each of the following in the shortest
way:
11. L. '12. 2vlz:3V2.
VS 3V5~4\/2
Which is the greater:
'^13. 3\/3or2\/7? 15. 2^6? or 3^?
14. VS or v^? 16. ^ or v^?
Find the square root of
17. 33+20A/2. 18. 107 + 12V77.
330 SCHOOL ALGEBRA
Simplify:
VIX +
^ Solve:
23. 2 + Vx + 3  Vx 2 + 3.
yx — 4
25 Vg+4 ^ Vxf 2
2Vi  1 2Vx  3
Find the value of each of the following in the shortest way:
27. ( Vo + x) ( Vo + x) ( Vo  x) ( Vo  a?).
28. (2V^  V^) (2\/x  Vv) (^Vx + Vi) (2\/5 + Vy).
29. Simplify (^ + Vr:=^«) (x  a/43F«)
(x + V4  x«)« + (a? \/4"^^»)«
Va*  x^ + 
30. Show that : — , ^' "" ^ reduces to ^'
(a*  x«)*
31. Show that — ^^^ reduces to
—x — {a—x)
^  ^ (2axa?)*
x«
. Show that 7==^ reduces to
RADICALS 331
2 4 a/3
33. Find the value of each of the following: 6 ■ ^ i
5
1^2 >5 J_ J_ JL 5 5/ 7~V3 \ 1
34. Find the value of:; when a « 2 and r « V2 + 1. Also
when a = f and r = ^/2 — 2. *
35. Simplify 2^/^ + V^O  6vT + ^/^  2^  2^^.
27>v/3J (V!_J_
36. Simplify — ^. Also _V3__V3^
V3  1 V3  1
37. find the value of the following to three decimal places by
extracting only one square root:
V28 4 3VT  V? + 2^/63.
38. Make up and work an example similar to Ex. 37.
39. Does V9 X 16 equal 3x4?' Does a/9 + 16 equal 3+4?
40. Does Vo* + ft* equal V^ + V^? Can you prove your
statement by an algebraic method? Does \/a*6* equal \/o' V6*?
41. By use of radicals factor x^ — 2. x' — 5. a? — 2. a; — 5.
aJ* h 2a;  2.
42. Simplify ^7^^ . Also ^' " ^
x2  2 a;  V6
43. Solve  +s^^ 5, 1 _ A_ = 6.
a; 2a;— y x 2x —y
44. In transposing a term in an equation, why do we change the
sign of the term?
45. Solve . 3Vf + 5Vy = 11.
6Va;  3A/y = 9.
*«• simplify ^^^r^^^
47. Having given V^ = 1.732 +, find the value of \/!03 to three
decimal places in the shortest way. Also v27. ^^27.
48. Factor o^ + a' by the use of radicals (see Art. 101).
332 SCHOOL ALGEBRA
49. In like manner factor x' + 1. x* 4 a*.
50. How is the product of two monomial radicals obtained, if
the radicals are of different orders? Give an example.
51. Show which is greater, 2^3 or 3^2.
52. Show that 3^3^/2 equals S^V^.
53. Simplify 3^^^ ^ 2v^  W^ + ^y ^^
54. Also 4Vl8 + 2 Vi  vC5 + \/i2i.
55. If a:  ^ ^^^ , find the value of x« + 3a: + 2.
56. Find the value of Va6 • "^d^ • 'i^.
57. Multiply ^ + 2'^2 by VJ. '
58. Collect as many instances as you can of men who have ex
celled in the study of mathematics and who have become distin
guished as statesmen.
59. Distinguished as generals in war. As scientists. As writers.
EXERCISE 108
Oral
Reduce to'simplest form:
^3' Vs' ^°' ^ 3' V3' Va ^ a ^ 6' ^ 16
2 1 ^/lV2_ 1
VS" 1 ' V3 + V2 ' 2V2 + 3 '
3. \ + ^,2\/3 + '^y V^ + V^25,2  'i/l.
4. VI8+V6, ('v/3+V^)(V3— v/2), V4 + V/5 X V4'v/5.
5. (v^)«, (v^)», (V2)', ( V2)", (\/3)".
6. Which is greater, ^/2 or ^^7 'ij^ or \/3?
7. Solve V^TT = 3.
8. Give the product of V^^V + a/» — y and \/x+y — Vjt^.
Of 3\/8 and 5 V^. Of 3\/2 and 4v^.
RADICALS 333
9. Expand (\/3 \/2)*. Also (Vx + \/8 4 «)*.
10. Simplify: 2A/i 3vT, 3VJ, sVi, Vi, 2Vf.
iiT?^ 1.2,1 1,2,1 x2,o,y'^ 1
U. Factor —H +5, I + +1, iH2 + ^» o^oT
X* xy y^ X X* y* or 27
12. What is meant by the root of an equation? The solution of
an equation?
13. In adding fractions we retain the conunon denominator, but
in clearing an equation of fractions we drop the conmion denomi
nator. Why this difference?
1* In what respect is ^^~ a simpler expression than ^^?
At sight give the value of x in
15. 3+\/i=4. 18. a+Vx^c. 21. V^o.
16. v^+T=4. ^^ \l^^' ^ V^c.
17. V^HhS 6, 20. y/? ^6. 23. y/i  5.
CHAPTER XVm
IMAGINARY QUANTITIES
«
218. An Imagiaary Quantity is an indicated even root of
a negative quantity; as V— 4, V— 3, and V— a.
The term " imaginary " is used because, so long as we con
fine ourselves to plus quantity and to its direct opposite,
minus quantity, there is no number which multiplied by
itself will give a negative number, as —4, for instance. All
the quantity considered hitherto, that is, all positive or nega
tive quantity, whether it is rational or irrational, is called
real quantity.
If we extend the realm of quantity outside of positive and
negative quantity, imaginary numbers are as real as any
others, as will be shown in the next article.
A complex number is a number part real and part imagi
^ nary; a s 3 + 2\/— 1 and
a + bV^.
,^/Zl 219. Meaning of V^.
If OA  + 1, and OA' is
• A of the same length, but lying
' ' in the opposite direction from
+ 1 0, Oil' =  1.
Hence, we regard the opera
tion of converting a plus quan
tity into negative quantity as
W equivalent to a rotation through
an angle of 180°. If we divide
this rotation into two eiqual rotations, ea^h of these will be a roti^
tion through 90**.
334
1
IMAGINARY QUANTITIES 335
Hence, V—l must be equivalent (geometrically) to the result of
rotating the plus unit of quantity through 90**. Hence, \/^ on our
figure will be represented by OB.
Hence, it is easy to see, also, that \/^^ x V— 1 = —1.
We thus perceive that the introduction of imaginajy quantity
enlarges the field of quantity considered in algebra from mere quan
tity in a line to quantity in a plane. This gives a vast extension to
power of algebraic processes and introduces many economies in
them, as will be found by the student who pursues the study of
mathematics extensively.
In taking up the subject for the first time, we consider only a few
of the first properties of imaginaries, so called.
220. The Fundamental Principle in treating imaginaries is
that V^ X V^ = 1.
Using t as a symbol for V^, this principle is, i X t = — 1',
ori* = — 1.
Considering this matter algebraically, if we use the law
of signs in the most general form,
(V^)2 = V^ X ^/^ = ^/I = =b 1
Now, if we extract the square root of + 1, we shall not have
V— 1. But if we extract the square root of — 1, we shall have
Hence, we must limit the product V—l X V^ to —1.
Likewise, V^ X V^ = VaV^ x Vb^/^^
= VaA/6(V^)2 =  V^
Or, using the symbol i, ai Xbi = — a6.
321. Powers of y^^.
(v'^=T)«= 1 ..i^^ 1
(V^HT)* = (a/^)*V^*=  V^ .. i» =  1 (SeeOA'ofite
. , / , / lire in Art.
(^31)4 . (V 1)V 1  + 1 .. t«  1 219.)
336 SCHOOL ALGEBRA
Thus, the first four powers of V— 1 are V— 1, — 1,— V^ 1,
+ 1; and for the higher powers, as the fifth, sixth, etc., these four
results recur regularly. The same fact is clear from the figure in
Art. 219.
222. Operations with Xmaginaries. It follows from Art
220 that, in performing operations with imaginaries^ we ti^e
all the ordinary laws of algebra, with the exception of a Umi
tation in the use of signs^ which may be mechanically stated
as follows:
The product of two minus signs under the radical sign of the
second degree gives a minus sign outside the radical sign. But
in diwidmg first indicate the dimsian and afterwardfraJtunUdixe
the demminator.
Ex. 1. Add V^, 3 + 2V^, 72v^^^.
3+2 V 1 =  3 + 2/^
7  2>/^^l6 = 7  8\/^
4  3^^ Sum
Ex.2. Multiply 2 V^ + 3a/^ by 3 V^  5\^.
2^/334.3^176
3V^5>/^
6(3) 9\/l8
+ 10 Vg + 15 VT2
 18  27\/2 f lOVe + 30V3 Produa
Ex. 3. Divide 2\/6 by V^.
2V6 2^/6 V32
2V 12 >
Ex. 4. Extract the square root of 1 + 4V— 3.
1 + ^V^^ = 1 + 2 V  12
IMAGINARY QUANTITIES 337
The two numbers which multiplied together give — 12, and added
together give 1, are 4 and — 3.
. • . V^l + 4 V"^ = Vi + V^ = 2 + V^ Root
Let the pupil work the above examples using i instead of —4.
EXERCISE 109
Collect:
1. 7a/^+3V^^^^10a/^.
2. V 12  V 27 + 2V 3 + V^^^.
3. dV^  3 V^ + 4V50  V200.
4. 2V^^  SaV^ +  V 16a*  f V 36a».
a
5. a + fr'V^^  6  a'/^  a/^^ + 2V^a.
6. (a26)\/^(2a + 6)V^.
7. Express in tenns of i the results obtained in Exs. 16.
Multiply :
8. V^ by v^. 12. 2^/^^ by 2a/^^.
9. V^ by 2V^. 13. 6V^ by 2V^.
10. V^byV7. 14. Vxyby Vyx.
n. A/^^byV18. 15. aVT^by~V(al)«.
16. ^/"^ + V^ by '/^  2V^^.
17. 3\^^  2a^^ by 2\/^^ + 3\^^.
18. 2'v/22A/^^by3V2 + 3\/"^.
19. 3\/"^  V^^ by 2\/3  V^^.
20. v^V^+V^by V^+ V^+ V^.
21. V^+ V^by a/S + 'Z^.
22. a;  2 + V^ by X  2  a/^.
23. aV—a + bV b by aV^^  feV^.
ai. a:l\/in^bya;l + V^.
338 SCHOOL ALGEBRA
25. Multiply X by a: —^ •
26. In the shortest way find the value of
* {V^+^/^) (V3 + V2) (V^V^) (V5V2).
Divide:
27. VlSbyV^. 29. 6V 15 by 2V^.
28. V 12 by V"^. 30. 8a/ a» by ^2\/a.
31. 2\/^^^ 4V 15 + lOVsO by 2\/^.
32. aV^^2aV 6a a^Vs^ by  aV^.
Express^ with rational denominators:
33. L_. 3^_3V35a/^
3V2 2/^ 3/^
34. 2zi^^. 3a
2+a/^
Va;
 1 + Vl 
■«
2\/1
(1
— a; — Va; ■
V1)*
1
35. ^51^; 1— ^ 39.
2v^ + V2 23V1
36 « + ^^^^ 40 3V^+2V^V^
o6V^ ■ 3V2_2\/r7+/Iio
Find the square root of
41. 36/^. 44. 12a/1038.
42. 12V'^. 45. 2924^/^.
43. 12^66. 46. 7 + 40'/^.
47. By uSe of i (or V— 1), factor o* + 6*. Also o* + 6*.
ar' + 2. oV + 6^. .a?+l. x» + l.
48. For what values of x is V2 — x imaginary? V2  a^?
49. Find the value of (a/^)^ (X{V^^)*, {V^f,
('/=l)^ (V3T)3, (V^)», (a/TT)!.
IMAGINARY QUANTITIES 339
Ex. Simplify 3(V^ + 2)^ (2\/^  1)«.
Substitute i for V 1
3(i + 2)*  (2i  1)«  3t« + 12i + 12  4i« + 4t  1
«  i» + 16i + 11
«(!) I iGyCrr 4, 11
«12 + 16\/1 Arw.
Simplify:
50. t^ + 3i*  2i8. 52. t« X i* X 3t«.
51. i^5i + 4t*. 53. (tl)»(i_i)«+3(il).
54. (a/3T  l)»  (^/^  l)« + 2('v/^  1).
55. (a/^T 2) (3 V^ + 1)  (V'^ 3)»  (V^)\
56. (V^  1)* + 3(V^  1)» + 4('S/^  1)2.
57. If « = ^^^— ^, find the value of 3a;«  6a; + 7. Of
a:*  5a:2 + 2a;  1.
58. Simplify i^+^ Also i^+2, i^+S i^+^
59. Who first discussed imaginary quantities and when?
Who first put the use of these quantities on a scientific basis?
Who invented the symbol i f or V— 1?
60. Make up and work an example similar to Ex. 2. To
Ex. 17. Ex. 51.
EZEBCI8E 110
Obal
1. State which of the following are imaginary: V 3, — VI^
 Vf, V^, V'^ \^^16, ^~\,
2. Give the product of each of the following: \/— 2 X V— 3,
#/^ X \/3,  a/^ X ^3, V^ X V«; ' Va X V"«>
VSXV^, i'Xi*; f*Xi».
340 SCHOOL ALGEBRA
3. Give the product of (a/ 3\/2)('v/3 V 2),
4. Of 3i X 2i,  6i X 3i,  2i» X 3i,  4i« X 2i«.
5. Does (a + x)" equal a" + x*?
6. Does V« + a? equal v^a 4'C^?
7. Is {axy equal to a"x"?
8. Is 'V^ equal to '{J^'^^?
9. What advantage is it to know principle contained in Ex. 7?
10. That contained in Ex. 8?
11. Which is greater, (30" or 3» • 3i»? How many times greater?
At this point, if the teacher prefers, the class may review by use
of Exercise 157 (p. 475).
y CHAPTER XIX
1/
^QUADRATIC EQUATIONS OF ONE UNKNOWN
QUANTITY
223. ITeed and Utility of Equations of the Second Degree.
Ex. One basketball team has won 5 games out of 17 played,
and another team has won 6 games out of 12 played. How
many straight (i. e. consecutive) games must the iBrst team
win from the second in order that their averages of games
won may be equal?
Let X » the required number of games.
Then 5+^ «
17 + X 12 + X
Sunplifying this equation we obtam,
x* + llx = 42
Hence, in order to find the value of x^ it will be necessary to solve
an equation of the second degree.
Why do we now proceed to make definitions and rules?
224. A quadratic equation of one unknown quantity is
an equation containing the second power of the unknown
quantity, but no higher power.
A pure quadratic equation is one in which the second
power of the unknown quantity occurs, but not the first
power.
Ex. 5a? 12 = 0.
A pure quadratic equation is sometimes termed an incomplete
quadratic equation.
341
342 SCHOOL ALGEBRA
An affected (or complete) qnadfatic equation is one in
which both the first and second powers of the unknown
quantity occur.
Ex. ix^rx + U =0.
PuHE Quadratic Equations
225. Solution of Pure ftuadratics. Since only the second
power, Qd^, of the unknown quantity occurs in a pure quadratic
equation, in solving such an equation, we
Redvjce the given equation to the form o? = c;
Extract the square root of both members.
a^ _ 12 0^2 _ 4
Ex. 1. Solve — =
3 4
Clearing of fractions, 4x* — 48 = 3x* — 12
Hence, x^ = 36
Extracting the square root of each member,
a; = + 6, OT  6.
That is, since the square of + 6 is 36, and also the square of  6 is
36, X has two values, either of which satisfies the original equation.
These two values of x are best written together. Thus,
X » =^ 6 Roots
Check. For a; = 6
x'~12 ^ 36  12 ^ 24 ^g
3 3 3
X*  4 364 32 Q
— i r"T'^
Ex. 2. Solve
Check. For x =  6
x«  12 3612 Q
3 3—''^
j«4 _ 364 _g
7? h X? ^ a
03^ a^ ^ hx^  6* .
x2 = a + b
a; = rfc Va \b Roots
Let the pupil check the wprk.
PURE QUADRATIC EQUATIONS 343
EXERCISE 111
Solve:
1. 5ar^ = 80. 8 ^ L = A.
2. 3x^5 = a? + 3. ' 4a? So? 12
3. Ja^  1 = i  3a?. ^ _J 1 36 .
4. lfa? = a?~4f. 2a: 1 2a:+l
5a^ a^ 10. aa? + a^ = 5a' — Soa:*.
^' T"" "¥* 11. aa? + c = 6.
a? 5 fa? a: + 2a , x2a 26
6. = = ' — = 12. ;r H — r = —•
7 5 a2aa: + 2a5
14. (ax + by + {ax by = 106^.
15. (x + a)(xb) + (x a) (« + &) = 2{a^ + 6« +ab).
16. 3(2ar  5) (a: + 1)— 2<3ar + 2) (2a;  3) = a;  9.
28 2
17. If a? = , find the value of x when « =
1 + 3d 7
18. The square of a certain number increased by 9 equals
twice lie square of the same mmiber diminished by 27.
Find the number.
19. State Ex. 1 as a problem concerning a number.
20. Also Ex. 2. 21. Also Ex. 3.
22. The product of a niunber by one third of itself equals
12. Find the number.
23. A certain field contains 256 sq. rd. and the field is foiu*
times as long as it is wide. Find the dimensions of the field.
24. A certain field is four times as long as it is wide. If
each of its sides is increased by one half, its area is increased
180 sq. rd. Find the dimensions of the field.
344 SCHOOL ALGEBRA
25. Who first formed the idea of absolute or independent
negative numbers (see p. 461). How was negative number
used before this? How did the Arabs treat it?
26. Make up and work an example similar to Ex. 2.
To Ex. 14. Ex. 18. Ex. 23.
27. Practice oral work with fractions as in Exercise 58
(p. 190).
Affected Quadratic Equations
226. Completing the Square. An affected quadratic
equation may in every instance be reduced to the form
An equation in this form may then be solved by a process
called completing the square. This process consists in adding
to both members of the equation such a number as will make
the lefthand member a perfect square. The use of familiar
elementary processes then gives the values of x.
Thus, to solve a;» + 6x = 16
take half the coefficient of x (that is, 3), square it, and add the
result (that is, 9) to both members of the original equation. We
obtain
a;2 + 6x + 9 = 25
or, (x + 3)2 = 5»
Extract the square root of both members,
xf 3 = =*= 5
Hence, x = — 3 =*= 5
That is, a;=345=2 1„,
Also, x=35 = 8t *^^
Hence we have the general rule:
By clearing the given equation of fractions and parentheses,
transposing terms, and dividing by the coefficient of oc^, reduce
the given equation to the form ac^ + px = g;
AFFECTED QUADRATIC EQUATIONS 345
Add the square of half the coefficient of ^ to each member of
the eqwUion;
Extract the square root of each member;
Solve the resulting simple equatimis.
Before clearing an equation of fractions, it is important to
reduce eaxih fraction to its simplest form,
Ex. 1. Solve 6ar^  14a: = 12.
Dividing by 6, a:* — Jx =» 2
Completing the square, x^ ix \ {iY = 2 + f  =» ^
Extracting the square root, a? —  = =*= ^g^
x^i^^
a; = 3, or § Roots
Check. For x =  f
6x»  14x = 6 X J + 14 X f
1+^ = 12
Chbck. For X Z
to*  14r = 6 X 9  14 X 3
= 5442 = 12
Ex. 2. Solve ^7? = 2(1 + 2z).
Clearing, 3x2 = 2 + 4ic
Transposing, 3x2 — 4x = 2
Hence, x* — f x = §
x«  Jx + (S)2 = ^
,..^io.lVTo^
3 V 9
Let the pupil supply the check.
2a: — 2
Ex. 3. Solve = 3.
ar^1
2x — 2
If we fail to reduce the f ractioh — ^ — r to its simplest form before
clearing the equation of fractions, we obtain
2x  2 = 3x«  3
3x«  2x  1 =
Whence, x =  i, 1
346
SCHCX)L ALGEBRA
Check. For x  — J
2a;2 »2 «
x«l i 1
Check. For x1
2x 2 2 2
x«l"ll"o
Hence, x » 1 is not a root.
The root x » 1 was introduced (see Art. 129) by a failure to
reduce the given fraction to its simplest form. .*
Let the pupil give the correct solution of Ex. 3.
2 6
13.
xl
BXEBCI8E tia
Solve:
1. a? + lftc = 24.
14.
Sx + 5 « 2ar5
. — — s ^ _.
2. a? 8a: 20 = 0.
a; + 4 a:2
3. a?5x = 6. .
15.
2a:l x + i 1
4. aJ» + llx + 24 = 0.
a; + 3 2a:3 2
5. ar* + 4a; = 7. ^
16.
a: = 4 —  + — •
X 2 2x
6. 5a:*6a;»8.
a: 5— a: 15.
7. 2a?6aj=7.
17.
5—3! » 4
8. 3a? + 7x = 26.
1 X 4
9. 2a; + 3ja?«4.
18.
« + 2 a:2 3
10. 35 = 2«*3x.
2a; + l , «!
u. 3a? = ia: + 2f,
19.
■ h X = •
X1 ' 3
+ ^^2^
20.
21.
+
3x
3a:  1 2a:  5
3a: + 1 ^ a; + 3
5a; + 4 4x + 6
=
. l(a:+l)?(2a:l) = 12.
a:3,3a:l  la:,3 + 2a:
23. —z: 1 :; — =1 : j
/
24.
2a: 3
2a:  1 3a:  4
x + 1 x1
= 1
4a:
4a: 14
l^.a:^
AFFECTED QUADRATIC EQUATIONS 347
x3 5a; 7 ^ 2a? + 5 
3ar2"^49x* 2 + 3x
2a; 1 1 3a; a;  7 .
a; + l x + 2 a;l
27. «2 + 2a; = 1. 29. 2a? + 5a; =  4.
28. 3a;*  5a; =  1. 30. da;* _ go; + 5 = 0.
31. 3a;(a;+l)(a;2)(a;+3)=2+(la;)*.
a;* + a; — l.a:* — a;— 1 «.
32. V^=^ —  + ^ —  —  = 2.
ar« + a; + 1 ^ a?  a; + 1
33. ^^^3 + ^+1.
1 — X 1 — a?
. ''+a;3 ^ a;3
35. ii«?+i = ^^±20:r+l.
a; + 2 a; + 2 ^
36. . ^^^^ .+
4(a; + 5)(x8) (a; + 4)(a; + 5) a; + 4
3:c — 3
37. Take the equation ^—\ — 6 = and, by separat
ing the fraction into two parts, form an equation which
contains an extraneous root, the appearance of the equation
giving no indication of this fact.
38. Make up and work an example similar to Ex. 37.
39. The square of a certain number increased by 4 times
the number equals 45. Find the niunber.
40. State Ex. 1 as a problem concerning numbers. Ex. 7.
41. Find two (X)nsecutive numbers such that the sum of
their squares is 41.
42. Practice the oral solution of equations as in Exercise
64 (p. 209).
348 SCHOOL ALGEBRA
227. Literal Qnadratio Equations are solved by the
methods employed in solving quadratic equations with
niunerical coefficients.
Ex. 1. Solve "2 " "e "" 6^^ "^ ^^*
Clearing, 3x^ — ax  ax \ c^
Hence, 3a^ — 2ax = o*
. 2a a»
x' X = —
3 3
"o^d)'
4fl^
9
X = a, —  RooU
o
Let the pupil check the work.
Ex. 2. Solve (a  b)h?  (a^ ^V)z=  ab.
a  6 {a by
a*
_.x l/ g + b V ^ l/ a+6y _ 4a6
"^y^ 4Va  6/ 4\a  6/ 4(a  6)«
0+6 a 6
X —
2(a6) 2(a6)
a: r, r Roots
a b a 6
Let the pupil check the work.
EXERCISE 113
Solve:
1. x^ + 4ax = 12o2. 7. 2aV + ox = 3.
2. ar^ + 46a: = 21fe2. g 7c2^i0aca;+3a**0. •
3. ar^ + Sea;  10c2 = 0. 5^. q
4. a^ = 6a262  5a6a:. ^ ^"^ 7 " ^'
5. 6a:2 = 1262 + 6a:. ^ ^ ^ 3^
6. 3a:2 +4^^ = 15^^^ l^ ^"^ a ~ 4a*'
AFFECTED QUADRATIC EQUATIONS 349
11. 0^ — (a + i)x= — a. 16. as? — a^x +x a.
12. 2a^ + abx = 1SI>^. 17. JL^+?L+1^^±
13. a* + 2a: — Zax = 6a.
14. 3aV+a(366)a:=56.
15. aba?+{a^+b^)x+ab=0. 19. 4(0:^  i) = 6(4aj « 6).
a — a; a; a— a:
18. X =* a.
X — a
20. (a + 6)a:2_(^_5)a:^ = 0.
a + 6
21..a6:r^ = ^[a:(a + 6)l]. ,
22. ^ti=?t*+ "
25.
X c a + b'
23. a(a:2 _ 52) ^ 5(aJi _ j2 + ^) + ^ ^ 0.
24. (a + c)a:2  (2a + c)a: + a = 0.
a; + 1 a + 1 26. oar^ + 6a: + c = 0.
s? a* 27. ar^ + pa: + g = 0.
28. Show that the sum of the roots obtained in Ex. 27
equals —p. Also that the product oi these roots equals q.
29. How many examples in Exercise 45 (p. 155) can you
now work at sight?
228. The Factorial Method of solving equations was ex
plained in Art. 104. Sometimes this method must be supple
mented by the method of completing the square.
Ex. 1. Solve a:* + 1 = 0.
Factoring, (a: + 1) (a;*  a: + 1) =0
a: 4" 1 = 0, gives a: = — 1 Root
Alflo, x*a; + l«0
Whence,^ a? —x ^  1
X « i * l^/^S RooU
Let the pup3 check the work.
350 SCHOOL ALGEBRA
The factorial method of solution is especially helpful in
solving certain literal quadratic equations.
Ex. 2. Solve (ab)x^'{a^b^)x + ab=Ohythe factorial
method.
We obtain [(a  b)x  a] [{a  6)x  6] 
Hence, x «= r, r Roots
a —b a—b
Let the pupil check the work.
This example is the same as Ex. 2 solved on p. 348. On compar
ing the solutions, we observe that at least three fourths of the labor
of solution is saved by use of the factorial method.
E2XaCISE 114
Solve:
1. a? + 8x + 7=0. lO.Qfa^x + l^O.
2. a?5a:=84. ll: (2a:  l)(Cr^a:2) =0.
3. 6x2 + 7a. ^ 90. 12. 3(a:2  i)  2{x + 1) = 0.
4. ix^10x + Z=0. 13. 5(ar^  4) = 3(x  2).
5. 24ar^ = 2a: + 15. 14. 7(a?*16)53a:(ar^4)=0.
6. SaV + lOax = 8. 15. 3xix^l) + 2(ar  1) =0.
7. a:* = 16. 16. a:^ _ 27 = 13a:  39.
8. a:* = 8. 17. 2a:« + 2ar^ = a: + 1.
9. a:<5a:2+4 = 0. 18. 2a:« + 6ar^ = Sa:^ ^ g^. . 3,
19. Find the six roots of a:®— 1 = 0.
20. Find all the values of v 1.
21. Find all the values of \/8.
22. Obtain a complete solution of the equation x' = 8.
23. Solve a? + (a + 6)a: + ofc = by the method of
Art. 226.
Solve the same equation by the factorial method.
AFFECTED QUADRATIC EQUATIONS 35i
About how much shorter Is the latter method than the
former?
Solve by the factorial method:
24. a^ + cx + dx + cd — 0.
25. abcQi^  (a^V + c^)x + abc^O.
26. ^ + « + * «
a — X X a — X
27. 3i?2bx + V + xb^0.
28. a(b  6)2? + b{c  a)x + c(a  6) « 0.
29. (4a2  96«) (x2 + 1) » 2a:(4a« + 96*).
a + jb + x x + b
31.
+ 46 o~46 _46
a: + 26 xib'" a'
a + 6
Form the equations whose roots are
33. 2,3. 35. 2,3. 37. 2,3,4.
34. 2,3. 36. 0,2. 38. V2, V2, 0.
39. Solve (a + 6) V  (o*  62)x  oft = by the method
of completing the square (Art. 226). Now solve it by the
factorial method. Compare the work in the two processes.
Why do we not solve all quadratic equations by use of the
factorial method?
40. Find all the roots in the solution ol t? ^ 16a:.
How might one of these roots be lost by careless use of
Ax. 4 (p. 96)?
41. Work again Exs. 6268 of Exercise 75 (p. 246).
42. How many of the examples in Exercise 114 can you
work at sight?
352 SCHOOL ALGEBRA
Equations in the Quadratic Form
229. Simple Xrnknown Quantity. An equation oontain
ing only two powers of the unknown quantity, the index of
one power being twice the index of the other power, is an
equation of the quadratic form. It may be solved by the
methods already given for affected quadratic equations.
Ex. 1. Solve a^5x^= 4.
Adding (f)' to both members will make the lefthand member a
perfect square. Thus,
a:*  5x« + ()«  J
Hence, x'  f = =*= 
X* =4, or 1
a; = =*= 2, ±1 Roots
Let the pupil check the work.
This equation might also have been solved by the factorial
method.
Ex. 2. Solve 21^^  3V^ = 2.
Using fractional exponents,
2x"*  3x"* = 2
Whence, x^  f x"* = 1
x"*=2, i
Whence, a;*  i,  2
X = J, — 8 Roots
Let the pupil check the work.
230. Compoimd Unknown ftuantity. A polynomial may
be used in place of a single quantity as an unknown quantity
Ex. 1. Solve 2Vx + 12 + Z\^x + 12 = 14.
EQUATIONS IN THE QUADRATIC FORM 353
This equation may be written, 2(a; + 12)* + 3{x + 12)* « 14
Let ix + 12)* = y; then (x + 12)*  y*
Hence, substituting, 2y^ f Sy = 14 ,
Whence, 2/ f= 2, or  J
a: + 12 = M*^
a; = 138^ Ro<4
. • . >^a; + 12 = 2
a; + 12 = 16
x « 4 22ao<
Let the pupil check the work.
Ex.2. Solve a:2 _ 7a: + Vo^  7a: + 18 = 24.
Add 18 to both sides,
x^  7a; +.18 + Va;'7x + 18 = 42
Va;*  7a; + 18 = 2/ then2/«+y=42
Let
Whence,
2/ = 6 or  7
Hence, V^~7a;+18 = 6
a;* 7a; +18 =36
X = 9, 2 /^oote
Let the pupil check the work.
Va;*7a;+18 = 7
a;«7a;+18 =49
X =J(7* Vl73) Roots
SXERCISE 116
Solve:
1. a?*17ar^ + 16 = 0.
2. 4iC* 13x2 + 9 = 0.
3. 27a;« = 35a:5  8.
4. 3a:*5x* = 2.
5. 27a;3 + 19a;* = 8.
6. 3a;* = 4a;* + 4.
7. 2^ = ^ + 1.
8. 3a;"* + Sa;"^ = 2.
9. 6x^  a;"* = 12.
10. 9a;~* + 4 = 13a;"*.
11. 3v^  5Vx =  2.
12. 5^ = 8^^ + 4.
13. 7^f^ 4^^ = 3.
14. 3V2x  2v^2i = 1.
15. (a;l)2 + 4(a;l) =21.
16. 2(a^Sy7(a^S)=S0.
17. &{x^+iy+13(a?+l) = 2S,
18. 2V2a:3+5^2a;3=7.
864 SC^OOL ALGEBRA
19. (x + 2)*  ^J+2 = 2.
20. (3«  2)*  4(3a;  2)* + 3 = 0.
22. Z{Zi^ 2x+ 1)  4V3a:«  2x + \ = 15.
23. 2(2r' + 3ar  4)«  3(2a^ + 3a;  4)   1.
24. x^iflx 3a/«» + 7a; + 1 = 17.
25. 6(a;» + x)  7y3i(x+ir^'2 = 8.
26. 3a?  7 + 3V'3a?  16a; + 21 = 16a;.
27. ^33;  2a?  (3x  2a;*)* 2 = 0.
28. (a;  a)'  ^{x  o)* =  2o*.
29. 3a:~*  7a;* = 4. 31. 16a;* 22 = 3a;~*.
30. 3a:* = 8a;*  10. 32. 2a? Va?2a:3=4a;+9.
33. 5(2a?  1)*  4 = ^(2a?  l)».
34. Make up and work an example similar to Ex. 1. To
Ex. 12. Ex. 19.
35. Practice oral work with radicals as in Exercise 108
(p. 332).
Radical Equations
231. Badioal Equations Besulting in Affected Quadratic
Equations. If an equation is cleared of radicals by the meth
ods given in Art. 215 (p. 325)^ the result is often a quadratic
equation.
Ex. Solve VZx + 10 + Vx + 2 = VlOx + 16.
Squaring, 3a; + 10+2V(3x +10) (a;+2) + a ?+2 = lOx + 16
Hence, V(3a: + 10) (x +2) = 3a: + 2
Squaring again, 3a;* + 16a; +20 = 9a?* + 12a; +4
6a;*  4a; = 16
x=2, t
RADICAL EQUATIONS 355
Substituting these values in the originAl equation, the only value
that verifies is « » 2, which is the root. The other value, x » — ^,
is not a root of the original equation, but is introduced by squaring
in the process of clearing the equation of radical signs. It satisfies
the equation,
V3xf 10  Va?4 2 = VlOa; + 16
EXERCISE 116
Solve:
1. X  1  VSx5 =0. 4. 3a:  2\/6i 6 = 0.
2. 2a:+lV7a; + 2=0. 5. V3x+1 2\/2x+3 = 0.
3. xVSx^e. 6. 2+\/2a:+7V5aj+4 = 0.
7. V3a; + 7  Vx + 1  2Va; 2 = 0.
8. A/2a;+l  2V« + Va:  3 = 0.
9. Vxa^ + Vx + 2€?  Va: + 7a* = 0.
10. 3 V5T17  2V^?T41 + V?+T = 0.
11. Va; + 4 4 V3a:+1  Vftr+l = 0.
12. 2V5x  A/2a; 1 = f •
A/2a:  1
3V2^  5 _ 9  2 V2g
3 + a/2x"' V2X3'
3 Vl2a; ^
14. , z = 0.
2 + Vl2^
15.
Vg _ V'g + 2 5_Q
Vx + 2 Vi 6
16. Va; + f + V^4a:  i = VSa; + J.
17. Vl2a:2_3._g + vl2?T"x^ = V24ar^  12,
a; + Va:*  a * a;  ^o?  o* ^_ .
18. . , = 8var — a\
x  Va:*  o* x+Va?a^
19. VS+1 + V'2a: + 3  V5a;+1 + Va: + 6^. ,
366 SCHOOL ALGEBRA
20. The square root of a certain number^ plus the square
root of 1 increased by twice the number, equals 5. Find the
number.
, 21. State Ex. 1 of this Exercise as a problem concerning a
number.
22. Similarly state Ex. 3. Ex. 6.
23. Practice oral work with imaginaries as in Exercise
110 (p. 339).
232. Other Methods of Solving Quadratic Equations, besides
those given in the preceding part of this chapter, may be
used. One of these methods may occasionally be used to
advantage for some special purpose.
233. Hindoo Method to Avoid Fractions in Completing the
Square. After simplifying the equation,
Multiply through by four times the coefficient of oi?;
Add to both sides the square of the coeffiment of x in the siwr
plified eqimtion.
The reason for this process is evident, since if ax^ + 6x = c
is multiplied by 4a, we obtain
4a V + 4a6ar = 4ac
The addition of V gives on the lefthand side 4aV + Aabx
+ 6*, which is a perfect square.
Ex. Solve 3a^ — 2a: = 8 by the Hindoo method.
Multiply by 4 X 3, or 12,
36x»  24a; = 96
To each member add the square of the coefficient of x in the
original equation; that is, add ( 2)', or 4.
36x« 24x44 = 100
Hence, 6a;  2 « =*= 10
6a;  12,  8
X » 2,  1 RooiM
Let the pupil check the work.
QUADRATIC EQUATIONS 357
EXERCISE 117
Solve:
1. a? + 5a: = 6. 9. («* + 3)«  7(x« + 3) =60.
2. 3sr*a; = 2. lo. 4aj* lOla;^ h 25 = 0.
3. ear* + 5a; = 4. u. 6'^llV^ = 10.
4. 8a?2a; = 3. u. 3(x2)« + 5(x2) = 12.
5. 4a:* + 4x = 35. ,1 „ , 1
13. X + — = a + — •
6. 16a:*40ai2 +9=0. 2x 2o
7. 6ar'aa; = 2o«. 14. (al)ar' + (a + l)x= 2.
8. 4o*ar' + 5oa: = 21. is. (o*fc*)ar'+(a*+6*)x=a6.
234. Use of rormtila. Any quadratic equation can be
reduced to the f onn
Solving this equation by use of Art. 226,
^ "" y 2a
By substituting in this result, as a formula, the values of a,
6, c in any given equation, the values of x may be obtained.
Ex. Solve 5x2 + 3a:  2 = by use of the formula.
Here a = 5, 6=3, c =  2.
Substituting for a, 6, c in the above formula,
a^ = 10 10 ^
Let the pupil check the work.
EXERCISE 118
Work the examples in Exercise 117 by use of the fonnula.
35S
SCHOOL ALGEBRA
EZEBCISE 119
Review
Solve:
1. 6x«+x = L
2. 9a;«  4 = 2a^.
3. V^  v^ = 2.
4. x*  16iF = 0.
5. 3VS  12x"* = 5.
6. £j:l1 « 2i  ^
8.
x+1 _ o+l
\/x Va
x1
7. V3x =
— ViC2. 14.
Va^ 2
15. Vs^n  V3FTT Vx^
16. 5(a; + 2)* = 3(a; + 2)» + 2.
17. 3Vic + l  5^JT1 = 2.
«.202(,+)".3(.+).
I 20. ^^;  W2 ^ 2vg y/e
I '4Va:2V2 3^/3^5^/6*
9. a; + 2=Ut
6 X a
10. \/4x3 = 1 HV'^qn;
11. 3a;*  7a;i = 6.
12. X*  27a; « 0.
13. 5a;i + 6a;~* = 11.
a; 3 a; +3 6
a; 4 » +4 7
0.
21. 4a;«  7^2x2 + 3a;  2 = 19  6a;.
j4a;3 3a;4 _2f a;2 2a;l l
3 U + l X 1 J "^
22. 1
a;
23.
25.
abx^
o2 6*
1
+ 1 =
4
(a» H 6')a;
35
36
0.
24.
8
+:^i
Vx» l^a;*
^=.1+1 + 1
a;ha+6 a; a ft
26. oa;*" + 6a;" + c == 0.
27. (x« + 6x)2  2(a;2 + 6a;) = 35.
QUADRATIC EQUATIONS . 359
29. x« + (i/2)« = fi*. 32. y»4.a«+y =2a2/+a+6.
30. 32/* =361 4 Vy'  7. 33. a;«  1 = (1 aj) a/2 4x.
34. From T = 27ri2 (/?  H), find 22 in terms of the other letters.
Find the value of a; in the shortest way when
35. Yx^ = ¥(225) + V(64).
36. 3.1416x2  3.1416(441) + 3.1416(400).
37. Who, as far as we know, first solved a quadratic equation,
and at about what time? (See p. 462.)
38. How have the different cases in the solution of a quadratic
equation been classified at different times?
39. Write (but do not solve) an equation of each of the principal
kinds treated in this chapter.
40. Work again Exercise 24 (p. 99).
EXERCISE 120
^ 1. Find two consecutive numbers the sum of whose squares
is 61.
2. There are two consecutive numbers such that if the
larger be added to the square of the less the sum will be 57.
Find the numbers,
3. There are two numbers whose difference is 3, and if
twice the square of the larger be added to 3 times the smaller,
the simi is 66. Find the nimibers.
4. Seven times a certain number is one less than the square
of the munber next larger than the original number. Find
the number.
5. Find the number which increased by its reciprocal
equals \^.
v,^
360 SCHOOL ALGEBRA
^ 6. Find three consecutive numbers such that their sum is
15 less than the square of the smallest.
7. If the length of a rectangle exceeds the width by 5 yd.
and the width be denoted by x, express the length and the
area in terms of x.
^ 8. The area of a given rectangle is 36 sq. yd., and the
length exceeds the width by 15 ft. Find the dimensions of
the rectangle.
^ 9. The length of a certain rectangle is twice its width.
The rectangle has the same area as another, If times as
wide, and shorter by 4j ft. Find the length of the first
rectangle.
10. A rectangular garden contains onehalf an acre and
the length of the rectangle exceeds its width by 2 rd. Find
its dimensions.
11. A square garden contains 100 sq. rd. By how much
must its sides be lengthened in order that its area be doubled?
12. A rectangle is 30 X 40 ft. By what per cent must the
length and width be increased in order that the area be in
creased by 528 sq. ft.?
13. A rectangular park is 80 X 100 rd. By adding the
same amount to its length and width the area of the park is
to be increased by 50%. What is the amount added to each
dimension?
14. A rectangular lot is 8 rd. long and 6 rd. wide, and is
surrounded by a drive of uniform width, which occupies f
as much area as the lot. Required the width of the drive.
15. A farmer has a wheat field 80 rd. long and 60 rd. wide.
How wide a strip must be cut around the outside of the field
ii> order to cut 15 A.?
QUADRATIC EQUATIONS
361
8"
8"
^ " J
86"
SuG. Draw a diagram of the field with an inner rectangle showing
the uncut part left.
If the width of the strip cut is x, what are the dimensions of the
inner rectangle? What is the difference between the area of the
enl^ field and that of the inner
rectangle?
16. An open box 8 in. deep
and to contain 3200 eu. in. is to
be formed by cutting out small
equal squares from the comers of
a square sheet of tin and folding
up the sides. Find the length of
a side of the square sheet of tin.
17. A number of men bought
a yacht costing $2800 and each
pmchaser paid 7 times as many dollars as there were pur
chasers. How much did each man pay?
18. One baseball nine has won 5 games out of 13 games
played, and another baseball nine has won 9 games out of 15.
How many straight games must the first nine win from the
second, in order that the average of games won by the two
nines shall be the same?
19. One ball nine has won 6 games out of 18, and another
has won 12 out of 13. How many straight games will the
first team need to win from the second in order that the per
cenfiage of games won by the first team shall equal half that
of the games won by the second?
'20. The numerator of a given fraction exceeds its denom
inator by 2. Also the given fraction exceeds its reciprocal
by yf . Find the fraction.
21. A cistern is filled by two pipes in 18 min. ; by the greater
alone it can be filled in 15 inin. less than by the smaller.
Find the time required to fill it by each.
/S
y
1
I?
362 SCHOOL ALGEBRA
»
22. A cistern can be filled by 2 pipes in 1 hr. 33j min., but
larger alone can fill it in 1 hr. and 40 min. less than the
smaller one. Find the time required by the less.
23. A number of two figures has the units' digit double
the tens' digit, but the product of this number and the one
obtained by inverting the order of the figures is 1008. Find
the number.
24. The lefthand digit of a certain number of two figures
is I of the right digit. If the product of this number and the
number obtained by inverting the order of the digits be in
creased by twice the original number, the sum is 800. Find
the number.
25. A man can row down a stream 16 mi. and back in 10
hr. If the stream runs 3 mi. an hour, find his rate of rowing
in calm water.
26. Two trains run at uniform rates over the same 120
mi. of rail; one of them goes 5 mi. an hour faster than the
other, and takes 20 min. less time to run this distance. Find
the rate of the faster train.
27. One number is f of another, and their product, plus
their sum, is 69. Find the numbers.
28. Find two numbers whose product is 90 and quotient
2^.
29. Find two numbers whose difference is 4 and the sum
of whose squares is 170.
30. Find two consecutive numbers, the difference of whose
cubes is 217.
31. If the side of a square is 2 ft., how much must this
be increased to increase the area of the square by 153
sq. in.?
QUADRATIC EQUATIONS 363
32. A farmer has a rectangular wheat field 160 rd. long
and 80 rd. wide. How wide a strip must be cut around the
outside of the field in order to leave 30 A. uncut?
33. Two workmen can do a piece of work in 24 hours. In
how many hours can each do it alone^ if it takes one of them
20 hours longer than the other?
34. An open box 6 in. deep and to contain 864 cu. in. is
to be formed by cutting out small equal squares from the
comers of a square sheet of tin. Find the length of a side
of the square sheet of tin.
35. It takes a man 5 hr. to row up a stream 8 mi. and back.
If the stream flows at the rate of 3 mi. per hour/ what is the
rate of the man in still water?
36. A bin is to be constructed to hold 9 T. of coal. If
the bin is to be 5 ft. deep and twice as long as it is wide, and
if 40 cu. ft. are allowed for 1 T., what will the dimensions of
the base of the bin be?
37. The walls and ceiling of a room together contain 104
sq. yd. The room is twice as long as it is wide, and its ceiling
is 9 ft. high. Find the length and breadth of the room.
38. If a carriage wheel 11 ft. in circumference took xV ot
a second less to revolve, the rate of the carriage would be 1
mi. more per hour. At what rate is the carriage traveling?
39. Find two consecutive nimibers the sum of whose
squares is a.
40. Find two numbers whose difference is b and the sum
of whose squares is c.
41. The area of a given rectangle is p, and the length of
the rectangle exceeds the width by q. Find the dimensions
of the rectangle.
364 SCHOOL ALGEBRA
42. Why are we able to solve many of the problems m
the Exercise by algebra and not by arithmetic?
43. If the side of a square is a and an error e is made in
measming the length of one of its sides, what is the error,
E, in its area when the area is computed from the side as
measured?
44. Make up and work three examples similar to such of
the examples in this Exercise as you think are most interest*
ing or instructive.
45. How many examples in Exercise 31 (p. 121) can you
now work at sight?
EXERCISE 121
1. In * = ^gP, if fif = 32 ft. and s = 1000 ft., find t
Do you know the meaning of this example as applied to a falling
body?
2. In £ = — , if E = 500, m = 20, find v.
2
3. In 5 = ^^^  vt, find t/it s = 200 ft., g = 32 ft., and
© = 60 ft.
Do you know the meaning of this formula as applied to a bodv
sent upwards from the earth with a velocity of 60 ft. per second?
4. In h = a + vt^ gf, it h = 500 ft., a = 100 ft,
t^ = 200 ft., g = 32 ft., find t
Gan you discover the meaning of this example as applied to a
body sent upward from the earth from a point 100 ft. above the
surface of the earth with a velocity of 200 ft. per second?
5. In X = ttB^, if Z = 43560 sq. ft., and tt = ^\ find K
6. In f = 7rR{R + i), if T = 220, tt = \\ and L = 20,
find /J.
QUADRATIC EQUATIONS 365
7. In * = ^gf, find the value of t in terms of s and g.
8. In « = «< — 16<^, solve for t
9. In A = a + i?< — 16f^, solve for <•
10. In r = 7rR{R + i), find iJ in terms of T and i.
U. How many seconds will it take a body to fall from
rest a distance of 1000 ft. (resistance of air neglected)?
12. If a bullet is fired upward with an average velocity
of 2400 ft. per second, how long will it be before the' bullet
reaches a height of 1^ mi.?
13. If an arrow shot over the top of a steeple reaches the
groimd.in 6 sec. from the time the arrow left the bow, how
high is the steeple?
SuG. Use the formula of Ex. 1, letting t = one half of 6.
14. If the steeple of Ex. 13' were 200 ft. high, how many
seconds would it be before the arrow returned to the earth?
15. Using the formula 8 = ^g^ (where g = 32.2 ft.), find
the distance a body will fall from the end of 5.2 sec. to the
end of the 7 sec.
16. On the moon, g = 5.4 ft. Find the difference in the
distance which a body falls on the earth in 5 sec. and the
distance it falls on the moon in the same time.
17. Make up and work an example similar to Ex. 12.
To Ex. 14.
18. How many examples in Exercise 35 (p. 131) can you
now work at sight?
CHAPTER XX
SIMULTANEOUS QUADRATIC EQUATIONS
235. ITeed and Utility of Simultaneous Equations Involving
Quadratic Equations.
Ex. A rectangular park is known to contain 1^ acres.
The path which leads across it diagonally is measured and
found to be 26 rods long. Find the dimensions of the park.
Let X = no. rd. in length of park.
y = no. rd. in width of park.
^ li acres = 240 sq. rd.
Hence, a^+y^ 2& (1)
xy ^240 (2)
By solving (1) and (2) x and y can be determined (see Art. 242).
Try to solve the problem by use of only one unknown, as
X. Even if you succeed in getting a solution, you will find
the method awkward and inconvenient.
236. Quadratic Equations Containing Two Unknowns. The
general quadratic equation containing two unknowns is
as? +bQcy + cy^ + dx + ey +f = *
By giving a, 6, c, etc., different numerical values including
zero, this general equation may be made to take many
special forms.
What values must we give a, 6, c . . . . respectively in ord» to
obtain the equation 5x^ + 3xy + 2y' = 5, from the general equation?
The absolute term of an equation is the term which does
not contain an unknown factor, as / in the above general
equation.
366
SIMULTANEOUS QUADRATIC EQUATIONS 367
Simnltaneons quadratic equations is a brief, term for
simultaneous equations whose solution involves quadratic
equations.
Thus, the equations stated in Art. 235 are simultaneous quad
ratic equations.
In general, the combination of two simultaneous quad
ratic equations by elimination gives an equation of the
fourth degree in one unknown, which cannot be solved by
the methods of this book. Two simultaneous quadratic
equations can be solved by elementary methods only in cer
tain special cases.
237. A Homogeneous Equation is one in which all the
terms containing an unknown quantity are of the same
degree.
Thus, 3x^ — 5xy^ + y' = 18 is a homogeneous equation of the
third degree. What is the degree of the equation xy = 6?
General Methods of Solution
Case I
238. When One Equation is of the Pirst Degree, the Other
of the Second, two simultaneous equations may always be
solved by the method of substitution.
Ex. Solve '\2x3y = 2 (1)
x^ 2xy= 7 (2)
Eliminate y, since y occurs only once in equation (2).
From(l), y=^^ (3)
(2a;— 2\
"~3 — / "^
Hence, 3x«4x2+4a;=  21
a;»4x=21
a;= — 3 7]
Substitute for x in (3), y=h 4J ^^^^
r
368
SCHOOL ALGEBRA
Check.
For a: = — 3 and y— — i
2a: 33/= 6+ 8= 2
a;«2a:2/=9 16=7
Check.
For a; =7 and y = 4.
2a;3y= 14^12=2
x2 2x2/= 49 56= 7
EXEBCISErl22
Find the values of x and y:
m
1. 33?2f=  5.
x + y3 = 0.
2. a:  22^ = 3.
a^ + ^y^ = 17.
3. 2ar^ + xy = 2.
3a; + y = 3.
4. x^3y^l =0.
a: + 22/  4 = 0.
s. X — 3y = 1.
7xyx^ = 12.
6. 2a: + y + 3 = 0.
3ar^  72/2 = 6.
7. 2a: + 52/ = 1.
2a^ + Sxy = 9.
8. gX 2?/ ~ a
{xyy = f 7.
9. ar^  3a:t/ + 21/2 = 0.
ar^ + y2 = 52.
11. 92/2  62/  5 = 3a;.
91/ + a; + 5 = 0.
3 2 9
12. = .
y x xy
2x ^10 32/ _
y xy x
2x
y
13. — =
x + y
X
a; — 2^ + a = 0.
14. ar^ + 22/^  3a; = 30.
^ = 2.
a;
15. an/ = 12.
X «.
 = 3.
16. 11 = a; + 2(2/  1).
6 = (a; + l).
2a; + 82/ = 7.
17. 3a;  52/  1 = 0.
27? +Zxy 52/2  6a; + 72/ = 4.
18. 4a;2 _ 4a.y  ^2 _^ 3. ^ 3^ _ j^
4a;  2  52/ = 0.
SIMULTANEOUS QUADRATIC EQUATIONS 369
19. The sum of two numbers is 5 and the square of the
first number increased by twice the square of the second
number is 22. Find the numbers.
20. State Ex. 1 as a problem concerning numbers. Ex. 2.
21. How many examples in Exercise 48 (p. 163) can you
now work at sight?
Case II
Si39. When both Equations are Homogeneons and of the
Second Degree, two simultaneous quadratic equations may
aitoays be solved by the svbstitution y = vac.
Ex. Solve Q?xy\y^=2\
y^ — 2xy = — 15
Substitute y « «x, x«  tw^ + 1)*^^ = 21 (1)
t^^2vx^ = lb (2)
From (1), , x2 = ^ _^l_^ ^ (3)
From (2), ^^^ ^' = t^^ ^^^
Equate the values of x^ in (3) and (4),
21 _ 15
I V \}^ tf^ 2v ,
Hence, • 21 1;^ _ 42^ = _ 15 + ist;  \f^
36t;2  57t; =  15
12t;2  19t; =  5 . ' • t; = f , i
It ^ 2 21
If v = T, x2 =
xr 1 , 21
y= «x=(±4)= ±5
Hence, x = =*= 4, =*= 3V3
2/ = * 5, =*= V3
Let the pupil check the work.
Two simultaneous equations of the kind treated in Case II
may also be solved by eliminating the absolute term between
y =.vx =(=fc3\/3) = =fc\/3
RooU
370 SCHOOL ALGEBRA
them and factoring to find the value of one unknown in
terms of the other and then proceeding as in Case I.
EXERCISE 128
Find the values of x and y:
1. :i? +Zxy = 28. 6. 2f Axy +Z7?^ 17.
0^2/ + 4y2 = 8. 1^7?= 16.
2. 2a? + a^y = 15. i. 7? +xy + 2y^ = 74.
a?  2/2 = 8^ 2a:2 + 2a:y + 2/2 = 73.
3. a? + 3^2/ = 7. ^. 27? + Zxy +f ^ 14.
f +xy ^^. Zt? +2xy Ay^ =%.
4. 2ar^ = 46 + 2/2^ 9. 4^2/  ar* = 6.
«2/ + 2/2 = 14. 13a:2 _ 313^+ 15^2 ^ 2\. '
5. 3a:2 ^ y2 =, 12. 10. a? + «2/ + 22/^ = 44.
bxy ^7? = 11. 2ar^  a:2/ + 2/2 = 16.
Also solve the following miscellaneous examples:
U. 32/l=a:. 14. 10a: + 2/ (102/ + a:)=9.
52/2 _ 3^8 = 1. (10a; + 2/) (102/ + a:) = 736.
12. 2^2  32/2 := 6. 15. a:  2/ = 0.
3a:y — 4^ = 2. 5a:y + y2 = 54.
13. a: + 2/ + 3a:2/ = 83. 16. ar^ = 5 + 2xy.
3a;  2/  1 = 0. a:2 + 2/^ = 29.
17. Point out the examples in Exercise 128 (p. 380) which
come under Case I. Under Case II.
18. Point out the homogeneous equations in Exs. 1015
of Exercise 122.
19. Express Ex. 1 above as a problem concerning two
numbers.
SIMULTANEOUS QUADRATIC EQUATIONS 371
20. Work the example of Art. 235 (p. 366) by the method
of Art. 239.
21. Find a nmnber consisting of two digits such that if
the number is multiplied by the lefthand digit, the result
will be 260. But if the number is multiplied by the right
hand digit, the result will be 104.
22. Make up (but do not solve) an example in each of the
cases studied thus far in this chapter.
23. How many examples in Exercise 51 (p. 172) can you
now work at sight?
Special Methods op Solving Simultaneous Quadratics
240. The methods of Cases I and II are the only 'general
methods which can be used in solving all simultaneous quad
ratic equations of a given class. Besides these, however,
there are certain special methods which enable us to solve
important particular examples.
Examples which come directly under Cases I and II are
often solved more advantageously by one of these special
methods.
The special methods apply with particular advantage to
synmietrical equations.
241.. A Symmetrical Sanation is one in which, if y is sub
stituted for Xy and x for y, the resulting equation is identical
with the original equation.
Tbus^ each of the following is a symmetric^ equation:
7^ + SxV + y' = 18
x+y = V2(
xy^6,
372
SCHOOL ALGEBRA
Case III
242. Addition and Subtraction Method (often in connec
tion with multiplication and division). In this method the
object is to find first, the values of x + y and » — y, and then
the values of x and y themselves.
Ex. 1. Solve
a; + y= 7 (1)
. 3^2/ = 12 (2)
Here we have the value oix \y given, and the first object is to
find the value oi x — y.
Square (1), x* + 2xj^ + 2/» = 49 (3)
Multiply (2) by 4, 4x2/ = 48 (4)
Subtract (4) from (3), x*  2xj/ 4 2/* = 1 (5)
Extract square root of (5), x — y = =*= 1 (6)
Add (1) and (6), divide by 2, ^ = ^ or 31 ^ •.
Subtract (6) from (1), divide by 2, y = 3 or 4J ^^'^
Let the pupil check the work.
Ex. 2. Solve
Divide (1) by (2),
Square (2),
Subtract (3) from (4),
Hence,
Subtract (5) from (3),
But
Hence,
a:^ + y3 = 65
x + y = 5
(1)
(2)
(3)
x^ —xy + 2/^ = 13
x* + 2x2/ H 2/* = 25
3x2/ = 12
X2/ = 4 (5)
X* — 2xy + y2 = 9
. • . X — y = =*= 3
X +2/ = 5
x=4,ll
y  1, 4/
Roots
Ex. 3. Solve
1+1 = 11
X y
7? 2/^
Squaring (1),
^++^ = 121
^ xy y^
(1)
(2)
(3)
Subtract jng (2) from (3),
xy
60 (4)
SIMULTANEOUS QUADRATIC EQUATIONS 373
(6)
Hence, * — ± = st 1
But, from (1),
Hence, adding,
Subtracting (4) from (2), 1  A +1 = 1
•c *^ y
X y
X y
2
X
= 12, 10
. • . a: = i i
Let the pupil check the work.
Roots
EXERCISE 124
Find the values of x and y:
1. a: + 1^ = 13.
Qcy = 36.
2. a? + 2^ = 25.
x + y= 1.
3. x + y= 10.
xy = 21.
4. a? + xy + f = 21.
x + y=  1.
5. a?xy + f^S7.
x' + xy + f^lQ.
6. a? + 2/2 = 2§.
3a^ = 2J.
7. a + y + 1 = 0.
iw/ + 3i = 0.
8. Q? + y^=9. '
x + y = 3.
x + y^l.
10. 0:8 + 2/3218 = 0.
a:2  a:y + y2 = 109.
11. x^ + Sxy + f = 2l.
7? — xy + y^ = 12j.
12. xy  6a2 = 0.
^ + y^ = xy { 7a^.
13. a:^ + 2/3 =, 2a3 + 6a.
a:^ — ary + y^ = a^ + 3.
14. ? + ^ = 2i
2/ a^
a: + 2/ = 5.
15. a:3 + 2/3 =, 224.
ar^y + xy^ = 96.
— 6 = 0.
xy
374 SCHOOL ALGEBRA
17. i + i = 3i. 20. 7? + f = ^.
^ ]? ^' a;  y = 4.
1 + 1 = 2. 21. a:3_2^ = 98.
18. x8 + 2/8 ^ _ a^^ 22. a:2 + y2 = 5(a2 [^ fc^).
19. a?* + a^y2 + y4 = 4^^^ 23. Z^^^xy + Zf = 13.
Solve the following miscellaneous examples:
24. X2/ + 5. ^^ y + x_g
1 + 1=1. ^
x y 6' ar^2i/2 = i8^
25. ar^ + xy = 6. 28. a:8  2/3 == 7^
6t/2 = 8 — a:y. a: — 1/ = 1.
26. Q?^f^h. 29. K^  2/) = ic  4.
a + y = 3. a:3/ = 2a: + y + 2.
30. X + y = 2a.
0^2 + ^2 _ 2^2^
31. Find two nmnbers such that their sum is 14 and their
product is 48.
32. State Ex. 1 as a problem concerning two unknown
nmnbers. In like manner state Ex. 2. Ex. 8. Ex. 17.
33. Make up and solve a problem concerning two unknown
nmnbers such that the solution involves quadratic equations.
34. Point out the symmetrical equations in the examples
in this Exercise. ,
35. Make up (but do not solve) an Example in Case I;
one in Case II; and five different examples in Case III.
36. Practice oral work with fractions as in Exercise 58
(p. 190).
^
SIMULTANEOUS QUADRATIC EQUATIONS 375
Ex. Solve
Case IV
243. Solntion by the Substitutions, x^a{b and y^ab.
0^ + ^ = 242 (1)
. x + y^2 (2)
Substitute x=^a\h, y^ a— bin {I) and (2)',
Then, (o+6)» + (a 6)« 242
a^b+ab=>2
From (3), 2a«+20a»6»+ 10a6*= 242
From (4), 2a = 2
Divide (5) and (6) by 2, and substitute 1 for a in (5)
1+106»+56*=121
Hence, 6 = =*= 2, * ^/^
But a  1
Hence, . a:=a+6=» 3,1, l=fc\/— 6
y^ab^ 1,3,1=fV^
Let the pupil check the work.
.(3)
.(4)
.(5)
.(6)
RooU
EXERCISE 126
Solve:
1. of + j^^2U.
x + y ==4.
2. ar* + y* = 82.
x + y = 2.
3. 3l^ + V^ = 211.
x + y = 1.
4. ar* + 2^ = 257.
X — y = 5.
5. of y^ = 2.
X — y = 2.
6. ar* + 162^ = 97.
x + 2y == 5.
Work the following miscellaneous problems:
7. x + 4y = 14.
y2 + 4x=:2y + ll.
9.
y X 4
5aj2 + 3y2 = 15 + 4a^.
a:  2y = 2.
10. 31? + xy + y^ = 7.
a^ + a^ + y* = 91.
376
SCHOOL ALGEBRA
11. Solve Ex. 1. by dividing the first equation by the
second.
12. Make up (but do not solve) an example illustrating
each of the cases studied thus far in this chapter.
Case V
244. XTse of Compound Unknown Qnantities. It is often
expedient to consider some expression {as the sum, differencey
or product of the unknovm qumvtities) as a single unknown
quantity y and find its value y and hence the value of the unknown
quardities themselves,
V + y2=18a:y . . . . (1)
a:y = 6 ■ . . (2)
Multiply (2) by 2 2xy ^ 12 (3)
Add (3) and (1) x* + 2x2/ + y^ = 30  x  y W
Let X +y ^v
Then from (4), t^^ = 30  »
t;» + » = 30
V ^  6, 5
Ex. Solve
Hence, x + y = — 6
xy ^Q
/,x =  3 =*= V3
2/ = 3=F V^.
Let the pupil check the wor
Roots
also X +y ^ 5
xy =6
,\x =3, 21 p^^
y = 2, 3/ ^^^
EXERCISE 126
Solve and check:
1. x + y+Vx + y = 6.
xy = 3.
2. arV + xy ==6.
x + 2y =  5.
3. xy+\
x + y
/xy = 6
= 5.
•
SIMULTANEOUS QUADRATIC EQUATIONS 377
4. V^ + ^ = 6.
^ y y
X  2y = 2.
5. X — y + Va; — y = 6.
6. a:^ + y2 + a + y '= 24;
ary = — 12. .
7. ar* + y2 = x  y + 50. 9. (a:  i/)^  3(a:  y) = 40.
«y = 24. a:V  3a:y = 54.
8. a^y + 7a:y =  6. lo. a:^ + ^2 + 3. ^ 5^ ^ g^
5a? + «y = 4. a:y  2y =  2.
11. xV(a:* + y*) = 70.
a:V + a:*hy* = 17.
Also solve the following miscellaneous examples:
12. yjZl^i ^^ a: + y + Va: + y = 12.
X ' ' a:y = 20.
«y = 8. 17. a? + y2 = 125. • •
13. xy + 3y220 = 0. icy = 22.
a:^ _ 3a^ + 8 = 0. is. ^+^ = 3.
14. xV + 3a:y18. 2a:2 _ ^ ^ 20.
a; + 2y = 5.
a? . y^ 7
15. — + ^ =  .
19. ? + f=l.
a
y X 2 ? + *=4.
ic + y=l. X y
,20. Make up (but do not solve) an example illustrating
each of the cases studied thus far in the chapter.
378
SCHOOL ALGEBRA
Case VI
245. Factorial Solntions. The solution of a set of simul
taneous quadratic equations is often facilitated by the use
of factoring.
Thus, the solution of Ex. 9 of Exercise 122 (p. 368) may be short
ened by factoring in the first equation. The two following equations
will then be obtained:
Uxy) (x 2y) 0
1 2x + 3y = 7
Since the first equation is satisfied either when a? — y =« or
X — , 2y = 0, we obtain the following two sets of equations as equiv
alent to the original single set or system:
X —y 0
>+3y=7
xl]
Let the pupil check the work.
Whence,
Ana,
Whence,
x  2y =
2r+3y =7
x =21
yij
Am,
The solution of this example by the factorial method
requires less than one fourth the labor involved in the
sqlution by the method of Art. 238.
In general, if 4, 5, C, and D are algebraic expressions
integral, with reference to x and y, and if
\A'B'C = Q
Z) =
the given set of equations is equivalent to the following three
sets:
Z) =
5 =
2) =
C =
2) =
Hence, let the pupil state the sets of simple equations whose
solution is equivalent to the solution of
(x  2y) (3a: + y) (x  3t/) 
I 5xy 0
SlMULf ANBbtJS QXTADRAtlC EQUATIONS 379
246. Faotorial Kethod of Solution Aided by Bivirion.
It A, B, C, and D have the meaning given in Art. 245, and if
AC = B'D
AB
it is evident that this system of equations is satisfied either
when
or
5 =
C = Z)
A = B
Hence, the last two sets form a system equivalent to the
original system.
Note that the equation C = D is obtained by dividing the mem
bers oi A'C = 5 • D by the corresponding members of A = B.
Ex. Solve a? + x==9y^ (1)
x'+l^Qy (2)
• ■ ■
Writing equation (1) in the form x(x^ + 1) = 6^(7^), we obtain
the two systems which follow:
x« + 1 = 62/
Whence, x = 2 =»= \/3
Let the pupil check the work.
Roots
a:2f 1 =
Qy =
Whence, x
y
0
'1
Roots
EXERCISE 127
Solve by the factorial method:
2y = 4.
5xy + 4y^
31/ = 1.
a? + 3xy + 2f
x(x — y) = 0.
a? — 3xy — 4i/2
(ar2)(y2)
3a? — 4xy + y^
1. ar —
2. X —
3.
= 0.
= 0.
0.
0.
0.
5. (a;2)(y + 3) = 0.
(x  2y) {x + 2y) = 0.
6. \^:j?=^f. '
1 + a; = y.
7. y{x + 3) = 9a?  1.
2/ = 3x — 1.
8. ^ + y = 9a:^.
y2 + 1 = 6x.
380 SCHOOL ALGEBRA
9. Practice the oral solution of simple equations as m
Exercise 64 (p. 209).
10. Solve Ex. 1 of Exercise 123 (p. 370) by eliminating the
absolute term in the two equations and applying the factorial
method to the resulting equation.
11. In the same way work Ex. 2 of Exercise 123 (p. 370).
Also Ex. 3.
EXERCISE 128
Revibw
In solving a system of simultaneous quadratic equations, the first
thmg to notice is the degree of each equation.
Find the values of x and y:
1. 2x  52/ = 0. 9. (x  1) (y  2)  0.
x»  32/2 = 13. (2x 52/) (3a: 2/ +1) 0.
2. xf 2/ =2. 10. x^+y^ = 17.
?+?=6 x+2^=3.
^ y ' H. xy +2x =5.
3. 2x«  X2/ = 28. 2xy y ^3.
x» 4 22/» = 18. 12. x* + 2x  2/ = 5.
4. x  2/ + Vx^  12. 2x2  3x + 22/ = 8.
a; +2/ = 11. 13. (x + 2) (22/  1)  35.
5. x» + 2/* = 91. r xy x y =7.
x + 2/ = 1. ^ 14. x«2/*  5x2/ + 6 = 0.
6. 1+113 5X+32/14.
*X2/ ' \I 15. x«/»=« 3xy.
i36 a;2/«2.
^ 16. (x +2/)'  (a; +2^)  20.
^. x« + 32/2  28. 2x«  3x + 42/  14.
8.
x2+X3/+22/«16. a:! y« 3 .
2x X +y y X 2
y ' ^ ' 1 + 1,4!.
X —y +a 0. X y 2
SIMULTANEOUS QUADRATIC EQUATIONS 381
18. x^+i/^+x+ZylS,
»y — y =» 12.
19. ari + jr* » 2.
ar« + r* = 2i.
20. ar^^y^ '^^ 13.
6x2/ ~ !•
21. X* + y* = 13(6* H 1).
a; +2/ = 56 — 1.
22. a;« + y> + X + 2/ = 14.
xy \x \y = 5.
23. 3x*  35 = 5xy  7j/«.
2x* — 35 = y* — xy.
24. tt + ar + ar* = 65.
a + ar' = ^ar,
25. 11 = a + 2(n  1).
36=?(a + ll).
26. X*  y* = 2.
X  2/ = 26.
27. 12=3.
X 2/
X* y*
28. X + 2/  65.
'V^ + v^y = 5.
29. X + Vxy 4 2/ = 14.
X* + X2/ + y' =84.
30. x*+42/*+80*=15x+302/.
X2/60.
31. X y ^ V^{ Vy.
a;*  2/* « 37.
32. x*42/* 4.
33. tX^^y^ ^ 17.
xy » 2.
34. x» + 2/*  a^y + 7.
X — y = xy — 5.
35. X* = 4(a2 + 6*  y*).
xy = 2ab,
36. xi 4 2r^ = 5.
(x + i)i + (2/ + i)*=H.
37. x 4 :=2/(aJ 2).
2/  8 = x(2/  2).
38. 2xi + 52r* = 4.
X* — 2xV"* + 2r* = 1
39. xy + X + y = 5.
x*y H X2/*  — 84.
40. ax ^hy .= 0.
(ax 2) (62/ +3) = 2.
41. x»+x =92/.
x2 + 1 = 62/.
42. x« + 2/* = 3x2/  4.
x*+2/^ =272.
43. a(x — o) = 6(y — 6),
xy = ox + 62/.
44. X* + aV = ^dh^\
3x + ay = 5.
45. X2/ = a*.
x« = 6*.
2/« ■= c*
46. xy + x« = a.
xy + 2/2 = 6.
xz +yz =^ c.
47. 3^«2
aj+2/ 3
xz 3
X +z 4
2/g ^5.
2/ +« 6
382 SCHOOL ALGEBRA
48. Make up (but do not solve) an example of each of the
and principal subf onns in each case, treated in this chapter.
EXERCISE 129
1. The sum of the squares of two numbers is 58, and their
product is 21. Find the numbers.
2. Separate 32 into two parts such that their product shall
be 112.
3. Two numbers when added produce 5.7, and when
multiplied produce 8. Find the numbers.
4. What are the two parts of 18 whose product exceeds
8 times their difference by 1?
5. The sum of two numbers increased by three times their
product is 83; also three times the less number exceeds the
larger number by 1. Find the niunbers.
In working the following examples concerning rectangles, draiw a
diagram for each rectangle considered.
6. The area of a rectangle is 84 sq. ft. and the distance
around it (perimeter) is 38 ft. Find the length and breadth
(dimensions) of the rectangle.
7. The diagonal of a rectangle is Vf . If each side of the
rectangle were increased by 1, the area would be increased
by 3. What are the sides?
8 The area of a rectangular garden is 1200 sq. yd. If
tho width were increased by 5 yd. and the length by 10 yd.,
the area would be 1750 sq. yd. Find the dimensions of the
rectq^ngle.
9. The area of a (double) tennis court is 312 sq. yd., and
the perimeter is 76 yd. Find the dimensipiis of the court
in feet.
SIMULTANEOUS QUADRATIC EQUATIONS 383
10. If the dimensions of a rectangular field were each in
creased by 3 rd., its area would be 140 sq. rdc; but if its width
were increased by 8 rd. and its length diminished by 2, its
area would be 135 sq. rd. Find its actual dimensions.
11. A rectangular lot containing 270 sq. rd. is surrounded
by a road 1 rd. wide; the area of the road is 70 sq. rd. Find
the dimensions of the field.
12. A hall of 90 sq. yd. can be paved with 72fO rectangular
tiles of a certain size, but if each tile were 3 in. shorter and
3 in. wider, it would require 648 tiles. What is the size of
each tile?
13 . The area of a given rectangle is 800 sq. ft. If the length
of the rectangle were increased by 20% and the width by
4 ft., the area will be increased by 44%. Find the dimen
sions of the rectangle.
14. If .a train had traveled 6 miles an hour faster, it would
have required 1 hour less to run 180 miles. How fast did
it travel?
Sua. Let x = the number of miles the train travels per hour at
first, and y = the nimiber of hours it travels. Then what will rep
resent the number of miles per hour and the number of hours at
the second rate?
15. A gentleman distributed S9 equally among some boys.
If he had begun by giving each boy 5 cents more, 6 of them
would have received nothing. How many boys were there?
16. A number of men agreed to buy a boat for $7200,
but 3 of their number died, and each survivor was obliged
to contribute $400 more than he otherwise would have done.
How many men were there?
i7* A certain club owes a debt of $400, but is informed by
the tDcasuier that if 5 new members are admitted, the asses»'
384 SCHOOL ALGEBRA
ment to meet the debt will be $4 less per member. How
many members has the club?
18. The price of photographs is raised $3 per dozen, and
customers consequently receive 10 photographs less than
before for $5. Find the old and new price for a single photo
graph.
19. A certain number of eggs cost a dollar, but if there
had been 10 more eggs at the same price, they would have
cost 6j5 a dozen less. What was the price of a dozen eggs?
20. A given fraction when reduced to its lowest terms
equals f . Also if 3 is subtracted from the numerator of the
fraction, the fraction is the same as if 6 had been added to
its denominator. Find the fraction.
21. The niunerator of a given improper fraction exceeds
its denominator by 1. Also the given fraction exceeds its
reciprocal by yV«' Kiid the fraction.
22. The sum of the numerator and denominator of a cer
tain fraction is 8, and if 2^ be added to each term of the frac
tion, its value will be increased by y^. What is the fraction?
23. A baseball nine has \^on f of the games played. If
it should play 16 more games and win half of them, its aver
age of games won would be  of what it would be if it should
play 8 more games and win all of them. How many games
has it played, and how many has it won?
24. A certain number of two figures, when multiplied by
the left digit, becomes 56; but when multiplied by the right
digit, it becomes 224. Find the number.
25. Make up and work an example similar to Ex. 24.
26. A man finds that he can row 12 miles down stream in
2 hours, but that it takes him 4 hours to row 6 miles down
SIMULTANEOUS QUADRATIC EQUATIONS 386
stream and back. Find his rate in still water and the rate
of the stream.
27. A crew rowing at  their usual rate took 32 hours to
row down stream 48 miles and back to startingplace. Had
they rowed at their usual rate it would have taken 18 hours
for the same circuit. Find their rate and that of the stream.
28. Two trains traveling toward each other left, at the
same time, two stations 240 miles apart. Each reached the
station from which the other started, the one 3 hours, and
the other if hoiu^, after they met. Find their rates of run
ning.
29. The difference of two numbers is 5, and the difference
of their cubes is 215. Find the numbers.
30. Divide the number 12 into two parts such thfit the
sum of the fractions obtained by dividing 12 by the parts
shall be f .
31. Find two munbers whose product is 42, such that if
the larger be divided by the less, the quotient is 4 and the
remainder 2.
32. In placing telephone poles between two places, it was
found that if the poles were placed 10 ft. further apart than
was originally planned, 4 poles less per mile were needed.
How far apart were the poles placed at first?
33. A girl has 12,000 words to write. K she uses a type
writer she can write 25 words more per minute than she can
with the pen, and it will take 8f hours less to write the 12,000
words. What is her rate per minute with the pen?
34. Two square plots contain together 610 sq. ft., but a
third plot, which is 1 ft. shorter than a side of the larger
square, and 1 ft. wider than the less, contains 280 sq. ft.
What are the sides of the two squares?
386 SCHOOL ALGEBRA
35. Find two fractions whose sum is equal to their product,
and the difference of whose squares is f of their product.
36. A man finds that he can row 8 miles up stream in 4
hours, but that he can row 8 miles down stream and back in
5 hours. Find his rate in still water and also the rate of the
stream.
37. The area of a given rectangle is 2400. If its length
were increased by 50% and its width by 20 linear units,
the area of the rectangle would be increased by 125%.
Find the dimensions of the rectangle.
38. The hypotenuse of a right triangle is 20 and the sum
of the other two sides is 28. Find the length of the sides.
39. The fore wheel of a carriage makes 28 revolutions
more than the hind wheel in going 560 yd., but if the drcum
ference of each wheel were increased by 2 ft., the difference
would be only 20 revolutions. What is the drciunference
of each wheel?
40. Find two numbers such that their sum is a and thdr
product 6.
41. Why are we able to solve many of the problems in this
exercise by algebra and not by arithmetic?
42. Make up and work an example similar to Ex. 6. To
Ex. 17. Ex. 24,
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Graphs of Quadratic and Higher Equations
847. Oraph of a Quadratio Equation of Two Unknown
ftuantities.
Ex. 1. Construct the graph of y » a? ^ Sx + 2.
The graph obtained is the curve ABC.
A curve of this kind is called a parabola.
The path of a projectile, for instance that
of a baseball when thrown or batted (re
sistance of the air being neglected), is an
arc of a parabola.
X
y
2
1
2
3
2
1
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X
y
1
6
2
12
etc.
etc.
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388
SCHOOL ALGEBRA
It will be noted that the above method of graphing is the
same as that given in Art. 148 (p. 255)^ but that here it is
sometimes advantageous to let x have fractional values as
hf i> lV> f > etc. The observant pupil will also find methods
of abbreviating the work in certain cases.
In general, it will be found that the graph of a quadratic
equation of two unknown quantities is a curved line, and, in
particular, either a circle, parabola, ellipse, or hyperbola.
Ex. 2. Construct the graph of 4aj* — 9y^ = 36.
X
y
imag.
1
imag.
2
imag.
3
4
*L7
5
=fc2.6
6
*3.4
e1
be.
For negative values of x, the values of y are the
same as for the corresponding positive values of x.
Hence, the graph is a curve of two branches, ABC
and A'B'C'f of the species known as the hyperbola.
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GRAPHS OF QUADRATIC EQUATIONS 389
EXERCISE 130
Graph the following:
rn. y = a? — h xB. y^ = 4z.
L^y = a?2xS. 14. f  a? = 9.
3. y = a? — 4x + 4:. 15. a:^ — / = 9.
4. y = a:^ + 3a; — 4. 16. ary = 4.
5. y = ioi^. 17. xy = 1.
6. y = a? + 1. le. ocy = —2.
, 7. a? + y^ = 16. 19, X + ocy = 1.
8. x2 + y2 = 9. 20. x'+iy^y^b.
9, y^ = 4x — a?. 21. 92/2 — ar^ = — 9.
10. 9f + ix^ = 36. 22. f = ix + 4:.
11. a:^ + 16^2 = 16. 23. x^ — xy + y^ = 25.
12. 16x2 + y2 = 16. 24. 2/2 = 4.
25. a:2 _ 4a. + 3 = 0.
SuG. Show that whatever the value of y, x always = 1 or 3;
hence the graph is two straight Unes parallel with the yajoa.
26. Make up and work an example similar to Ex. 1. To
Ex. 7. Ex. 8.
Ex. Solve graphically
248. Graphic Solution of Simultaneous Quadratic Equations.
x + y = 1.
Constructing the graph of x^ + y^ ^ 25, we obtain the circle
ABC (p. 390). Constructing the graph of x + y = 1, we obtain the
straight line FH.
Measuring the coordinates of the points of intersection of the
two graphs, we find the points to be (4, —3) and (—3, 4).
These results may be verified by solving the two given simul
taneous equations algebraically.
390
SCHOOL ALGEBRA
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249. Special Cases; Imaginary Boots. Construct the
graphs of a? + y^ = 4 and a; + y = 3. You will find that
these two graphs do not intersect. Then solve the given
equations in the ordinary algebraic way. You will find
that the roots are imaginary. If you treat the equations
a? + y^ = 1 and 43^ + 9y^ = 36 in the same way, you will
obtain a similar result.
In general, imaginary roots of simidtaneous equaticms cor
respond to points of nonr4ntersection of the graphs of the given
equations.
Remember that in solving a pair of simultaneous eqiiations, the
nimiber of values of x (and also of y) is equal to the sum of the de
grees of the two equations. Hence, if two simultaneous equations
are both of the second degree, their graphs should intersect in four
points; and if their graphs are found to intersect in only two points
for instance, the other two points must correspond to imaginary
roots.
GRAPHS OF QUADRATIC EQUATIONS 391
The pupil may illustrate this by graphing and also solving alge
braicsJly y^ ^ 4x and x* +y^ = 25.
EXERCISE 131
Solve both graphically and algebraically:
1. y^ = ix. 13. xy ^ — 2.
y — X = 0. x + y = 1.
2. ^ = 4a;. u. 31? + y^ = 25.
y = 2x. xy = 12.
3. y^ = 4a;. 15. x^ + 2/2 = 25.
, x + y = Q. xy = 1.
/i. f ^ X. 16. x + y =  2.
\
x + y = 2. . xy = —S.
5. y^^2x. i7. 2/24a; = 0. ^
y = X  4. 3f + 2x^ = U. \
= 25.\
2y — X = 5, \ y = 2x.
t. a? + ir' = 25.\ 18. a? \^ f  lOx = 0.
/
7. ar* + 1^* = 25. 19. ic* + jr* = 16.
"^^^. ^ ia;  5 = g / ar^ + 4t/2 == 43,
8. a:* + y* = 25. . 20. By*  2a:2 = 12.
y = X. 3? \y^ = \&.
9. a:2 + y2 = 25. 21. 3a^ + 2^* = 3.
3y — 4a; = 0. ' y = a; + 2.
10. a:?/ = 3. "" 22. y — a: — 6 = 0.
x + y = 4. a? = Qy — y^.
11. a;y=3. 23. 4a:2 _ 92^2 = 3g^
a: + y = 2. a:2 + 2/2 = 1.
12. 33/ =  2. 24. 3^2 + 2^ _j_ a 4. 3y = 18.
a; + y =  1. xy  y = 12.
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392 SCHOOL ALGEBRA
25. Solve graphically a? + y = 7.
x + y^ = 11.
Also try to solve this pair of equations algebraically.
26. Make up and work an example similar to Ex.* 1. To
Ex. 6. Ex. .10.
250. Graphic Solution of a ftuadratic or Higher Equation
of One Unknown ftuantity. By substituting for y in the
first equation, the pair of equations y = a:^ — 3a: + 2 and
y = reduces to a:^ — 3a: + 2 = 0. Accordingly, the graphic
solution of an equation like a:^ — 3a: + 2 = is obtained by
solving graphically y = a:^ — 3a: + 2 and y = 0.
In other words, the roots of a quadratic equation of one urir
knovm quantity , asi? + fea: + c = 0, are represented graphically
by the abscissas of the points where the graph ofy = aa? + bx+c
meets the xaxis.
Ex. Solve graphically ar^  3a: + 2 = 0.
The graph of y = a:^ — 3a: + 2 is the curved line ABC of the
figure in Art. 247 (p. 387).
This curve crosses the xaxis at the points (1, 0) and (2, 0).
.*. a: = 1, 2 Roots
The same results are obtained by solving the equation a:* — 3a: + 2
= algebraically.
This method of solution applies also to a cubic equation or
to an equation of one unknown quantity of any degree.
Thus, to solve the equation a:* — Sa;^ + 5x — 2 = 0, graph the
equation y = a:* — Sa:* + 5a; — 2. The abscissas of the points where
this graph crosses the a;axis have the same value as the roots of
the given equation a:* — 3a;2 + 5a: — 2 =0.
EXERCISE 132
Solve both graphically and algebraically:
1. ar^  4 = 0. • 3. 4a:2 ±^^^^y
.2. a:^ _ 3a. _ 4 =, 0, 4. ar^  6a: + 9^il^
GRAPHS OF QUADRATIC EQUATIONS
393
5. ar^  4a; + 1 = 0. 6. a:^  2a: = 0.
7. a:^ _ 2a;  1 = 0.
SuG. Make the algebraic solution by the factorial method.
B. a?2? 6x = 0. 9. a;3_2a;2_2a; + 4 = 0.
10. a!^5a? + 4: = 0.
11. Make up and work an example similar to Ex. 1. To
Ex. 2. Ex. 10.
Some Applied Graphs
251. Wider Application of Graphs. Besides their use in
ordinary algebra, graphs may be used to represent the prop
erties of a great variety of functions, in particular those occur
ring in the various departments of science and in business life.
Sometimes it is found convenient to use a different scale
in laying off magnitudes on one axis from that used on the
other axis.
EXERCISE 133
1. Graph C = f (F — 32), making the scale on the Caxis
one half as large as that on the Faxis.
2. A thermometer reads as follows at different hours during
the day:
Hour ....
7 a.m.
8 a.m.
9 a.m.
10 A.M.
11 A.M.
12 a.m.
1p.m.
2 p.m.
Temperature .
50°
51°
54°
59°
65°
71°
75°
78°
Hour ....
3 p.m.
4f.m.
5 p.m.
6 P.M.
7 P.M.
8 P.M.
9 p.m.
10 P.M.
Temperatiu^ .
78°
77°
71°
65°
60°
57°
55°
51°
Construct a graph showing the relation between the tem
pierature above 60° (taken as plus) and that below (taken as
minus), and the hour of the day. Then point out some facts
to be learned from this graph.
1 1
394
SCHOOL ALGEBRA
3. Graph F = p, and on the graph obtained measure the
value of V when P = 1.5.
4. Construct the graph of ^ = 16.1^ making the ^scale
but one tenth as large as the ^scale.
5. The average temperatui:e on the first day in each month
for the last thirty years in New York City has been as fol
lows. Graph these data.
New York
Date .....
Jan. 1
Feb. 1
March 1
April 1
Mayl
June 1
Temperature ;
31°
31°
35°
42°
64°
64°
■
Date
Julyl
Aug. 1
Sept. 1
Oct. 1
Nov. 1
Dec. 1
Temperature
71°
73°
69°
61°
49°
39°
The corresponding temperatures in London were as follows :
London
Date
Jan. 1
Feb. 1
March 1
April 1
May 1
June 1
Temperature .
37°
38°
40°
45°
50°
57°
Date
Julyl
Aug. 1
Sept. 1
Oct. 1
Nov. 1
Dec. 1
Temperature .
62°
62°
59°
54°
46°
41°
Graph these results on the same paper with the graph of
the New York temperatures and then compare the two curves
of annual temperature, and give three facts which may be
inferred from these ciu^es.
6. The following table shows the number of years which
a person having attained a certain age may expect to live.
Construct a graph of life expectancy from the data.
GRAPHS OP QUADRATIC EQUATIONS
395
Age in Years
38.7
2
47.6
4
50.8
6
51.2
8
50.2
10
48.8
20
41.5
30
Life Expectancy in Years
34.3 1
Age in Years
40
27.6
50
21.1
60
14.3
70
9.2
80
5.2
90
3.2
100
2.3
Life Expectancy in Years . .
From this graph determine your life expectancy at the
present time, and also that .of several acquaintances of
various ages.
7. Graph the growth of the population of the United States
using the following table:
Year
1790
1800
1810
1830
1830
1840
1850
Millions .....
4
5
7
10
13
17
23
Year
1860
1870
1880
1890
1900
1910
Millions
31
39
50
63
76
92
From your graph determine as accurately as you can the
population in 1815. In 1835. In 1895. In 1905.
From your graph determine as nearly as you can in what
year the population was 35 millions. 70 millions. 80 millions.
8. The following table gives the amount of $1 at simple
interest, and also at compound interest at 4% for 5, 10, 15,
20, etc. years. On the same diagram draw (1) a ^raph of
the amounts at simple interest (2) a graph of the amounts at
compound interest.
' Years . . .
$1
1
5
10
15
20
25
30
35
Amounts at
Simple Int.
$1.20
$1.40
$1.60
$1.80
$2.00
$2.20
$2.40
•
Amounts at
Com. Lit.
1.22
1.48
1.80
2.19
2.67
3.24
3.95
396
SCHOOL ALGEBRA
The amounts of $1 at 5% for the same periods of time at
compound interest are $1, $1.28, $1.63, $2.08, $2.65, $3.39,
$4.32, $5.52. On the same diagram make a graph of these
amounts.
9. The following table gives the pressure for various
velocities of the wind:
Velocity of wind
in mi. per hr.
10
1.5
20
2
30
4.5
40
8
50
60
18
70
80
32
90
100
50
Pressure in lb.
per sq. ft.
12.5
24.5
40.5
Graph the above table of facts. From this graph deter
mine as exactly as you can the pressure when the velocity of
the wind is 25 mi. per hour. 45 mi. 65 mi.
10. From the same table determine approximately the
velocity of the wind when the wind pressure is 5 lb. per square
foot. 101b. 301b.
11. Graph y = or. 12. Graph y = o;^.
13. Construct the parallelogram whose sides are the
graphs of the equations 32/ — 4a: — 13 = 0,32/ — 4a: + 19=0,
2/ = 3, 2/ = — 1. Find the coordinates of the vertices of this
parallelogram, and also its area.
14. Who first invented graphs, and when?
15. Graphs, or geometric pictures of numerical data, take
many different forms beside the linear graphs treated in this
book. For instance, the density with which the national
banks are distributed over different parts of the country may
be indicated by dots scattered over a map. Collect examples
of as many different kinds of graphs as possible.
CHAPTER XXn
GENERAL PROPERTIES OF QUADRATIC EQUATIONS
252. Character of the Boots Inferred from the Coefficients.
It is important to be able to infer at once from the
nature of the coefficients of an equation whether the roots of
the equation are equal or unequal, real or imaginary, positive
or negative.
Any quadratic equation may be reduced to the form
a«^ + 6x + c = 0, in which a is positive.
Solving da? + bx + c = 0, and denoting the roots by fi, r^
(read, r subone, r subtwo), we obtain
 6 + V6«  4ac  6  Vb^  ^ac
fi , r2 =
2a 2a
From these expressions wd infer that
1. IfV — iac is positive, the roots are real and unequal.
2.1fh^ — 4ac equals zero, the roots are real and equal.
S.Ifb^ — 4a€ is negative, the roots are imaginary.
The roots are rational if 6^ — 4ac is a perfect square or
zero.
Since the character of the roots is thus determined by the
value of 6^ — 4ac, this expression is termed the discriminani
oi a>s? + bx + c = 0.
Ex. 1. Determine the character of the roots of the equa
tion, 01? — 2x — 1 = 0.
We have, a«l, 6= — 2, c=— 1
'624ac4+4=8
Hence, the rootig ^x^ r^ wi unequal,
?97
398
SCHOOL ALGEBRA
Ex.2. Qfa?2a; + l = 0.
Here, o= 1, 6= 2, c1,
6«4ac=44«0.
Hence, the roots are real and equal.
Ex. 3. Of a?  ac + 2 = 0.
Here, a « 1, 6 =  2, c = 2,
6«4ac=48=4.
Hence, the roots are imaginary.
The results obtained in Exs. 1^ 2, and 3 may be con
veniently illustrated by
means of graphs.
It is found that the graph
of y = X* — 2x — 1 is the
curve (1) and crosses the a>
axis at the two points A
and B (correq)onding to the
two roots of 7? —2x — 1 =0).
The graph of yx^—2x
+ 1 is the curve (2) and
meets the xaxis at only one
point (corresponding to the
two equal roots of x* — 2aj
h 1  0).
The graph of y=x*— 2x
+ 2 is the curve (3) which
does not meet the a>axis at
ail (which illustrates the fact
that the roots of the equation x^ — 2x + 2»0are imaginary).
253. Determining Coefficients so that the Boots shall satisfy
a Given Condition. It is often possible so to determine the
coefficients of an equation that the roots shall satisfy a given
condition.
Ex. Find the value of m which will give equal roots for
the equation (m — 1)qi? + ttix + 2m — 3 = 0.
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PROPERTIES OF QUADRATIC EQUATIONS 399
By Art. 252, 2, in order that the roots may be equal, ¥ —4ac =0.
In the given equation, a = m — 1, b^m, c = 2»i — 3.
/. m« 4(m 1) (2m 3) =
m«8m»+20w 12=0
7m«20m=12
w» = 2, f Ans.
Check. Substituting these values for fn in the original equation,
in each of which equations the roots are equal. /
EXERCISE 184
Without solving, determine the character of the roots in
1. a:^^ 53.^6 = 0. ^ ^ 9q^ + 4:
7. a: = —
2. 3ar^  7ar— 2 = 0.
12
3. 4ar^ = 4a:  1. 8. ea:^ + ^^^ = lOx.
4. 3x2 ^ 2a; + 1 = 9. a ^ 2 (^ ^ 1^^
5. 2a:2^5^ + 3^0. /^o. 35ar + 18 + 12a:2 ^ 0.
X 1 11. iar^ = 2a;  3.
. • 3 " • 12. 7a;2 + i = 53.^
13. Determine by inspection the nature of the roots in
(1) ,a?4x + 2 =
(2) x^4:X+A =
(3) x^4x + 6 = a
Verify your results by use of graphs.
14. Make up and work an example similar to Ex. 13.
15. In 3a;2 — 2a; + 1 =0, determine the character of the
roots by solving the equation. Now determine their char
acter by the method of Art. 252. Compare the amount of
work in the two processes.
'  w^ > ^ '■
400 SCHOOL ALGEBRA l<i '^'
Determine the value of m ior which the roots of each equa
tion will be equal:
16. 2a:^ — 2a: + w = 0. 19. 4«^ + ^ = wx.
17. ma:^ — 5a: + 2 = 0. 20. (m + l)a:^ + ma: = 1.
18. 2ar^  ma: + 12^ = 0. • 21. (m+ l)ar* + 3m=12a:.
22. (m + \)q? + (m — l)a: + m + 1 = 0.
^ 2ar^ + 3a: + 2  1 ' ^ ^
^ 23. ■ ■ — = — •
7 + 3a: — a:^ m
24. What is meant by a root of an equation?
25. In Ex. 16 for what values of m are the roots imaginary?
Real and unequal?
26. Answer the same questions for Ex. 17. Ex. 20.
27. Show that if one root of a quadratic equation is imag
inary the other root must be imaginary also.
28. State and prove a similar fact concerning irrational
roots.
29. In a:^ — 6a: + c = 0, substitute a value of c which
makes (1) the roots of the resulting equation equal; (2)
also another number which will make roots imaginary; (3)
real, unequal, and rational.
30. How many of the above examples can you work at
sight?
254. Eelation between Boots and Coefficients. In Art. 252
a method was obtained of inferring from the coeflScients
of a quadratic equation, the (pmlitative nature of the roots.
A more exiact, or qiuintitative, relation between the roots and
coefficients will now be obtained. This relation will enable
us in any given equation to determine the sum or product
PROPERTIES OF QUADRATIC EQUATIONS 401
(and often other functions) of the roots, without the laboi" of
solving the equation.
Dividing both members of oc^ + 6a: + c = by a, we
obtain an equation in the form a? + px + 9 = 0.
Solving this equation and denoting its roots by a and /3,
2 "^" '^ 2
Adding the roots, a+/3=— — ^= — p
Multiplying the roots, ai5 = ■ ^^~ = 9
Hence, in general,
(1) Thje sum of the roots of x^ + px + q = O equals — p.
or the coefficient of x with the sign changed;
(2) The product of the roots equals the knovm term q.
Ex. 1. Without solving the equation, find the sum and
also the prodiict of the roots of 5(1 — 2x) = Sx^.
The given equation reduces toa:*H~a;— =0.
o o
10 5
Hence, sum of roots = — «> product of roots = — 5 Ans.
— 1 sir V— 3
Ex. 2. Form the equation whose roots are «
The roots are
Hence, sum of roots
2 ' 2
2 2
Product of roots  l(3) ^l + 3^ ^
4 4
Hence, a:*h«+l = is the required equation.
Checks for the above example may be obtained by solving the
equations obtained.
\
% —  • 
402 SCHCX)L ALGEBRA
265. Factoring a Quadratic Expression. Any quadratic
expression may be factored by letting the given expression
equal zero, and using the property stated in Art. 254.
Ex. Factor Sx^ix + d.
Take 3(x*x + t) =
Solve x^ix+i^O
2=fc V^^^
Whence, x =
3
Hence, the factors of 3x* — 4a; + 5 are
EXERCISE 136
Find, by inspection, the sum and product of the roots in
each of the following equations:
1. ar^ + 3a: + 6 = 0. e. a^a?  ax + 2 = 0.
2. ar^a: + 7 = 0. 7. 5a:4ar^=l.
3. x^5x = 10. 8. 3  7a: = 11a:*.
4. 2x2 _ 6a; _ 3 = 0. Q X _x  a
, ^ a:5a:2 a 4a: + 1
5. a: + l = r'
xl 10. 1 2ca:2aa:2 = 3c.
Form the equations whose roots are
11. 2, 3. 17. .08, .2.
12. 3, 2. 18. a6, a.
13.  1, 5.
14. 5, .04.
V^l
19. T. • ^ 2
^^  2 =b ^/zr2
15. 1, }. 20. 1 + ^^,1^2:^*
2
". f, i 21. 3^ Vs. 25. Ja=bcVb.
26. Form an equation whose roots are 2 =*= t (see p. 335)
PROPERTIES OF QUADRATIC EQUATIONS 403
27. Form an equation whose roots are 3, i, — i. ,
28. If one root of the equation a^ — ^^ — f = is 3,
find the other root in two different ways.
29. If one root of the equation 3a:* — 4a; + 2 = is
, find the other root in three different ways.
30. Form the equation in which one root is — f and the
product of the roots is — f .
31. Make up and work an example similar to Ex. 28.
Also to Ex. 30.
Factor:
32. 3a? 10a: 8. 35. ar^ + 146a:.
33. a:* + 2a:  1. 36. 25a:2 + 2  30a:.
34. ar^  a:  1. 37. 3a:  33:* _ i
38. If f and 8 represent the roots of 3a:^ — 8a: + 5 = 0,
find, without determining the actual roots, the values of
r + 8; rs; r^ + 3^; rs; r^3^; j^ + s^;  + ; ; ^i'
T 8 r 8 r* 8r
39. Find the values of the sanie expressions for the equa
tion 2a:2 « 9a. + 7 = 0. Also for 6ar^  a:  12 = 0.
40. Find the values of the same expressions for the equa
tion aa? + 6a: + c = 0. Also for the equation a? + px + q =0.
41. If m and n represent the roots of the equation lOa:^ + 9a:
— 7 = 0, form that equation whose roots shall be mn and
. 1.1
nu+ n, m — n and — I —
m n
42. In oa:^ + 6a: + c = 0, if c = a show that one root is
the reciprocal of the other.
404 SCHOOL ALGEBRA
43. Fpr what value of p does the equation a? + (7p 
3)x — (52? + 10) = have a zero root? Find the other
root.
Suo. If one root is zero, what does the product of the roots
equal?
44. How many of the examples in this Exercise can you
work at sight?
45. Practice oral work with exponents as in Exercise 93
(p. 303).
CHAPTER XXIII
RATIO AND PROPORTION
Ratio
356. The Ratio of two algebraic quantities is their exact
relation of magnitude. It is the indicated quotient of the
one quantity divided by the other, expressed either in the
form of a fraction or by the symbol : placed between the two
quantities.
Thus, the ratio of a to 6 is expressed a8 f, or a« a : 6.
257. Utility of Ratios. Ratios have the same uses as
fractions (see Art. 110, p. 166). These uses are extended by
selecting important kinds of ratios, naming them (see Art.
259), and working out their properties once for all. Also
properties of equal ratios are worked out once for all and
stated in such a form as facilitates their application to prob
lems.
258. The Terms of a Ratio are the two quantities com
pared. The antecedent is the first term. The consequent is
the second term.
The terms of a ratio ^must be expressed in terms of a common
unit. Thus, to express the ratio of 3 qt. to 2 bu. either the quarts
must be expressed as bushels or the bushels as quarts.
If two quantities, as 5 in. and 2 bu., cannot be expressed in terms
of the same unit, no ratio between them is possible.
405
406 SCHOOL ALGEBRA
259. Kinds of Batio. An inverse ratio is a ratio obtained
by interchanging antecedent and consequent.
Thus, the direct ratio of a to 6 is a : 6; the inverse ratio of the
same quantities is & : a.
A componnd ratio is one formed by taking the product of
the corresponding terms of two given ratios.
Thus, ac :bd\s the ratio compounded of a : 6 and c : d.
A duplicate ratio is formed by compoimding a ratio with
itself.
Thus, the duphcate ratio of a : 6 is a* : 6*. In like manner, the
triplicate ratio of a : 6 is a^ : 6^.
A commensurable ratio is a ratio that can be expressed in
terms of two integers.
Thus, •=! equals o ^ r» or —  . Or  is a commensurable ratio.
An incommensurable ratio is a ratio which cannot be
expressed in terms of two integers.
Thus, ^ equals h^l^Jl . The fraction in the numerator can
5 5
not be completed so that the numerator and denominator can be
. expressed as a ratio of integers in terms of the same unit; hence
~— is an incommensurable ratio.
5
The properties of inconoLmensurable ratios are obtained
from those of conmiensurable ratios by an indirect method
not discussed in this book.
260. Fundamental Property of Eatios. // both antecedent
and consequent of a ratio are multiplied or divided by the same
quardity, the wlue of the ratio is not changed,
^ . a ma
tor, smce a ~ ~T
a : b has the same value as ma : mb.
RATIO 407
EXERCISE 186
Simplify each of the following ratios:
1. lft.8in.:4in. «• ^\:{x + l)K
2. 7^:f. 6. a^V" . {ha)\
{a + hT a^ + V
3. 1 gal. : 1 qt. Ipt. ^ qj^ . g^
4. 37i%:12i%. a (aS)iO:a«.aio.
9. Find the ratio of 4a:^ to (4a:)^, when a: = 6. When a; = J.
10. In the year 1910 an automobile went 1 mi. in 27^ sec.
How many feet per second was this?
11. Find the ratio of the area of a rectangle 3 yd. long and
2 ft. wide to the area of a rectangle 2 ft. long and 18 in. wide.
Find the ratio of a: to y from
12. 7a; — 3y = 4r + 2/. ^^ 3a; — 2y __ a
13. 4a; — 5y : 5a; — 4y = f . 4a; — 3y. h
15. Q?\'Qy^ = 5ocy.
16* A horse can pull 2 tons on a level macadam road, 15
tons on a level iron track, and 70 tons on a canal. Find the
ratio of each pair of these numbers.
17. The death rate in London in the year 1700 was ap
proximately 80 per thousand. In the year 1908 it was 14.3
per thousand. In a population of 5,000,000 people how many
lives are saved per annum by the diminished death rate?
18. "What is meant by the specific gravity of iron (or of
any material)? If a cubic foot of water weighs 62.5 lb., find
the weight of brick whose specific gravity is 2.3 and which
fill a wagon 6' X 3' X 2'.
19. If the weight of the human brain is ^ the weight of
the body; while the blood in the brain is  of all the blood in
the body, the density of the blood in the brain is how many
408 SCHOOL ALGEBRA
times that in the body as a whole? Of that in the rest of
the body?
20. If a ratio is less than unity, does adding the same
quantity to both the terms of the ratio increase or diminish
the value of the ratio? By how much if the ratio is t and c
be added to both terms?
21. Answer the same questions if the given ratio is greater
than unity.
22. Find out, if you can, what is meant by a nutritive
ratio and make up and work three examples concerning such
ratios.
23. Make up and work an example similar to Ex. 3. To
Ex. 10. Ex. 11.
24. How many of the examples in this Exercise can you
work at sight?
Proportion
261. A Proportion is an expression of the equality of two
or more equal ratios; as ^ = ;»> or a : 6 = c : d.
The above proportion is read "aisto6ascisto d"
262. Terms of a Proportion. The four quantities used in
a proportion are called its terms, or proportionals.
The first and third terms are the antecedents.
The second and fourth terms are the consequents.
The first and last terms are the extremes.
The second and third terms are the means.
Ina :b = c :d, dis Si fourth proportional to a, b, and c.
RATIO 409
263. A Continued Proportion is one in which each con
sequent and the next antecedent are the same; as
a:b = b :c == c :d = die.
In the continued proportion a : 6 = 6: c, 6 is called a mean
proportioned between a and c; c is called a third proportional
to a and b.
Two proportions of the form x :y = a :b, and y :z = b :c
may be combined in the form x :y :z ==^ a :b :c.
264. Equal Products made into a Proportion. // the
product of ttoo qujardities is equal to the product of two other
quantitieSy either two may be made the means, and the other two
the extremes of a proportion.
For, if ad^bc
Dividing by 6d, h~ 1 (^^' 1^* ^)
.'. a:b = c:d
265. Fundamental Property of Proportion. For algebraic
purposes, the fundamental property of a proportion is that
The product of the means is equal to the product of the extremes.
For, if
then
a lb
a
b
= c :d
c
"d
Multiplying bj
rbd,
ad
= bc
(Art.
15,5)
In like manner
,if
a :b
= 6 :c
62 =
ac
.'.b =
Vac
This property
enables us '
to convert a proportion
into
an
equation, and to solve a given proportion by solving the
equation thus obtained;
410 SCHOOL ALGEBRA
SXEBGX8E 187
Find a mean proportional between
1. 3dt^ and 12cf. 2. 3i and 2^. 3. (a  xY and (a + x)\
^ 3a?  5a:  12 , Bar* + 4a:
3ar^ + 5a: Ba:^ _ 4^. _ 15
. 2V6 + 5V3 ,3V6~4V3
5. — and = •
3 V2  4 8 V2 + 20
Find a fourth proportional to
6. 2a, 36, 4ac. 8. f , f , j^ij.
7. a:*, xy, 3a?. 9. a — 1, a, 1.
Find a third proportional to
10. X and 5. 12. (a + 1)* and a* — 1.
11. .4 and .08. 13. a and 1.
a a
14. How many answers to each of Exs. 15? To each of
Exs. 69? Exs. 1013?
Solve and check:
■
15. 2a: + 3 :3a: 1 = 3a:+l :2a: + l.
16. a: + 5 :3  a: = 10 + 3x :a:  10.
17. 3a: + 5 : 5a: + 11 = 7  a: : — 3a:.
18. x^A:x^x + 3=^x + 2:2x + 3.
19. What number must be added to each of the terms of
f to make the value of the fraction f ?
20. A baseball nine has won 17 games out of 25. How
many straight games will it have to win to make the games
won equal J of the games played?
21. What number added to each of the numbers 3^ 7, 13,
and 25 will give results which are in proportion?
' RATIO 411
22. The horsepower generated by a stream falling over
a dam is proportional to the height of ti^e dam. If on a
certain stream a dam 5 ft. high generates 200 H. P., how much
higher must the dam be made in order to get 280 H. P. ?
350 H. P.?
23. What short way is there of determining whether a
given proportion is correct? Illustrate by examples.
24. Test the following proportions: 1 : — 3 = —3:1.
1:1 = 1:1.
25. Convert each of the following into a proportion:
(1)3X4 = 6X2. (3)a? = a262.
(2) pq = ab. (4) 15 = ar^.
26. Separate a : 6 : c = 4 : 5 : 6 into two proportions.
27. Combine a : 6 = 2 : 3 and 6 : c = 3 : 5 as a single state
ment.
28. Make up and work aH example similar to Ex. 27.
29. Separate 1200 into two parts which shall be in the
ratio of 2 to 3.
30. Separate 1200 into three parts which shall be in the
ratio of 3^ 4, and 5.
31. Make up and work an example similar to Ex. 30.
32. That a door may look well, its height should be to
its width approximately as 7 : 5. If a door is to be 6 ft.
9 in. high, how wide should it be?
33. In a certain year the profits of a ^ven business
were $39,260. Divide these profits into two parts which
shall be as 9 to 4. Also into three parts, as 8, 3, 2.
34. In the steepest part of the Mt. Washington railway
(Jacob's Ladder), there is a rise of 13 in. in one yard of track.
What would be the rise in a mile of track at the same rate?
412 SCHOOL ALGEBRA
35. If the area of Rhode Island is 1250 sq. mi., of New
Jersey 7800 sq. mi., and of New York 49,000 sq. mi., by how
much does the area of New Jersey differ from a mean pro
portional between the other two areas?
36. The lengths of the Hudson, Ohio, and Mississippi
rivers are respectively 280, 950, and 3160 miles. By how
much does the length of the Ohio differ from a mean propor
tional between the lengths of the other two rivers?
37. If the weights of a man, horse, and elephant are re
spectively 150 lb., 1000 lb., and 2J T., how much does the
last of these numbers differ from a third proportional to the
other two?
38. If the diameter of the moon, the distance of the moon,
the diameter of the sun, and the distance of the sun are taken
as 2160, 240,000, 860,000, and 93,000,000 mi. respectively,
how much does the last number differ from a fourth propor
tional to the other three nimibers in the order given? By
what per cent does it differ?
39. In sterling silver, the amoimt of the silver is .925 of
the entire weight of the metal. 500 ounces of pure silver
will make how many ounces of sterling silver? (What other
metal is added to pure "Silver to make sterling silver, and
why is it added?)
40. If a given piece of ground can be divided up into 60
building lots each 30 ft. wide, how many lots 40 ft. wide
would it make?
SuG. If X denote the number of lots 40 ft. wide,
30x60=40xa;
or, a;:60= 30:40
This problem can be solved either from the equation or from the
proportion.
A pi'oportion of this kind is termed an inverse proportion.
RATIO 413
41. If 2400 shingles 4 in. wide are needed in building a
house, how many 3 in. shingles would be needed?
42. If 15 yd. of cloth 36 in. wide are used in making a
dress, how many yards 48 in. wide would be needed?
43. If a trolley company reduces the hours of its conductors
from 12 to 10 hours per day, by what per cent must it in
crease the number of its conductors?
44. Make up and work an example similar to Ex. 12. To
Ex.22. Ex.41.
45. Practice oral work with radicals as in Exercise 108.
266. Tranflformations of a Proportion. Before converting
a proportion into an equation, the proportion may often be
simplified by the use of one or more of the following principles :
li aib = c :d, then
1. a:c = b :d (called aliernaticm).
2. b :a = d :c (inversion).
3. a + b :b ^ c+d:d (addition).
4. a — 6:6 = c — d:d (subtraction).
5. a + b:a — b = c + d ic—d (addition and subtraction).
For, from a:b— cid^we have ad be (Art. 265);
whence we obtain 1 and 2 (Art. 264).
Also T = J, whence ^f 1 = t;+ 1 (Art. 15, 3).
a a
Whence, ^=^^±^
'be
Let the pupil prove 4 in like manner, and obtain 5 from 3
and 4.
Ex.1. Sdve ix?2x + 3:a? + 2x3 = 2x^x'3:2x^
+ X + 3.
2x* 4x^
By addition and subtraction.
•4x + 6 2z  6
414 SCHOOL ALGEBRA
.. x+3=4a; 6
X = 3 Ans.
The factor 2x^ also gives the roots z^0,0.
Check. For x » 3.
g«2g+3 ^ 96+3 _^ 6 ^1
a;«42x3 9+63 12 2
2x«a;3 1833 ^12^1
. 2x«4a;4"3''l8+3 + 3"24"2
Let the pupil check ihe work for x » 0.
Ex.2. Solve ^/^+^_zl ^^.
Vx + 1  Va?  1 2
By addition and subtraction, "^ ^"^ = , "^^
2\/x^ 4a;3
Simplif3ring and squanng,
x1 16x224x+9
By addition and subtraction, etc., 5 = 16x« 8x45
•^ 1 16x4
Hence, 16x« 4x= 16x« 8x+5
X = J Ans.
Let the pupil check the work.
267. Given some proportion (or equality of several equal
ratios), a,s a :b = c :d, a, required proportion is often readily
Or C
proved by taking r = 3 = t (hence, a = br, c = dr), and
a
svbstituting for a and c in the required proportion.
Ex. Given, a : 6 = c : d,
prove 2oP + Bofc^ :2a3  8062 = 2cj8 + 3a? :2c»  3c(i«.
Let r"°j~^ .*.a"«6r, c»<ir.
a
Substitute in each ratio the values a ^hf^c ^dr.
RATIO 415
J 2g»f3ay 2bh*+Sb^ yr(2r« + 3) ^ 2r»43 .
' 2a»  3a6» " 26»r»  36V " 6«r(2r«  3) 2r2  3 '
J J 2c« h3cd« ^ 2(;Pr»+3(iV ^ cPr(2r» + 3) 2r»+3
2c»  3cd* 2d»r»  3cPr cPr(2r«  3) " 2r«  3'
each equal to the same expression.
Hence, a given expression may be proved to be identical
with another expression either (1) by reducing the first ex
pression directly to the form of the second or (2) by reducing
both expressions to a common third form.
268. Composition of Several Eqnal Batios. In a aeries of'
eqwd ratios y the sum of all the antecederUs is to the sum of ail
the consequents as any one antecedent is to its consequent.
^. a c e g
Let each of the equal ratios equal r.
rrn a C C
.\a = br, c^dr, e^fr, g ^ hr
Adding the last series of equalities,
a + c + e + g^{b + d+f+h)r
. a+c+e+g _ ^
^'b+d+f+h b
.*• a + c + e + g :b + d+f+ A « a :&
EXXBCISE 138
Solve:
1. a? + 2x'l:Q!? + 2x + 5 = 2x + l:2x5.
2. a:»3a? + 5:a:» + 3ar^5 = a* + 2:a?2.
3. 2a:»8a*3a:+l:2a:»10x* + 3a;l=:r* + ll:«*"ll.
416
SCHOOL ALGEBRA
V
4. VSx + 1 :2V2a:  1 = Vx  1 : VsT+l.
^ x+ Vl2a  X Va,+ 1
6. — ^ — = •
X  Vl2a X Va1
/ g 2a?  So? + X + 1 __ Za?  a? + 5x  IS
' 2a?S3?xl 3a?a?5x+13
^ 3 + V2a; + 3 4+^3:+!
5 V2x + 3 ~" 4  Vx + 1
g a/5+ V5 + g _ V2 + Va;2
V5  a/5 + a; V2  Va;  2
g 3a + \/4a;  3a^ ^ a+Vx + a^
5a  V4x  3a2 3a  V^To^
10. 8y — 6a; : a: + y — 1 = 5 — 3a: : 4 — y = 7 : 4.
3^1 a: + z/:ya: = a:l.
la:y — 3 : a: — 1 = a + 2 : 1.
If a : 6 = c : d, prove
12. a^:(^ = ab:dc. 13. a^ li^ = 0^ + c^ :62 +
14. ac:hd== {af c)^ :(b + d)\
15. (ac)2:(6d)2 = a2 + c2:62 + d2^ ^^
16. a : 6 = Vo^+l? : V^TS^.
17. 2a2 + 3a6 : 3a6  46^ = 2c2 + Scd : 3cd  4^.
18. a2a6 + 62:?^ ?: = ^^  cd + (P :^^ — ^.
a c
If a, 6, c, d are in continued proportion, prove
19. a : c — d = b^ :bd — cd.
■
20. a :c = a2 + 6^ + c2 : 62 + c2 + (P.
21. a:d = a^ + 2¥ + S(?:b? + 2(? + ScP.
/
RATIO 417
l{(a + b) (c ^ d) + (b + c) (d — a) ^ cd — ab, prove
that a : 6 = c : d.
23. If (a + 6  3c  3d) (2a  26  c + d) = (2a + 26 
c — d){a — h — Zc\ 3d), prove that a : 6 = c : d.
24. Fmd two numbers in the ratio of 2 to 5, such that when
each is increased by 5 they shall be as 3 to 5.
25. Find two niunbers, such that if 7 be added to each
they will be in the ratio of 2 to 3; and if 2 be subtracted from
each, they will be in the ratio of 1 to 3.
26. Separate 32 into two parts, such that the greater dimin
ished by 11 shall be to the less, increased by 5, as 4 to 9.
27. Separate 12 into two parts, such that their product
shall be to the sum of their squares as 2 to 5.
28. Is a proportion true after the same number has been
added to all of its terms? Give a numeriqal illustration.
29. If a : 6 = c : d, prove that a6 + cd is a mean propor
tional between a? +c^ and 6^ + d^.
30. If a : 6 = c : d) prove that 6 : a = : j»
c a
31. A and B are in business and their respective shares of
the profits are as 2 to 3. If the profits for a certain year are
$16,000, and during the year A takes out $1200 and B $1000,
at the end of the year how much of the profits does each
receive?
32. During the American Civil War, in the Northern
armies 224,000 men died of disease and 110,000 of wounds
received in battle. Owing to improved sanitary methods in
the RussoJapanese war, in the Japanese armies 27,000 men
died of disease while 59,000 died of wounds. Approximately
how many Kves were saved in the Japanese armies by the use
of improved sanitary methods?
418 SCHOOL ALGEBRA
33. An active walker goes 4 mi. an hour. Sensation travels
along a nerve at the average rate of 120 ft. per sec. Find
the. velocity of a rifle bullet which is a third proportiQi^al to
the velocities just named.
34. The velocity of the earth in its orbit is 18 mi. per sec.;
of a message on a submarine cable 2480 mi. per sec.; and of
light 186,300 mi. per sec. How much is the middle one of
these velocites from a mean proportional between the other
two?
35. The sun's distance from the earth is 93,000,000 mi.
and light from a star travels 5,900,000,000,000 mi. in a year
(called a " light year '*), show that the following proportion
is approximately correct:
1 inch : 1 mile = sun's distance : 1 light year.
«
36. If the rear and fore wheels of a wagon are respectively
a and b feet in circumference, how many rotations does the
rear wheel make while the fore wheel rotates p times?
. 37. If —  — = = , and X, y, and z are unequal,
I m n
show that Z + m + n = 0.
38. If (a + 6 + c + d) (a  5  c + d) = (a  6 +cd)
(a + & — c — d), prove that a:h =^ c:d,
39. If a : 6 = c : d = c :/, prove that
c :d = Va? + 4ac + 6c2 : Vfc^ + 46d + 5(P.
40. If a, h, c, and d are in continued proportion, prove that
g^ __ g^ + fc^ + c^ _ g^  c^
W W + (^ + ^ V(P'
41. Solve Exs. 2630 (pp. 241242) by aid of the graphical
method given in Ex. 2, pp. 260261, and by the principles of
proportion.
RATIO
419
42. Make up and work an example similar to Ex. 2. To
Ex. 13. Ex. 31.
43. Practice oral work with imaginaries as in Exercise 110
(p. 339).
Meaning op the Ratio Forms * > > §.
a *
269. The Meaning of ~ has been made clear in Exs. 30
32, p. 79. The same result has also been made evident in
the process of constructing certain graphs.
In general,
Zero divided by any number (except zero) gives zero.
270. Infinity is a number (or quantity) greater than any
assignable (or definitely expressible) number.
The symbol for infinity is oc.
The symbol = means " approaches."
271. The Meaning of g is made clear most readily by the
use of graphs.
1
Ex. Gri^ph y — —
X
X
y
4
i
2
i
1
1
i
2
i
4
J
8
etc.
2
J
1
1
i
2
i
4
i
8
etc.
r
•
1^
\
\
<B.
*s
►—
,
A
X
>
<r^
^
P
c\
(■
00)
Y'
• 420 SCHOOL ALGEBRA
Graphing y =  f or the positive values of x, we obtain the branch
ABC (p. 419). ^ ' .
From the diagram it is evident that as the value of x becomes
smaller, that of y Lor  J increases, and as x ±: 0, y ( or  j = oc.
If y s ~ is graphed for the negative values of x, in like manner
X
as
X = 0,j/f orj =  «.
Hence, in general,
As the wlue of the denominator of any fraction approaches
zero (the value of the numerator being finite), the wlue of the
fraction approaches infinity (or negative infinity).
This statement is often abbreviated into the form j^ = *•
272. The Meaning of g is best shown algebraically.
aj2 — 1
Ex. Find the value of — when x = 1.
x — l
If we substitute the given value of x directly in the given fraction,
x«  1 1  1
we obtain r « :; t = ^
X — I 1—1
If, however, we simplify the fraction before substituting and then
substitute 1 for x, we obtain '
X — 1 X —1
Hence, in this case ^ represents the value 2.
Show in a similar manner, that when x = 1, the value of
— is 3. Also that the value of r is 4.
a: — 1 Q x — \
Hence, the value of ^ varies with circumstances, or as it
is usually expressed
. .
Q is indetermiruUe.
RATIO 421
273. The Meaning of ^ is best shown algebraically.
Ex. Find the value of r 5 ^ — r, when x =^ 1. .
a; — 1 ar — 1
By direct substitution. —  +  —  =  +  r= ^ (see Art. 271).
X— 1 a;* — 1 0°°
But if the given expression is simplified before the substitution
for X is made,
we have i^ + ^ = i X ?^ = a; + 1 = 1 + 1 = 2.
x — l x' — 1 X — 1 1
Hence, in this case ^ stands for the value 2.
Show, in like manner, that oc may stand for 3, 4, or any number.
^ We express the result arrived at as follows:
oc is indeterminate.
The above results for the meaning of *>;:> ^ might be
a ^
obtained by purely logical methods, but the thorough dis
cussion of these methods lies beyond the scope of this book.
EXERCISE 189
 and thus find the value of ^
X
1. Graph y =  and thus find the value of ^ and q^'
2. Graph y = • In this process what special ratio
aj — 1
is evaluated?
X
3. Graph y = ^ and make a similar inference.
By an algebraic process find the value of
a:^  4 , « a^  1 , .
4. ■ when X = 2.
X'2
5. when X = 2.
a:2
Find the value of
a:^
8.  when a: = 0.
^.
a
V¥ in,
jIX Ml X*
,
7.
o*
a2
a2
when o=
=2.
9.
1
when X
= 0.
422 SCHOOL ALGEBRA
1
10. x^ when x 0. U. 7* when a; = 0.
12. when ar = 0. When a; = 1.
x1
13. 5X0f 14. 3X0 + f
3
15. 5a: h x{x — 1) when a: = 1. Also when a;=0.
a;
16. aXO + fcXOcXO.
17. <^whena; = 0.
18 — when x^ a.
V a — a:
■ . . 3 + 7~**
19. What is the limit of ^ i when n is increased
indefinitely?
20. Find the value which — — it— — — approaches
2a
when 6 = 0. When c = 0. a = 0.
21. Find the same values for "" ^ "" ^^^ "" 4gg.
2a
22. Who invented ex as the sign for infinity?
23. Make up and work an example similar to Ex. 2. To
Ex. 8. Ex. 10.
24. Make up and work an example similar to Ex. 14. To
Ex. 18.
25. How many of the examples in this Exercise can you
work at sight?
CHAPTER XXIV
THE PROGRESSIONS
274. A Series is a succession of terms formed according to
some law^ as
1, 4, 9, 16, 25, . . .
2, 4, 8, 16, 32, . . .
275. Utility in an Algebraic Treatment of Series.
Ex. If a car going down an inclined plane travels in suc
cessive seconds, 2 ft., 6 ft., 10 ft., 14 ft., etc., how far will it
go in 30 seconds?
The direct method of solution would be to set down the 30
niunbers involved and add them. But by investigating the laws
of the series involved and expressing these as formulas, it will be
found (see Art. 278) that this long addition can be converted into
two short multiplications and much labor can thus be saved.
The algebraic study of the laws of series will enable us to save
labor in various ways, and to obtain other important results.
AnrrHMETiCAL Progression
276. An Arithmetical Progression is a series each term of
which is formed by adding a constant quantity, called the
differeTwe, to the preceding term.
Thus, 1, 4, 7, 10, 13, • • • is an arithmetical progression (de
noted by A. P.) in which the difference is 3.
Given an arithmetical progression, to determine the dif
ference: /rom any term subtract the preceding term.
Thus, in the A. P., i,  J,  3,
the difference = — I — J " ~" I
423
424 SCHOOL ALGEBRA
277. (luantities and Symbok. In an A. P. we are con
cerned with five quantities:
1. The first term, denoted by a.
2. The common difference, denoted by d.
3. The last term, denoted by /.
4. The number of terms, denoted by n.
5. The sum of the terms, denoted by s.
278. Two Fnndamental Formulas. Since in an A. P.
each term is formed by adding the common difference, d, to
the preceding term, the general form of an A. P. is
a, a + d, a + 2d, a + 3d, • . . .
Hence, the coefficient of c2 in each term is one less than the
niunber of the term.
Thus, the 7th term is a + 6d,
12th term is a + lid,
nth term is a + (w — l)d.
Hence, l = a+{nl)d (I)
Also,
9 = a+(a + d) + (a + 2i) + + {li) + l . .(2)
Writing the terms of this series in reverse order,
8 = l+(ld) + il2d) + + (o + d) + o . .(3)
Adding (2) and (3),
= n(a + 1)
.•., = (o + (4)
If we substitute for I in (4) from (1),
* = 52a + (n  1)41 (5)
ARITHMETICAL PROGRESSION 425
Hence, combining results, we have the two fundamental
formulas for I and s,
L Z = a+(nl)i
s = [2a + (n  m
Thus formula I substitutes a multiplication for successive
additions of the common di£Perence; and formula II substi
tutes a multiplication for the addition of the successive
terms.
Ex. 1. Find the 12th term and the siun of 12 terms of the
A» Jr., Oy Of ij "~ 1, ~~ o, . . • • •
In this series a = 5, d = — 2, n » 12.
From I, /= 5+ (12 1) ( 2) = 6 22=  17.
From II, «=J/(517) = 72 /Sum
Ex. 2. Find the sum of n terms of the A. P.,
a + b a — b a — Zb
, , >
2 2 2
Here a=2±^, d=6, n=n.
Substituting in the fundamental formula, s = [2a + (n — \)d]y
«^a+6+(nl)(6)]
= So+(2n)6] /Sitm
EXERCISE 140
1. Give the value of d in Exs. 215.
2. Find the 8th term in the series 3, 7, 11,
3. Find the 9th term and the sum of 9 terms in 7, 3, — 1, . . .
4. Find the 20th and 28th terms in 5, ^?, V,
426 SCHOOL ALGEBRA
5. Find the 16th and 25th tenns m  13^,  9,  4^ . . .
6. Fmd the 7th and 10th terms and the sum of 10 tenns in
the series i,h^f
7. Find the 18th term and the sum of 18 terms in the
series 3, 2.4^ 1.8^
8. Find the 30th term of the series 1, 4, 7, 10, .... . by
successive additions of the common diflference. Now find
the 30th term by use of the formula. About how much shorter
is the second process than the first?
Find the sum of the series: '
9. 3, 8, 13, to 8 terms. 5^ ^v] ^
10. 3,  3, 7 9, to 9 terms. <^ ^ \\ ^ f
11. 2i, 3J, 5, .... to 14 terms. . ^ Aj ^ s^^
^ — J» i» i» • • • • to 38 terms.  X '^ i
13. — i, — f , — iij • • • • to 55 terms,
14. 5V2  2a/3, 4\/2  3\/3, to 11 terms. '•,
11 ( ^^'
15. 3a , 2a, a + ,.... to 12 terms. ^ ^ ^/ ^
16. 1, 4, 7, 10, .... to n terms. )
17. .6, 3, 0, — 3r — 6, i . . . to p terms.
18. 5, 3, 1, — 1, .... to n terms.
19. 2a; — y, a: + y, 3y, . . . . to r terms. '
20. Find the smn of the first 30 odd numbers by writing
them down and adding them. Now find their sum by. use
of the formula. Compare the amount of work in the two.
processes.
"* 21. How many strokes does a clock make in striking each
hour of the day?
22. If a man saves $100 in his 20th year, $150 the next
/
.3 Srn
ARITHMETICAL PROGRESSION 427
year, $200 the next, and so on through his 50th year, how
much will he save in all?
23. If a body falls 16.1 ft. in one second, 3 times as far
in the next second, 5 times as fat in t&e third second, and so
on, how far will it fall in 6 seconds? In 15 seconds?
24. If the velocity of a falling body at the end of 1 second
is 32.2 ft. per second, at the end of the next second is 64.4 ft.,
at the end of the third second is 96.6 ft., what'is it at the end
ofx 10 seconds?
«
^ 25. A body rolling down an inclined plane goes 6 ft. in the
first second, three times as far in the next second, 5 times as
fap.in the 3d second, and so on. How far will it go in 10
seconds?
26. State in general language the first of the formulas
obtained in Art. 278.
Sua. '' The last term equals the first term increased by," etc.
27. State the second formula of Art. 278 in general
language.
■»
28. State the third formula of Art. 278.
29. Make up and work an example similar to Ex. 9. To
Ex. 16. Ex. 22. Ex. 25.
^30. . How many examples in Exercise 114 (p. 350) can you
' f,now work at sight?
••
279. Given Any Three of the Five Quantities a» d, I, n, «,
to find the Other Two;
' The method, in general, is as follows:
^ If the three known quardities are found in one oj the formulas
of Art. 278, substitute the three given values in the formula and
solve the resuiting equaiion;
^
428 SCHOOL ALGEBRA
The remaining unknown quantity may then he found hy use
of one of the other formulas of Art. 278;
If the three given qimrdities do not occur in one of theformvlas
of Art. 278, substitute in two of these formulas and solve hy
elimination.
Ex. 1. Given Z = 13, « = 49, n = 7, find a and d.
Since the letters Z, «, n, and a all occur in the formula
n
« =  (a \l)y substitute the values of Z, s, and n in this formula.
(\ ^
y Hence, 49 = l(a 4 13)
98 = 7a + 91 ■ a = 1 Ans,
From Z = a + (n  l)d 13 = 1 + (7  l)d
whence, d = 2 Ans.
Ex. 2. Given d = 2, 1 = 21, s == 121, find a, n.
Since d, Z, and « do not occur in one formula, we
Substitute for cZ, Z, 8 in Formulas I and II,
^ \ ^ 21 = a + (n  1)2 (1)
\^:^^ ^^ " 121 . !^(?L±21) (2)
'''^ '~ /. a2n=23 (3)
an + 21n = 242 (4)
Substitute for a in (4) from (3),
n(23  2n) + 21n = 242
Whence, ^ = 11] ^ns ^ "^^ ''
Hence, from (3), a = 1 'h y * , L^
EXERCISE 141
Find the first term and the sum of the series when
1. d = 3, / = 40, n = 13. 2. d = f , Z = 18i n = 33.
Find the first term and the common difference when
3. ^ = 275,Z = 45,n= 11. 4. « = 4,Z = 10,n = 8.
5. * =  246, Z =  34, n = 17.
6. * = 9, Z = 2f , n = 9.
7. 5 =  '^s Z =  4, n = 47.
ARITHMETICAL PROGRESSION
Find n and d when
8. a = 5,/= 15,* = 105.
9. a = 19, /= 21,5= 21.
10. a = ^,l= f , 8 = 2J.
11. a = 3J, Z = 9i, « = 48.
Find a and n when
12. s = 10,d = 3,l = 8. 13. ^ = 10, d = 3, Z = 4.
14. Z =  8, d =  3, * =  3. r
15. z =  1, d =  ^V, * =  ¥.
How many consecutive tenns must be taken in the series
:^ 16. 1, 1, 2 .... to make the sum 45?
17. f, I, 4, .... to make the sum — 1?
18. f , i, 1 . . . . to make the sum 4.5?
19. A body roUing down an inclined plane goes 6 ft. the
first second, 18 ft. the next second, 30 ft. the third second and
so on. In how many seconds will it have traveled 486 ft.?
20. Make up and work an example similar to Ex. 1. To
Ex. 12. Ex. 19.
21. How many examples in Exercise 45 (p. 155) can you
now work at sight?
280. Arithmetical Means.
Ex. Insert 9 arithmetical means between 1 and 5.
We have given a= 1, Z= 5, n = 11. Hence, we find d=i.
The required means are therefore If, If, 2, Ans,
In case only a single arithmetical mean is to be inserted
between two quantities, a and 6, this one mean is found most
readily by use of the formula —z — For if x denotes the
required mean, the A. P. is a, x, 6.
<^
430 SCHOOL ALGEBRA
Hence, x — a — b — x
2x = a + b
a + b
X = — r —
EZEBCISS 142
Insert
1. Four arithmetical means between 7 and — 3.
2. Seven arithmetical means between 4 and 6.
3. Thirteen arithmetical means between f and — f .
4. Fifteen arithmetical means between — 4 and 9.
5. The arithmetical mean between 2^ and — 5f .
6. The arithmetical mean between x + 1 and x — I,
7. Find the A. M. between t and — — : — and
b a X + y X — y
8. If the height of Bunker Hill Monument is 221 ft., of
the Washington Monument 555 ft., and the length of the
Olympic 882 ft., by how much does the middle one of these
numbers differ from the arithmetical mean between the other
two?
9. Make up and work a similar example concerning 12^
mi., 49 mi., and 100 mi., which are the lengths of the Simplon
Tunnel, the Panama Canal, and the Suez Canal respectively.
10. Rome was founded 753 B. C. and fell 476 A. D. How
far is the latter number from being an arithmetical mean
between the former and the number of the year in which
Columbus discovered America?
11. Ether boils . at a temperature of 96° F., alcohol at
167°, and water at 212°. How far is 167° from being an
arithmetical mean between the other two temperatures?
^^m
ARITHMETICAL PROGRESSION 431
12. Show that if twice one number equals the sum of two
other numbers, the three numbers may be arranged as
an A. P. ^y
13. State the formula x = —  — (of Art. 280) in general %
language.
14. Work at sight such examples on pp. 101102 as the .
teacher may indicate.
3
A
\
281. Miscellaneoiu Examples. ^ ^
Ex. 1. The 7th term of an A. P. is 5, and the 14th term is  ^ "^
— 9. Find the first term.
By the use of Formula I (Art. 278),
the 7th term is a + 6d, and the 14th term is a + 13d.
.. af 6d5 (1)
a + 13d   9 (2)
Subtractmg (1) from (2), 7d =  14 (
d =  2 ^ :
Substitute for d in (1), a — 12 » 5  i ""
a = 17 Ans,
Ex. 2. The sum of five numbers in A. P. is 15, and the
sum of the 1st and 4th numbers is 9. Find the numbers.
Denote the numbers by
a;  2y, re  y, x, ir + y, x\2y
Add, &c = 15 (1)
Also, (»22/)+(x+i/) 9
.. 2a;  1/ = 9 . (2)
From (1) X = 3; hence, from (2), y = — 3.
«
Hence, the numbers are 9, 6, 3, 0, — 3, Ans,
Similarly, in dealing with four unknown quantities in A. P., we
denote them by
«  3y, a;  y, x\y, z+Zy
432 SCHOOL ALGEBRA
EXERCISE 143
Find the first two terms of the series wherein
1. "The 4th term is 11 and the 10th is 23.
2. The 6th term is  3 and the 12th is  12.
3. The 7th term is   and the 16th is 2.
4. The 5th term is c — 36 and the 11th is 36 — 5c.
5. Find the smn of the first n odd nmnbers. State the
result obtained as a rule in general language.
6. Set down the first 20 odd numbers and find their sum
by addition. Now find their sum by the formula result
obtained in the preceding example. Compare the amount of
work in the two processes.
7. Find the sum of the first n nmnbers divfeible by 7.
8. Make up and work an example similar to Ex. 6, but
showing the utility of the result obtained in Ex. 7.
9. Which term in the series 1^, Ij, 1^, • • • is 18?
10. The first term of kn arithinetical progression is 8; the
3d term is to the 7th as the 8th isto the 10th. Find the series.
11. Find four numbers in A. P., such that the sum of the
first two is 1, and the sum of the last two is — 19.
12. Find four numbers in A. P. whose sum is 16 and pro
«
duct is 105.
13. A man travels 2 nai. the first day, 2f the second, 3
the third, and so on; at the end of his journey he finds that
if he had traveled 6 mi. every day he would have required the
same time. How many days was he walking?
14. The sum of 10 numbers in an A. P. is 145, and the sum
of the fourth and ninth terms is 5 times the third term. Find
the series.
ARITHMETICAL PROGRESSION 433
15. If the 11th term is 7 and the 21st tenn is 8f , find the
41st term of the same A. P.
16. In an A. P. of 21 terms the sum of the last three terms
is 23, and the sum of the middle three is 5. Find the series.
17. Required five numbers in A. P., such that the sum of
the first, third, and fourth terms shall be 8, and the product
of the second and fifth shall be — 54.
•v^is. The sum of five numbers in A. P. is 40, and the sum
of their squares is 410. Find them.
19. The 14th term of an A. P. is 38; the 90th term is 152,
and the last term is 218. Find the number of terms.
20. How many numbers of two figures are there divisible
^ by 3? By 7? How many numbers of three figures are
divisible ty 6? By 9?
21. How many numbers of four figures are there divisible
^y 11? Find the sum of; all the numbers of three figures
divisible by 7. ' ' .
22. Jf a car starts at the top of a hill and runs down an
inclined track 2 ft. the first second, 6 ft. the next second,
10 ft. the next, etc., and reaches the bottom in 12 seconds,
how long is the track?
23. Sulphur fuses at a temperature of 239° F., tin at 442°,
and lead at 617°. By how much dpes 442° differ from the
arithmetical mean between the other two temperatures?
24. Copper fuses at a temperature of 2200° F., gold at
2518°, and iron at 2800°. Treat these temperatures in a
way similar to that used in the preceding example.
25. The heights of Mt. Washington, Pike's Peak, Mt.
McKinley, and Mt. Everest are respectively 6290 ft.,
14,147 ft., 20,464 ft., and 29,002 ft. Find the difference
t
434 SCHOOL ALGEBRA
between each of these numbers and the corresponding term
in an A. P. whose first term is 6290 ft. and common diiSerence
7500 ft.
26. In an A. P. whose second term is 14,200 ft. and conmion
difference 7600 ft.
27. If a body falls lOyV ft. in the first second; three
times this distance in the next; five times in the third,
and so on, how far will it fall in the 30th second? How far
will it have fallen during the 30 seconds? In how many
seconds will it have fallen 6433 J ft.?
28 If a, b, c, d, are in A. P. prove: (1) that a + d — b + c;
(2) that cJc, bk, ck, dk are also in A. P.; and. (3) that a + k,
b + k,c + k,d +JcB.TemA.'P. State this problem without
the use of the symbols, a, b, c, d, k.
29. Make up and work an example similar to Ex. 1. To
Ex. 7. Ex. 22.
30. Practice oral work with fractions as in Exercise 58
(p. 190).
Geometrical Progression
282. A Geometrical Prognression is a series each term of
which is formed by multiplying the preceding term by a con
stant quantity called the ratio.
Thus, 1, 3, 9, 27, 81, .... is a geometrical progression (or G. P.)
in which the ratio is 3.
Given a geometrical progression, to determine the ratio:
divide any term by the preceding term.
Thus, in the G. P.,  3, ,  f ,
the ratio «= — ^ = —
3 2
GEOMETRICAL PROGRESSION 436
283. Quantities and Symbols. The symbols a, I, n, s aie
used, as in A. P. Besides these, r is used to denote the
ratio.
284. Two Fundamental Fonnnlas. Since in a G. P. each
term is formed by multiplying the preceding term by the
common ratio, r, the general form of a G. P. is
a, ar, ar", ar^, ar*,
Hence, the exponent of r in each term is one less than the
number of the term.
Thus, the 10th term is ar^.
15th term is ar^^,
nth. term, or I = ar"^"^ (1)
In deriving a formula for the smn, we know, also,
8 = a + ar + ar^+ ....^ ar"^'^ .... (2)
Multiply (2) by r,
rs = ar + ar^ + at^+ . . . . + ar"^"^ + af". . (3)
Subtract (2) from (3),
rs — 8 = ar"* — a
.•.* = ^. ' (4)
Multiply (1) by r, rl = ar""
Substitute rl for ar"" in (4),
rl — a /v
*=rT (^^
Hence, collecting the results obtained in (1), (4), (5), we
have the two fundamental formulas for / and 8:
I. I = ar""^
II. 8 = r
r — 1
rl — a
436 SCHOOL ALGEBRA
Ex. 1. Find the 8th term and sum of 8 terms of the G. P.,
1,3,9,27......
In this case, o = l, r=3, n=8
From I, Z = 1X3^=2187
Fromll, 8 = ^ ^^^^\ " ^ = 3280 Sum
Ex. 2. Find the 10th term and the smn of 10 terms of the
G. P., 4, — 2, 1, — J,
Here a = 4, r =  §, n = 10
Hence, Z = 4(  §)» =  t^t =  lir
* ryiTi 128^"*^
EXERCISE 144
Give the value of r in Exs. 215.
1. Find the 6th term in the series 2, 6, 18,
2. Find the 7th term in 3, 6, 12,
' 3. Find the 6th and the sum of 6 terms in 45, — 15, 5, . . .
4. Find the 5th and the sum of 5 terms in 81, — 54, . . . .
5. Find the 7th and the sum of 7 terms in Ij, — f , ....
6. Find the 9th term in the series 2,_2\/2, 4,
7. 15th term of t, 75, tx, .
tr tr
8. nth term of p, , ^, ^,
V
Find the sum of the series j
9. 3, — 6, 12, .... to 6 terms.
10. 27,  18, 12, to 7 terms. ^ IX
11. — f , If , — 2, .... to 9 terms. <\ *'^^
12. , — , ^i, . • , • to 8 terms. a
GEOMETRICAL PROGRESSION 437
13. — F=, 1, V3, .... to 8 tenns.
V3
14. V2  1, 1, V2 + 1, to 6 terms.
15. 1, 2, 4, 8, .... to n terms.
16. The following is a series of specific gravities: cork, .25;
oak wood, .75; aluminmn, 2.5; iron, 7.5; platinmn, 21.5. By
how much does each term of this series differ from the cor
responding tenn in a G. P. whose first term is .25 and whose
ratio is 3?
(What is meant by specific gravity?)
. 17. If the average age of parents be taken as 30 years,
find the total nmnber of a person's ancestors in a period of
600 years.
18. The population of the United States in the year 1900
was 76,300,000. If this should increase 50% every 25 years,
what would the population be in the year 2000?
19. If a man saves $300 each year for 10 years, what is
the amount of his savings in 5 years at compound interest at
5 per cent? In 10 years?
20. A ship was built at a cost of $70,000. Her owners at
the end of each year deducted 10% from her value as esti
mated at the beginning of the year. What is her estimated
value at the end of 10 years?
21. A grain of wheat when planted produced a stalk on
which were 30 other grains. The next year each of the grains
was planted and produced similar stalks. If this process
were continued, at the end of 10 years how many bushels
would be produced in the last crop if 1 quart contains 2000
grains?
22. Make up and work a similar example concerning the
438 SCHOOL ALGEBRA
amount of corn produced from one grain^ using probable
numbers.
23. If 32 nails are used in shoeing a horse, make up and
work an example concerning a man who paid a blacksmith
for shoeing a horse at the rate of J0 for the first nail driven, •
Jjlf for the second nail, 1^ for the third, etc.
State in' general language:
24. The first formula obtained in Art. 284.
25. The second formula. The third formula.
26. Make up and work an example similar to Ex. 4. To
Ex. 15. Ex. 20.
27. Practice the oral solution of simple equations as in
Exercise 64 (p. 209).
285. Given Three of the Five Quantities a, I, n, s, r, to
determine the Other Two.
Use the same general method as that given in Art. 279
(p. 427), for A. P.
Ex. 1. Given a =  2, n = 7, Z =  128; find r, s.
From I,  128 =  2r*
Hence, r* = 64, f = =»= 2
FromII,ifr = + 2, s = ?^^=iy^f^^^ = 256+2 = 254
Tf. 9 ^_ (2)(128)(2) _ 256+2 _
Ifr=2, s ^^^j^ ^3 W>
Hence, there are two sets of answers; viz.,
r = +2, 3 =2541 ^
r= 2, s= 86 J ^'^'
Ex. 2. Given, a = , r= ^,5 = rh; find /, n.
The most convenient method of solution is to find, first Z, then n.
rl — a
Substituting in the formula s =
r1
^=^i,whenceZijAna.
GEOMETRICAL PROGRESSION . 439
Using i=ar*S  irk = l(  i)""'
Whence, ~ Tk X i = ( i)**""*, and n  6 Ana.
Let the pupil check these results.
EXERCISE 145
Find the first term and the sum when
1. ^ = 6, r = 3, 1! = 486. 2. n = 8, r = 2, Z = 640.
3. 7i = 8,r= f, /= Mi
4. n= 7,r = iV6,/ = 3.
Find the ratio, when
5. a ==  2, Z = 2048, n = 6.
6. a = 9, Z = ^, J = 23^,
7. a = 2ff, Z= ii w = 6.
8. a =  16i, Z = tV, <J =  12tV
Find the number of terms when
9. a = 2, r = 2, * = 62. 11. a = , Z = j^, r = i.
10. a = 4, r =  ^, * = 2f . 12. a = 3, Z = 96, s = 63.
13. a = 18, r = , 5 = 12f .
How many consecutive terms must be taken from the
series
14. 2, 4, 8, .... to make the sum 62?
15. f , i, I, . . . . to make the sum \Wt
16. 5i,  8, 12, to make  22?
17. Make up and work an example similar to Ex. 1. To
Ex. 13.
18. How many examples in Exercise 83 (p. 273) can you
now work at sight?
440 SCHOOL ALGEBRA
286. Oeometrio Means.
Ex. Insert 5 geometrical means between 3 and 7^.
We have given a = 3, Z = ^i^, n = 7, to find r. "^
Solving by Art. 285, r = i *^ ^ ] ^ ) ^ ,
Hence, the required geometrical means are, i y ;
In case only one geometrical mean is to be inserted be
tween two quantities, a and 6, this one mean is found most
readily by using the formula Vab. For if x represents the
geometrical mean between a and b, the series will be
a, X, b
Hence,  = , .\ 7? = ab, x—Vdb
a X
SXEBCISE 146 —
^
Insert
1. Three geometrical means between 8 and J.
2. Three geometrical means between f and ^.
3. Six geometrical means between ^ and — ^.
4. Four geometrical means between — \ and 3584.
5. Six geometrical means between 56 and — iV
Find the geometrical mean between
6. 4jandf. 7. 3f and 6f .
8. 28a*x and 63aay*. 10. .7 and .343.
^ a\x J yVd? U. .5 and .125.
9. .— g— and ^—='
(?\y XV (? 12. .005 and .125.
13. 5V2 + land5\/2 1.
14. Insert 6 geometrical means between — and
16 V^
GEOMETRICAL PROGRESSION 441
15. Insert 7 geometrical means between ^ and — •
8 ,n*
'nr 2
16. Is a mean proportional between two numbers the same
as the geometric mean between the nim[iber8?
17. State the formula x = Voft (of Art. 286) in general
language.
18. Make up and work an example similar to Ex. 2. To
Ex. 12.
19. How many examples in Exercise 85 (p. 280) can you
now work at sight?
287. Limit of the Sam of an Infinite Decreasing Geomet
rical Progression.
If a line AB
C D
A I B
^  — ^— ■ ■ ^ — ^ — ^ u ^ ■^^■^^■^^
is of unit length, and one half of it (AC) is taken, and then
one half of the remainder (CD), and one half of the re
mainder, and so on, the sum of the parts taken will be
2+4+8 + 16 + 35"+
This is an infinite decreasing G. P. in which r = ^. The
sum of all these parts must be less than 1, but must approach
closer and closer to 1 as a limit, the greater the number of
parts taken. This illustrates the meaning of the limit of
an infinite decreasing G. P.
In general, to find the limit of an infinite decreasing G. P.
we have the formula
8=^ Ill
1 — r
Formula II of Art. 284 may be written, s = •
1 — f
442 SCHOOL ALGEBRA
Then^ as the number of terms increases,
/ approaches indefinitely to
rl " "
/. arl " " a0 = a
^ a^ rl ,, « a
1  r 1  r
c
1 — r
Ex. Find the sum of 9, —3, 1, — , .... to infinity.
Here a = 9, r =  .
9 9 27 „
288. Bepeating Decimals. By the use of Art. 287, the
value of repeating decimals may be determined.
Ex. 1. Find the value of .373737
.373737 = .37 + .0037 + .000037 +
Here a = .37, r = .01
.37 .37 _^37 .
• • * *" 1  .01 "* .99 " 99 '^^•
Ex. 2. Find the value of 3.1186186
Setting aside 3.1, and treating the remaining terms as a G. P«,
a « .0186, r = .001
. .0186 ^ .0186 ^ 186 ^ 62
• • * 1  ,001 .999 9990 3330
/. 3.1186186.  3tV + jih « 3;A ^na.
EXERCISE 147
Find the sum to infinity of the series
!• 2, f , f, 6. 1^, y^, ff,
a. 2,  1, 1, 7. 2\l  li li,
3.  9, 6,  4, . a 6, 3\/2, 3,
y 4J, 2J, 1, . ■ V21' 'V2 + 1''
• • •
GEOMETRICAL PROGRESSION 443
10. iV2 + iV5 + iV2,
^. i+(i_i) + (i_iy+
U. Give the ratio in the G. P. in each of the following:
(1) .333 (2) .272727 (3) .356356. (4) .79127912.
(5) .5333 (6) In Exs. 1321.
find the values of
A
13. .63 ^14. .417 15. 5.846
16. 3.52424 19. 1.02727
17. 1.4037037 20. 1.027027
18. 3.215454 21. .30102102
22. Find the first term in an infinite decreasing geometrical
progression whose smn is f and whose ratio is — \.
23. If the velocity of a sled at the foot of a hill is 60 ft.
per second and this velocity should be diminished by one
third each second as the sled moves out on the horizontal,
how far would the sled move before coming to rest?
24. Make up and solve a similar example concerning a
car which ran down an inclined track out on a horizontal
track.
25. If a ball, dropped from a height of 80 ft., rebounded
40 ft., and, on striking the ground again, rebounded 20 ft.,
and so on, how far would it travel before coming to rest?
26. Make up and work an example similar to Ex. 25.
27. State the formula a = in general language.
28. Make up and work an example similar to Ex. 3. To
Ex. 16.
29. Practice oral work with exponents as in Exercise 93
(p. 303).
444 SCHOOL ALGEBRA
289. Hisoellaneons Problems.
Ex. Find four numbers in G. P., such that the sum of the
first and fourth is 56, and of the siecond and third is 24.
Denote the required numbers by a, avy ar^, cof*.
Then o+ar** 56
< ' ar + ar* « 24
Or, a(l+r») =56 (1)
ar(l+r) =24 ^ (2)
Divide (1) by (2), l^L±Jl « 7
T O
Hence, 3  3r + 3r* = 7r
3r*  lOr ^ 3
r=3, ori\
And a = 2, or 54
Hence, the numbers are, 2, 6, 18, 54 ? >
Or, 64, 18, 6, 2i '^^•
EXERCISE 148
Find the first two terms of the series in which
1. The 3d term is 2, and the 5th is 18.
2. The 4th term is f and the 9th is 48.
3. The 5th term is 6 and the 11th is ^.
Determine the nature, whether Arithmetic or Greometric,
of each of the following series:
Aii4 ■7336
5. 4> 6* ^* 8' 37> 4^> '^Jf
^ 4> ^» j^i 973, 5^, 4^2^,
10. Divide 65 into 3 parts in geometrical progression, such
that the sum of the first and third is 3 J times the second part.
11. There are 3 numbers in G. P. whose sum is 49, and the
sum of the first and second is to the smn of the first and third
as 3 to 5. Find them.
\
GEOMETRICAL PROGRESSION 445
V 12. The sum of three numbers in G. P. is 21, and the sum '^, ~^^,
of their reciprocals is x^. Find the numbers. j^^ V^^
*^ 13. Find four numbers in G. P., such that the sum of theN ^
first and third is 10, and of the second and fourth 30. ;
^ 14. Three numbers whose sum is 24 are in A. P., but if 3,
4, and 7 be added to them respectively, these sums will be in \
G. P. Find the niunbers.
^/l5. The siun of $225 was divided among four persons in ; #
such a manner that the shares were in G. P., and the differ p ^.
ence between the greatest and least was to the difference ij^^i^
between the means as 7 is to 2. Find each share. ' ^ ^ ^
12
16. Find the sum of —7= — , v 2, ■~;f= . . . od infinitum.
~ 17. There are four numbers the first three of which are in
G. P., and the last three are in A. P.; the sum of the first and
last is 14, and of the means is 12. Find the numbers.
18. If the series f, § Hj. V,. be arithmetical, find the 102d
\iterm; if geometrical,^find the sum to infinity. /
1 ^19. Insert between 2 and 9 two niunbers, such that the
first three of the four may be in A. P., and the last three in
G. P.
20. Prove that the series ^2  1, 3\/2  4, 2(5\/2  7)
.... is geometrical; that its ratio is 2 —a/2; and that its
sum to infinity is unity.
21. The cost per ounce of mailing different kinds of mail
matter is given in the following series: 2^, Ij/S, ^jf, \^, y^ff.
How far does each of these numbers differ from the corres
ponding term in a G. P. whose first term is 2ji and whose
ratio is ?
f
i
446 SCHOOL ALGEBRA
22. If the areas of Rhode Island, New Jersey, New York,
and Texas are respectively 1250, 7815, 49,170, and 265,780
sq. mi., how far are these numbers from forming a G. P. of
which the second term is 7815 and the ratio 6?
23. On p. 396 a table gives the amomit of $1 in different
periods of time at simple interest and also at compound in
terest. Which of the two series of numbers forms an
A. P. and which a G. P.?
24. If an air pump at each stroke removes J the air in a
receiver, what fraction of the air is left at the end of 10
strokes?
25. If the amount of air in a receiver is indicated by the
height of a mercury column in a tube attached to the receiver,
and this height is 30 in. at the start, what will the height of
the mercury be at the end of the 10 strokes?
26. There were 2,500,000,000,000 tons of coal in the United
States in the year 1910, and 3,000,000,000 tons were consumed
between the year 1900 and 1910. If the consumption of coal
should double every decade, tell to the nearest decade how
long the coal in the United States would last.
27. Work again Exercise 76 (p. 249), or similar examples
suggested by the teacher or pupils.
CHAPTER XXV
THE BINOMIAL THEOREM
For Positive Integral Exponents
290. The Binomial Formula. The results obtained by
insi)ection in Art. 166 (p. 276) may be combined in a formula
as follows:
, n(n — 1) (w — 2) o 3 .
We shall now give a proof of this formula for all positive
integral values of n.
291. Proof of the Binomial Formula for Positive Integral
. Values of «. This proof may be conveniently divided into
three parts.
I. By 'actual multiplication it is found that, for any definite
value of w, as n = 4,
(x + a)* = x< + 4x»a 4 Ox^a* + 4xa» + a*.
That is, the binomial formula is true when n = 4.
II. We shall now prove the general principle that if the binomial
formula is true for any power, as the Mh, it is true for the next
higher power, the A; + 1 power.
We write out the formula for the A;th power and multiply both
Eddes by x + a.
447
448 SCHOOL ALGEBRA
(X +o)» x*+fcr*»a+*^x»*a'+^^^^i^x»*(i»+. • •
z+a y x+a
 1 X ^
(i+o)«»  x«» + (fc + l)x»a +[^j^ + A;]x*»o»
^L 1X2X3 ^ 1X2 J ^
or, (x + o)**"^
 x^i+(Jb+l)x*a+^^^x»'a'+^^^!^^x*V+ • • •
This is the result which would be obtained by expanding (x + a)^^
by the formula.
Hence, we have proved that if the binomial formula is true for
any power, as the Arth, it is true for the next higher power, the k il
power.
III. But by actual multipUcation (in I) the binomial formula
was shown to be true for (a; + a)*, or the 4th power. Hence by the
general principle just proved (in II), the formula must be true for
the next higher power, the 5th. In like manner, it must be true for
the 6th, etc., to the nth power.
The method of proof used in this Article is called rnathematicd
induction.
282. When a is negative, c?, cf, etc., are negative; hence,
^ ^ 1X2
1X2X3 ^ ^^
This formula may be proved by changing a into  a in
the proof given in Art. 291.
THE BINOMIAL THEOREM 449
BXEBCISE 149
1. Write out the formula for (x + a)'.
2. For (x + a)»+^ 3. For (x + a)*~^
4. How many terms are there in the expansion of (x+a)*?
Before the pupil attempts the proof of the /ollowing laws, each
law should be illustrated numerically till its meaning id thorbughly
understood. ^
5. By mathematical induction prove that
1 + 2 + 3+ +n =(n + l)
SuG. (1) We have 1+ 2 + 3 = i(3 + 1), or 6
(2) If 1+2+3 + .... +A;=Ka; + i)
adding A; + 1 to each member,
1+2+3 + .... +A; + (Aj + 1) =(a; + A + *; + 1
(fe + l)(A;+2)
2
(3) Hence, etc.
By mathematical induction, prove that
6. The sum of the first n even numbers equals n{n + 1).
7. The siun of the first n odd numbers equals n\
8. 12 + 22 + 32+ +n2 = n(7i+l)(2n + l).
9. 22 + 42 + 62+ + (2n)2 = f n(n + 1) (2n + 1).
10. 1« + 23 + 33 + + ^8 = \n\n+ 1)2
= (1 + 2 + 3 + +n)\
11. a** — 6** is always divisible by a — 6 when n is an
integer.
12. Make up and work two examples to be solved by the
method of mathematical induction.
450 SCHOOL ALGEBRA
293. Key Number and rth Term. In memorizing the
binomial formula, it is helpful to observe that a certain
nmnber may be regarded as governing the formation of each
term of the formula. This number is one less than the num
ber of the term. , .
Thus, for the third term we have ^ a;**~V, in which
1 X ^
there are two factors in the niunerator of the coeflScient; two
in the denominator; the exponent of a: is n — 2, and that of
a is 2. Hence, we regard 2 as the key number of the term.
The number 3 occurs in a simUar way in the formation of
the fourth term; 4, in the fifth term, and so on.
For the rth term, the key number would be r — 1.
^®^^' ^ n(n 1) (nr + 2)^_^, ^i
rth term = ^^ r^^ ■ —  ^^^dT^
r1
294. Examples;
f^ 1 V
Ex. 1. Expand I — — grr ) • •
(  rY  (   *"*y
(x 2 \*
Ex. 2. Find the sixth term of (  — . — j .
The key niimber for the sixth term is 5. Hence we obtain
^"^*^""° 1X2X3X4X5 V2J \r " )
1 2* 3» 27 ^
THE BINOMIAL THEOREM 451
Ex. 3. Find the term in f a* tj which contains a?".
We must first find the number of the term and then the term
itself.
The rth term of (x^  2x"*)" = (coeff.) (x*)"^* (  2x^)'^\
For the required term, the x's collected must = x".
Hence, (a;«)"'(a;"*)'i  x^
24  2r  ^J = 12
Whence, r = 5 '^'
6th term  ^^^^^(a^)ni ( . 2x'^i
 5280x» ilrw.
EXERCISE ISO
1. Change each of the given expressions in Exs. 615 to a
form in which it can be most readily expanded by the bi
nomial formula.
Expand:
11. (^x y Y 14. (r'a: + 2)».
\V^ 2v^y_ 15. (2ar + a;^».
13. (da^Vb  b^i^)*. 17. (o* + 2aa;  a?)* .
Find the
18. Sixth term of (a  2arO".
2, 19. Eighth term of (1 + xVy)".
'■' t
452 SCHOOL ALGEBRA
20. Find the seventh and eleventh terms of {7? — y Vi)^*.
21. Find the sixth and ninth terms of (^d^h — 2Va)".
Find the ratio of
22. The third to the fifth term in the expansion of
(
'+'tT
23. The tenth and twelfth terms of fx^ + ^Y^
24. Find the middle term of (3a*  x'i^)^.
Write the formula for
25. The r + 1st term of (x + a)\
26. The r — 1st term. For the r + 3d term.
27. The rth term of (x + a)*+^
X ) •
Q^ j •
30. Tenn containing^ in (I + V?)".
(2\^
7? ) .
32. Term containing x in I yV^ + V / *
33. By use of the binomial formula find the value of
(1.1)" to three decimal places.
SuG. Expand (l+.l)".
Find the value of
34. (1.2)". 35. (1.3)«. 36. (2.2)».
J I THE BINOMIAL THEOREM 453
I I Q2)*jind the coeflScient of a:* in ( a; ) •
r ^ U \ xJ ^
V 4I> M the shortest way find the 98th tenn of /^2ay='\ .
<3gL Expand (x + a)""*^* to 4 terms.
^) Expand (x + a)*^ to 5 terms,
@) Expand (1 — l)** by the binomial theorem.
^) Prove that in the binomial fonnula the smn of the
coefficients of the odd terms equals the siun of the coefficients
of the even terms.
43. Prove that the siun of the coefficients of the terms in
the expansion of (a + by^ is 2^°. That the smn of the co
efficients in the expansion of (a + by is 2*.
44. Who discovered the binomial theorem and when?
(See p. 464.) Find out all you can about this man.
45. State the advantages or utilities in the binomial
theorem.
46. Make up and work three examples similar to such of
the above as the teacher may indicate.
47. Practice oral work with exponents as in Exercise 93
(p. 303). <4
■ X ""\^
I
CHAPTER XXVI
HISTORY OF ELEMENTARY ALGEBRA.
296. EpocIiB in the Development of Algebra. Some knowl
edge of the origin and development of the symbols and
processes of algebra is important to a thorough under
standing of the subject.
The oldest known mathematical writing is a papyrus roll,
now in the British Museum, entitled " Directions for Attain
ing to the Knowledge of All Dark Things." It was written
by a scribe named Ahmes at least as early as 1700 b. c, and
is a copy, the writer says, of a more ancient work, dating,
say, 3000 b. c, or several centuries before the time of Moses.
This papyrus roll contains, .among, other things, the begin
nings of algebra as a science. Taking the epoch indicated
by this work as the first, the principal epochs in the develop'
ment of algebra are as follows:
1. Egyptian : 3000 B. G.1500 B. C.
2. Greek (at Alexandria): 200 A.D.^0 A.D. Principal
writer, Diophantus.
3. Hindoo (in India): 500 A.D.1200 A.D.
4. Arab: 800 A.D.1200 A.D.
5. European : 1200 A. D. Leonardo of Pisa, an Italian,
published in 1202 a. d. a work on the Arabic arithmetic
which contained also an account of the science of algebra as
it then existed among the Arabs. From Italy the knowledge
454
toSTORY OF ELEMENTARY ALGEBRA 455
of algebra spread to France, Germany, and England, where
its subsequent development took place.
We will consider briefly the history of
I. Algebraic Symbols.
II. Ideas of Algebraic Quantitt,
III. Algebraic Processes.
I. History of Algebraic Symbols
296. Symbol for the Unknown Quantity.
1. Egjrptians (1700 b. c): used the word hau (expressed,
of course, in hieroglyphics), meaning ".heap."
2. DiophantuB (Alexandria, 350 a. d.?): 9', or 9°'; plural,
99.
3. Hindoos (500 a. D.1200 a. d.) : Sanscrit word for
" color," or first letters of words for colors (as blue, yellow,
white, etc.).
4. Arabg (800 A. D.1200 a. d.) : Arabic word for " thing "
or " root " (the term root, as still used in algebra, originates
here).
5. Italians (1500 a. d.) : Radix, R, Rj.
6. BombeUi (Italy, 1572 a. d.) : vl; •
7. Stifel (Germany, 1544) :A,B,C,
8. Stevinns (Holland, 1586) : (D
9. Vieta (France, 1591) : vowels A, E, I, 0, U.
10: Descartes (France, 1637) : x, y, z, etc.
297. Symbols for Powers (of x at first) ; Exponents.
1. Diophantus: Svvafii^, or S" (for square of the unknown
quantity); kv/So^, or /c" (for its cube).
2. Hindoos: initial letters of Sanscrit words for " square "
and " cube."
456 SCHOOL ALGEBRA
3. Italians (1500 a. d.) : " census " or " zensus " or " z "
(for 7?); " cubus " or " c " (for a?).
4. Bombelli (1579): <Li, vj^, v3y, (for x, a?, a?).
5. Stevinus (1586): ®, ®, ®, (for x, a?, t?).
6. Vieta (1591) : A, A quadratuSy A cubus (for a:, 7? s?).
7. Harriot (England, 1631): a, aa, doa.
8. Herigone (France, 1634) : a, a2, a3.
9. Descartes (France, 1637) : x, a?, a?.
Wallis (England, 1659) first justified the use of fractional
and negative exponents, though the use of fractional expo*
nents had been suggested earlier by Oresme (1350), and the
use of negative exponents by Choquet (c. 1500).
Newton (England, 1676) first used a general exponent, as
in a;**, where n denotes any exponent, integral or fractional,
positive or negative.
298. Symbols for Known ftnantities.
1. Diophantus: fiovaBe: (i. e. monads), or /j!*.
2. Regiomontanus (Germany, 1430) : letters of the alphabet.
3. Italians: d, from dragma.
4. Bombelli: v?;. •
5. Stevinus: ®.
6. Vieta: consonants, B, C, D, F, , . ^ .
7. Descartes: a, b, c, d,
Descartes possibly used the last letters of the alphabet, x, y, 2, to
denote unknown quantities because these letters are less used and
less familiar than a,b,c,d, , which he accordingly used to de
note known numbers.
299. Addition Sign. The following symbols were used:
1. Egyptians: pair of legs walking forward (to the left), A.
2. Diophantus: juxtaposition (thus, ab, meant a + 6).
HISTORY OF ELEMENTARY ALGEBRA 457
3. Hindoos; juxtaposition (survives in Arabic arithmetic,
as in 2f , which means 2 + f )
4. Italians: plies, then p (or e, or <^).
5. Germans (1489): h, +, +.
300. Subtraction Sig^.
1. Egyptians: pair of legs walking backward (to the right),
* thus, ZV_; or a flight of arrows.
2. Diophantus: ^ (Greek letter yjr inverted).
3. Hindoos: a dot over the subtracted quantity (thus, mn
meant m — ft),
4. Italians: minus, then M or m or de.
5. Germans (1489): horizontal dash, — .
The signs + and — were first printed in Johann Widman's
Mtercantile Arithmetic (1489). These signs probably originated in
German warehouses, where they were used to indicate excess or
deficiency in the weight of bales and chests of goods. Stifel (1544)
was the first to use them systematically to indicate the operations
of addition and subtraction.
301. Multiplication Sign. Multiplication at first was
usually expressed in general language. But
1. Hindoos indicated multiplication by the syllable bka,
from bharita, meaning " product/' written after the factors.
2. Oughtred and Harriot (England, 1631) invented the
present symbol, X .
3. Descartes (1637) used a dot between the factors (thus,
ab).
302. Division Sign.
1. Hindoos indicated division by placing the divisor under
the dividend (no Une between). Thus, "^ meant c f d.
2. Arabs, by a straight line, Thus, a — 6, or a I 6, or •
b
458 SCHOOL ALGEBRA
3. Italians expressed the operation in general language.
4. Oughtred, by a dot between the dividend and divisor.
5. Pell (England, 1630), by 4.
303. Equality Sign.
1. Egyptians: Z □ (Also other more complicated sym
bols to indicate different kinds of equality).
2. Diophantus: general language or the symbol, K
3. Hindoos: by placing one side of an equation i^mlediately
under the other side.
4. Italians: cb or a; that is, the initial letters of cBqualis
(equal). This symbol was afterward modified into, the form,
» , and was much used, even by Descartes, long after the
invention of the present symbol by Recorde.
5. Recorde (England, 1540) : = .
He says that he selected this symbol to denote equality because
" than two equal straight lines no two things can be more equal."
304. Other Symbols used in Elementary Algebra.
• Inequality Signs, > <, were invented by Harriot (1631).
Oughtred, at the same time, proposed ~I3, _I] as signs of in
equality, but those suggested by Harriot were manifestly superior.
Parenthesis, ( ), was invented by Girard (1629).
The Vinculum had been previously suggested by Vieta
(1591).
Radical Sign. The Hindoos used the initial syllable of
the word for square root, Ka, from Karania, to indicate
square root.
Rudolph (Germany, 1525) suggested the symbol used at
present, V, (the initial letter, r, in the script form, of the
word radix, or root) to indicate square root, /W to derApte
the 4th root, and AW to denote cube root.
HISTORY OF ELEMENTARY ALGEBRA 459
Girard (1633) denoted the 2d, 3d, 4th, etc., roots, as at
present, by V^ V, V, etc.
The sign for Infinity, oo , was invented by WalUs (1649).
305. Other Algebraic Symbols have been invented in recent
times, but these do not belong to elementary algebra.
Other kinds of algebra have also been invented, employing
other systems of the symbols.
306. General niustration of the Evolution of Algebraic S3rni
bols. The following illustration will serve to show the
principal steps in the evolution of the symbols of algebra:
At the time of Diophantus the numbers 1, 2, 3, 4, ... . were de
noted by letters of the Greek alphabet, with a dash over the letters
used; as, a, ^, y, . . . .
In the algebra of Diophantus the coefficient occupies the last
place in a term instead of the first as at present.
Beginning with Diophantus, the algebraic expression,
a? + 5x — 4:, would be expressed in symbols as follows:
S"a ^SejfifiB (Diophantus, 350 a. d.)
I2 p.5 RmA (Italy, 1500 a. d.).
lQ + 5iV4 (Germany, 1575).
l(^p.5(^m.4^ (BombeUi, 1579).
1(2) + 5®  4(0) (Stevinus, 1586).
\Aq + 5^4® (Vieta, 1591).
' laa+5a 4 (Harriot, 1631).
Ia2 + 5al  4 (Herigone, 1634).
a? + 5a:  4 (Descartes, 1637).
307. Three Stages in the Development of Algebraic Sym
bols.
1. Algebra without Symbols (called Rhetorical Algebra).
In this primitive stage, algebraic quantities and operations
were expressed altogether in words, without the use of sym
460 SCHOOL ALGEBRif
bols. The Egyptian algebra and the earliest Hindoo, Arabian,
and Italian algebras were of this sort.
2. Algebra in which the Symbols are Abbreviated Words
(called Syncopated Algebra). For instance, p is used for plus.
The algebra of Diophantus was mainly of this sort. European
algebra did not get beyond this stage till about 1600 A. d.
3. Symbolic Algebra. In its final or completed state,
algebra has a system of notation or symbols of its own, inde
pendent of ordinary language. Its operations are performed
according to certain laws or rules, " independent of, and dis
tinct from, the laws of grammatical construction."
Thus, to express addition in the three stages we have pluSy
p, + ; to express subtraction, minus, my—] to express equality,
oequaliSy (F, =..
Along with the development of algebraic symbolism, there
was a corresponding development of ideas of algebraic quan
tity and of algebraic processes.
II. History of Algebraic QuANnrr
308. The Kinds of Qnantity considered in algebra are
positive and negative; particular (or numerical) and general;
integral and fractional; rational and irrational; commensur
able and incommensurable; constant and variable; real and
imaginary.
309. Ahnies (1700 b. c.) in his treatise uses particulary pos
itive quantity, both integral and fradioncd (his fractions, how
ever, are usually limited to those which have a imity for a
numerator). That is, his algebra treats of quantities like 8
and J, but not like — 3, or — f , or V2, or — a.
310. Diophantns (350 a. d.) used negative quantity,, but
only in a limited way; that is, in connection with a larger
HISTORY OF ELEMENTARY ALGEBRA 461
positive quantity. Thus, he used 7 — 5, but not 5 — 7, or
— 2. He did not use, nor apparently conceive of, negative
quantity having an independent existence.
3U. The Hindoos (500 a. D.1200 a. d.) had a distinct
idea of independent or absolvie negative quantity, and used
the minus sign both as a quality sign and a sign of operation.
They explained independent negative quantity much as it is
explained today by the illustration of debts as compared
with assets, and by the opposition in direction of two lines.
Pythagoras (Greece, 520 b. c.) discovered irrational quan
tity, but the Hindoos were the first to use this in algebrJa.
312. The Arabs avoided the use of negative quantity as
far as possible. This led them to make much use of the pro
cess of transposition in order to get rid of negative terms in
an equation. Their name for algebra was "al gebr we'l
mukabala," which means " transposition and reduction."
The Arabs used 8urd quantities freely.
313. In Europe the free use of absolute negative quantity
was restored.
Vieta (1591) was principally instrumental in bringing into
use general algebraic quardity (known quantities denoted by
letters and not figures).
Cardan (Italy, 1545) first discussed imaginary quantities,
which he termed " sophistic " quantities.
Euler (Germany, 170783) and Gauss (Germany, 1777
1855) first put the use of imaginary quantities on a scientific
basis. The symbol i for V — 1 was suggested by Gauss.
Descartes (1637) introduced the systematic use of variable
quantity as distinguished from constant quantity.
462 SCHOOL ALGEBRA
III. History of Algebraic Processes
314. Solution of Equations. Ahmes solved many dmpk
equations of the first degree, of which the following is an ex
ample:
" Heap its seventh, its whole equals nineteen. Find heap."
In modem symbols this is,
Given = + a; = 19; find x.
The correct answer, 16, was given by Ahmes.
Hero (Alexandria, 120 b. c.) solved what is in effect the
quadratic equation,
{{dJ' + ^d^s.
where d is unknown, and s is known.
DiophantoB solved simple equations of one unknown quan
tity, and simvltarveous equations of two and three unknown
quantities. He solved quadratic equations much as is done
at present, completing the square by the method given in
Art. 226. However, in order to avoid the use of negative
quantity as far as possible, he made three classes of quadratic
equations, thus,
as? ^rhx = c,
ao!? + c =bx,
ax^ = bx + c.
In solving quadratic equations, he rejected negative and
irrational answers.
He also solved equations of the form ax"^ = bx\
He was the first to investigate indeterminate equations, and
solved many such equations of the first degree with two or
three unknown quantities, and some of the second degree.
HISTORY OF ELEMENTARY ALGEBRA 463
The Hindoos first invented a general method of solving a
quadratic equation (now known as the Hindoo method, see
Art. 233). They also solved particular cases of higher de
grees, and gave a general method of solving indeterminate
equations of the first degree.
The Arabs took a step backward, for, in order to avoid the
use of negative terms, they made six cases of quadratic equa
tions; viz.:
aa? = bx, aa? + bx = c, '
ax^ = c, aa^ + c = bx,
bx = c, aa^ = bx + c.
Accordingly, they had no general method of solving a quad
ratic equation.
The Arabs, however, solved equations of the form aa^^'+bxP
= c, and obtained a geometrical solution of cubic equations
of the form 7? + yx \ q = Q.
In Italy, Tartaglia (15001559) discovered the general so
lution of the cubic equation, now known as Cardan's solution.
Ferrari, a pupil of Cardan, discovered the solution of equor
tions of the fourth degree,
Vieta discovered many of the elementary properties of an
equation of any degree; as, for instance, that the niunber of
the roots of an equation equals the degree of the equation.
315. Other Processes. Methods for the addition, sub
traction, and multiplication of polynomial expressions were
given by Diophantus.
Transposition was first used by Diophantus, though, as a
process, it was first brought into prominence by the Arabs.
The word algebra is ah Arabic word and means " transposi
tion " {al meaning " the," and gebr meaning " transposition").
The Greeks and Romans had a very limited knowledge of
464 SCHOOL ALGEBRA
fractions. The Hindoos seem to have been the first to reduce
fractions to a conunon denominator.
The square and cube root of polynomial expressions were
extracted by the Hindoos.
The methods for using radicals, including the extraction
of the square root of binomial surds anid the rationalizing of
the denominators of fractions, were also invented by the
Hindoos.
The methods of using fractional and negative exponents
were determined by Wallis (1659) and Sir Isaac Newton.
The three progressions were first used by Pythagoras (569
B. C.500 B. c.)
Permutations and combinations were investigated by Pascal
and Fermat (France, 1654).
The binomial theorem was discovered by Newton (1655),
and, as one of the most notable of his many discoveries, is
said to have been engraved on his monument in Westminster
Abbey.
Graphs of the kind treated in this book were first invented
by Descartes (France, 1637).
The fundamental laws of algebra (the Associative, Com
mutative, and Distributive Laws; see Arts. 316317) were
first clearly formulated by Peacock and Gregory (England,
1830^5), though, of course, the existence of these laws had
been implicitly assumed irom the beginnings of the science.
Students who desire to investigate the history of algebra in more
detail should read the second part of Fine's Number System of
Algebra, Ball's Short History of Ma^iematics, and Cajori's History
>/ ElemerUary Mathematics.
APPENDIX
Fundamental Laws of Algebra
316. The following Laws of Algebra have been used in the
preceding pages without formal statement:
A. The Commutative Law (or Law of Order).
1. For addition, a+b = b + a.
2. For mvUipliccUion, ab = ba.
3. For division, a^6Xc = aXc5&.
B. The Associative Law (or First Law of Grouping).
1. For addition, a + 6 + c = a+(6 + c) = (a + b)
+ c.
2. For midtiplication, abc = a{bc) = (ab)c.
C. The Distributive Law (or Second Law of Grouping).
1. For midtiplicaiion, a{b + c) = ab + ac. Hence, in
versely, ab + ac = a(b + c).
2. For division, =  + .
a a a
Who first formulated the laws of algebra? (See p. 464.)
^ 317. Utility of the Laws of Algebra. The laws stated in
Art. 316 are methods adopted for arranging and grouping
algebraic symbols so as to decrease the amount of work and
to increase the importance of the results attained.
Thus, in the following example we are able to eliminate the
parenthesis by use of the Distributive Law and to collect terms by
use of the Commutative and Associative Laws.
465
466 SCHOOL ALGEBRA
Ex. 6(aj+y) + 3(xy+«) + 2(a;+2y«).
= 6x4 62/+ 3x 32/+ 3«+ 2x+ 4y2z
« Qx+ 3z+2x{ 62/ 32/+ 42/+ 32 22
= lla;+ 72/+ z Ans.
The use of these laws enables us to diminish the 23 symbols used
in the first expression to the 8 symbols used in the last expression.
It should be noted that by changing the laws stated in Art. 316,
kinds of algebra different from that presented in this book, and
adapted to other uses, may be devised. Thus, in a certain important
kind of algebra ab — — 6a, not ba.
Even in arithmetic the conmiutative law holds only in a limited
way. For, while 5x7= 7 X 5, 57 does not equal 75.
Detached Coefficients
318. Examples.
Ex. 1. Multiply a? + Za^x  2a3 by a:^  4aa^^ + So*.
l+b+3 2
14hOf 3
1+0+3 2
4012+8
+ 3 + 0+96 .
14+311 + 8+96
Hence, x*  4ax« + Zch^  1 la»x» + 8a*x2 + 9a«a:  6a« Fradvxi
Let the pupil work this example in full and compare the labor in
the two processes.
Ex. 2. Divide a?*  io^f + 8a^  By* by a? + ^ry  j/*.
1 + 02+83 11 + 21
1+21 12+3
21+8
24+2
3+63
3+63
Hence, x^ — 2xy + 3^^ Qvxjiie^ni
n
APPENDIX 467
EXERCISE 161
By use of detached coefBcients, work such examples in
Exercises 17 (p. 68) and 21 (p. 85) as the teacher may indicate.
Factor Theorem
319. niustratioiui. The method by which an expression
is prepared for division at sight and hence for factoring, as
explained in Ex. 3, of Art. 99 (p. 150), niay be carried fur
ther and then abbreviated.
Ex. 1. Factor it^  is?  s^ + 16x  12.
We may test this expression as to its divisibility by a: — 1 by
splitting terms in succession thus,
a:*4x»x*f 16x 12
= x*a^Sx^+Sx^Ax^+4x+12x12
= x«(x l)3x^{x l)4a;(a; l)+12(a; 1)
Hence, aiL  1 is a factor of the original expression.
This result might have been obtained in a shorter way by ob
serving that, as this last expression reduces to zero when x — 1, the
first expression, might be tested as to its divisibility by a; — 1 by
substituting 1 for x and noting whether the expression reduces to
zero. This last test may be further abbreviated to a matter of
noting whether the algebraic siun of the coefficients of the terms is
zero.
Ex. Determine by inspection whether a? + a? — 6x^ — 4x
+ 8 is divisible by ar — 1.
Summing the coefficients, we have
l+l.64+8=0;
hence, a: — 1 is a factor of the given expression.
In like manner, if an expression is divisible by a; + 1, the smn of
the coefficients of the even terms must equal the siim of the coeffi
cients of the odd terms.
468 . SCHOOL ALGEBRA
320. Factor Theorem. If any rational integral expression
containing x becomes equal to zero, when a is substituted for x,
then x^a is a factor of the given expression.
For, let E stand fot any rational integral algebraic expression.
If ^ is divided by x — a till a remainder is obtained in which x
does not occur, denote the quotient by Q &nd the remainder by B,
Then
E = Q(xa) +R
Let rr » a, then
= Q(0) + R (since ^ = when x = a)
.*. jB =0.
Hence, E = Q(x — a), or x — a is a factor of E
Ex. Factor ic^  12x + 16.
By trial we find that x«  12a; + 16 =
When X = 2
.'. X — 2 is a factor of x' — 12x + 16.
By division x»  12x + 16 = (x  2) (x* + 2x  8)
= (x 2) (x  2) (x + 4) Fadors
Note that the only numbers which need be tried as values of x
are the factors of the last term of the given expression. This follows
from the fact that the last term of the dividend must be divisible
by the last term of the divisor.
EXERCISE 162
Factor by use of the factor theorem:
1. a? 4. 8. ^a^Ax^lixe.
2. x23a;28. 9. So? + 8qi? + 3x  2.
3. a^U^ + 3(a6). lo. 2a^ + x^  Ux^ + 5a: + 6.
4. (a&)2 + 3(afc). 11. 6a:*13x345x22x + 24.
5. a? + 5a6. 12. x^ 2x^+1.
6. 2x2 + 7x  15^ 13. a:8 _ g^ ^ 25.
7. 2x3x2_7a. + 6^ 14, X*  28x2 + 33a:  90.
APPENDIX 469
15. Prove that a:'* — y" is always divisible hy x — y.
16. Prove that x** + y" is divisible hyx + y when n is odd.
17. Show that (1 — xy is a factor of 1 — a: — x'* + x*'^^
18. Make up and work an example similar to Ex. 5. To
Ex.7.
H. C. F. AND L. C. M. Obtained by Long Division
321. H. C. F. by Long Division. For polynomials that
cannot be readily factored, the H. C. F. is found by the same
general method that is used in arithmetic to determine the
G. C. D. of large numbers.
Ex. Find the G. C. D. of 65 and 117.
65)117(1 .. G. C. D. of 65 and 117 is 13
65
52)65(1
52
13)52(4
52
In appl3dng the above method to algebraic expressions, note, for
instance, that the H. C. F. of x' — 4 and x* — 3x + 2 is the same
as the H. C. F. of 5x(x^  4) and 2a{x^  3x + 2); the H. C. F. in
either case is a; — 2.
Or, in general, if m, n,P, and Q are algebraic expressions, the
H. C. F. of P and Q is the same as the H. C. F. of mP and nQ, pro
vided that m has no factor which is a factor of Q and n has no factor
which is a factor of P.
This property of algebraic expressions enables us to simplify the
process of finding the H. C. F. by multiplying or dividing one of
the algebraic expressions by an expression which is not a factor of
the other expression.
Ex. 1. Find the H. C. F. of 4a:«  4ar^  5a: + 3 and 10a?
 19a: + 6.
To render the first expression divisible by the second, we may
multiply the first expression by 5, which is not a factor of the second
expression.
470
SCHOOL ALGEBRA
The work may then be conveniently arranged as follows:
iac219a;+6
10a;«  15x
 4x+6
 4a; +6
5
2ac»  20x2  25aj + 15
2(^  38x2 ^ i2x
18x2 37X + 15
5
90x2  185x + 75
90x2 _ 171a. + 54
7l 14X+21
H. C. F = 2x  3
2x
5x2
The work may be shortened by the use of detached coefficients
(see Art. 318).
Ex. 2. Find the H. C. F. of 8aJ*  Sa:^  lOa:^ ^ gx and
30a;*  b77? + ISa:^
8x*  8x«  10x2 + 6x = 2x(4x»  4x2  5x + 3)
30x^  57x« + 18x2 _ 3a;2(i0x2  19x + 6)
The H. C. F. of 2x and 3x2 ig ^^
The H. F. of 4x3  4x2  5x + 3 and 10x2  19x + 6, by the
method of Ex. 1, is found to be 2x — 3.
Hence, the complete H. C. F. is x(2x — 3) Ans.
322. L. C. M. by Long Division.
Ex. Find the L. C. M. of 182 and 299.
By the division method, the G. C. D. is foimd to be 13.
Then, since 182 = 13 X 14, ^99 = 13 X 23,
13 113X14 , 13X23
14 , 23
.. L. C. M = 13 X 14 X 23.
Similarly, to find the L. C. M. of two algebraic expressions
which cannot be readily factored, we first find the H. C. F. of
the two expressions by the division method.
Ex. Find the L. C. M. of 4ar^ + 3a;  10 and ^ + 1^
 3a;  15.
APPENDIX 471
We first find the H. C. F. by the division method; this is 4x — 5.
Then 4x^ +3x  10 ^ (4z  5) (x + 2)
4a*\7x*  3x  15 = (4a;  5) (x^+^x+S)
:. L.C.M. = (4x  5) (x +2) (x» +3x +3) Ans.
EXERCISE 163
Find the H. C. F. and L. C. M. of
1. 2x^'X'SmdAs?ix^Zx + 5.
2. 6ar^  a:  12 and 6a^  ISo^  ga: + 18.
4. Z3!?93? + 9x 3, 6a:8  6ic2  6a: + 6.
5. 6a:*  Sa:^ + 6ar^ + 5x, 2a:*  Oa:^  9a?  2a:.
6. Oa:^ + 3a:*  3a:' + 12ar^, 18a:* + 42a:3 + 6ar*  24a:.
7. 3a:3 + 7ar^  5a: + 3, 23? + Sa?'7x + 6.
8. a:*  a:»  ar^ + 7a:  6, a:* + a:'  5a:2 + 133. _ 6.
9. 2a:'  16a: + 6, 5a:« + 15a:5 + 5x + 15.
10. 2a:5 + a:* + 2a:'  ar*  1, 5a:* + 2a:' + 3a?  2a: + 1.
11. 3a:* + 2a?y + 2xh/^ + 5xf  2y*, 6a:* + a?y + 2xh/^
+ 2xf  y*.
12. 3a:5 + 2a:*  8a?  3a? + ^^ ^^ _ iQa:* + 14a?  11a?
+ 4a:.
The H. C. F. (or L. C. M.) of three or more expressions may
be obtained by finding that of two of them; then find the H. C. F.
(or L. C. M.) of this result and another of the quantities; the last
H. C. F. (or L. C. M.) thus obtained is the one required.
13. a?  a?  a:  2, a?  2a? + 3x  6, 2a?  3a?  a: 2.
14. 2a:*  14a? + 12a:, 2a:* + 6a?  32a? + 24a:, 6a?  30a?
+ 42a?  18a:.
472 SCHOOL ALGEBRA
Cube and Higher Roots
323. Cube Boot of Polynomials. A general method for
determining the cube root of any polynomial which is a
perfect cube may be found by studying the relation between
the terms of a binomial — or, in general, of a polynomial —
and the terms of its cube (as a + 6, and its cube, c? + 3a*5
+ 3a6^ + 6'). This relation stated in the inverse form gives
the method for extracting the cube root.
The essence of this method consists in writing o? + ZcFb
+ 3a62 + fcs in the form a^ + h{U^ + 3a6 + 6^).
Ex. Extract the cube root of a:* + 3a:^ — 5a:^ + 3a: — 1.
Ix' +x — 1 Root
a* 4 3x«  5x3 + 3x  1
^
3(a;2)2 = 3x*
3(x2)a; + «« = +^+ x^
Complete divisor = 3x* + Sx* + x^
3(x2 + a;)2 = 3x* + 6x»43x*
3(x2Ha;)(l)+(l)«« 3a;^3x+l
3x* 5x«
3x« 4 3a:* + x*
Complete Divisor = 3a:*H6a:^— 3x+l
3a:*6a:«+aFl
3x* 6a:»+agl
Let the pupil state this process as a formal rule.
EXERCISE 164
Find the cube root of
1. a^ + Qdhc+l2ax^ + %^. 2. 27  27a + 90^0'.
3. a«  3a*  3a* + lla^ + Ca^  12a  8.
4. 12a?*  36a: + 64a:«  6a?  8 + 117a?  144a?.
5. 95a3 + 72^4 _ 72a2 + ISa* + 15a + a«  1.
6. 114a?  171a?  27  135a: + 8a?  60a? + 55a?.
7. a»3a? + 6a:7 + 4 + A
a: a? a?
^ y ^ 2f 4y*, 2y* %f f'
APPENDIX
473
324. Cube Soot of Arithmetical irnmben. The same
general method as that used in Art. 323 can be used to ex
tract the cube root of arithmetical numbers. The process
is slightly different from the algebraic process, owing to the
fact that all the niunbers which compose a given cube are
united or fused into a single number.
Thus, (42» = (40 + 2)»  40» + 3 X 40* X 2 + 3 X 40 X 2«.H2»
. 64000+9600+480+8
74088
Reversing this process, we obtain a method of extracting the
cube root of a number.
Ex. 1. Extract the cube root of 74088.
Trial Divisor,
Complete Divisor,
7408842 Root
40*= 64
3 X 40« = 4800
3 X 40 X 2  240
2»= 4
10088
10088
= 5044
Ex. 2. Extract the cube root of j^ to 4 decimal places.
y% = .416666666666 +
.416666666 + [.7469+ Root
343
3X(70)« 14700
3 X (70 X 4)  840
4« = 16
15556
3 X (740)2  1642800
3 X (740 X 6) = 13320
62= 36
1656156
3 X (7460)2 = 166954800
73666
62224
11442666
9936936
1505730666
1502593200
3137466
474 SCHOOL ALGEBRA
The first three figures of the root are found directly. The last
figure is then found by division of the remainder, using three times
the square of the root already found as a divisor. The number of
figures of the root that may thus be found by division is two less
than the number of figures already found.
Let the pupil state the above process as a rule.
EXERCISE 156
*• Find the cube root of
1. 3376. 4. 43614208. 7. 344324.701729.
2. 753671. 5. 32891033664. a .000127263527.
3. 1906624. 6. 620688691.125. 9. 0.991026973.
Find to three decimal places the cube root x)f
10. 75. 12. 6.6. 14. 7t\. 16. iV M. 1t5.
11. 6. 13. 3f . 15. 19h 17. i^j. 19. 82V.
Compute the value of
20. ^5 + 2^5. 21. hVlO2Vl6.
22. ^3^08  2VI.935.
Visualize the following objects by the aid of cube root:
23. 150,000,000 cu. yd. of earth.
24. 40,000,000,000 feet of lumber.
25. 60,000,000 tons of iron (taking 480 lb. as the weight
of one cubic foot of iron).
26. Make up and work an example similar to Ex. 12. To
Ex.23. Ex.26.
325. Higher Eootfl Obtained by Successive Extractions.
By the law of exponents, the square of the square of any
quantity gives the fourth power of the quantity. Hence, re
versing the process, the fourth root of a quantity is the square
APPENDIX 475
root of the square root of the quantity. Similarly, the sixth
root of a quantity is the square root of the cube root of the
quantity. The eighth, nirUh, tenth .... roots of a quantity
may be found by similar methods.
Ex. Extract the fourth root of
81a^ + 108a^ + Ma^ + 12a + 1
Obtain first the square root of the given expression, which is
9o2 + 6a + 1. Extracting the square root of this, we obtain 3a + 1,
the fourth root of the original expression.
EXERCISE 156
Find the fourth root of
I. 130321. 2. 3418801. 3. 90. 4. .8.
5. 1  12a6 + 54a262  lOSa^b^ + 81a^b\
6. ar*  2a? + f a?  Ja; + yV
7. 64ar^56ar* + 16a? + x* + 16  32a?+16a?  8a;^ + 64a:.
Find the sixth rdot of
a 7529536. 9. 1544804416. lO. 15.
II. a? + 1215a? + 729  1458x + 135a?  540a?  18a?.
12. 4096a?23072a;io + 960a;8160a? + 15a?  fa? + ^\.
EXERCISE 167
Review of Algebra to Quadratics
1. If a; = 2, y =  3, « = — J, find the value of
3xy  y{x + 4z)  xz(iy + 6x) + 3yz{x +y)(y + 2z).
2. Fmd the value of  H ^ ^/T^TZ
when X = J.
3. Find the niunerical value of the following expressions when
o « 3 and 6 =  3: 3a«  562; (^v^Z^)2; (^5)6.
476 SCHOOL ALGEBRA
4. Simplify
(a46)(a6){(a+6c)^,(6ac) + (6 +ca)} (a6c).
5. Divide x^+x^+ax'^+bxS by a;^ + 2a;  3, and find
what values a and b must have if thctre is to be no remainder.
6. Factor:
(1) X*  9. * (4) x^  jr^.
(2) a;*+27. (5) J Sb^K
(3) 4x  y\ (6) 25n'^  y7«.
7. Factor: (1) a:y — 1 — a; + y.
(2) o* 4 6* 4 <i*  2o6 4 2ac  26c.
(3) o*a;*+a*a;* + l.
(4) 7(p ~ 1)»  27(p  1) + 18.
a Factor:
(1) 3a;  8x*  35. (4) aM  3a* 4 5a;*  15.
(2) Kb*  19a;*  56. (5) 60  7 V3a  6a.
(3) 12a;* 4 5a;*  72. (6) 15a;  2\/^  24y.
9. Find the H. C. F. and the L. C. M. of
a« 4 a*6* + a*6*62 an^j ^^  a*6*  a*6*  h^.
10. The H. C F. of two expressions is a{a — b), and their
L. C. M. is a^b{a 4 6) (a — 6). If one expression is ab{a^ — 6*), find
the other.
11. Find the L. C. M. of a;* 4 ax h a^, a^  a», and x^  a*.
12. Find the L. C. M. of a;« 4 xyi  22/, 2a;* 4 5xyi 4 2y, and
2x* — opy^ — y.
13. Simplify
wi 4~ w 1 wi — w
w — n m \n ___ ^ __^
m4w m— n w w
w — n m \n
Write the foUowing expressions with positive exponents and
simplify:
14. (o  6)i 4 (o 4 6)"*.
^. . (g 46) (fl  6) I (g 6) (g 46)"^
1  (g2 4 6*) (a 4 6)^
APPENDIX ,477
16. o (1  a*) (o + o») (1 + a»)' (o« + 1)».
17. Showthatr^(!Lpi) + n(nl)(n2) . (n + D n(n  1)
18. Simplify ^^' , +
(la;«)* (la;2)*
19. Show that , ,. , r + tt — w n + 7 wl — \ ^'
(a 6) (c o) (6 c) (a 6) (c o) (6 c)
20. What must be the value of n in order that (2a + n) *•
(3n + 69a) may be equal to g^^, when a = ^?
21. The equation ax^ + 6xy + cy* « 5 is to be satisfied when
^ = 1, y = 1; when X = 2, 2/ =  1; and when a; =  1, y =» 2.
Find the values of a, h, and c.
Solve:
22. ?  5y = 13. 24. ax + 6y + C2 = 1.
? fex+cy+a« = l.
' ■ + iy = 4. ex + ay + 62 = 1.
X
23. 4x  — 3. 25. gx  r6 = p(a  y).
3x + f=6J. a \ b/
26. Given A = 2, .B  2A = 0, C  2B + 3^ =  3, D^2C
+ 3B =  1, and E  22) + 3C = 0; find the values of B, C, D,
BjidE,
H n
27. Extract the square root of 4x" + 9x"*+ 28  24x""2  IGx^.
28. Obtain the square root of a* + 6* + 2a6(a2 +¥) + 3a*6«. ;
Simplify:
29. 8"* + 25*  rt)» + 13°  (A)~*.
30. (x*")"^ • (x'»)'«+i • (x»«y*^
1 _ aib* . (1 + o^6*) \ ^ a'&+5aV666
6*Va a6 a"*5 a*6 + 3a^ y 6  54
35. (.09)"*. Also (.064)1
478 SCHOOL ALGEBRA
36. Find the ratio of ~^_^^ to £1^^.
37. Simplify ^^^^ 4—^'
n _n
38. Multiply a;« + x^ +1 by ar« x « + 1.
39. If p = jf ?  5^\ show that (1 + P*)* = ^l
\y» y / 2yxy
40. Simplify ^50+ ^9 ^Vi+ V27+ ^^27 V64.
41. Find to 3 decimal places the value of —r=\
V^ \/2l
42. Simplify V(w  nYa + V(m~+n)^ — Vow* + V«(l "^Y
43. Which is the greatest, VI, \/, or Vf ?
44. From (c x)\/(^^x^ subtract i/^lL?.
V c — a;
45. Show that every even power of i is real (use t** as represent
ing any even power of i). Also show that every odd power of i is
ima^nary.
V6
46. Does — r — equal a/— 3?
^ 47. Given ^3= 1.73205. Compute the value of V^ in the
shortest way.
^ cj. rr Va^x2 + 2x2(a2x2)"*
48. Sunphfy ^^ ;^— ^ ^•
49. Find the value of i* ^ i*. Of i**»+i. Of 1 s i».
50. A certain shelf will hold 20 geometries and 24 algebras, or
15 geometries and 36 algebras. How many geometries alone, or
how many algebras alone, will the shelf hold?
51. A baseball nine has won .625 of the games it has played. If
it has won 8 more games than it has lost, how many games has it
played?
52. A certain solution is 45% alcohol. Water equal to what frac
tional part of the solution must be added to change it to a 25%
solution?
APPENDIX 479
53. A certain paint is half oil and half pigment. Oil equal to
what fractional part of the given amount of paint must be added
to make the oil equal to 60% of the whole?
54. Solve ^±^ (x"^) = 36.
('¥)=
55. Find the algebraic expression which, when divided by
X* — 2x + 1, gives a quotient of x' f 2x + 1 and a remainder of
X 1.
56. Arrange V5> V^24, v 11 in descending order of magnitude.
57. Solve ?waj+ = 1, nx \ — = 1.
y V
58. Factor «*" + x^^ + j/"**.
59. If a cijertain kind of cloth is 27 in. wide and loses 2% in width
and 5% in length by shrinking, how many yards must a dressmaker
buy in order that i^ter shrinking it she shall have 20 sq. yd.7
60. Sunplify ^'"^^^"^^ (3x  2)* 5 (3x  5).
2x +
3x 5
61. Express algebraically: 5 times the cube of a is divided by
the fraction whose numerator is 6 times the square of 6, and whose
denominator is the square of the difference between x and twice the
cube of y. Also express in words h^ — —r^ •
62. The United States 5^ piece (or nickel) is 75% copper and
25% nickel. If a mass of nickel and copper weighing 80 pounds is
90% copper, how many pounds of nickel must be added to it to
make it ready for coinage into hi pieces?
63. Separate 200 into three such parts that the first divided by
the second gives 2 for a quotient and 2 for a remainder; and the
second divided by the third gives 4 for a quotient and 1 for a re
mainder.
64. Simplify (Sv^i)'.
1 12 2
65. Extract the square root of x* +  + 2+ — — —
X^ QC* X* X
66. Find that number which, when divided by 3, is equal to one
quarter of the simi of itself and 24.
480 SCHOOL ALGEBRA
«
67. Divide 2a;*y»  5a; V* + 7x*jr^  6a;* + 2x*y by xV* 
68. In the year 1910 the record for the baseball throw was 426
ft. 6 in., which was 8 ft. 2} in. more than 17 times the record for
the running long jump. What was the latter record?
n3 n1
69. Show that (n  1)^(<2 + a*) « + (^ + a^) ^ reduces to
»3
(rU*+a«)(^+o2)2 .
70. Solve the following equations for x and y:
ax +hy 1, bx —ay ^1.
71. Simplify 16* X 2* X 32* ]
72. Multiply y/± by y/^.
73. Define a literal equation. Quadratic equation. Root of an
equation. Absolute term. Degree of an equation.
74. A baseball player has been to the bat 150 times in a given
season and made an average of .280 hits. How many more times
will he need to bat to bring his average up to .375, provided that the
number of base hits he makes in the future equals half the number
of times he bats?
75. Solve ^—^ + ^"^^ 1 0.
a:»8^2a;«+4x+8 x2
76. Simplify the product of (ayxr^)^, (hxy^^)^, and (y*a«6*)*.
77. Find the numerical value of the following expression when
a = 5, 6=3, c = — 1, d =  2, and x ^0
2 V3 +2d + a (3c  rf)x
3 Vfl +b —ex —c 7ad — Va6c
78. Simplify i 1= ^
P + V 3
79. Reduce ^  "" V ^ to an equivalent fraction having
V2 + V3  V5
a rational denominator, and find its value to two decin:ial places.
APPENDIX 481
80. Find the value of x^ _ 6x + 14, if a; = 3 + V— 5.
81. The planet Venus is said to be in conjunction when it is in
a line between the earth and the sun. If it takes Venus 225 days
to make one revolution about the
sun, how long is the interval be
tween two successive conjunctions 2 ^
of Venus?
82. The interval between two
successive conjunctions of the planet Mercury is 116 days. How
long does it take Mercury to make one revolution about the sun?
83. Simplify
^" \5x + 7y)
84. Solve ^2 ^2J^ ^=0.
X^ +x , 1 — X^ X^ —X
85. Multiply Vmr^ — ^s/mr^ ^/n + ^sfrrF^ y/n — Vri^ by mT^ +n*.
86. The natural waterpower of the United States is 75,000,000
H. P. This is 5,000;000 H. P. more than 10 times the waterpower
of Niagara Falls. Find the latter. Make up and work a similar
example concerning the fully developed waterpower of the United
States, if the latter is 230,000,000 H. P.
87. Simplify [(a^+«)^* (a^*)*] ^' •
88. Find the values of x, y, and z which satisfy the simultaneous
equations x + 2^/ = 3, 3y + 2 = 2, and 22 4 3a; = 1.
89. If a number of two digits is divided by the sum of the two
digits, the quotient is 4. If the digits are interchanged, the resulting
number will be greater than the original number by 36. Find the
niunber.
90. Factor x^ 2ax¥ \ 2ab.
91. In a^{x^ — yz)~^ introduce o^ into the parenthesis without
changing the value of the expression.
92. Given V = ^ttR^  J7rr», ir = ^^, R ^ 21, and r = 14, find
V in the shortest way.
This example illustrates the utility of what algebraic principle?
482 SCHOOL ALGEBRA
93. If 1/ = 7i—, find the value of tin terms of the other letters.
94. If ether boils at a temperature of 35f°C, at what temperature
on the Fahrenheit scale will it boil?
95. E xtract t he square root of 7 + 4\/3. Of 3 + \/5. Of
2a f 2>v/o*  ^.
96. A sinking fund is a fimd accumulated to meet a debt by
setting aside a certain siun, the sum set aside to accmnulate by
compound interest. If
C = number of dollars in the debt, a = sum set aside annuaUy,
n = nimiber of years, r = rate of interest,
then it may be shown that
r
If a city wishes to take up $2,500,000 worth of bonds at the end
of 4 years, how much must it set aside each ye^u*, the rate of interest
being 5%?
97. Solve 2 + Vx  V2x +7 = 0.
98. Simplify
(g h)"^ {b c)^ + {b c)^ (c g)^ + (c ~o)^ (o h)'^
(g 6) (6 c) + (6 c) (c g) + (c g) (g 6)
99. Rationalize the denominator of 7= and find the
yc
value of the result, when g = f , 6 = 20, and c = 5.
 100. Find the value of Vg»6"*  4g *6"* + 6  4g"i6* + a"^6*.
1^. a vr x^ — {y — «)* , y* — (x — z)^ , z^ — (z — yY
102. Simplify l+n»'n« ^ «^
1  g» g«  1
103. A sum of $1050 is divided into two parts and invested.
The simple interest on the one part at 4% for 6 years is the same
as the simple interest on the other at 5% for 12 years. Find how the
money is divided. .
104. Simplify 3VI + V40 + Vl 4=' \ \
APPENDIX 483
105. Factor ^  3^ + 2.
106. Solve {x  o) (6  c) + (a: — 6) (c — a)  for x,
107. Simplify — •
a
6i
c
108. Find the values of x and y which satiny simultaneously the
2 13 2
following equations: + = 2,  =5H —
X y X y
109. Solve for X, 2/, and 2: ox •\hy = c^cx + az ^ b,hz + q/ = a.
110. Find the greatest common divisor of aV —2acxz 6V +c*«*
and a*x* + 2a&r2/ + 6^*c*«*.
Ul. Solve/.L^^l+l+^V^=3^23.
Hurii)
\x X /
^Uoi lil la:V3 1^'
112. Simplify [(x + yY + {x y)^ [(x + yY  (x  y)*^.
113. At what time between 7 and 8 p. m. are the hands of a clock
opposite each other?
114. Solveforx, 2/, andgix +y = xy,2x +2z =X2,32 +3y=yz.
115. The indicated horsepower of a steam engine is found by
use of the formula
TI p  P^^
^ ''• " 33,000
where p = average steam pressure in pounds per square inch,
I — length of the piston stroke in feet,
a = piston area in square inches,
n = niunber of revolutions per minute.
In an engine whose piston area is 402.12 sq. in., and the length
of whose stroke is 2 J ft., find the indicated horsepower (to the
nearest unit), when the steam pressure is 40 pounds per square
inch, and the number of revolutions is 30 per minute.
Also solve the above formula for n. For p.
484 SCHOOL ALGEBRA
116. Write a statement of the advantages in representing num
bers by letters in algebra.
117. Which is greater, 2\^ or S^SV^^
118. Free the following equation from radicals and find the
value of X when g => 0:
aJ =  k + V fc' + V¥^'
1X9. Solve A_.=7, i^S.
120. Factor ac{a — c) — ab(a — 6) — bc{b — c).
121. If 19 pounds of gold and 10 pounds of silver each lose one
poimd when weighed in water, find the amount of each in a mass of
gold and silver that weighs 106 pounds in air and 99 pounds in
water.
122. From [m(3m — p) 2n(4n —Sp)]x + [m(p — m) — p(2n + v)]y
take 3 \p f 2n  ^)  (2m  3p) 1 x  [p(p  m) + 2n(2n + p)]y.
123. Show that ^ j^ ^^ + v^ = x* +2.
X*  1 x^ + 1 x*  1 x* + 1
124. A man having 10 hours at his disposal made an excuraon,
riding out at the rate of 10 miles an hour and returning on foot at
the rate of 3 miles an hour. Find the distance he rode.
125. Find the value of 1"^ + 1"*+ 1*  1".
126. Find the value of x which satisfies the equation
3a;4 _ x^ + 1 ^ 2x+3 _ /^ , ox
a:«3x+2 x1 x2 ^ "*■ ^'
127. Find the numerical value of (5.1)^ to 4 decimal places.
128. Simplify \ + ,
X + yx^  1 X — V aJ* — 1
130. Does Va^ = o^? Does VoM^ equal a+b?
131. Solve for x and y: — ^^ + S— = 2a, ^^^ = 1.
a+b a — 6 4ao
APPENDIX 485
132. Find the square root of the product of oj* — 1, a?  Sa; + 2,
and a;2 — a: — 2.
133. Fmd the value of ] 7^ t^! when x = V2  1.
1 + 2x H x*
134. Rationalize the denominator of —  .
V3 + V5 + V 5  VS
135. Simplify (2\/  !)♦.
136. An automobile ran 100 miles in 4 hr. and 30 min. In
the second half of the journey the speed was 5 miles per hour
greater than in the first half. Find the speed in each half.
137. Simplify {a:*2/"^ (a:* + 2/*) 5 {x^ + vh) (x +y  x^y^) and
find its value when x » 18 and y  2,
13a Solve ^+A=o+6, +a«+6».
ox ay X y
139. Divide 1 by Vx — 1 to four terms and extract the square
root of the quotient to three terms.
140. Find the highest common factor of a:* + a;V + V* a^d x* + y*.
141. Reduce —  + t tt^ + t rrr to a common de
X X + 1 (x h 1)* (x + 1)^
nominator and arrange the terms of the numerator according to the
ascending powers of X.
142. By finding the value of t in the first equation and substi
tuting in the second, climate t between the equations v = u + gt
and 8 — vt { ig^. Hence find s when ^ = 32, t; = 10.4, and u = 2.2.
3 1
143. From ■p= subtract 7= and express the result as
5  2\/S S + VS
a fraction having a rational denominator. \
1
144, Simplify x
X* +x —
1 X J _1
X
145. If V^ = VEy^ and x = ^/2y, find x and y.
146. Shnplify 2^/24 • 3^^ • 4v^.
347. Extract the square root of 1.672 to four decimal places.
1
486 SCHOOL ALGEBRA
148. Solve for a:: V^ ~ 8 ^ V^ ^^ ^
Vi  6 2 + yi
149. Divide ^i^ 4xy+ 4y^ + 42/» by 4^ + 2^/^ +2y.
150. Simplify (6x« 6) (§^) •
151. A gave B as much money as B had; then B gave A as much
money as A had left; finally A gave B as much money as B then had
left. A then had $16 and B, $24. How much had each originally?
152. Simplify 16+ j^±^ + ^24^;p.
{x — a X ja x^ +a*}
153. Factor (a  b)x^ +2ax + {a^h).
' 154. A man and two boys do a piece of work in 24 da. which could
have been done in 12 da. by three men and one boy. How long
would it take two men and two boys to do the work?
155. Solve sf^^^ + J^Z''^ t^^—tt+I
t(x  1) J(l + X) ggi
<'^) '
156. Collect in the shortest way
+
157. Find the value of
«**  1 «'» + 1 a:**  1 a;** H 1
4Vi2l2\C5
6Vl^
158., Simplify /^,.
159. Solve ^=1 +^^ ^= V5.
ya —x^ \a \a — x — ya ^
160. Show that the following polynomials do not have a common
actor: x« + 2x  8, x« +x«  3x + 1.
161. Solve — jr  rr » 0.
2x 4 X 2 2x 2
Vl — tV^
162. Rationalize the denominator of =»
3
APPENDIX 487
\ <
163. Extract the square root of
9^ 25 ^4 6^ 15
164. Extract the square root of a:* + 4x' + Sx* + 8a; — 21 until
a numerical remainder is obtained, and thus show that the original
expression equals (a:* + 2x + 2)'— 25. Hence obtain the factors of
the original expression. Treat in like manner y* + Sy* + 1 ly* + 6y — 8.
X ^
if
^ " ^ y i EXERCISE 168
'V ^* Review, Beginning with Quadratics ; . ^ " u
Solve: \ > ^ ' ^l ^%1
1 ..2.1 .2.1 3. x2 + l.&r;ill.5=C, '^'^
1. a:«+=a»+. ^ ^ ^
2. a:« +4a;  2Vl = 0. ** •^'•+ ^ " '^' ^ ^^
5. Find the two values of x which satisfy the equation ^
4^x 1 + 2Vx 1 1=0. ;^.^
6. By writing a numerical quadratic eqliation, as (x — 2)* = 9,
in the form (x — 2)* — 9 « 0, show that the solution by completing
the square may be reduced to the factorial solution.
7. Solve a;+ = 1 + Kir— e ' r^ ' 0
X 1 _ ^^
t r
8. By letting o, 6, c, etc., have si)ecial values, convert aa^ + bxy \
'\cy^ +dx +ey +/ = into j. ^^ " ^
(1) a homogeneous equation of the second degree. »,;,,. ^"^"^^ •
(2) a symmetrical equation of the second degree.
(3) a homogeneous symmetrical equation of the second degree.
Solve: \1
Qxy = 1. X y 5
10. VxlTy =3. 12. x^ +xy +x ^ 14.
xyS. y*+xy+y =28.
488 SCHOOL ALGEBRA
13. 1 +1 = o. 16. x+y + V^ = 14.
1 . 1
a;
1 . 1
;i +i  6. 17. VgTT+y = 6.
V?T22y2 + X* = 22.
yi + 1^2 " ^ 18. VxTy + Vx y = 4.
<2>2 j2 s Q
14. x^+xy^^6. ^^ ,J ^^
SxV + &ry  3. 19. i/5 + Jy = i2.
11 V y V a; 3
x+y = 10.
20. V^ — Va^ —y  11.
V^ — yV^y = 60.
the equations whose roots are 1 ^^ i. Also i ^^ iu
, 22. Find by inspection the sum of the roots of 3a;' —2x + 1=0.
I Find also, the product of the roots. Verify your result by solving
/the given equation. About how much shorter is the first process
than the second?
•
{ 23. What must be added to each of the terms of a' : 6^ to make
1 the resulting ratio equal to a : 6?
24. What munber must be subtracted from each of the num
^'^i^era, 9, 12, 15, and 21, so that the remainder shall form a proportion?
25. If a box car 36' X 8J' X 8' has a capacity of 60,000 lb., by
how much must the length be increased to make the capacity
100,000 lb.?
26. When z = 25, solve the following system of proportions:
x:y :z:w =3:4:5:6.
27. The rates of two boys traveling on bicycles are as p to 9.
If the first boy rides a miles in a given time, how far does the other
boy travel in the same time?
^ 28. For what value of x will the ratio x^—x + lia^ix + lhe
/ equal to 3:7?
29. If => = =: i: prove ^ = ^, \, , — ^r^
30. If a : 6 = c :d, show that ab + cd ia a mean proportional
^ — between a^ + c^ and 6* + (?.
r
APPENDIX 489
31. If a :h =' c :d, and a is not equal to 6 or c, prove that it is
impossible to find a number x (other than zero) such that a— z:c^ x
= h — X :d — X,
32. If 5 = ^= != A:, prove that ^^!^^^^^ A;.
a b c V^a«+g6Hr^
33. Find the sum of 30 terms of the A. P. 3, 5, 7, ... . by the ad
dition of successive terms. Now find this sum by the use of one of
the formulas. of Art. 278. Compare the amount of work in the
two processes.
34. Prove that the differences between the squares of successive
integers forms an A. P.
35. Prove that equimultiples of the terms of an A. P. form
another A. P.
36. Obtain a formula for the nth term of the A. P. 9, 7, 5, ... •
Also for the n + 2d term.
37. If the hours of the day were niunbered from 1 to 24, how many
times would a clock strike in striking the hours during one day?
38. Show that the sum of n consecutive integers is divisible by
n, if n is odd but not if n is even.
39. Find the sum of n terms of 1, 26, 46*, 86*, ....
40. If each stroke of an air pump removes § of the air in a re
ceiver, what fraction of the air will be left in the receiver after 10
strokes?
41. Find a G. P. in which the sum of the first two terms is 2f
and the sum to infinity is 4^.
2x*  ^ J by finding all the terms
up to the 7th. Now find the 7th term by the method of Art. 294.
Compare the amount of work in the two processes.
43. Expand (V^Tfl  \^x  1)*.
44. Find the 98th term of (3a  2by^.
45. Find in a short way the sum of the coefficients of the terms in
the expansicin of (2a  6)8. Of (2a + 6)^.
46. JBina the two middle terms of ( 2^/^ j ] • '
V
u
490 SCHOOL ALGEBRA
— — 9J » fi^d the coefficient of x*.
48. Find the ratio between the sixth term in the expansion of
f 1 + « ) fl^d the fifth term in the expansion of ( 1 f —J •
49. Solve Vx + i + V^2 = \/2x+3.
50. Find the simi of all positive integers of three digits which are
divisible by 9.
51. What is meant by an irrational root of an equation? By an
imaginary root?
52. Solve a;2 + a;  4aa; + 3a*  5a  2 = 0.
53. Solve xV  a^2 = 12, x»  ?/» = 63.
54. Solve v^a;2 + 12 + Vx^ + 12 = 6.
55. The sum of 5 terms of an A. P. is — 5, and the 6th term is
— 13. What is the common difference?
56. What is the ratio of the mean proportional between a and b
to the mean proportional between a and c?
57. Form an equation whose roots are ^ —
58. A boat crew rowing at half their usual speed row 3 mi. down
stream and back again in 2 hr. and 40 min. At full speed they can
go over the same course in 1 hr. 4 min. Find in miles per hour the
rate of the crew and of the current.
^ 59. Solve x^\xy+y^ = 1, 2x^ + Sxy + 4y^ = 3.
60. Show that the sum of the squares of the roots of the equation
a;2  5a; + 2 = is 21.
61. The mean annual rainfalls at Phoenix (Ariz.), Denver,
Chicago, and New Orleans are 7.9 in., 14 in., 34 in., and 57. 4 in. re
spectively. By how much do these numbers differ from the corres
ponding terms in a G. P. whose first term is 7.9 in., and whose ratio
is 2?
62. Solve in the shortest way — '■ — I 1 — = 1 — »
•^ x^8 ^a;6 ^a:+6 ^a;+:8
63. If a : 6 = c : d = 6 :/, show that a»+c»+e':6»fcP+/* =
a>ce : hdf. O
64. Solve yx^Vy =5, x+y = 13. ' ^' ~
/
APPENDIX 491
By use of the binomial theorem find the ratio of the 5th term
to the 7th term in the expansion of (1 — V2i)^«
66. Solve ^P3+^^.
67. Derive the formulas for the nth term and for the sum of n
terms of a geometrical progression in terms of the first term and the
common ratio.
\/€8. Solve n\x^ + 1) = a' + 2nH.
69. Which term of the series J, ^, , etc., is 8?
•
 70. Solve x*  3a;  6 V^J* 3a;3+2=0.
71. Show that the roots of the equation x*+ax — 1 ^Oare
real and imequal for any real value of a.
/ 72. Solve (x+iy=4 + (ll)(l+l).
' 73. Find four numbers in A. P. such that the smn of the first
' and third shall be 18, and the sum of the second and fourth shall
' be 30.
74. Find the G. P. whose simi to infinity is 4 and whosci second
term is }.
75. A rectangular park is 100 rods long arid 80 rods wide. By
what pier cent must its dimensions be increased in order that its
varea shall be doubled?
76. The difference between the reciprocals of two consecutive
nimibers is ^V Find the numbers.
77. Find the sum to infinity of — 3 + J — i^y • • •
78. Given K = irip, and C = 2ir72, eliminate U and find K in
terms of C
79. Given B = irRL and T = irUiJEt + L), eliminate R and find
T in terms of & and L.
80. By use of an A. P. find the sum of aU the numbers between
1 and 207 which are divisible by 5.
81. A man sold a horse for $96 and in doing so gained as many
per cent as the horse cost him dollars. What did the horse cost
hhn?
82. Solve x + y + Va; +2/ = 20, a:?/ = 63.
492 SCHOOL ALGEBRA
83. Given I  distance in feet between two adjacent supports
of a trolley wire,
8 = sag of tie wire in feet,
t — tension of the wire in pounds,
w — weight of wire in pounds,
V = actual length of wire between two adjacent supports,
and (^) * ^ "q7 * ' ' ^^^ (^) ^' "^ ^"*" qT"'
(1) Find the value of I in equation (a). Also in equation (6).
(2) Eliminate I between the two equations.
84. Expand (2Vs  W  2)* and simplify.
85. The sum of the first seven terms of a G. P. is 635, and the
ratio is 2. What is the fourth term?
86. Two boys start on bicycles at the vertex of a right angle and
ride along its sides at the rate of 6 and 8 miles per hour, respectively.
How many hours will it be before they are 100 miles apart?
87. Determine by inspection the roots of the equation
ax{bx  2) (x*  9) =0.
88. If a and P are the roots of the equation px^ + qx +r ^0^
find the values of a + yS, a — /3, and a)8 in terms of p, q, and r.
89. A man finds that it takes him 2 hours less to walk 24 miles,
if he increases his speed 1 mile per hour. What is his usual rate?
90. If a : 6 = 6 : c, prove that a +h :h +C ^b^ icu^.
91. Insert four geometric means between 160 and 5.
92. How many terms of the A. P. 42, 39, 36, must be taken
to make 315?
93. Express the repeating 'decimal .3232 .... as a fraction.
94. Solve (a;* + i)» = 27.
95. Solve 9qi^ + 25y^ = 148, 5xy ^S.
96. The first term of a geometrical series is 2 and the sum of the
fourth term and three times the second term is equal to four times
the third term. Find the series.
97. Solve (»*  x) (a: + 2) = 0.
APPENDIX 493
a;« I/* _
98. Solve ±.  it = 8i, a;  y  2.
y X
99. If — ?— = ^^ = — ^, show that a:  y +« = 0.
6fc c+a a —0
100. Solve a;(a;  j^) = 0, x« + 2x1/ + ^ = 9 by the factorial
method as far as possible.
101. In the same way solve
(2/+ar7)y =0, (y +a;  3) (y +2a; 4) = 0.
102. Solve X* + 3x"* = 4.
103. The hyi)oteniise of a right triangle is 20. The sum of the
other two sides is 28. Find the length of the sides.
104. Solve x2y»  lOxy +24=0, x+2/=5.
105. The sum of the first seven terms of an A. P. is 98, and the
product of the first and seventh terms is 115. Find the common
difference. ^
106. Solve x» + xy + 2/* « 133, x  Vxy + y = 7.
107. Find the sum of the odd integers between and 200. How
many of these are not divisible by 3?
108. Find the values of x and y which will satisfy the following:
x+i = 1, 2/+i=4.
y X
109. In an A. P., given a = §, i = — 2J, s  —4, find n and d.
110. If the speed of a train should be lessened 4 miles an hour,
the train would be half an hour longer in going 180 miles. Find the
rate of the train.
111. Plot the graphs of the following system of equations:
X* + y^ = 4, 3x — 22/ = 6. From thQ graphs find the approximate
values of x and y that satisfy both equations.
112. Solve 9x  3x2 + 4Vx2^^3xT5 = 11.
113. The sum of an infinite G. P. is 4 and the first term is 6.
Find the ratio and the siun of 4 terms.
114. Solve Vx I ^3  V3x f x^ = ^3.
115. What is meant by an extraneous root of an equation? Give
an example of an extraneous root.
494 SCHOOL ALGEBRA
116. At his usual rate a man can row 15 miles down stream in
5 hours less than it takes him to return. If he could double his rate
his time down stream would be only 1 hour less than his time up
stream. What is his rate in still water?
117. Given S = 4nB^, and V = jiriB', eliminate R and find V in
terms of S,
118. Solve 5aj*  ac*  14 = 0.
119. If the 6th term of an A.P. is 9, and the 16th term is 22),
find the 25th term and the sum of 30 terms.
120. What two numbers, whose difference is A, are to each other
as a: 6?
121. If =  ^ = , find the value oiz hy +z.
a —0 —c c — a
122. What distance is passed over by a ball which is thrown 60
feet vertically upward and at every fall rebounds \ the distance
from which it fell?
123. Solve x* + y* + 2{x +y) =12, xy {x+y) = 2.
124. What niunber added to both numerator and denominator
of ^, and subtracted from both numerator and denominator of 3,
a
will make the results equal?
125. Find the tenth term of — J, — J, J, ... . and the sum of the
series to ten terms.
126. Solve x^  2\/7x +2=0.
127. Find in a short way the sum of the coefficients in the ex
panded form of (2x — Vy)".
128. li a :b » 6 : c = c :d, show that 6 + c is a mean propor
tional between a +b and c + d.
129. If a, 6, c, and d are in A. P., show that o H (f = 6 + c.
130. Solve x+y + Vx +y = 12, x y + Vx y = 2.
• 131. Solve (x + 1 + xr^) {x 1+ xi) = 5J.
132. If a boy runs 100 yd. in 10 sec. how much does his velocity
differ from a mean proportional between the velocity of a man walk
ing 4 mi. per hour and an express train going 60 mi. per hour?
APPENDIX
495
133. The following table gives the normal or average height of
a boy and girl at different ages:
Age in years
3
6
9
12
15
18
21
Height of boy
Height of girl
2'11"
2'11"
3'8"
3'7"
4'2"
4'2"
4'7i"
4'9"
5'2i"
5'li"
5'6r
6'3i"
5 8i
6'3r
Graph the above facts as two graphs on one diagram.
From these graphs determine as accurately as you can the normal
height of a boy and of a girl at 10 years of age. At 14 years.
134. The sum of the first ten terms of a G. P. is equal to 244
times the sum of the first five terms, and the sum of the fourth
and sixth terms is 135. Find the first term and the common ratio.
Sua. Show that 1^°  1  244(r»  1), etc.
135. Insert between 1 and 21 a series of arithmetical means
such that the smn of the last three shall be equal to 48.
x' \ ^
137. Prove that either root of the quadratic equation x* — g =
is a mean proportional between the roots of a;*+ pxH g = 0.
138. Simphfy {^/a+b + Va  &)• + (\/a+6  \/flf^A
139. Solve a?+J/=5— a:2/> x \y = —
xy
140. find the sum of n terms of the series
(..)+(fg)+gg)
I . • . •
141. The formula used for determining the elevation of the
4BF2
outer rail of a railroad track on a curve is as follows: E = , p ,
where E = elevation of outer rail in inches
B = width of the track in feet
R = radius of the curve in feet
V = maximum speed in miles per hour of a train taking
the curve.
Find E when B = 4 ft. 8J in., R = 425 ft., F = 20 mi. per
hour. Also when F = 60 mi. per hour.
142. Solve the formula for F. From this result determine
the hiaximum speed at which a train can take the track when
^  5 in.
LOGARITHMS
326. The Logarithm of a number is the exponent of that
power of another number, taken as the base, which equals
the given number.
Thus, 1000 = 10«. Hence, log 1000 = 3, 10 being taken as the base.
Again, if 8 is taken as the base, 4 = 8s. Hence, log 4 = .
If 5 is taken as the base, log 125 = 3, log ^ =  2, etc.
The base is sometimes stated as above; but when desir^
able, it is indicated by writing it as a small subscript to
the word log.
Thus, the above expressions might be written,
logiolOOO = 3; log84 = i; log^ 125 = 3 ; log^ A =  2 ; etc.
In general, by the definition of a logarithm,
number = (base)^^**'"*^",
or JV= I?, Hence, log^ N= I.
327* Uses of Logarithms. One of the principal uses of
logarithms is to simplify numerical work. For instance, by
logarithms the numerical work of mvltiplyirig two numbers
is converted into the simpler work of adding the logarithms
of these numbers.
To illustrate this principle, we may take the simple case
of multiplying two numbers which are exact powers of 10,
as 1000 and 100. Thus,
1000 = 10«
100 = 10^
Henee, 1000 x 100 = 10* = 100,000,
the multiplication being performed by the addition of exponents*
496
APPENDIX ' 497
Similarly, if 384 = lOaM^W
and 25 = 10i«9T»*+
To multiply 384 by 25,
Add the exponents of 102*8488+ and 10i'»''»*+, thus obtaining
108.98227+^
Then get from a table of logarithms the value of 10898227+^ yiz. 9600.
In like manner, by the use of logarithms, the process of
dividing one number by another is converted into the
simpler process of subtracting one exponent, or log, from
another. The process of involution, also, is converted
into the simpler process of multiplication ; and the extrac
tion of a root into the simpler process of division.
We can save labor still further, through the use of
logarithms, by committing to memory the logs of numbers
that are frequently used, as
2, 3, ... 9, TT, V^, , V2, V3, etc.
By the use T)f the slide rule^ the practical use of loga
rithms is reduced to sliding one rod along another and
reading off the number at one end of a rod.
It will be a useful exercise to teach the class the use of the slide
rule ill connection with the study of this chapter.
328. Systems of Logarithms. Any positive number, ex
cept unity, may be made the base of a system of logarithms.*
Two principal systems are in use :
1. The Common (or Decimal) or Biiggsian System, in
which the base is 10. This system is used almost exclu
sively for numerical computations.
2. The Natural or Napierian System, in which the base
is 2.7182818"^. This system is generally used in algebraic
processes, as in demonistrating the properties of algebraic
expressions.
498 SCHOOL ALGEBRA
EXERCISE 169
1. Give the value of each of the following: loggS,
logs 27, log4 64, log4 j\, logg i, logg ^, log^o j\, logjo .01,
logio .001.
2. Also of logg 32, log2 ^\j, log2 yl^, log4 8, logg 16.
3. Simplify logg 4 + logg 9 + logio .1  logg l.
4. Write out the value of each power of 2 up to 2^ in
the form of a table.
Thus, 21 = 2, 22 = 4, 2« = 8, etc.
5. By means of this table, multiply 32 by 8, perform
ing the multiplication by the addition of exponents.
6. . In like manner, convert each of the following mul
tiplications into an addition : 32 x 16, 64 x 32, 1024
X 16, 512 X 64.
7. Convert each of the following divisions into a
subtraction : 1024 f 16, 512 * 64, 32,768 s 1024.
8. Convert each of the following involutions into s^
multiplication : (32)8, (64)2, (32)*.
9. Convert each of the following root extractions into
a division: v^, Vvm, ViOyB.
10. Make up two examples like those in Ex. 6. In
Ex. 8. In Ex. 9.
11. Construct a table of powers of 3 and make up
similar examples concerning it.
12. How many of the above examples can you work at
sight ?
329. Characteristic and Mantissa. If a given number, as
384, is not an exact power of the base, its logarithm, as
'2.58433'**, consists of two parts: the whole number, called
the characteristic^ and the decimal part, called the mantissa.
APPENDIX 499
To obtain a rule for determining the characteristic of a
given number (the base being 10), we have :
10,000 = 10*, hence log 10,000 = 4 ;
1000 = 108, hence log 1000 = 3 ;
100 = 102, hence log 100 = 2 ;
10 = 101, hence log 10 = 1.
Hence, any number between 1000 and 10,000 has a
logarithm between 3 and 4 ; that is, 3 plus a fraction.
But every integral number between 1000 and 10,000
contains four digits. Hence, every integraKnumber con
taining /owr figures has 3 for a characteristic.
Similarly, every number between 100 and 1000, and
therefore containing three figures to the left of the decimal
point, has 2 for a characteristic.
A number between 10 and 100 (i.e., a number contain
ing two integral figures) has 1 for a characteristic.
Every number between 1 and 10 (that is, every number
containing one integral figure) has for a characteristic.
Hence, the characteristic of an integral or mixed number
is one less than the number of figures to the left of the dedr
mal point,
330 Characteristic of a Decimal Fraction.
1 = 100. .. log 1 = 0;
.1 = ^=101. .•.log,.l = l;
•'^ = l4=l^«=^^" •••log.01 = 2;
•^^^ = i^ = ii3==^^"' •••1^^001= 3, etc.
Hence, the logarithm of any number between .1 and 1
(as of .4, for instance) will lie between — 1 and 0, and
hence will consist of — 1 plus a positive fraction.
500 SCHOOL ALGEBRA
The logarithm of every number between .01 and .1 (as of
•0372, for instance) will be between — 2 and— 1, and hence
will consist of — 2 plus a positive fraction ; and so on.
Hence, the characteristic of a decimal fraction is negative^
and is numerically one more than the number of zeros be
tween the decimal point and the first significant figure.
There are two ways in common use for writing the
characteristic of a decimal fraction.
Thas, (1) log .0384 = 2.58433, the minus sign being placed over
the characteristic 2, to show that it alone is negative, the mantissa
being positive.
Or (2) 10 is added to and subtracted from the log, giving
log .0384 = 8,58433  10.
In practice, the following rule is used for determining
the characteristic of the logarithm of a decimal fraction :
Take one more than the number of zeros between the deci^
mal point and the first significant figure^ subtract it from 10,
and annex — 10 after the mantissa.
EXERCISE 160
Give the characteristic of
1. 452
6. .08267
u.
7
2. 16,730
7. 1.0042
12.
6267.3
3. 767.5
8. 7.92631
13.
.000227
4. 64.56
9. .007
14.
100.58.
5. 9.22678
10. .0000625
IS.
23.7621
16. How many figures to the left of the decimal point
(or how many zeros immediately to the right) are there
in a number, the characteristic of whose logarithm is 3 ?
2? 5? 1? 0? 4? 810? 710? 910?
17. Can you make up a rule for fixing the decimal point
in the number which corresponds to a given logarithm ?
APPENDIX 501
18. If log 632 = 2.8007, express 632 as a power of 10.
19. If 257 = 102.4099, what is the log of 257 ?
20. If a number lies between 9000 and 20,000, what
will its characteristic be ?
21. If a number lies between 10,000 and 100,000,
between what two numbers must its logarithm lie ?
331* Mantissas of numbers are computed by methods,
usually algebraic, which lie outside the scope of this book.
After being computed, the mantissas are arranged ia
tables, from which they are taken when needed. In this
connection, it is important to note that
The position of the decimal point in a number affects only the
characteristic^ not the mantissa^ of the logarithm of the number.
Thus, if log 6754 = 3.82956
RT^id 1 08*82956
log 67.54 = log 5^^ = log t^ = log 10182966 = 1.82956.
^ 100 ^102 ^
In general, log 6754 = 3.82956
log 675.4 = 2.82956
log67.54 = 1.82956
log 6.754 = 0.82956
log .6754 = 9.82956  10
log .06754 = 8.82956  10, etc.
332* Direct Use of a Table of Logarithms ; that is, given a
number^ to find its logarithm from a table. From the follow
ing small table of logarithms, the student may learn
enough of their use to understand their algebraic proper
ties. The thorough use of logarithms for purposes of
computation is usually taken up in connection with the
study of trigonometry.
In the table (see pages .504, 505), the lefthand column is a column
of numbers, and Ls headed N.
The mantissa of each of these numbers is in the next column op
posite. In the top row of each page are the figures 0, 1, 2, 3, 4, 5, 6,
7, 8, 9.
502 SCHOOL ALGEBRA
To obtain the mantissa for a number of three figures, as 364, we
take 36 in the first column, and look along the row beginning with 36 till
we come to the column headed 4. The mantissa thus obtained is .5611.
If the number whose mantissa is sought contains four or five figures,
Obtain from the table the mantissa for the first three figures, and also
that for the next higher number, and subtract^
Multiply the difference between the two mantissas by the fourth (or
fourth and fifth) figure expressed as a decimal ;
And ADD the result to the mantissa for the first three figures.
Thus, to find the mantissa for 167.49,
Mantissa for 168 = .2253
Mantissa for 167 = .2227
Difference = .0026
Since an increase of 1 in the number (from 167 to 168) makes an
increase of .0026 in the mantissa, an increase of .49 of 1 in the number
will make an increase of .49 of .0026 in the mantissa.
But .0026 X .49 = .001274 or .0013  .
Hence, .2227
13
Mantissa for 167.49 = .2240
Hence, to obtain the logarithm of a given number,
Determine the characteristic by Art. S'29 or Art. SSO;
Neglect the decimal pointy and obtain from the table
CpP' 604,505) the mantissa for the given figures.
Exs. Log. 52.6 = 1.7210. Log. .00094 = 6.9731  10.
Log. 167.49 = 2.2240. Log. .042308 = 8.6264  10.
EXERCISE 161
Find the logarithms of the following numbers :
1. 37 6. 175 11. .0758 16. .7788
2. 85 7. 32.9 12. 5780 17. .04275
3. 6 8. 4.75 13. .00217 18. 234.76
4. 90 9. .08 14. 63.21 19. 5.6107
6. 300 10. 1.02 15. 3.002 20. 7781.4
APPENDIX 503
333. Inverse Use of a Table of Logarithms ; that is, given
a logarithm^ to find the number corresponding to this loga
rithm^ termed antilogarithm :
From the tahle^ find the figures corresponding to the man
tissa of the given logarithm ;
Use the characteristic of the given logarithm to fix the
decimal point of the figures obtained.
Ex. Find the antilogarithm of 1.5658.
The figures corresponding to the mantissa, .5658, are 368.
Since the characteristic is 1, there are 2 figures at the left of the
decimal point.
Hence, antilog 1.5658 = 36.8
In case the given mantissa does not occur in the table,
obtain from the table the next lower mantissa with the corre
sponding three figures of the antilogarithm ;
Subtract the tabular mantissa from the given mantissa ;
Divide this difference by the difference between the tabular
mantissa and the next higher mantissa in the table ;
Annex the quotient to the three figures of the antilogarithm
obtained from the table.
Ex. Find antilog 2.4237.
.4237 does not occur in the table, and the next lower mantissa is
•4232. The difference between .42:^2 and .4249 is .0017.
Hence, we have antilog 2.4237 = 265.29
4232
17)5.00(.29
If a difference of 17 in the last two figures of the mantissa makes
a difference of 1 in the third figure of the antilog, a difference of 5 in
the mantissa will make a difference of ^ of 1 or .29 with respect to
the third figure of the antilog.
504
SCHOOL ALGEBRA
N.
1
2
8
4
6
6
7
8
•
10
0000
0043
0086
0128
0170
0212
0263
0294
0334
0374
11
414
453
492
531
569
607
645
682
719
756
12
792
828
864
899
934
969
1004
1038
1072
1106
13
1139
1173
1206
1239
1271
1303
335
367
300
430
14
461
492
523
553
584
614
644
673
703
732
15
1761
1790
1818
1847
1875
1903
1931
1959
1987
2014
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
279
17
304
330
355
380
405
430
455
480
604
629
18
653
577
601
625
648
672
695
718
742
765
19
788
810
633
856
878
900
923
945
967
989
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
21
222
243
263
284
804
324
345
365
385
404
22
424
444
464
483
602
522
641
660
679
598
23
617
636
655
674
692
711
729
747
766
784
24
802
820
838
856
874
892
909
927
945
962
26
3979
3997
4014
4031
4Q48
4065
4082
4099
4116
4133
26
4150
4166
183
200
216
232
249
265
281
298
27
314
330
3'46
362
3/8
303
409
426
440
466
28
472
487
602
518
633
648
664
679
694
609
29
624
639
654
669
683
698
713
728
742
757
30
4771
4786
4800
4814
4829
4843
4867
4871
4886
4900
31
914
928
942
955
969
983
997
6011
5024
5038
32
6051
5065
5079
5092
6105
6119
6132
145
169
172
33
185
198
211
224
237
260
263
276
289
302
34
815
828
840
353
366
378
391
403
416
428
35
5441
5453
5465
5478
5490
5502
6514
6527
6639
5651
36
663
575
587
599
611
623
635
647
658
670
37
682
694
705
717
729
740
752
763
776
786
38
798
809
821
832
843
855
866
877
888
899
39
911
922
933
944
956
966
977
988
999
6010
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6n7
41
128
138
149
160
170
180
191
201
212
222
42
232
243
253
263
274
284
294
304
314
325
43
335
345
355
365
375
385
395
405
416
426
44
435
444
454
464
474
484
493
603
613
622
45
6532
6542
6551
6561
6571
6680
6590
6699
6609
6618
46
628
637
646
656
665
675
684
693
702
712
47
721
730
739
749
758
767
776
785
794
803
48
812
821
830
839
848
867
866
875
884
893
49
902
911
920
928
937
946
965
964
972
081
50
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
51
7076
7084
003
101
110
118
126
136
143
152
52
160
168
177
185
193
202
210
218
226
236
53
243
251
259
267
276
284
292
300
308
316
54
324
332
340
348
366
364
372
•
380
388
396
N.
1
2
3
*
5
6
7
8
9
APPENDIX
605
N.
1
2
3
4
5
6
7
8
9
56
7404
7412
7419
7427
7436
7443
7451
7469
7466
7474
66
482
490
497
606
613
620
528
636
643
661
67
669
666
574
582
.689
697
604
612
619
627
68
634
642
649
657
664
672
679
686
694
701
69
709
716
723
731
738
746
752
760
767
774
60
7782
7789
7796
7803
7810
7818
7826
7832
7839
7846
61
863
860
868
875
882
889
896
903
910
917
62
924
931
938
946
952
959
966
973
980
987
63
993
8000
8007
8014
8021
8028
8036
8041
8048
8056
64
8062
069
075
082
089
096
102
109
116
122
65
8129
8136
8142
8149
8156
8162
8169
8176
8182
8189
66
195
202
209
216
222
228
235
241
248
264
67
261
267
274
280
287
293
299
806
312
319
68
325
331
338
344
351
367
803
370
376
382
69
388
396
401
407
414
420
426
432
439
446
70
8461
8457
8463
8470
8476
8482
8488
8494
8500
8606
71
613
519
625
631
637
643
649
555
661
667
72
673
679
685
501
697
603
609
615
, 621
627
73
633
639
645
651
657
663
669
676
681
686
74
692
698
704
710
716
722
727
733
739
746
75
8761
8756
8762
8768
8774
8779
8785
8791
8797
8802
76
808
814
820
825
831
837
842
848
854
859
77
865
871
876
882
887
893
899
904
910
915
78
921
927
932
938
943
949
954
960
965
971
79
976
982
987
993
998
9004
9009
9015
9020
9025
80
9031
9036
9042
9047
9053
9058
9003
9069
9074
9079
81
085
090
006
101
106
112
117
122
128
133
82
138
143
149
154
159
166
170
175
180
186
83
191
196
201
206
212
217
222
227
232
238
84
243
248
253
258
263
269
274
279
284
289
85
9294
9299
9304
9309
9316
9320
9326
9330
9335
9340
86
345
350
356
360
366
370
376
380
385
390
87
395
400
405
410
415
420
426
430
435
440
88
446
450
466
460
466
469
474
479
484
489
89
494
499
604
609
613
618
623
628
633
638
90
9542
9647
9652
9657
9562
9566
9671
9676
9681
9686
91
690
695
600
605
609
614
619
624
628
633
92
638
643
647
652
657
661
666
671
675
680
93
685
689
694
699
703
708
713
717
722
727
94
731
736
741
745
750
754
759
763
768
773
95
9777
9782
9786
9791
9795
9800
9806
9809
9814
9818
96
823
827
832
836
841
845
850
864
869
863
97
868
872
877
881
886
800
804
899
903
908
98
912
917
921
926
930,
934
939
943
948
952
99
966
961
966
969
J974
978
983
987
991
996
N.
1
%
3
4
5
6
7
8
1
9
1
606 SCHOOL ALGEBRA
EXERCISE 162
Find the numbers corDesponding to the following >
logarithms: ^
1. 1.6335 7. 0.6117' 13. 0.4133 ^
2. 2.8865 8. 9.797310 14. 1.4900 ^
3. 2.3729 9. 7.904710 15. 3.8500
4. 0.5775 10. 8.631410 16. 1.8904 ^
5. 3.9243 u. 7.7007^10 17. 2.4527
6. 1.8476 12. 6.100410 18. 9.640210
19. Write log 17 = 1.2304 as a number equal to a power
of 10.
20. Make up and work a similar example for yourself.
334. Properties of Logarithms. It has been shown
(Arts. 162, 179, 180, 181) that
when m and n are commensurable. By the use of suc
cessive approximations approaching as closely as we please
to limits, the same law may be shown to hold when m and
n are incommensurable. It then follows that
(1) log oi = log a + log h (3) log a" = /? log a
(2) log (l^ = log a log i (4) logVa = i5^
Proof : >
Let a — lO". .*. log a —m,
h = 10». .•. log b = n. ^
ah = 10*+*. .*. log a6 = m + n = log a + log 6 . . . . (1)
 = 10"»». . .•. log f^j = m n = logo logft .... (2)
QP _ lO'**. .".log ap = pm = p log a (3)
V?=10^.' ...logVa = ^ = ^^S^ (4)
P P
The same properties may be proved in like manner for a system of
logarithms with any other base than 10.
1
N
APPENDIX 507
335* Properties Utilized for Purposes of Computation*
I. To Multiply Vumbers,
Add their logarithms^ and find the antilogarithm of the
9um. This will he the product of the numbers.
II. To Divide One Number by Another,
Subtract the logarithm of the divisor from the logarithm of
the dividend, and obtain the antilogarithm of the difference^
III. To Raise a Number to a Required Power,
Multiply the logarithm of the number by the index of the
power. Find the antilogarithm of the product.
I Y. To Extract a Required Root of a Number,
Divide the logarithm of the number by the index of the re
quired root. Find the antilogarithm of the quotient.
Ex. 1. Multiply 527 by .083 by the use of logs.
log 527 = 2.7218
~ log .083 = 8.9191  10
aiitilog'l.6409 = 53.7+, Product '
The following form is the arrangement of work used by many
practical computers. It has the advantages of brevity and of
showing all the steps in a complex logarithmic computation.
527 log 2.7218
.083 lo g 8.9191  10
Product 53.7 log 1.6409
Observe that " 527 log 2.7218 " is read « 527, its log is 2.7218."
Ex. 2. Compute the amount of 1 1 at 6 % for 20 years
at compound interest.
The amount of ^ 1 at 6 % for 20 years = (1.06)».
1.06 log 0.0253
20
ilfw.» 3.21 log 0.5060
Computing (1.06)^ by direct multiplication, will make clear the
amount of labor sometimes saved by the use of logarithms.
608 SCHOOL ALGEBRA
Ex. 3. Extract approximately the 7th root of 15.
15 log 1.1761, ^ log 0.1680
Root 1.47 log 0.1680
336* Cologaritlim. In operations involving division, it is
usual, instead of subtracting the logarithm of the divisor, to
add its cologarithm. The cologarithm of a number is ob
tained by subtracting the logarithm of the number from
10 — 10. Adding the colog gives the same result as sub
tracting the log itself from the logarithm of the dividend.
The use of the cologarithm saves figures, and gives a
more compact and orderly statement of the work.
The cologarithm may be taken directly from the table
by use of the following rule :
Subtract each figwi e of the given logarithm from 9, except
the last significant figure^ which subtract from 10.
Ex. 1. , Find colog of 36.4.
log 36.4 = 1.5611
colog 36.4 = 8.4389  10
Ex. 2. Compute by use of logarithms ^'^ ^ ^^'^
2V576 X3.78
8.4 log 0.9243
32.4 log 1.5105
2 log 0.3010 colog 9.6990  10
676 log 2.7604 \ log 1.3802 i colog 8.6198  10
3.78 log 0.5775 colog 9.4225  10
Ans. 1.5 log 0.1761
EXERCISE 163
Find, by use of logarithms, the approximate values of
1. 75 X 1.4 4. 831 X .25 ^ 336.8
2 98x35 ■ ' "^^^^
2. 9.8 X 3.6 g^ ^gj^
3. 15.1 X .005 * 13.4 • .0049
APPENDIX 509
78.9 10. .48i(1.79) 3.51 x 67
98.7 97.7
42.316 „ 1.78 X 19 „ 12.9 '
U. 7r7r= 18.
.06214 23.7 4.7x9.1
14. 47.1 x 3.66 X .0079
15. .0631 X 7.208 X. 51272
16. 4.77x(.71)t(.83)
523 X 249
767 X 396
18. (2.8)« 23. Vl9 28. •v'. 00429
19. (1.082y» 24. </BW 29. (2.91)*
ao. (8.57)* 25. </TM 30. </W
ai. (.96)^ 26. V^ ai. VWxV^
28. (.795)« 27. ^/61 32. </19*V46
33. VMSx \^:0765 35. s/J X </^
34. 2*x7* 36. ■s/ixVsxVrf
37.  (8.12)8 ha/( 42.8)''
38. \/. 000479 !r"vT0568
39. ^CMT ^^ JB7.56 X 26.5
^67x618 ^22.7x16.78
By the use of logarithms :
41. Find the amount of $1250 at 6 per cent compound
interest for 12 years. Also make the computation with
out the use of logarithms. What fraction of the work is
saved by the use of logarithms ?
42. Find the amount of $25 at 5 per cent compound in.
terest for 500 years.
43. Find the amount of $ 300 at 6 per cent for 50 yearSi
interest being compounded semiannually.
510 . SCHOOL ALGEBRA
44. Find the amount of $300 at 6 per cent for 50 years,
interest being compounded quarterly.
45. Find the radius of a circle whose area is 100 sq. yd.
46. Find the radius of a sphere whose volume is 20 cu. ft.
(User=7ri28.)
47. A given parallelogram is 12.7 ft. long and 8.9 ft.
high. Find the side of a square whose area is equal to
that of the parallelogram.
48. Compute Vl5 to three decimal places without the
use of logarithms. Now obtain the same result by the use
of logarithms. Compare the amount of work in the two
processes.
49. Find log ^^ X a/100 without the use of tables.
50. By use of logarithms, find the value of V6^ — (?
when I = 276.5, c = 172.4.
51. How many years will it take a sum of money to
double itself at 5 % compound interest ? At 7 per cent ?
52. If the area of a lot is 401.8 sq. ft. and the length is
52.37 ft., find the width.
53. The diameter of a sphericaL balloon which is to lift
a given weight is calculated by the formula
^=a/^
w
.6236(A  a)
where D = diameter of the balloon in feet.
A — weight in pounds of a cubic foot of air.
Q = weight in pounds of a cubic foot of the gas
in balloon.
W— weight to be raised (including weight of
balloon materials).
If ^ = .08072, a = .0066, TF= 1250 lb., find D.
' APPENDIX 511
54. Also in Ex. 63, if A = .08072, a = .0056, D = 35.5,
find F:
55. In warming a building by hotwater pipes, the
required length of pipes 4 in. in diameter is determined
by the formula
X^C^OCy Ox. 0045(7
FT
where L = length of pipes in feet.
P = temperature (F.) of the pipes.
T= temperature required in the building.
t = temperature of the external air.
C7= number of cubic feet of space to be warmed
per minute.
Find L when P = 120° F., « = 40.5° F., 2^= 61.5° F.,
and a = 35.6 X 102.
56. Make up and work three examples similar to such
of the above as the teacher may indicate.
MATERIAL FOR EXAMPLES
Formulas
F6rmulas used in the following subjects may be made the
basis for numerous examples.
■
I. Arithmetic
p = br
i = prt
a — 'p + prt
r
1
II. Geometry
K = ioV3
K = p(6 + V)
C = 2irR
K = irRL
T = irRiR + L)
T = 2irR{R +H)
V = ttR'H
V = ^ttR'H
^= 180
K =
V =
= Vsis  a) (a  6) (s  c)
= ^H{B + b+ VBb)
In. Physics
v = gt
s = vt + igfi
r
^= 2
012
MATERIAL FOR EXAMPLES 513
JB = — ij = ^^
2a g + s
^_ 4PP 1 = 1 + 1
bhfm S V V'
t^^Jl H = .24Citt
iJ C = i(F32)
IV. Engineering
H. P. = oonfu\ (horsepower in an engine)
5 = — and /' = / + ^ (sag in a suspended wire)
E = (elevation of outer rail on a curve)
IF = = — k (weight a beam will support)
L
L = ^^ — ^^ ^— (length of hotwater pipe to heat
a house)
D^PL
T = (tractive force of a locomotive)
W "
G — — ■ — (no. gal. water delivered by a pipe)
D = \ ~^7jj^ (diameter of a pump to raise a given
amount of water)
^1 W . ...
D = V goofi/j_/Ti (diameter of balloon to raise a given
weight)
514
SCHOOL ALGEBRA
lMPORTAi«rr Numerical Facts
Areas
8q. Mi.
Rhode Island 1250
New Jersey 7815
New York . 49,170
Texas 265,780
United States 3,025,600
North America 6,446,000
Land surface of earth 51,238,800
Great Britain and Ireland 121,371
France 207,054
Europe 3,555,000
Astronomical Facts
Planet
Diameter
Distance from Sun
Time of Revolu
Synodic
Period
in Days
in Miles
in MilHon Miles
tion about Sun
Mercury
. 3030
36
88 da.
116
Venus '
7700
67.2
225 da.
584
Earth
7918
92.8
365 da.
Mars
4230
141.5
687 da.
780
Jupiter
86,500
483.3
11.86 yr.
399
Saturn
73,000
886
29.5 yr.
378
Uranus
31,900
1781
84 yr.
369
Neptune
34,800
2791
165 yr.
367
Sun's diameter 866,400 mi.
Moon's diameter 2162 mi.
Moon's distance 238,850 mi.
Distance of nearest fixed star, 21 millions of millions of
miles (or 3.6 light years).
MATERIAL FOR EXAMPLES
515
Dates (a. d. unless otherwise stated)
Kome founded . . 753 b. c.
Battle of Marathon 490 b. c.
Fall of Jerusalem . . 70
Fall of Rome .... 476
Battle of Hastings . . 1066
Printing with movable
type 1438
Fall of Constantinople 1453
Discovery of America 1492
Jamestown founded. . 1607
Declaration of Indepen
dence 1776
Washington inaugurated 1789
Battle of Waterioo . . 1815
Telegraph invented . . 1844
First transatlantic cable
message ...... 1858
Telephone invented . . 1876
Battle of Manila Bay . 1898
Distances
From New York to Miles
Boston 234
Buffalo 440
Chicago 912
Denver 1930
San Francisco .... 3250
From New York to Miles
Philadelphia 90
Washington 228
New Orieans 1372
Havana 1410
London 3375
San Francisco to Manila 4850
New York to San Francisco via Panama 5240
London to Bombay via Suez 6332
Heights of Mountains
Feet
Mt. Washington .
. ^6290
Pike's Peak . . .
. 14,147
Mt. McKinley . .
. 20,464
Mt. Everest . . .
. 29,002
Feet
Mt. Mitchell .... 6711
Mt. Whitney. . . . 14,501
Mt. Blanc 15,744
Acongua 23,802
516
SCHOOL ALGEBRA
Heights (or lengths) of Stbuctubes
Fed
•
Bunker Hill Monument .
221
Olympic ....
. 882ft
Washington Monument
555
Deepest shaft . .
. 5000ft.
Singpr Building (N. Y.) .
612
Deepest boring .
. 6573ft
Metropolitan Life
Simplon Tunnel .
. 12imi.
Building
700
Panama Canal
. 49 mi.
Eiffel Tower
984
Suez Canal . . .
. lOOmL
Lengths of Rivebs
Miles Miles
Hudson 280 Mississippi 3160
Ohio 950 Rhine 850
Colorado 1360 Amazon 3300
Missouri ...... 3100 Nile 3400
Rainfall (mean annual)
Inches Inches
Phoenix (Ariz.) ... 7.9 New York 44.8
Denver 14 New Orleans .... 57.4
Chicago 34 Cherrapongee (Asia) . 610
Records (year 1910)
100yard dash 9f sec.
Quartermile run 47 sec.
Mile run 4 m. 15f sec.
Mile walk 6 m. 29^ sec.
Running high jump 6 ft. 5f in.
Running broad jump 24 ft. 7^ in.
Pole vault . 12 ft. lOjin.
MATERIAL FOR EXAMPLES 617
100yard swim . 55f sec.
lmile swim 23 m. 16f sec.
100yard skate . . 9^ sec.
1miIe skate , 2 m. 36 sec.
1 mile on bicycle 1 m. 5 sec.
1 mile in automobile 27^ sec.
1 mile by nmning horse 1 m. 35f sec.
1 mile by trotting horse in race . 2:03 J m.
Throw of baseball 426 ft. 6 in. *
Drop kick of football 189 ft. 11 in.
Transatlantic voyage (from N, Y.) 4 da. 14 h. 38 m.
Typewriting from printed copy. . 123 words in one minute
Typewriting from new material . 6,136 words in one hour
Shorthand 187 words in one minute
Cost 1 lb. radium $2,500,000
Com crop per acre 255f bu.
Milk from 1 cow (1 year) .... 2'Z,432 lb.
Butter from cow (1 year) ..... 1164.6 lb.
Resources (crops, etc., year 1910)
(All these figures are approximate estimates.)
Coal lands in U. S 400,000 sq. mi.
Coal m U. S 2,500,000,000,000 tons
Iron ore in U. S 15,000,000,000 tons
Waterpower of Niagara. .... 7,000,000 H. P.
Natural waterpower in U. S. . . 75,000,000 H. P.
Possible waterpower in U. S. (de
veloped by storage dams, etc.) . 200,000,000 H. P.
Redaimable swamp lands in U. S. . 80,000,000 acres
Lands in U. S. reclaimable by irri
gation 100,000,000 acres
518 SCHOOL ALGEBRA
National forest reserves of U. S. . 168,000,000 acres
Corn crop of U. S 3,000,000,000 bu.
Wheat crop of U. S . 700,000,000 bu.
Cotton crop of U. S 13,000,000 bales
Temperatures (Fahrenheit)
Normal temperature of the human body 98.7^
Ether boils at 96° Temperature of arc light 5400°
Alcohol boils at 173° (approx.)
Water boils at 212° Average change of temperature
Sulphur fuses at 238° below earth's surface 1° per
Tin fuses at 442° 62 ft. (increase)
Lead fuses at 617° above earth's surface 1° per
Iron fuses at 2800° (approx.) 183 ft. (decrease)
VELOcrriEs
•
Wind 18 mi. per hr. (av.)
Sensation along a nerve .... 120 ft. per sec*, (av.)
Sound in the air 1090 ft. per sec. (av.)
Rifle bullet 2500 ft. per sec. (av.)
Message in submarine cable . . 2480 mi. per sec.
Light 186,000 mi. per sec. (approx.)
Weights
Boy 12 years old 75 lb. (av.)
Man 30 years old 150 lb. (av.)
Horse 1000 lb. (av.)
Elephant 2 tons (av.)
Whale 60 tons (approx.)
1 cu. ft. of air 1 j oz. (approx.)
1 cu, ft. of water 62.5 lb.
MATERIAL FOR EXAMPLES 519
Specific Gravities
Air ^iir Stone (average) ... 2.5
Cork 24 Aluminum , 2.6
Maple wood 75 Glass 2.63.3
Alcohol 79 Iron (cast) 7.4
Ice 92 Iron (wrought) .... 7.8
Sea water ...... 1.03 Lead 11.3
Water 1 Gold 19.3
Clay ........ 1.2 Platinum 21.5
Miscellaneous
Heart beats per minute — Frog ..... 10
Man ..... 72 '
Bird . . . : . 120
Smallest length visible to unaided eye. . . tto inch
Smallest length visible by aid of microscope las.ooo inch
Accuracy of work in a machine shop . . . to"o"o inch
Accuracy in most refined measurements . . lo.odo.ooo inch
Dimensions of double tennis court .... 78' X 36'
Dimensions of single tennis court .... 78' X 27'
Dimensions of football field ....... 160' X 300'
Standard width of railroad track 4 8
Weights and Measures
Avoirdupois weight, 1 ton = 2000 lb.; 1 lb. = 16 oz. = 7000 gr.
Troy weight, 1 lb. = 12 oz. = 5760 gr.; 1 oz. = 20 pwt. =
480 gr.
Long measure, 1 mi. = 1760 yd. = 5280 ft. = 63,360 in.
Square measure, 1 A. = 160 sq. rd. = 43,560 sq. ft.; 1 sq. yd
= 9 sq. ft. = 9 X 144 sq. in.
520 SCHOOL ALGEBRA
Cvbic measure, 1 cu. yd. = 27 cu. ft. = 27 X 1728 cu. in.
Dry measure, 1 bu. = 4 pk. = 32 qt. = 64 pt.
Liquid measure, 1 gal. = 4 qt. = 8 pt.; 1 pt. = 16 liquid oz.
* Paper measure, 1 ream = 20 quires = 480 sheets.
Metric system, 1 meter = 39.37 in.; 1 kilometer = .621 mi.
1 liter = 1.057 liquid qt.; 1 kilogram =
2.2046 lb.
1 hectare = 2.471 A.
1 kilometer = 10 hectometers = 100 decame
ters =. 1000 meters = 10,000 decimeters =
100,000 centimeters = 1,000,000 millimeters.
At the option of the teacher, the pupil may insert on the
blank pages at the end of the book other important formulas
or numerical facts, particularly those which are important
in the locality in wUch the pupil lives.
INDEX
'PAGE
Abbreviated division . . . 123
multiplication 109
Abscissa of a point .... 252
Absolute term 366
value 28
Addition 39
in a proportion 413
Affect^ quadratic equation 342
Aggregation, signs of . . . 12
Ahmes 454, 460, 462
Algebra 7
derivation of 461
Algebraic expression ... 21
numbers 28
sum 39
Alternation 413
Antecedents 408
Arabs . 454, 455, 457, 461, 463
Arithmetic formulas . . . 496
Arithmetical means .... 429
progression 423
Astronomical facts .... 498
Axes 252
Axioms 18
Binomial expression . . '.
surd . .
theorem 447,
Bombelli 455,
Cardan 461,
Characteristic. . . . 498,
Choquet
Coefficient
literal
niunerical
of a radical
Commensurable ratio . . .
Common difference ....
21
322
464
456
463
499
456
16
17
17
306
406
424
PAQB
Common factor 160
multiple 162
Comparison, elimination by 228
Completing the square . . 344
Complex fraction 186
number . 334
Compound ratio ..... 406
Conditional equation ... 94
Consequents ' 408
Continuation, sign of . . . 13
Continued fraction .... 187
proportion 409
Coordinates of a point . . 252
Cube root • 464, 472
of numbers 473
Dates, important 499
Deduction, sign of ... . 13
Degree of an equation ... 93
of a radical 306
of a term 67
Descartes . . 455, 459, 461, 464
Detached coefficients . . 68, 466
Diophantus 455460,462, 463
Discriminant 397
Dissimilar terms 40
Distances, important . . . 499
Distributive law ...... 62
Division 76
Duplicate ratio 406
Egyptians .... 454468, 460
Elimination 224
Engineering formulas . . . 497
Entire surd 306
Equation 52, 93
conditional 94
fractional 196
integral 196
521
522
INDEX
PAGE
Equation — Contimied
linear 255
literal 207
numerical 207
Equivalent equations . . . 204
Euler 461
Evolution 279
Exponent 17
fractional 289, 290
law of 272
negative 289
Extraneous root . . . 204, 328
Extremes of a proportion 408
Factor theorem . . . 467, 468
Factorial method of solution
157, 349, 378
Factoring 134, 402
Factors 16, 134
Formula method of solution 357
Formulas, important . . . 496
Fractions 166
Functions 251
Gauss 461
Geometrical progression . . 434
Geometry formulas .... 496
Germans .... 457, 458, 461
Girard 458
Graphs 36, 251, 464
of quadratic equations 387
Gregory 464
Harriot 456,457,458
Heights of mountains . . . 499
of structures 500
Herigone 456
Hero 462
Higher roots 425
Highest common factor . . 160
by long division .... 469
Hindoo method 356
Hindoos 454, 455, 457, 458, 460,
461, 463, 464
History of algebra .... 454
Homogeneous equation . . 367
expression 67
PACnt
Identity 94
Imaginary nimiber . . 279, 334
Important numerical facts . 498
Improper fraction .... 174
Independent equations . . 224
Index law 272, 280
Indicated roots 305
Inequality 266
conditional 268
of same kind 266
signs of 266
xmconditional 268
Infinity 419,459
Inverse ratio 406
Inversion 413
Involution 272
Italians . . 455459,461,463
Key number 450
Laws of algebra . . . 464, 465
for + and — signs ... 32
Limit 441
Linear equations .... 255
Logarithms 496
Lowest common denominator 177
Lowest common multiple . 162
by long division .... 469
Lowest terms ...... 169
Mantissa 498, 501
Mean proportional . .
Means of a proportion
Measures
Members of an equation
of an inequality . .
Mixed expression ...
Monomial
Multiplication ....
409
408
503
52
266
168
21
58
Negative quantity .... 28
Newton 456, 464
Numerical facts, important 498
Numerical value 22
Order of operations
Ordinate of a point
Oresme
22
252
456
INDEX
523
PAGE
Origin 252
Oughtred 457, 458
Parenthesis .... 12, 47, 458
Peacock 464
PeU 458
Perfect square and cube . . 134
Physics formulas 496
Polynomial 21
Positive quantity 28
Power 17
Prime quantity 134
Principal root 279
Progressions 464
Proportion 408
Pure quadratic equations 341
Pythagoras 464
Quadrants 253
Quadratic equation .... 341
properties of . . . . . . 397
Quadratic surd 322
Badical equations . . 325, 354
Radicals 305
addition of 312
division of 317
involution of 320
multipUcation of ... . 314
Radicand 305
Rainfall 500
Ratio 405, 434
Rational root 305
Rationalizing a denominator 318
Real nimiber 279, 334
Recorde 458
Records, athletic, etc. . . . 500
Regiomontanus 456
Repeating decimals .... 442
Resources 501
Rhetorical algebra .... 459
Root 17, 93, 279
principal 279
Series 423
Sign of a fraction ..... 168
Signs from arithmetic ... 12
PAGE
Similar radicals 306
terms 39
SimpUfication of a radical . 307
Simultaneous quadratic
equations 366
Solution of an equation . . 52
by factorial method 157, 349,
378
Specific gravities 503
Square root ... 18, 282, 464
Stevinas 455, 456
Stifel 455
Substitution, elemination by 227
Subtraction 44
in a proportion 413
Surd 305
SymboUc algebra* 460
Symbols, classes of ... . 12
Symmetrical equation . . . 371
S3rncopated algebra .... 460
Tartaglia 463
Temperatures 502
Term 21
of a fraction 167
of a proportion 408
of a ratio 405
Third proportional .... 409
Transposition 52
Trinomial 22
Triplicate ratio 406
UtiUty of algebra .... 8, 248
of algebraic processes 29, 39,
44, 47, 52, 62, 109,
123, 134, 160, 166,
223, 251, 341, 405,
423, 465
Variable 251
Velocities 502
Vieta . 455,456,459,461,463
Wallis 456, 464
Weights 502, 503
Widman 457
Zero exponent 292
ANSWEES
EXERCISE 1
1. 56, 28. 4. 24, 12 quarts.
2. Daughter, $8000; son, $4000. 6. $12.40, $6.20.
3. Man, $96.60; boy, $32.20. 9. 11,250,000 bales.
10. Tenant, $4000; owner, $2000.
11. New York, 49,200; Mass., 8,200 sq. mi.
12. 200 and 40. 15. .0036 and .0009.
13. 4.84 and 2.42. 16. $90, $30.
14. i and A. 17. 4f and f .
19. Lowest part, 12 ft.; middle, 24 ft.; top, 96 ft.
20. $1000, $2000, $3000.
21. Hat, $7; coat, $14; suit, $21. I
22. Niece, $12,000; daughter, $24,000; wife, $48,000.
23. Cement, 3,375: sand, 6,750; gravel, 16,875 cu. ft.
25. Nitrate of soda, 500; ground bone, 500; potash, 1000 lb.
26. Lime, 1901?; potash, 952/t; sand, 2857i?r lb.
27. Boy, $9.90; adult, $19.80. 30. .0062, .0124, .0186; A, A, A
28. 20, 40, and 60. 31. 30, 30, 60, 120.
29. 20, 40, and 60. 32. $94.74, $284.21+, $1421.05+
33. 35? lb.
EXERCISE 2
19. (1) 6; (2) 3; (3) J; (4) 4; (5) 8; (6) 1; (7) 9; (S) 11; (9) 10; (10) 21
35. Walter, 25 marbles; brother, 35 marbles.
37. 9 hr. 14 min. 42. 7 and 8.
38. 15t, 12}. 43. 10, 11, and 12.
39. 35. 44. 121, 391 sq. mi.
40. 7,258. 45. 6290 ft.
11
SCHOOL ALGEBRA
EXEBCISE 3 •
16. (1) 40; (2) 72; (3) 324; (4) 9; (5) 18; (6) 26; (7) ; (8) 9.
17. (9) 80; (10) 125; (11) 0; (12) 4.
23. 10; 32. 28. 2».
26. 2, 4, 8, 32, 128. 29. 1200 sq. ft.
27. 81, 125. 30. $24; $14.
31. State, $360; county, $720; township, $720.
32. Pedestal, 155 ft.; statue, 151 ft.
33. Charcoal, 500 lb.; sulphur, 500 lb.; niter, 1000 lb.
34. 800,000,000 bu.
1. 17.
2. 4.
3. 0.
4. 23.
5. 7.
6. 18.
7. 2.
8. 29.
9. 6.
10. 9.
11. 3.
12. 0.
13. 17.
14. 18.
15. 12.
16. 16.
31. f.
17. 12.
32. i.
18. 5.
33. 1.
19. 9.
34. f.
20. 37.
35. J.
21. 108.
36. 7.
22. 21. .
37. H.
23. 21.
38. J.
24. 63.
39. 24.
25. 2.
40. 15.
26. 15.
41. 1.
27. 26^
42. ^.
28. *.
43. i
29. 3.
44. W.
30. 3.
45. 4.
EXEBCISE 6
46. 2 + V2.
47. 0.
48. When a; = 3.
49. When a; = 2.
50. When a; = 2, 3.
51. When x = 3.
55.
X
21 + 1
1
3
2
5
3
7
5
11
i
2
i
IJ
1.5
4
1. 66; 60.32.
2. 180; 183.976.
3. 374.5; .3456; 105.
4. $37.50; $2052.05.
5. 314.16.
6. 10.
7. 257.28; 100.6.
8. 402 ft.
9. 35'*; 37}^
10. 1482f .
11. 78.54.
16. Daughter, $14,400; son, $9,60a
17. $31.20.
18. Tenant, $1890; owner, $2520.
ANSWEBS 111
19. Owner, $4125; other, $2475.
20. 21, 105; 36, 90.
21. .004t, .023i; .008, .02.
22. Township, $10,800; county, $5,400; state, $1,800. '
23. Gravel, 2000 lb.; sand, 1000 lb.; cement, 500 lb.
EXEBCISE 6
2. 2^; 7. 3**; 11^
3. 7^. 8. $9000.
4. 5°; 13*; 30**. 9. 21°.
5. 4°; 7°; 16^ . 10. $50; $25; $50.
la 15 games.
16. Defeated candidate, 6,105; winning candidate, 6,315.
18. Walter, 30; brother, 53. 21. J, ^.
19. 81, 19. 22. 234 mi.
20. 1.07, 3.33. 23. 555 ft.
EXEBCISE 7
1. 3. 6.  4. 11. 13. 16.  6.
2. 2. 7. 2. 12. 9. 17. 2.
3. 0. 8. 6. 13. 5. 18. 62''.
4. 2. 9. 1. 14. 5. 19. +5.
5.6. 10. 1. 15. 2.5.
22. Man, 240; boy, 80. 24. Yoimger, $8.20; older, $16.40.
23. $1880; $1340. 25. $3.84 and $8.84.
26. Zinc, 400; tin, 800; copper, 3400 lb.
27. 3100 mi. 28. 2266} lb.
29. State, $4000; county, $8000; township, $12000.
30. $3000, $5000^ $6000. 31. 612 ft.
9
12.  6a:*. 18. 2x.
14. 4(a + 6). 20. 3a«  2x*.
16. VoTx. 21. a f 6.
17. i7T7«. 22. 2aJ»llj/».
1. 5.
5. 5a;.
2. 6.
6. 5a
3. 2x.
. 7. 7a:».
4. 4x.
11. 4aa;.
iv SCHOOL ALGEBRA
23. 2&y*. 35. 2.
""24. 7x« + 2i/«. 36. x'* ^a:*  7a? + 2.
26. w»  mn  2n«. 41. $3000; $6000; $9000.
^ 29: ia^ h 4x2/«. 43. 11. 12, 13.
30.  2a;  4y H 2z. 44. 25, 26, 27, 28.
31.  xy + 2aa; + 2/»  3x*. 46. 716,555 sq. mi.
EXERCISE 10
1. 4o6. 4. &r. . 7. 2(a + 6). IL s»5a;.
2. 4aj. 6. «». . 9. Va+aJ. 13. 3ai» 7.
14. 6a:« + 7a?8. 22. 1  2a;  2a:» + a:* + a:*.
15. a + 46  4c H d. 23. m  3(i  a; + 3c.
16. 8 + a; + 7a;». 24.  3a;* + 3a;» 4 4a^  6x.
19. l2a; + 2a;« + a;» + 3a;*. . 26.  2a;« + 4aj«  a:« + 2.
20. 12a?/»  a;V  9a;*2/. 27. Oy.
28. 3ai; a; + y;  3a* + 2a6  6».
31. 4a;»  2a;  2. '  35. 3 in.
32. 2a;» + 6a;» 2x  4. 36. $10.10, $14.70.
33.  2x»  2a;a  8a;  2. 37. $6.20, $18.60.
34. 4a;»  4a;* + 8a; + 4. 38. $1300, $1600, and $2100.
39. $1200, $1500, and $2300.
40. $3529iV, $17641* , and $705}f
41. $857^, $17141, and $3428^.
EXERCISE 11
1. 5a  6.
9. 4.
17. 2c  &  (f.
2. a; + 1.
10. 4a;  1.
18. 3a;  2a;».^
3. 1  a;.
11. 0.
19. 7a:» + a;«
2a;l.
4. L
12. a  1.
20. 2.
"
5. 2a; + 1.
13. 0. '
21. 2y..
6. a;H3y.
14^ 2  2x.
22. 3a;.
7. 1  2a;.
15. X + 1.
25. 4.
27. 3.
8. 9a;  1.
16. 6.
26. 5.
28. H.
ANSWERS
EXEBCI8E 12
1. a^  (ar«  3aj + 1).
2. a  (6  c  d).
3. 1  (  2a h a» + 1).
4. 1  (a« + 2a5 + 6*).
5. a^ (4x + a:* + 4).
6. o«6»  (2cd + c« + (P).
7. 4x*  (9a:«  12«y + 4i)«).
8. x*4x» + 4a;«(4a; + 4+aj»)
9. (m + 2)x  (n + 4)y + (3 + n)z.
10. (1  a  6)a:  (1  6 + a)y  (2 H a  c)^.
11. (  7  2a + 2b)x + (12  c  Qd)y  (10  36)^.
12. (6  3cd)y  (3acH 4ai  2c + 5a)x  (5ai + 4)2.
13. (3  a  c)«»  (2  a +c)x« f (1  2a  c)x  5.
14. (  a + 1  26)a?  (1  6 + a)s»  (1+ a f Zb)x + 3a.
15. ( 6*  a»)a^ + (a»  26*  c)a^  (a  36 ^ 3c)x  a  c.
EXERCISE 13
1. 3.
6. 2.74,
11. .3.
2. 5.
7. 1.
12. f.
3. 3.
8. 1.
14. f
4. 3.
9. i.
15. 12.
5. 4.
10. «.
16. 6.
22. 8.9,
7.5.
24. 13, ]
17. 3 J, 8 J.
18. 4, 9.
19. 3i, 8i.
20. 8.9, 7.5.
21. 8.9, 7.5.
25. 18, 20, 22; 8, 10, 12, 14, 16.
EXEBCISE 14
1. 8.
2. 12.
3. 297.28.
4. 36; 144.
'^ 6. 2x*3a;» + 2a?7a;2.
6. 3V3  SV2  1.
7. 6a^ + x«  5x + 2.
8. 2a* + 33^ + 7xy»2y».
9. 12j;  3.
10. 12a;.
11. (1 +a  2c)a*  (3 +c)a:«  (1 +a  3c)a;»  (2a + 5)a;* + 2.
12. 1 + 2a  (1 + 2a + 36  c)a;  (1  2a  36)a;»  (1 + 2a  36)a;»
13. 6. 14. 3. 15.  J. 16. J.
17. 'ar«2az«U3ax+2a»a<.
VI
SCHOOL ALGEBRA
18 79. 23. lia^  lix.
19. s» + 3aa; f 5aK 24. x>. iz  f.
20. 6aJ»  4aa;  6a«. 25. .95a* + .45a + .3.
2L 4. 26. 1.23a« + 2.12a + .6.
22. 6,405,000 sq. mi. 27. 3(a: + y) + (y + «).
28. 3a»  10a5 + 3a«6«.
29. First, z^ — a^ \ x — 1; second, — x> + 4a; — 1; third, — a^ + x
+ 14; fourth, — a:* + a; — 1.
30. First,  a;* + 2a:>  3a; 4 1; second, a;*  3a;  9; third, 2a;»3a;
6; fourth, 2x*  2a;  a.
31. 2a;»  2a^y + Qxy* + 5y». 33.  2a:» + 2a;2/« + 3y».
32. 4a;»  2a;«2/ + 2a;2/» + 72^. 34. 6a;»  23^  3/.
36. 17,480; 15,064 votes.
37. Lowlands, 44 mi. ; Culebra Cut, 5 mi.
EXERCISE 16
1.
20.
10.
20a*hccP.
21. 2\
40. 0.
2.
6a.
11.
l^cP.
23. x^+K
41. 0.
3.
15a5.
12.
16a;»2^2*.
24. a«a;»M«.
42. 16
4.
30aV.
13.
.8a*.
25. a»a;'»+».
43. 4.
5.
8a;».
14.
ia»a;«.
27. a»a;«»*.
44. 1.
6.
15a;«.
15.
.015a;«.
29. 10(a + h)K
45. 3.
7.
 12aV.
17.
^2^.
32. 21(a + 6)'H4.
46. A
8.
42aV.
18.
ia^.
33. 2«i.
47. 2.
9.
 21a«a;y.
19.
2«+i.
35. 27. 39. 0.
48. 1.
EXERCISE 16
1.
6a*a; + 9aa;*.
8. 2a;'»+»  3a;»+«.
2.
 153^ + 10a;2/«.
•
9. * 12a;'»+*  28a;»»+i.
3.
Sx'y*  2a;V.
10. 3a;~+i + 3a;».
4.
 21a«6a;2y ^ l2db^xyK
11. 3x^ + 5x7«.
•
5.
40a*(M  15amW.
12.  4a7'» + 14a«».
6.
— 7w*n + 7m*n + 21 w«n.
13. x'  1.48a;» + .204 x.
7.
24a;»y«  15a;*2/» 
Sxy*.
14. ia;«  iai»  x
■
ANSWERS Vll
15.  AoV + Aa'a^ + io*x.
16. 'Ax* + ia* H Ja*  ix\
18. 10(o + by  6(a + by  10(a + 6).
19. 21 (x  yY + 6(x  2/)»  18(x  y)K
22. 56. 24. 5.567. 26. 56. 28. 80.
23. 18. 25. 40. 27. 30. 29. $238.25.
3a 60. 31. 169.9. 32. 69.75.
33. Daughter, $10,500; son, $5500.
34. Iron, 460 lb.; aluminum, 158 lb.
35. 19. 37.  9. 38. 3.
EXEBCISE 17
1. 2x^  7x  4. 7. 32a»c  2a&<c».
2. ar*  7x  6. 8. iZai^ + x»2/»  14x2/».
3. 2x«  9x  35. 9. a» + &».
4. 12x»  25xy + 12/«. 10. x*  y*.
5. 28x< + xV  15y*. 11. 8x*2x»+x2l.
6. 30xV + ^  42. 12. 6x»  19xay + 21x2/»  lO^/*.
13. 2x»  5x*  2x» + 9x«  7x + 3.
14. 3xV  10x»2/» + 4xV + 6xy* + 2/».
15. x»  5x*y H 10x»2/*  lOxV + 5x2/«  /».
16. 4x« + 9x*  16x» + 22x«  21x + 6.
17. x»  x»  7x* + 3x» + 17x»  5x  20.
18. x»  6x*y + 9xV  1^.
19. a« + a«6» + &*.
20. 16x* + 36xV f 811/*.
21. x7  9x»2/« + 7xV + 13x»2/*  19xV + 8xj/«  y^
22.  x» + 2ax* h 8a«x»  16a»x«  16a*x + 32a«.
23. a» H 6* h x» + 3a6« + 3a«6. 26. ix»17ix+2.
24. a«6» + c^  aV  WP. 27. .la»  .23a6 + .126*.
25. ia«  16«. 28. 4.5x»  7.1x«  .4x f .24.
29. x»+i  x«i  6x'»*  2x + 4.
30. x»*+i  x*»  2x««i f 3x»»*  10x»»».
31. x«* + x»»»  x«« + x»i  x** + 7x»+i + 10x»+«.
Vlll SCHOOL ALGEBRA
34. 2aj* + 5a:»  8aj« 4 11a?  20.
35. 12a^  a;»  27a:*  3a; + 10.
36. Qs^ + 5x<y  16x»i/« + 14xV  6a^ + 2/*.
38. 35. 40. 50. 42. 60. 44. 60.
30. 35, 145. 41. 45, 50. 43. 24, 46. 45. $200.
46. 315, 85. 47. $780, $220.
EXEBCISE 18
1.  a». 5:  a;  2. 8. 10x« + 7a;  12.
3. (a; + y)* (x  y)». 6. a:^a;2. 9. 5aH24.
4. 16,  1. 7. 17a;  12. 10. 24a + 20.
11. a;». 21. 2a* + 4a;*  2x.
12. 4a;* 12a;» + 13a;»  6a; + 1. 22. a;» + 10a;  16.
13. 40a  24a&. 23. 0.
14.  20aj. 24. a;*  5a; + 8.
15.  6a;» + 13a;  4. 25. 3a;»  10a;y f /».
16.  2a;»  3a; + 6. 26. a;«  2«.
17. a;»  7a; f 6. 27. 4a*  aa; + 6a; + wy + cy.
18. 2a6 + 86*. 28. 0.
19. 2/*4a» + 2y2+z^. 29. 5a» + 2a*a;  lloa;* + 10a*.
20. 2a; + 1. 30. 6a;y.
32.12. 35.12. 38.1. 41.5.
33. 36. 18. 39. 26. ,42. 29.
34. 12. 37. 16. 40. 1. 43. 8o*.
44. 76p*. 52. 1}, i.
45.  2a*6* + 14a6  6. 53. $45, $55.
46. 27, 9.. 54. $10, $45, $45.
47.  6a + 296. 55. $13* , $28f , $28f , $28f
48. 50. 56. 80.
49. $100. 57. 22, 11, 17.
50. $920, $80. 58. 13, 14, 15, 16, 17.
51. .0012, .0003. 59. 15, 9.
60, Daughter, $940; each son, $1780.
61, 25, U, 62. 21, 22, 63. $26, $37, $35^
ANSWERS
IX
19
12.  16a. 16. ^.
13.  24y. 17. Va?.
14. 40m. 19. 2r«,47r,4r.
22. 5(a: + y)«; l6(x + y).
23. .2(a  6)«; .7(a  6).
24. a*«; a»«;  a*».
25.  3a»;  6a;  6a«; 2a*;  6a»+«.
26. a»+4; a*; a»+«. 29. 2«»; 2»»*.
33. 0. 34. 0. 37. 0.
1.
8.
4.
5xy.
2.
3««.
5.
— OJ.
3.
2a.
6.
7y«.
21.
4 ' ^'
2m9.
1. af« + 3a:.
2. &r — 2y.
9. 2x« + .4x30.
10. .04a*  ,68ab  1.66».
11. }x« + x + J^.
12.  Ja»6 + Ja»6« + {a6».
13. 3a^ + 2x»4.
KXEBCISE 20
3.  26 + 3ac. 6. 1 f m  m* + m».
6. Ssfi 2xy  I/*. 8. 3s» + 2a?  5.
14.  2x + 6a;*  3a*«.
15. X*  2a;» + 3«» + a;.
16. 2a:* + 4a* + a;« + 3a:.
17. 3a;'»i  2a;~ f 4a:»Hi . ^n+i,
18. 5(af6)+4.
19. (a;  y)«  .3(x  2/).
20. x — y. 24. 60.
22. a; f 1. 25. 120.
23. 90. 26. 84.
33. 14quaite]iB,7bills.
36. 2. 37. 9.
27. 300.
28. $300.
29. 600.
34. 15 each.
38. 2.
30. $600.
31. 125 nickels.
32. $37.50.
35. 17 each.
EXEBCISI 21
1^ 3« + l.
2. 2a? + 1. '
3. 4x — 5y.
10. s^hxy + t^.
11. 9i«  6a? + 4.
12. 3a;  7.
13. 25 + 20x + 16a;*.
4. 3a: + 7. 7.  6a? + 8.
6. 3a; — 5y. 8. 4a? + y.
6. 3a + 4c. 9. a + 26.
14. 4a«a;*  2axj^ + y*.
15. a*  3a? + 1.
M6. 7a?* + 8a? + 1.
17. 3a*  4aa? + a;*.
SCHOOL ALGEBRA
18. 22/»  42/« f y  1. 2L 2x»  x + 1.
19. c* + c>s» + X*. 22. 3a? + 4a:*2/ + 5x2/" + 2j/».
20: 2x>  3x« + 4a;  5. 23. 2x*  3a:»y  22/«.
24. a:* + 2a:>2/ + 4x2/2 H 82/».
25. a:*  2x»y 4 ^y^  Sxj^ + 16y*.
26. X* — a:*y + x*^ — xV + a?y* — 2/*.
27. 64x« + 16xV + 4xV + y*.
28. 2x»  5x  1. 39. 2x»  3x» + x  5.
29. 3x«  X  5. " 40. ia  i6.
30. 2x»  4x>  X + 3. 41. ix + iy.
31. 2x» + 3x«2/  4x2/» + 2/». 42. \a* + Ja6 + i6«.
32. 3a«  4c»6 + 3a6«  26». 43. ix^ + fa;  f.
33. X* + 2/*— 2*— X2 + xy— y«. 44. .4x — ,by.
34. c» + d* + n*cdcndn. 45. 2.4x  3.
35. 2/* + 22/» + 3y« + 22/ + 1. 46. 2x''  Zx^K
36. 2x* 4x» H 3x« 2x + 1. 47. 4x»'» + 3x«»  x».
37. 2x2/  2x2  32/2. 48. 4x'Hi '3a;» + x'^^
38. x»  3x f 1. 49. 3x»i +'2x»»«  3x»*.
52. X* + x»2/ + a?2/» + a:2^ H 2/* + ?l^.
X* ^ — y
53. Hx+x» + x»Hj— ^. 56. x + 5, x +.a, x + 2/.
54. l+ax + a«x>+ 58. 6 hr.
55. 15, 3a, 3x, 3(x + 2). 60. 4i hr., 36 mi.
61. 108 mi., 126 mi.
63. 5f hr. after second boy starts.
64. 2 hr. 56 min. after second train starts.
&^ 6 hr._ 68. 4 mi., 8 mi.
67. 8 hr. 69. 5 mi., 10 mi.
72. A, 24 mi.; B, 21 mi.
EXERCISE 22
«
1. 3.2x*  2.42x2/  .242/«. 8. 6.
2.  7.15a«  1.5a6  1.86*. 9. 18.
3.  3.8p2  .5p + 3.85. 10. 0.
4. 2.6x«  .5x + 2. 11. 8x* 18x» 13x> f 9x + 2.
5. 45. 12. 1 + 2x  3x«  x».
7.  22x + 54. 13. 10, 11, 12.
/ . 1
I
ANSWERS XI
14. Cement, 40Q lb.; sand, 800 lb.; gravel, 1600 lb.
15. 9t sec. . 25. 5a» (x  y)K
17. a* + X  1. 27.  5.
18. x»  3. 29. (1) 512; (2) 512; (3) 64.
19. 38. 30. 6x»  22a:« + 10a; + 10.
20. a + b; 3a 5& + 4c + 7d. 32. }x* ^aa* + ia«x« + fo*.
21.  7  3a; + 2y  z, etc. 33. a;*" + xV + 2/***
22. 2a + 36; 5. 34. 2  3a; + 3a;«  3a:» + 3a;* . . . .
35. a;>  1.
. 36. 4.8a;»  17.95a;»2/ + 18.45a;j/*  6.32/».
37. a;*  a;» + 2a;«  3a; + 5 +
38. 6a;  Jy  i. 42.  3.
39. 1.6a;*  2a;y + 2.4y». 43. 2z  x.
40. a* + 6« + c« ab ac be. 44. 0.
41. 2. 48. 54; 2; ^.
EXERCISE 23
1. 3* 10. 17 in. l9. 3. 28. 3.
2. 4. 11.  2. 20. 1. 41. 3.
3. 10. 12. 11. 21. 1. 42. 2.
4. 4. 13. 0. 22. 14. 43. 2^.
5. 2. 14.  f 23.  3. 44.  4.
6. 3. 16. 10. 24. 100. 45. 2i.
7.  5. 16. 6. 25. 4. 46.1.
8. 4. 17.  i • 26. 1. 47. 5.
9. 10 ft. 18. i. 27. 3. 48. 7.8 ft.
50. 8.4. 52. 2 yr. 3 mo.
EXERCISE 26
1. $63, 121. 5. A, $61; B, $39.
2. $48, $36. 6. Owner, $45; other, $22.50.
3. $48, $24, $12. 7. 23.22.
4. 36, 12. 8. 38.
9. \Tife, $9000; each daughter, $3000.
10. Wife, $12,200; each daughter, $2200.
11. Alcohol, 50; water, 62.5 lb.
xil SCHOOL ALGEBRA
12. 20, 21, 22. 15. 91 sec. 18. $26, $37, $35.
13. 150. 16. 11, 24. 19. 73, 19.
14. 21, 22. 17. John, 36; William, 60. 20. 22, 11, 17.
21. Horse, $67; oow, $27.
23. Limestone, 50; coke, 350; iron ore, 400. 25. 12 lb.
24. 87 yd., 174 yd., 99 yd. 26. 234 mi.
27. Passengers, $1410; freight, $1675.
28. Niece, $12,000; daughter, $16,000; wife, $36,000.
2Q. 15. , 33. 8 ft. 37. 7' X 12'.
30. 4' X 9'. 34. 5 yd. 38. 8 in. X 12 in.
31. 4 yd. X 6 yd. 35. 20' X 40': 40. 36' X 78'.
32. 6 in. 36. 20' X 60'. 41. 160' X 300'.
42. Boy, 15 yr. ; brother, 5 yr. 43» Man, 20 yr. ; brothjer, 10 yr.
44. 8 lb. 51. 8 min. 56. 6.
45. 8i lb. 52. 26' X 34'. 57. 4, 5, 6, 7, 8.
47. 13, 14, 15, 16, 17. 53. 30. 58. 8, 9.
48. 19, 21, 23. 54. }. 59. 8 yr., 12 yr.
49. 21 words. 55. 43. 60. 26i, 6}.
61. 27' X 78'; 36' X 78'.
62. Mon., 52; Tues., 104; Wed., 57; Thurs., 97.
63. $4. 65. 8 hr. after second boy starts.
64. 3 hr. after second boy starts. 66. 38 da.
26
1. n« + 2ny + y». 4. 9«»  12xy + V
2. c«  2ca: + a^. 8. 1  Uy^ + 49i/«.
3. 4a»  4x1/ f j/«. 9. 9x» + 30a;» +26jt«.
10. 36icV  1323^2' + 121y«z«.
11. 25a*»  30xV«"' + 92^2^.
12. 16x«2/1024n + 72a^y8»+.22n ^ gi^
13. ia* + h^'\ i2/». 18. 2.25m« . .06m + .0004.
15. .04a;» + .12xy + .(W 19. a« + 2a6 + 6» + 8a + 86+16.
16. .09a» + .024a6* + .00166*. 20. a«+2a6H6»6a66+9.
23. 9 4 6a + 66 +a* + 2a6 + 6*.
25. 4a*  4a«6 + 8a»c h 6»  46c + 4c».
26. a:* + 2xy + 2/* 2aa;  2ay  26x  26y + a« + 2a5 + 6».
ANSWERS XIU
28. 998,001. 31. 2,601. 34. 992,016.
29. 994,009. * 32. 1,006,009. 35. 99,940,009.
30. 99,960,004. 33. 9,4q9. 36. 9,840.64.
EZEBCXSE 27
1. a:*  2*. 6. a:*  4. 9. Ja^  J6».
2. j/»  9. 6. a«x*  6V. 12. .0025a*  .09&«.
3. 9a:»  I/*. 8. 4x«»  25y^. 13. Aic*  .492^.
14. o>«+«  i6»"« 27. a«  6» + 66a;  9a^.
15. a« + 2o6 + 6*  9. 28. a:* + a^. + y*.
16. X* + 2xy + j/«  a«. 29. a:^\a* + 1.
18. 16  x»  2x  1. 30. 4ic*  29a:« + 25.
19. 4x»  92/« + 302/  25. 31. 4a:*  29a:V + 1^.
20. a« + 2a6 + 6«  9. 32. a:*  3a:*y» + 1/*.
22. 16  a<  2a;  1. 33. a* + 2a5 + 6«  c« + 2c  1.
23. 4a;«  92/« + 30y  25. 34. a;* + y*  x*y*  L
25. a;* + 6a;»49a;*4. 35. a;» + 2a;y + i^s? 22 1.
40. 8096. 43. 999,975. 46. 9996 sq. ft. 61. 996,004.
41. 9991. 44. 1200. 47. 18.96. 62. 999,996.
42. 9975. 45. 292.40. 48. $48.91. 63. 9409.
64. 9991.
EXERCISE 28
1. 4a*h3/* + H4a;y + 4a;H2y.
2. a;* + 4y* + 42* — 4a;y + 4a» — Syz.
3. 9a;« + 4y« 4 25  12a;y  30a; + 20y.
4. 4a« + 6» + 9c*  4a6 + 12ac  66c.
5. a;* H 42/» + 9«»  4a;y  6a» + I2yz.
6. 16a^ + 9j/» + 1 + 24a;y SxQy.
7. a;*  2a* + 3a;»  2x + 1.
8. 4a* + 20a» + 13a«  30a + 9.
9. a;« + 2/* + 2? + 1  2a;2/ + 2a»  2a;  22/2 + 2y  22.
10. 4a;» + 92/» + 162« + 25 + 12a;y  16a»  20a; 2^yz  SOy + 40b.
11. 9a;*  24a;« + 22a;*  20a;» H 17a;»  4a; + 4.
12. ia;*  la* + ^a;«  ^ + 25.
13. ta*  }a* + Ja;* + ^a*  }a* + 9ai + 36.
14. .04a« + .096» + .25c« + .12a6  .2ac  .36c.
29. .0004a;*  .012a;* + .11a;*  ,Zx + .25.
Xiv SCHOOL ALGEBRA
EXERCISE 29 .
1. aj» + 7x + 10. 15. a« + ia T }.
2. x»  8x + 15. 16. a«6» + 4abx + Sx*.
3. a:*  3a;  28. 17. a«6«  2abx  3jj«.
4. a^ + 4a;  32. 18. a^  4xyz*  2U*.
6. a;*  6a;  7. 19. a;*»  25.
6. a;*  5a;» + 6. 20. (x + yy + S(x+y) + 15.
7. a;* + 4a;» + 3. 21. (a; + 2/)« + 2(a; + y)  15.
8. a*7ax SOs^. 22. (x + y)« + 2(a; +y)  15.
9. a;«  6a;y  7j/«. 23. (a + 26)* + 8(a + 26) + 15.
10. a* + .7x + .1. 24. (2a;+3y)«2a(2x+32/) 15a*.
11. a;* + fx H i. 25. a;»  a«  2a6  6».
12: a* + 5.02X + .1. 26. 4a*  a«  6o6  96*.
13. a» + .52o + .01. 27. 4a;«  a» + 6a6  96».
14. a:»  ia;  i. 30. 381 ft.
EXERCISE 80
1. 2a;» + llx + 12, 7. Sa;* + 34a;  7.
2. 2a;*  11a; + 12. 8. 3a;* +a;y  2^.
3. 2a;*  5x  12. 9. 12a<  lla*6  56».
4. 2a* + 5a;  12. • 10. }a;« + Ja; + i.
5. 6o« + 19a + 15. 11. 2a* + .la6  .066*.
6. 6a*  a  15. 12. }a* + 2aa;  fa*.
EXERCISE 31
19. a* + 2a6 + 6*  3^  2xy  i/^,
20. a* 4 a*a;* + a;*. 36. 8a6. 40. Zxy.
22. 16a* + 4a* + 1. 37. 8a;* 8x + 2. 41.  169a;«.
29. a*  6*. 38.  8a;. 42. c*  2c + 2a.
35. 2a* + 86*. 39.  6a + 5. 43. aV  a*j/ y»+ y*.
46. 72i. 51. 9900i. 56. 38,025. 61. 23.04.
47. 380i. 52. 56.25. 57. 990,025. 62. 9604.
48. 39,8001. 53. 380.25. 58. 94.09. 67. 4}.
49. 240i. 54. 9900.25. 59. 96.04. 68. 2.
50. 2450i. 55. 5625. 60. 92.16. 69. 9991 sq. id.
70. 9604 sq. rd. 71. 135.96. 72. 180.75.
i
ANSWERS XV
SZEBCX8E 32
1. a+x. 11. .2x  .3y. 28. 64, 46.
.2. 3 + 2a;. 16. x + 1  a. 29. 53, 47.
4. 5x + 62/». 16. a + 6  2c. 30. 109, 91.
6. 4x»  72/«. 18. o  6  c + 1. 31. 10.9, 9.1.
7. a6*  6c»d*. 19. 1  a  6 f c 32. 89, 71.
8. ix + iy. 27. 64, 46. 34. 12.
10. .5af .46.
XXEBOISE 33
1. o»  2a + 4. 10. la» ia6* + tV^*.
2. x« + a; + l. 14. c*c + ca; + l 2a;+««.
3. 93^ + 12z + 16. 16. 4 + 2a; + 22/ + X* + 2a:y +2/*.
6. 26 + &c» H a*. 16. 9x*  16a:V + 25y».
6. 9a*  3aV + 2^. 17. a«  2a + 1+ aa;«  a:« + x*.
8. .04a;« + ,2xy + 2/». 18. a;*  2x» + 2a;» + 2a; + 1.
9. ia;* + ia;y + li/*. 19. 4a;«  8aT/ + 42/* + 2a»  2yz + «*.
29. 23. 30. 23. 31. 23. 32. 17.
33. 63. 34. 47. 36. 90.
EXERCISE 34
1. a* — a*x + a*x^ — (u^ + x^,
2. a* + a'x + ah^ f oa?* + ic*.
3. 6«6»2/ + 6V6»2/» + 6Vb2^ + J/«.
4. 6« + &»yf &V + &V+.6V + &2/* + l/«.
6. a*  2a» + 4a«  8a + 16.
6. . a« + 2a» + 4a* + 8a» + 1 6a« + 32a + 64,
7. a;* + a;» + a;«+a; + l.
8. a;*a;*+a;»a; + l.
9. 16a;* + 8a;»y + 4aV + 2a;2/» + 2/*.
10. ai«  a^x H a^a^  aV + aWa^T^ + a*a!«a«a;'+a*x« aa;»Ha;i°.
11. x« + a;«2/» + aV + a^' + 2/".
12. 81 + 27a + 9a* + 3a» + a*.
23. 11. 24. 13. 26. 12.
26. 7. 28. 210.
XVI SCHOOL ALGEBRA
3L 3^ + dx+a* + x + a.. 42. 10.
32. a* + ox + a« + 6x + 5a. 43. 6.
35. x»  2ax + a» + 5x + 6a. 44. 1}.
37. (a«  6»)*. 46.  4.
38. (a«  46»)». 46. 6.6.
39. (9x»  4j/«)«. 47. 180, 540, 280.
40. 3a;» + 1^  21. 48. 12i mi.
41.  lla:« + 2ftr  13. 49. 612°.
50. 120 ft. per sec. Latter is 1 A times as great.
BZERCISK 36
1.
a^{2x + 5).
6. 3a«x«(a  5x).
9. a»x(l  2x).
2.
x(tx*  2).
7. 9x*(2x  3y).
10. Jx«(2x + 1).
3.
x(x + 1).
8. x»(l  X  i;«).
11. ia&(3a  206).
12.
ix»(5a  12x).
22. (a + 6) (7x + 5y).
13.
.2a:»(x + 2a).
23. xy(a + 6) [7x f 5(a + 6)y].
14.
.02o«»(l  20o).
24. 7(x 
y)«[3  2(x  y)].
15.
.6m (2n — m).
25. 3(2x 
 a)»[3  4(2x  a)*].
16.
3(a»  2aaj + 3a^).
26. 1694.
•
17.
2a;(l +2x 3x»).
27. 938.25. V
20.
2xV(2/«  4a;»»y + 3a*»). 28. 58,190.
21.
o'»yc«'»(l + lie).
29. 314f .
30.
517,000.
33 JL
a + 6 + c
'^' 3(x«  2)
31.
10 1
34 ^^
"^•206 + 30
a*
IT
a + 6* 2'
"'• 2(2*  3)
32.
10
a + h
35 ^2^
'^' a + 26
KZERCISE 37
38 ^^^ •
^•2(2pM«l)
1.
{2x + «)«.
3. (5x  1)«. .
6. c(7 + 26c)»,
2.
(4a  32/)«.
4. (X'  10j/)«.
6. o»(6 + 2)«.
7. x(y + 1)«.
12. 2y(2a  5x)^
■
10. x(2x + ll/«)*.
13. 2x«
(«  2)«.
11. a6(9a + 76)«.
16. (x»»
+ y)'. .
ANSWERS xvii
17. (a  6  c)«. 19. (8a  12  6)«.
18. (3x 4 3y + 22)«. 20. (5x  5y  12xy)«.
21. (a + 6 f c)«. 24. (.2a  .36)*. 27. a + 36.
22. (tr + 3y)*. 25. (5a  3a;)«. 28. 1  2a.
23. (ix f iy)«. 26. a  36. 29. 2a + 36.
EXERCISE 38
1. (x + 3) (a;  3). 9. xix + 3a) {x  3a).
2. (5 + 4a) (5  4a). 10. a^(x + 3a) (x  3a).
3. (2a + 76) (2a  76). 11. w(l + 8a) (1 ^ 8a).
4. (x + 2y) (x  2y). 12. ,2(11 + x) (11  x),
7. (1 + 8fn) (1  8m). 15.'(a» + a<) (a Hx) (a  x).
8. 3(x + 2y) (x  22/). 16. (a« + 96*) (a + 36) (a  36)
17. (x* + 2/*) (x« H j/«).(af + y)ix  y).
18. x(x* + l){x + 1) (x  1).
20. x(a* + 1) (a» + 1) (a + 1) (o  1).
21. (ISx'* + y) (15x*  y). 27. (.9x + .056) (.9x  .056).
22. (x + ly) (x  iy)' ' 28. (x» + y») (x»  j/»).
23. (fa + 36) (fa  36). 29. (x*'' + yV) (x»*  yV).
24. (.3x + Ay) (.3x  Ay). 30. (x + y + 1) (x + y  1).
25. (.la + .26) (.la  .26). 31. (x + y + 1) (x  y  1).
26. (.5y + i6) (.5y  i6). 32. (x  y + 3) (x  y 3).
33. (2x  2y + 5) (2x  2y  5).
34. (1 + 6x + 12y) (1  6x  12y).
35. •(4x + 2y + 1) (2y  2x  1).
36. (11a  86) (9a  26).
• 37. y(xV + 2») (x»y« + «*) (xV  «*)•
38. (9x« + 4y«) (3x» + 2y) (3x»  2y).
39. x(x» + 12y2?) (x«  I2ys?).
40. (a  6 + 2c +2) (a  6  2c  2).
41. (10x«  lOx  9) ( 10x« 4 lOx + 11).
42. a 26. 43. a + 26. 44. 3+a. 45.1+6.
9
EXERCISE 89
1. (X f 3)(x + 2). 3. (x + 3)(x2).
2. (a; + 2) (x  3). 4. (x + 11) (x  4),
\
xviil SCHOOL ALGEBRA
7. (x 4 Sy) {x  2y). 20. (x  8a) (x f 3a).
8. (x  Sy) (x + 2y). 21. (x«  8) (x + 1) (x  1).
IL (x  9) (x + 4). 24. x(x + 4) (x + 3) (x 4) (x 3).
12. (x» + 4) (x + 3) (x3). 25. (x"  8) (x» + 7).
15. (x  12) (x + 4). 26. (o5  13c») (afc + 2c»).
16. (x + 16) (x  3). 27. (x + a) (x + h).
17. (x  24) (x +2). 28. (x + 2a) (x  36).
18. (x  12) (x + 8). 29. (x + a + 6) (x 4 6 + c).
19. {xy  12) (xy  11).' 30. (x + a  c) (x + 6 + c).
31. (xy6)(xy + 3).
KZERCISB 40
1. (2x + 1) (x + 1). • 10. (2x+ 5) (x  2).
2. .(3x  2) (x  4). 11. (4x + 1) (3x  2).
3. (2x + 1) (x + 2). 12. (4x  1) (x h 3).
4. (3x + 1) (x + 3). 13. (5x  1) (x + 5).
5. (3x  5) (2x + 1). 14. 3x(x  2) (3x + 1).
6. (x + 3) (2x  1). 15. 2y(3x + 2) (x  X).
7. 2x(x + 4) (3x  2). 18. (8a  Q6) (4a + 56).
19. (2x + 3) (x + 1) (2x  3) (x  1).
20. (3x + 2) (x + 4) (3x  2) (x  4).
21. (4x + 3«) (3x  42).
22. 2x(6x  j/«) (2x + 9y»).
23. (5a + 46) (5a  46) (a« + 6»).
24. (8x»  W (2x« + y*).
25. (3X'' + y) (x*  3y).
26. (5a + 46) (a + 6) (5a  46) (a  6).
29. (a + 6 + 8) (a + 6  3).
30. (3x  3y  22) (x  y + 32).
31. (3x« + 6x + 4) (x + 3) (x  1).
32. 4x(x + 4) (x + 2) (x f 1) (x  1).
33. 2(1+ 3x) (2  x).
EXERCISE 41
1. (m  n) (m» f mn + n«). 4. (a + 26c) (a»2a6c h 46V).
2. (c + 2d) (4J^  2cd + 4d«). 7. (a6 + 1) (a«6»  a6 f 1).
3. (3  x) (9 +3x +x»). 8. (1  lOx) (1 + lOx + lOO**).
y
ANSWERS XIX
9. x(3x + o) (Qjc'  Sac + o*).
10. (8a;  y») (64x« + 8a:!/« + y*).
11. a(l 4 7a) (1  7a + 49a»).
12. (a + x)(aa;)(a»aa;+x»)(a« + ax+x»).
13. (a;* + y) (a*  y) (x*  a:«y + y«) (a:* + a;^ + !/■).
14. (a + 2n«) (a  2n«) (a« + 2an« + 4n<) (a«  2an* f 4n<:
15. 2a;(5  a^) (25 + 5a;« + a;*).
16. (2x« + y) (4x*  2a;V + 2/*).
17. (a + 6 + l)'(a« + 2a6 + 6»a6 + l).
18. (5 + 26  a) (25  106 + 5a + 46«  4a5 + a«).
19. (2cd)(4 + 2cf2d + c« + 2cd + (P).
20. (2a;  y) (13a;»  5a;y + !/•).
21. 2a;(2a;y*  Sz) (4a%* + 6a;y»2 + 92»).
22. (x + y) (a;*  a;»y'+ aV  a;y* + y*).
23. (a;y)(a* + a*y + aV + aV+*aV + a^ + y«).
24. (a« + wi*) (a<  ahn^ + wi<).
25. (a;* + y*) (a;»  a^ + y')
26. (a  26) (a« + 2a»6 + 4a*6« + 8a»6» + 16a»6* + 32a6» + 646»).
27. (a + x) (a"  a»x + a^  aV + a«a;*  €h^\(^7^ a^z* + a«x«
 ox* h x^o).
31. (3  x) (81+ 27x + 9x» + 3x» + x*).
32. (4  a 4 6) (16 + 4a  46 + a»  2a6 H 6»).
33. (2x  4y + 1) (4x«  16xy + 16y«  2x + 4y + 1).
36. (2x  a«) (16x* + 8a»x» + 4a*x« h 2a»x + a«).
37. (a»4y»)(a*aV + y*).
38. (2x* + y») (4x»  2xV + y*).
39. [8x  (a + 6)«] [64x« + 8x(a H 6)» + (a + 6)*].
KZERCISE 42
1. (a + 6)(x + y). 9. (y« + 1) (y + 1).
2. (x  a) (x + c). 10. (ax  1) (x  2a).
3. (5y  3) (x  2). 11. (x  y) (x  3).
4. (m  2y) (3a  4n). 12. (2  1)» (2 + 1).
5. x(a + 3) (a + c). 13. (6  1) (a  y).
6. (3a  5n) (a + 6). • 14. (x  1) (x» + 2) (x»  2).
7. x(x» + 2) (x + 1). 15. (x ^y)(fl^ 6).
8. 2x(x + a) (x  a) (x  1). 16. (x + 4) (x + 2)«.
'J \
XX SCHOOL ALGEBRA
17. (a + 3) (a«  3). 19. (x  1) (2x  1)«.
18. (aj  y) (2x + 2y  1). * 20. (a;  1) (a:* + 3* + 3).
lil. (x — y) (x + y + a^ + xy + ^).
22. (xy)(x«+xy + 2/»Hl).
23. (x  y) (x« + xy + !/•  X  y).
24. (x  y) (x« + xy + y*  X + y).
25. (x + 2) (1 + a) (x  2) (1  a).
26. (xy)(x« + a^ + y»fa; + y + l).
27. 4a(x  1) (x» + 2).
28. (3a  x) (3a •+ 2x) (a  x). 32. (2x  3) (4x  3) (x  2).
29. (x  2) (x + 3) (x  1). 33. (x + 2) (x + 1) (x 3).
30. (x + 3) (2x  5) (2x  1). 34. (x + 3) (x ~ 2) (x  4).
31. (2x + 1) (4x  3) (x + 1). 35. (x  2) (x  1) (x  5).
KZERCISE 43
1. (o + 6 + x)(a + 6x). IL (x + a6)(xa + 6).
2. (a 6 + 2x) (a 6 2x). 12. (z  a + y) (x  a  y).^
3. (a + x + y){ax — y). , 13. (a + y + a;) (a + y — x).
4. (3a + X + 2y) .(3a x 2y). 14. (a« + x« + y) (a«  x«  y).
6. (4a + a;y)(4ax + y). 15. (x 4y + 1) (a;  y  1).
6. (m + x + y)(mX' y). 16. (1 4 a;  y) (1  x + y).
7. (a H 6 + 2x) (a + 6  2x). 17. (cha h) (c a + h).
8. (a + 6 + 2x) (a + 6  2x). 18. (a h + c) (a b  c).
9. (a + 6 + 2x) (a + &  2x). 19. (a6 + 1 + x) (a5 + 1  x).
10. (x + a + 6) (x  a  6). 20. 2(2  1 + a") («  1  ^).
2L (x + 2yr6«)(x2y + 62).
22. (a f 6 + c + d) (a + 6  c  d).
23. (x  2y + a? + 1) (x  2y  3z  1).
24. (3a  26 + 6x + 1) (3a  26  5x  1).
25. (a  66 + 36x  1) (a  56  36x + 1).
EXERCISE 44
1. (c« + ex + x«) (c»  ex + «•).
2. (x« + x + l)(x«x + l).
3. (2x» H 3x  1) (2x»  3x  1).
4. (2a*  3a6  36«) (2a« + 3a6  36»).
1
ANSWERS
•
6. (3x« + 3a^ + 22/«) (3a:«  3xy + 2i/«).
6. (7c« + 9cd + 5d») (7c«  9cd + 5cP).
7. (4x« + X  1) (4x«  X  1).
8. (lOx* + a:  3) (lOo:* xZ),
9. (15a«6«+8a6 + 2)(15a«6»8a6 + 2).
. 10. 2(4o« 4 6a6 + 6«) (4a» — 6afe + 6»).
11. (a« + 2ad + 26») (a»  2a6 + 26»).
^ 12. (1 + 4a; + 8a:*) (1  4x + 8x*).
13. (xV + 6x2/ + 18) (a:*!/*  6x2/ + 18).
EXERCISE 46
»
10. 3x(x».H X + 1) (fc»  X + 1) (a; + 1) (x  1).
11. (2a 4 1) (a + 1) (2a  1) (a  1).
12. 2{x« + 2x + 2) (x»  2x + 2) (x> + 2) (x«  2).
13. (x + 9) (x  5).
14. (2x + a  1) (2x  a + 1).
 15. 5a(x« + x» 4 1) (x« + X + 1) (x  1).
16. 3x(3x + 4) (2x  3).
17. (x» + a» 4 2««) (x«  X2 + 22«).
18. (x * 1) (a * 3). 25. (a« 4 2) (a  IJ.
19. (U +x) (10  x). • 26. 2x(3x 4 1) (a;  1).
20. (3x  5/) (x 4 62/). 27. il ^ 5z + «»).
21. 7a(l * (0^)/ * 28. 2*(4  y) (16 442/4 2/*).
22. 2(3x4^ 4) (x 4; 1). 29. (t 4a 46) (1  a  6).
23. (x 4 2) (x 1) (x«x 4 2). " 30. (3a  5) (7a 4 6). '
24. 3a(l 4a) (1  a 4a*). 31. (x* 42/*) («»  xV 4 2/»).
32. (2x + 93») (4x»  18x2» 4 Slsfi). ,
33. 45x*(3t/« =*= 1).
34. (a» 4 5) (a 4 2) (a 2).
35. (c + dl)(c« + 2ai4d* + c4d4l).
36. (x2/)(a;2^ + 2).
37. (8x  92/) (3x 4 42/).
38., (x 4 22/) (x  2if)K
39. (a 4 3)(a 4 2) (a  3) (a  2).
40. («««*=« 4 1).
41. (a + 6 4 ^) (a 4 &  c) (a  6 4 c) (a  6  c).
^
/
xxil SCHOOL ALGEBRA
42. (7a; + 3y) {Zx  ly).
43. (2 + n) (16  8n H 4n»  2n» + n<).
44. 6x(x« +V) (x*  a^ + y*).
45. (m + n)(m«m»n + m*n«m«n» + mVmn» + n*).
46. io(2a; + y) (4a*  2xy + j/*). 49. (2a =*= 1) (a * 3).
47. (1 + a*) (1 + xy (1  x). 50. (x =*= 2) (a* =*= 2x + 4).
48. 6(x  3) (4  x). 51. (x + 1) (x + 2>(x  3).
52. y(2x  «*) (16x* + 8x»2^ + 4x*z* + 2xz« + «»).
53. (x* + 6xV + 2/*) (x + y)« (x  y)\
64. (x« + j/» + «») (x + z) (x  2).
65. (ax  y) (x  1) (x« + x + 1).
56. (a + 6) (a  76).
67. (a6+ xy) (a«2a6 + 6»ax + 6x + ay6j/ + x*2xy+ j/»).
58. a«6»(a  6)». 60. (x + y) (2a* + xy + 2y»).
69. (x< + a») (x«  a»x* f a«). 61. (a 6) (x + y) (a  ft+x + y).
62. (a6 + 2x + 2y)».
63. (a« + 1) (a*  a» + 1) (a =»= 1) {fl*^a + 1).
'64. (2a  35) (2a + 36 + 2). 68. (a =*= 6«) (1  x) (1 + x + x»).
65. (2a + 36 + 1) (2a  36  1). 69. (3 + a) (x =»= 3).
66. (x + y)« (x  y)«. 70. (a + 6)«.
67. (x  1)* (x + 2)«. 71. (a  x)».
72. (a6c — mnp) (ax — my).
73. (x + 2 + 2a  n) (x + 2  2a + n).
74. (x  2) (x + 5) (2a; + 1). 76. 9xV(x + yY (x  y)\
75. (a« + 26«) (a«  26» + 1). 77. 2(9x  y) (x + 3y).
78. (1 + X  x«) (1+ 2x H 2a* + x» + x«).
79. (1  x)« (1  2x  x»). 82. (a  36 + 3) (a  36 ~ 3).
80. (x + 1) (ax  c). 83. (x« + 4) (x + 2)« (ar2)«.
81. (x« :*= 9x + 1). 84. 9(1 * x) (7x« + 6x« h 3).
85. (x» ,3 H 7y) (x«  3  7y).
86. (xV  2 + 2x  y) (xV  2  2x + y).
87. (ax — 6my) {an + cmz).
•
EXERCISE 46
1. 2, 3.
3. 3, 4. 5. 4, — 3.
7. =fc3.
2. 2, ~ 1.
4. 3,  2. 6. * 4.
8. 0, ^ 2
ANSWERS
XXlll
9. 0, =fc 5.
10. 0, =fc 3.
11. 1, i.
12. 2,  f .
25. J, 1.
28. X*  7x + 12 t= 0.
29. a:« + 3a;  10 = 0.
30. x» + 10a; + 21 = 0.
34.  5, 9. 36. 3,  4.
35. 4,  10. 37. 0, 9.
42. 50' X 150^. 43. 617*".
13. 0, 3,  2. 17.  1, =»= 2. 21.  1, f .
14. 0,  2. 18.  1, ± 3. 22. =b 1.
15. 0,  a. 19. ± 1, =*= 2. 23. 0, =*= 3.
16. 0, =*= a. 20. 1, =fc 2. 24. 1, 2.
26. 2, 2. 27. =*= 2, ± 3.
31. a;»  a;«  4a; H 4 = 0.
32. a;*  2a; = 0.
33. a;»  5a;« + 6a; = 0.
38. 0, 25. 40. 8, 9.
39, 0, ± 5. 41. 1,2:
44. 49,179 sq. mi.
1. 2a&.
2. 5a;^.
3. 8a»a;*.
6. X — 3.
7. a;(2a; + 3).
8. a — X.
9. a;  1.
10. a(2a + 1).
EXERCISE 47
11. a; + 1.
12. 4aa?(a — a;).
13. x{x  1).
14. 2x  y.
15. x{x  2).
16. 6(1  a«).
17. 1 + a + a*.
EXERCISE 48
20. an^ifl  h)K
21. 3a;«(a;  y)K
22. xy,
23. (a;  y)».
24. 2  X.
25. a  3.
26. x(x  a).
1. 6a«6«.
2. 36a*a;V.
3. 12a6c.
4. 12a»6»c».
7. 2x(a;»  1).
8. 6a6(o + 6).
9. 14a;«(a;  3).
10. {2*  1) (a; + 1).
11. (a;«  2^) (a;  2v),
12. 6a;{a; + 1) (a;  1)*.
13. 15 afea%(a; + y) (a;  y)«.
14. x{x + 5) (a;  8) (a;  1).
15. a«  6».
16. 6a;«(a; + 1) (a;  1).
17. 12oft(a + 6) (o  6).
18. (2a; + 1) (a; + 1) &x  1).
19. 6a;(a;»  1) (a;  1).
20. 6a;(3a; + 10) (2a;7) (a;3).
21. 2a;(l + a;*) (1 + x) (1  x),
22. 14aV(a; + 1)» {x  1)».
23. 6a;»(3a; + 1) (a;  1) (3a;  1)».
24. (a;  1)» (x + 1)* (a; + 3)« (x3).
27. a*6*(o + h)* {a  by,
28. 18a»&»c»(c =t d) (a  d)».
29. a*&«(a + 6)» (a  6)«.
30. 36a;*(x + y? (x  y)K
31. a  6.
32. 36(o  6)».
xxiv SCHOOL ALGEBRA
33. (a + h)(a h) {x  y), 35. (x + 1) (x + 2) (a?  2)«.
34. (a + 6) (a  h) (x  y^. 36. x»(a + x) (a  a;)«.
37. 2, 3.
39. Lrigable land, 100,000,000 A.; swamp land, 78,000,000 A.
40. 2 da. 14 hr. 40 min. 41. 866,400 mi.
42. Son, $3600; daughter, $5600; wife, $10,800.
43. 2162 mi. 44. 39.37 in.
•
EXERCISE 49
, a+h+c
^' 3
5.^.
a
3 ^ .
^' 43.560
g fk + wy
c
M
2a
4.
3a;'
5.
4a;
6.
X
2Zax
tm
3xz
7.
4j/«
8.
3a
46*^
9.
1
2al
10.
1
2a'
11.
1
^ •
a
12.
1
so
*• *• *^ 5r 23. ^^^p^.
24 ^±i?.
a — o — c
1 +a — a?
13.
2a
3a;
14.
2(a; + 1)
3
15.
5
2(a;  y)
16.
a + 6
2(a  6)
17.
2
3a; 4y
18.
2&
3a'
19.
1
2x + 3y
20.
7a; + 8y
23C?
21.
a;« + 3a; + 9
a;3
22.
1
26.
27.
28.
29.
a;fal
3x+_4a
a;F^ "
a;2
a; + 3
x\2
30. a*  2/*.
31 '^^^'
gg a;y^g2
a; + y 'x — y « — y — ap — 2
3. "2^
ANSWERS
EXERCISE 61
'
^  1
3a
yf 2
10.
46«
7.1.
11.
x« + 3x + 9
a;3
^ a + 6 + c
12.
2 + a + b
2a + h
9 ^.
13.
yx^
a + h
XXV
y + 2z
3 + m ^ 1 ,„ 2 + a + 6 "^
4 — m
3 + :c
4.
18. f 19. 6. 20. t 21. 4. 22. 20. 23. 5.
EXERCISE 62
1. 6f. 2. 13i. ^ 3. lOH.
4.x 2 +  14. x«l§^
X x* + 2
5. 2x« + 3;^ 15. 3o+ ^^
2x ^. ^ . 3^, _ 26
6. 2a*x« + 1  y^ 16. x«  2x + 2 H • ^
5ax X + 3
7. x*4x + 5 ^ 17. 2a 26+ ^^
x4l ' aH6
n.x^x + 24^?:^. 20.1x + 2x>^^2^
x*fx — 1 l+x— X*
12;*'+*«+2x + 3 + J^±^ 21. 4  2« + a^  15^^
X'— X — 1 i5 f X — X*
EXERCISE 68
1. V. ^ x»  2x „ x 'x
6. r 9. ;;5i jr
2. ijA. x1 x*4x4l
/a»a + l 7 8x«y «* + 1
^ a 2x + l ^^' a2*
 x» Q a* + a& io ^H^
X — 1 a + 2o x + a
xxvi SCHOOL ALGEBRA
ItJ. ^zT • 10. J lO. rrr
2bc 4 X + 1
4 •• 1+x *"• x1
20. 3248 mi. 21. 27i sec.
BZEBCISE 64
1. If, H. a ^ 2ab
2. tt, tt, IJ. 2aW' 2a«6«' 2a*b^'
« 4x 15x ' 1 2a» 2a 3o
'^' 18' 18 ' ^ a»  a' ";S«"ir^' ^1 
g* 43?* 43? a; 41
(X 4 1) (x»  D' (x 4 1) (a^  1)'
X x(2x  3) 4j;«9
a;(4x«  9)' x{4a^  9)' a:(4x»  9)'
2x + 4 15X30 18
6(x«  4)' 6(x«  4)' 6(x«  4)'
BZEBCISE 66
, 19 ^ 3x + l ,. ' 3m« + l
1. ;r' 7. — 7n — • 14.
10.
11.
15.
ex 24 (m+l)(ml)*
^ 8x9 412a 8.1. ^ (a  3fc)«
2 1255 • 9. _^. 4(a  6)«*
1564c6a * ^ " f. ^^' ::^'
^^ 3x« 2x» + 3xl
4. ?^. n ?5a^; ^7 x(x^  D •
6a«6 11. j2 18. 0.
9J^0a^ y»3x'2»>6ya^ * 1
12ax« 12. g^^^^ . 19. ^^^72*
6.4±g. 13.^. 20.^.
flj — 5» X* — 4 X*— 1
x^h5x + 10 5x«y3y'
''^ (x + l)(x + 2)(x + 3)* ^ xix'^^y 25.0.
5x(x 4 3) x« + 4x13
^'^' (2x + 1) (2x 1) (x + 1) ' ^' 2(x*  1)
9R ^ 27 ^Q^
ANSWERS
XXVll
1
2ft ^*
+ 90r
9
"^ 6(x»
 9) (X
3)
30.
1
1
31.
0.
.^9
5
•
20.
1
1 x»
"^^^ 4/»  a^
35. 0.
(X  3) (a;  4)
36. 0.
13
39.
40.
6x
(a;  2) (x  3) (a?  5)
546
(a 3) (a 2) (6 2)'
41. 0.
7
42.
12a;(x + 1)
47. 0.
48. 1.
49. 0.
50. 1.
37.
38.
8(1  a*)
X
a;*  15a;  18
•xn.
{x^  9) (x  1)
44.
0.
45.
17a;«  42x 4 39
15(x»  9)
AR
x«  4a;  22
(a;  2) (x  3) (x  5)
51. 0. ' a;(a;» + l)'
EXERCISE 66
1 ^^ r
Zacy
4a;«
3. 1.
»*
4. 
5y»**2
5.
6.
7.
8.
7
3(x 4 1)
a;(2a;  1)
ab
2a l'
x* + 2a;  3
X
aix + 1)
2xH3\
^* 3(3a;  l)
(2x + D' l
(X H 1)« ;
10.
11.
12.
13.
14.
15.
a + x
•
aJ*(o — x)
o* + a H 1
■ ■ •
a
(x + 1) (2x  1)
(2x  3) {3x f 2)
2
x* — XJ/+2/*
16. 1.
''■^
18. 1.
1
19.
20. 1.
21. 1.
a
23. X 4 y.
24. 5±^.
a — x + 1
25. 1.
o*c4a6« + 6c«
26.
a + 6 f c
27. 1.
28. •
a
x\y
29.
m* + 4n*
4f7in
XXVIU SCHOOL ALGEBRA
c
9.
1
a?'
10.
a + x.
11.
x\y
zy
1. m^. 12. _L_. 23. 2z4±^
X 2a:* — 1 2
V
2.,,^^. 13.^^. 24. « + ^^
2x41 a + l 2a + 26
3. x + 1. 14 qc + 6cab 25. J.
12.
1
2a:«l
13.
d1
a + 1
14.
oc + 6c — a6
€tc + bc jab
15.
X — ai1
x + a1
16.
0.
17
abcd + 1
xu*
a» + l
20.
2al
a
21.
aL
99
6
4. • "' ' "', \" oa (a + 6 + c)«
26.
26c
_ o — 1
6. r^ ,„ „ ^ 1.
® ^Tl^ "• ^^li 28. 2«.
^ 4^ + 2x + l . jg 3. 29. ^^*
2x aj *
a; 41 .Q o(a + 1) 30. 1.
x(x + 3)' ^* a» + l * .
31.
l+2a;
32. 14.
33. 18.
2x 34. 102.
EZEBCI8E
1. 3a + 56  4c.
2. — 5 — a — 2xfy + «;0 — a — 2x + y + 2.
3. 9a« + 6« + c« + 4d«6a66ac + 12ad + 26c  46d  4ai.
7. 8o» + 20o + 9; 19. ¥.
8. 35. 20. ^.
' 3 ' "^21. 5 + 2a3a«,
10. 2, 3. 28 ^ ■" 26a; .
12. 32, 12, lOt. (x*  4) (a:»  9)
13. 11.33a». 4g»~26g'~26a;+144
.. (a*6«)» (x»4)(a;«9)
14. — . . — •
4«V gjj 8xy'
18. t, A. ""•x'V*
ANSWERS
r
X3
31 ^(2^
^^' 2(3x 
3)
1)
•
38.
l+a;*
x(l + X*)
32. 0.
33. ^ ^^
x + 2y
•
39.
40.
1
a?
1.
34 ^
'^' x(2  x) {x
3)
■ •
1
41.
2x
■MM ai^^ «
3
35 ^ .
^^•(l^)(^
4 •
2)(x3)
42.
43.
5x.
_1
">
37. 1.
48.
5240 mi.
EXERCISE eO
1
1. 2.
17.
3.
32. A.
47.
6f.
2. 3.
18.
10.
33.5.
48.
12. '
3. 2.
19.
«.
34. 7.
49.
.5.
4. 2.
20.
ft.
35. 3.^
50.
.2..
5.  1.
21.
5.
36. 4.
51.
15,200.
6. 13.
22.
4.
37. 2.
52.
.05.
7. 5.
23.
13.
38. ¥.
53.
.05..
8. 1.
24.
tt.
39. 3.
54.
3 yr. 6 mo.
9. V.
25.
7.
40. }.
55.
122, 212.
10. 2.
26.
4.
41. 3.
56.
2a
11. 5.
27.
2.
42. 8,
57.
14ft.
12. i
28.
t.
43. t
58.
19ft.
13. 2.
2?.
J.
44. 73.
59.
36f.
14. t.
30.
tt.
45. 5.
46. 30.
60.
10.
16. 0.
31.
0.
61.
12.
•
EXERCISE 61
1. ♦.
5.
9.
9. f
13.
5.
2. 3.
6.
i.
10. 23.
14.
7.
3. H.
7.
i^
11. 8.
15.
0.
4. 12.
8.
2.
12. i.
16.
3.
17
'. ♦:
20. 183 ft.
XXX SCHOOL ALGEBRA
« J
EXERCISE 62
8. 2,  !♦. 9. 5,1*. 10. 2, 2*.
11. 1*, J^^. 12. 3,  1*.
EXERCISE 03
1. 3o. ^ a 17. 17a.
4.
5.
ah
a6
2c
36
52a
36 + 2i
2ac
ob
2. 12.
3. 2.
12 . 91 o&(q  6)
• ^* 2 21. ^,_2a6«6»
13.
a« + 6«* 22.
6 ±ZJ?.. "* "^ "^ acab{hc
' "  '0.
a.
1^ ^ 23. 0.^
w ^* 3a«  1
06 ^^55 + 6^^* 26. §(1 ^ 2a  a«).
EXERCISE 66
1. 4. . 2a + 6
4.
5.
2
5!.
P
6. 5.
10. 24.
7. 69.
11. 30.
a. 60.
12. A; i; A.
9. 247.
13. A. 14. 45
EXERCISE 66
1. 120. 3. 336. 4. 120.
6. Tin, 37A; zinc, 76A; copper, 301 A: lb. 6. 27, 28.
7 1^ ^ 1156
lOO' 20* 100 ' . .
8. Owner, $2800; other, $2000.
9. State, $4000; county, $8000; township, $6000.
10. $360, $42.0; $330, $440. ^ 11. 33, 42.
ANSWEBS
XXXl
/
12. 15,000,000,000 tons.
13. India, 234,375,000; China, 421,875,000.
14. First, $30,000; second, $22,500.
15.
16.
17.
18.
19.
21.
41.
177Jcu.ft. 22. 11.
96. 23. 30 gal.
860 million. 24. 1^ gal.
10. 26. 4J lb.
8. 27. 9ft lb.
60. 28. 1501b.
1 54A: min. past 4.
[dSih niin. past 1.
29. 100 lb.
30. 80 lb.
31. i gal.
62. zrf •
9 X
33. 5f da.
42.
34. li da.
35. 6 da.
36. 4 da.
37. 28t min.
38. 36 min.
39. 169i!Vmin.
32i^ min. past 6.
54i^ min. past 10.
43.
5A and 38A min. past 4.
21A and 54iS: min. past 7.
44.
45.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
779  da. 46. 686 da.
398 + da. 47. 15 hr.
( 1st, each 12 hr.
A, 42 mi.; B, 40 mi.
2d, each 492 hr.
A, 1722 mi.; B, 1640 mi.
8 ft.
1100 ft.
2357 + ft.
50. 6 hr.
51. 19H mi.
48. 24 hr.
49. 108 min.
63. 36 lb.
64. $20,000.
65. 22} min.
66. 5A: and 38A: min. past 10.
67. 6.
68. 29} yr.
69. 12 da.
gold, 2i lb.; silver, 18f lb.
Aluminum, 35 lb.; iron, 451b. 70. 1} gal.
Copper, 51} lb.; tin, 48f lb. 71. 6.
r 46t bu. oats. 72. 15.
1 53} bu. com. 73 gp + 2, 2p + 3.
/ $3250 at 4%. ^hc
\ $1800 at 5%. 74. ^qj^
10; 14; 6; 24. (jbc
26, 27, 28. ^^' ha
mi.
ft.
16. 6} yr.
16. 4}%.
EXERCISE 67
17. (1) 122**; (2) 32°; (3) 4892*'.
18. 442 + ''; 617°; 1995 + ''; 2804°.
20. 60 lb.
xxxil SCHOOL ALGEBRA
EXERCISE 68
1. 1, 1. 5. 2, 1. 9. i, J. 13. 15, 10.
2. 1,  1. 6. 1,  J. 10.  J, 2. 14. 3,  4.
3. i,  i. 7.  3, IJ. 11. 3,  7. 15. 10,^ IQ.
4. 2, 3. 8. i,  i. 12.. 8, 9. 16. 12, 18.
17. 7, 5. 18. 12, 21. 21. Sugar, 6^; rice, 6^.
EXERCISE . 69
2. 1, 2. 4. 3, 2. 6. 3,  }. 8.  4, 3.
3.  1,  1. 5. 2,  1. 7. 5, 4. 9. 2, 6.
EXERCISE 70
2. 3, 2. 4. 3, 2., 6. 1, 3. 8. 5, 1.
3. 3},  4}. 6. 3, 6. ' 7.  2, 1. 9. H, h
10.  6f,  3f . 11. it, Y.
\
EXERCISE jri
1. 5, 12. 5. 3, 1. ^,.^^^9' 2, 4. 13. 18, 12. ^
2. 5, 2. 6. li^T 10. .2, .6. 14. 9,  1.
3. i^. 7.  J, f. 11. .015, .01. 15. 17, 6..
4. X  1. 8. 1,  1. 12. 2,  3. 16. 2,  1.
17. 2, 3.
EXERCISE 72
1. 2a, — o.
2. ^, 2a.
2/  & a  a'
o.
a5'a'6 o6'  a'6
4. m + n, m — n.
^ 26fl a2
5. ; >
a
6. a + 26, 2a 6.
an — frw an — hm ' b — d h — d
f. a h
o. r' •
6 a
9. 1. 1.
c d
10. w — w, n + m.
11 Q 2a + l
11.3, ^ .
12. a,  6. 13. a + 6, a
6.
m^sgmm
ANSWERS XXXIU
15. a, &. 17 « + 26 a2h ^
1«. 9 9
1A 1 1 2 ^
of 6 + c' + 6 + C 18. a, 6. 19. h,a.
20. a + 1, 6  1.
EXERCISE 73
1. 1,2,3. 4.2,2,2. 7.  3, 3i, 2.
2. 2,3,4. 5. li, U, IJ. 3 2, 3, 1,4.
_3. 3, 4, 7. 6. 2, 31,  4. . 9. 12, 18,  24
10. aH 6, a 6, 2a. :^ 11.6,40,20. "
12.:r=a + 6+c. 14. ^^3a^.
2=a + 6c. . 3/ « 2af 3b
13. a; = a  6 + 1. ^
2 « o + 6  1. 6
b + c f d  q a + h + dc ^
15. a; = J 2 4
y ™ 4 4
EXERCISE 74
1. r, 1. 6. J, i 8 i, 1.
2.*, J. 6 _2n_ __2n_. * m n
3. i,^. ''• If^" 1^* 9. ?, ^.
4. i,i. 7. a, a. «^ "* « 1.
iA 1 1 ^^ 26c 2ac 2ob
10. 1, 1. 15. — , r » rj'
11 I _ J b\c a + c a + h
12! 1', ^ i, i 16 25L., .JL., _^.
13. 2, i,l. i + ^ ^ + ^ '^"^^
14. }, J, I. 17. i,  1, 1.
EXERCISE 76
1. 9, 14. 2. 9, 12. 3. 2, 8.
4. Flour, 3^; sugar, 5^. 6. 57 pear trees; 43 apple trees.
6. Man, $3; boy, $2. 7. Silk, $1.80: *a*«. $1^
XXxiv SCHOOL ALGEBRA
9. Iron, 480 lb.; lead, 700 lb. 13. $4600; $5400.
10. First, 3 points; second, 1 point. 14. 67 and 172f.
11. First, 5 points; second, 3 l^' 50 — 3.
points; third, 1 point. 17. Wheat, $7; potatoes. $50.
18. Wheat, $8; com, $20; potatoes, $24.
19. !Flrstclass, $1.52; secondclass, $1.36.
20. Fourthclass, $1.02; fifthclass, $.81; sixthclass, $.70.
21. Com, 2,772,000,000 bu.; wheat, 737,000,000 bu.; oats, 1,007,
000,000 bu.
22. Copper, 550 lb.; iron, 480 lb.; aluminum, 156 lb.
23. .2875 in., .5025 in., .3625 in.
24. Eififel Tower, 984 ft.; M. L. B'ld'g, 700 ft.; Wash. Men., 555 ft
25. Nitrogen, 15ff; potash, 5^; phosphate, 5^.
26. Oats^ 46f bu.; com, 53i bu.
27. 20 lb. of 20^ coffee; 40 lb. of 32^ coffee.
29. 40 lb. of 75^ tea; 60 lb. of 50^ tea.
30. Cream, llf gal.; milk, 8^ gal.
31. $3250 at 4%; $1800 at 5%.
32. $2000 at 5%; $10,000 at 4%.
34. /' X 5". '*4. ^
35. 15' X 6'. 45. V, I.
36. 12 boys; $60. 46, i.
37. 90 mi. 47. 16, 81.
39. 13 played, 8 won. 4?. 21, 79.*
40. 68 cases, 50 successful. 49. 14, 54.
41. i. 50. 32, IS,
42. A. . 61. 5thr.; 17hrJ
52. A, in 24 da.; B, in 48 da.
53, A, 14A^ da.; B, ISA da.; C, 34* da.
57. 49. 58. 23. 5Q, 64. 60. 151.
61. Oarsman, 4 mi.; stream, 2 mi. per hr.
62. Oarsman, 5} mi.; stream, 1} mi. per hr.
63. Oarsman, 6 mi.; stream, 1} mi. per hr.
64. Cast iron, 450 lb.; wrought irbn, 480 lb.
65. From earth, 93»000,000 mi.; from Mars, 141,000,000 mi
ANSWERS XXXV
66. Tea, 50ff; coffee, 30ff.
67. 24 bu. from 1st; 16 bu. from 2d.
68. A, 170; B, $llO.
69. 80 lb. of 25^ spice; 120 lb. of 50^ spice.
7t). 480 mi. 71. i:
7o. = f 7Z — •
p_g qp
EXERCISE 77
11. (1) 5; (2) V45; (3) 13; (4) V34.
12. 25 sq. spaces. . 13. 42 sq. spaces. 14. 17ifiq.^ace8
EXERCISE 79
1. 2, 1. 3.  4,  2. 6.  4,  1. 7.  f, 0.
2. 1,  1. 4. 0, 0. 6. 2, 3. 8.  f.
9. f. 11. 2.9+, 3.3.
EXERCISE 80
1. 60mi.; 1:30 p.m. 6. 3:7.
3. 251f mi.; 5:17p. m. 8. 2:1. • \
5. 17: 10. 10. At end of 6 hr. 20 mi. from P.
4. A. ^
5. A.
6. ^ + ^
7.
8.
x(x]
l)»'
3(4a;
15)
5{2x
■3)
a^ + 1
a^1
23. a*
6».
95^
<T/Z
a — c
EXERCISE 81
10. 2.
16. 7, 10.
1/. ,» , , •
11. f
18. 3, 2, 1.
12. 3.
19. f , t.
13. 4.
^" a'6a6'' ab'a'b
14. i f.
15. h h
22. a1.
27. 11, 9, 18.
^28. 9y(4x« + 2x2/ + y»).
30. c — a{bj —a — h — c.
xxyvi
SCHOOL ALGEBRA
32. 7«.
33. 64.
34. 6061.
35. 2, 1.
36. 4, 3.
42. 174 
1. x<l,
2. x>3i.
3. x> —
a
4. X > 2.
5. a;>6.
6. x<i.
F.; 79  ^ C.
48. 8
49. 8
50. 8
2M'
^[2Z  (n
1)4
EXERCISE
'• ^^2a3b
8. x<6.
9. x>6and<7.
10. X < 2 and > 1 J.
11. X > 251 and < 301.
12. X > 59} and < 66 J.
af
n
r1
13. x>8§, y<3}.
14. x>llA,y<6tt,
15. Either 17 or 18.
16. 13.
26. 18.
28. 33; 8.
1. 49o*b*.
2. 25xV.
3. tx*i/".
^ 36a«6«
^ "isr*
6. 169x«"j/»
lOOz*
1
64c«d«*
10. 27x»j/».
9.
11. 8x«.
12. ixV.
13. 125x>V.
125c«d"
^^" 343x»'» *
16. 343a»6«c».
17. 121a»o68.
32x^Vo
20. ^^*7^\
21. 128x«i.
22. Aw»*.
23. ^^x^.
24. .0009.
27. .000000027*
28 ^^^.
81a*»*
29. 1^.
48. i2»; 2».
51. AV.
52.
53.
54.
55.
56.
57.
2
1
0'
1
 2
f
i
f
f
i
¥
2
2
1
 3
1
10
6
5
 2
2
2
. 30
24
19
7
V
i
f
«
f
"V
6
2
2
1
 5
12
7
10
6
11
18
20
14
30
24
35
47 1
ANSWERS XXXVll
BZEBCISE 84
r
»
1. a» + 5a*x f 10o»a^ f 10a««" + 5ox* + a!».
3. b» h 76«^ 4 21bV + 35&V + 356V + 216V + 762^ + y».
4. 6»  66»y + 166V  206>2/» + 166V  66y» +1^.
6. a« + Sa^y +28aV + 66aV +70aV + 56oV + 28aV +8ay» +y».
7. 32a» + 80a*6 + 80o»6« f 40a*6» + 10a6* + 6«.
8. 32a» + 40o*6 + 20a»6» + 6a«6« + f a6* + A6».
9. 1  fix + 16x»  20a:» + 16x*  ex» +a:«.
10. 32c»  80c*d» + 80c»d*  40c*d» + 10cd«  di».
11. 343  441«» + 189x*  27x«.
12. 243  ^c« + HV  Vc» + Hc»  Ac".
13. 64aj>«  192x" + 240a;»  160a:« + 60a:*  12a;« + 1.
16. 243a" + 136a« + 30a» + ¥a* + JVa« + ifir.
17. 266x" + 612a;" + 448a;" + 224a;" % 70a;» + 14x«+ ht^+h^+jh
18. 128H*ic + *Ha;»Wa5' + Wa:*!a5* + VA«»irATa;'.
19. a;* + 3a*  6a;* + 3a;  1.
20. a;«  9a;» + 24a;*  9a;«  24x*  9a;  1.
21. a« + 4a7c + lOaV + 16aV + 19a*c* + 16aV + lOoV + 4ac'+c^,
22. a^  Zjhf h 3a;»« + 3aV  ^xyz + 3aa«  y» + Z^z  Zysfl+ifi.
23. 8a;^  12a* j 42a;*  37a;» 4 63a;»  27a; h27.
24. H4a; + 2aJ»8a*6a;*f 8a* + 2a*4a;»+a;».
EXEBCI8X 86
1. 3a;2^.
8. 3a*.
14. 2L
2. 5a».
9. 5ys^,
• ^^' j*it
3. 12y\
10. }o6«.
16. 8a;«.
4. 4xy.
, 3a*a*
16. 2y».
. 6a*
^ 7?
6. ixi/^.
4y*»
17. 2a*2/
13 2^^.
18. 2a»a;'».
19. is^.
20. fa*.
22. JaJ*»y.
31. 75.
32. 36.
33. 216.
34. 42.
36. 676.
36. 1260.
XXXVm SCHOOL ALGEBRA
EXERCISE 86
1. ir«2x + L 8. 2n»3n»+4n5.
2. 1  a  a». 9. 3rr» + 4a:«V  4a:2^  32/».
3. 3x»  2a; + 1. 10. a; + w + 3.
4. 5f3x + x«. II. l+x + 2x*a*,
5. n»  2n» + 3. 12. 7x»  3«»  4a;  2.
6. 2a;" + 3a;*  2x  3. 13. ix  5.
^*' 2 3 6 3^4 ^^•^'"x ^""
15.2, 21.a.+ffL 26.a+?^^4.
16. a;»+a:J. oo ^ . 1 . ^ "*
17. Ja* Ja + 6. ^^ 2 +2^ + 0* ^ <>n^^
18. +3H. 23. ^+
27 2ayi65+
a; a 2 x a 3 9
19. a*fa; + f. 24. lj2x2a;»+... . ^' ^'^2x''^'^"
31. 9,  3. 33.  3,  5. 35. },  i. 37. 4,  2.
32. 3,  7. 34.  3,  5. 36. },  }. 38.  1,  9.
BZEBCISE 87
1. 85. 10. 3.2105. 19. 2.5819 +. 28. 1.4342 +.
2. 51. 11. .17071. 20. 1.2747 +. 29. .4532>.
3. 325. 12. 1230.321. 21. .3415+. 30. .8914+:
4. 427. 13. 2.6457+. 22. .2213+. 31. 2.0369 +.
5. 581. 14.. 3.3166 +. 23. 1.0031 +: 32. 7.071 + in.
6. 753. 15. 3.5355 +. 24. 6.0075 +. 33. 3.5355 + in.
7. 6012. 16. 1.8257 +. 25. 1.9318 +. 34. 250.3  yd.
8. 90.08. 17. 1.4529 +. 26. 1.3687 +. 35. 756.
9. 14.114. 18. .9486+. 27. 2.7262+. 37. ^91+.
38. V^or5.3851+; V73 or 8.5440+; V34 or 5.8309 +.
39. 515.5 + mi. 41. 456.9 + mi.
42. 14,896,509 + sq. ft.; side of square, 3859 + ft.
44. 173 + sheep on one side.
45. 10.3 + ft. 47. 188.9 + ft.
46. 15.63 + ft. 48. 582.7 + ft.
wm
ANSWERS XXXIX
tt
1. vs:
2. </?.
18 ^^*
4a'c*
3. 2y/a.
j^ 281*3*
4. 2a»V^.
32f.2*
7. aV^.
20. 9.
8. V9»V^.
21. 125.
9. J.
22. 8.
10. ci
23. 16.
11. 2x*.
24. 128.
14. 2x2/1.
25. 4.
16. ojM.
26. 81.
27. 64.
39. flHajt.
28. 36.
40. 2x*.
29. 27.
41. 256.
30. if.
31. W.
42. >,
32. a*.
43. y*P.
33. 2a*.
44. a**.
34. a*x«.
45. a^.
46. a^.
35. 4a^.
36. a^\
47. a!^«+«.
^o 2a»a:«
37. 7a*.
48.
c
38. y/^.
49. At.
89
1. a<r«.
16.
Tib*
43.
10a?
63. 1. \
2. a6»ci<i«.
17.
27.
a*
^. 7a"
3. 3X2ia<r^.
18.
t
44.
2a;.
54. ^•
3a;t
4. 2z'iy*.
19.
20.
21.
i.
216.
45.
46.
6a
xi
2.
55.^
22.
23.
47.
2
66. ^,.
9
10. ^ .
24.
25.
A.
A?ir.
48.
7a»
5x^y
1
57. a: *• .
68. {\)\
18a»
26.
1.
49
11. i.
27.
i.
3?y
59. 3i.
12. i.
28.
3 X 10».
50.
«v.
60. 2JHi
14. 25.
31.
1 xio«.
a*
61. 51.
15. i
32.
16 X 10«.
52.
3
62. 38.
rim—
3d
SCHOOL ALGEBRA
EXERCISE 90
1.
!»•.
2.
a^.
4.
1.
9.
0?
10.
16.
11.
*.
12.
1.
13.
X*
26*
14.
1.
15.
■ ■■ ■
8
16.
27a;<
17.
4a«
a?
19.
243 '
20. a*"*?.
21. oA.
22. o*.
23 3^
24. oT^.
25.
26.
27.
28.
27
126xi
4y»
9a;*
2/^
18.
25y
29.^
30.
2c
31. 1.
32.^
xy
33.
b»
a*xV
34! A.
35. rc«"»"+^
36. \^.
37. x».
38. 27.
40. aiftc*.
41.
tf
Vb
42.
1
43.
bd
44.
xyz.
45.
49.
4.
50.
9.
51.
i.
52.
i.
53.
32.
M.
1.
65.
1
56.
1
(3)
67.
81.
2. a + 1.
3. 9a; f 5a;V + ^V
EXERCISE 91
5. or* + a16 + 6».
6. x»2r* + 3 + 4arV.
4. 4x*  1 + lix"*. 7. 4a;*y  7x" V^ + 3x"*ir*.
8. 6x«7x*19x*+5a; + 9x*2a;*
9. 2  4a''*x* + 2a"*x».
10. 5  3x "V + 12x" V + 4x'y"*.
11. 5x'  3x* + 1.
12. 4x»  3ir*  2xy*. 16. a*  a*6* + b*.
13. x"*  2x"* + 3x"*  1. 16. 9a H 6a*y"*  3ir*.
14. X* + xV + y*.
ANSWERS xli
17. oj"*  ir* + a:*jr*. 24. x"* + 4x"*  1.
18. xV^. 3a:*y"* H2 4a;'V 25. 3a;"*  5ar»y  ?«"*»«,
19. 3a"*  2a"*x* + 4a%  x*. 26. 6a*6"*  1  6a"*6*.
20. 2o*  3a"*  a"*. 27. 3ar«  ^x'^y^ + y.
21. X*  2xV. 28. 3x*  4xy"*  2x"*y"'.
23. a"*  26* + 3o*&. 29. Jx"*y  Ix"* + 3ir*.
8. 1.
11.
1
hi
14.
1.
15.
aP.
16.
a*^K
EXERCISE 92
1.  2», 13J. 7. af\
2. First, 32 times.
3* Latter, 2" times.
5. 6a« + 2x*  3a«x. ^ ^'
6. x»**». 10. 8.
19. (x*  y*) (x* + x^y^ f y*).
20. a* + o*6* + a6 + a*&* + o*6* + &*^
21. 3i, . 29. 2». 2«. 2'+i. 30. 2^«.
31. x«  12x* + 48x"*  64x*.
32. x»  8x^ + 24x*  32x* + 16x*. 33. 27} sec.
34. 66t lb. of 18f& coffee; 33} lb. of 30^ coffee.
«
EXERCISE 9ft
1. 2V3. 6. 4tV2, 11.  IOV3. 16. 4a«x\/a.
2. SVS, 7. 2V?. 12. 2V'3. 17. 2a«xV2a.
3.  2V6. 8. 2V^. 13. 2a'i^Sa^, 18. 10a»V2a.
4.  6V7. 9. 3V^. 14. 56«V^. 19. 7xV V3x.
5. 2V5. 10. V^. 15. 3Vila. 20. OxVVTx!
21.  3ax\/35. 22. a(xr)Vx"^. 23. 7x(aHl)«Vx(a fl).
^
Xlii SCHOOL ALGEBRA
24. ^V3^. ^ V^ 33. ^VgS. 38. ^V^SJ.
39. tV^.
26. ^VtS. ^* *^' 34. JVS.
*>A i!/A 3L ViS. 35. V^.
20, tV6. 4a; g .^ 4ac^/7^i:
Oft o^v/^ 40. ^— Vl56cy2.
27. VIO. 3c . ^
28. jVaiO. 2a« 37. V6x». ^^ 2\^^^
42. 5(^^y) Vl5(x«/«). 44. a</a{a + h).
43. (j^V3^Tir. 45. ^Vi5^.
46. 3.4641 +, 5.19615+, 8.66025+, 15.58845+.
1. Vi2. 7. V^ io '/3i^ 16. Va«  4y«.
•
3. V432.
13. V 4x(3a;y) . 17. y/^Z^.
4.  V20. 10. Vf. 8
6. Vl6. 11. Vf. V 2 18.  Vx  1.
EXERCISE 97
1. Va. 5. V3a. 9. y/^aa^, 12. 6V}.
2 VS 6. VlOax*. ^^ ,
• *_ ^ 8, 10. V3^^. 13. Va  26.
3. Vo^. 7. V2a6V.
4. V7. a V^. 11 yf^' ' 14. V2(a26).
EXERCISE 98
1. V'^; VI2I. 2. V^; V^. 3. \/27; V^. 4. vT; V^.
5. VlOOOOOO, V^390625. 9. V^Sio*, V^, V^5?.
6. V^2i6, V^OO. 10. V(a;fy)«, V^(x  y)«.
7. V^; V^; 5^125. 11. V^, V^».
8. Va«, Va«, V^. 12. V^, Vc«*, V<aHf,
/
1
\
ANSWERS
xliii
13. Vs. 16. 3v^.
14. Vl6. 16. V^.
21. V3.
19. \^3.
17. 2\/§.
18. V^. 20. \/A.
22. 2V6.
1. 6V2.
2. V2.
3. IIV3.
4. 6V5.
5. 2V5.
6. 3V^.
7. 2V^.
8. 0.
9. tV6.
10. 5\/3.
11. 2\/ax.
12. }V^.
13. AV'e.
14. 4V6.
15. 2\/2c.
16. 0.
EXERCISE 99
17. 5acV6.
18. 66V^.
19. 8V2.
20.  I9V3.
21. 2V6.
22. 7VI5.
23. 0.
25. 5V7.
26.  5V3.
'27. 12y6+ 10 V5.
28. abVZq.
29. 12\/221\/3.
30. V^\/2 }V5.
31 (5a + 6)Vx.
32. 5ix + 2y) Va.
24. 6V36V^. 34. 5.6668 +. '
1. I5V3. 4. 7V5.
2. 12V^. 5. A.
3. 30\/2T. 6. iVs.
13. 2V6  4V3 + 8V6.
14. 48  I8V2 + I2V3.
15. t Vg  f Vis + 1 V21.
16. 2  4V^.
17. 62V6.
18. 2VI1.
19. Va«  X,
20. 7V6  12.
21. 2Vi5  6.
22. 28272V10.
23. 30 V6 + 54V5  34.
Vtj
10. V5466.
11. 1.'
12.
f
EXERCISE 100
7. \/72.
8. a:\/864^.
9. 3V^.
25. 96  I6V3.
26. xVE + V3a:*  3a;.
27. V3x2 4. 3x + a: + 1.
28. 2Va:*l  fix  6.
29. g.
30. aVax + x + aVa.
31. 25  7a;.
32. 2a;.
33. 25V3.
34. 36a;*  50a;  100.
35. 14,147 ft.
36. 1093.6 + yd.
2y
xliv
SCHOOL ALGEBRA
1. 3. tf ob
2. 2V2. ^
3. 2V2. ® *•
4. 1. 7.. V6.
16. V2T  6 ViS.
17. 2V7+4V66V6.
101
8. V2,
9. ttVlO.
10. iV^.
11. 3.
12. Vj.
13. V^.
14. \/26. .
15. 5V7I4.
18. 3V^+ 6V^ + 7V^.
19. fVTs" i\/5 + iy/m.
1. §\^.
2. iVo.
2V5
20. Va: + y.
EXERCISE 102
2\/3r V2
3.
4.
5.
14.
15.
16.
15
V62V3
^ II III ■ III ■ ■»■■■■■ «
6
2V7 + V35
14
^ f Vq6  126
4a  96
a; h VxTT  5
6a  6 + 5 V2?
6.
7.
8.
9.
■ ■ — — ■■■ I III •
6
\/l2\/l8
10. 13 + 7V3.
13\/630
6
I8 + V2
11.
11 6V^
18.
12.
13.
23
V5 + V3O
5_V305V6 46V5
19.
2V3V^
— a
14a 9
17. V3 + V2.
22. 2.12132+,
2$. 1.05409+.
24. 4.53556+.
2. X*.
3. a*«».
21.
25. .057735+.
26. .709929+.
27. 9.00996+.
20. Vx*  1  a^.
Va6 + 6»  Va6
4. 6.
5. V5?.
6. a;".
103
7. 8x».
8. a.
9. 9a?*.
28. .10104+.
29. 2.63224+.
30. .62034+.
10. 4flVS".
11. aV.
12. Vl^^
ANSWERS • xlv
14. 16, 27, 81, ley^, 64, 243, ^fiflVs, 243.
15. 49+20V6. 17 oQRQi 1971 V6
16. 89\/3109V2. / 2
EXERCISE 104
1. 2V^ 3. 9. 2Vl5 3 Vs. 17. 2  V3.
2. V^ + 2V6. 10. iV5 + V6. 18. 3 V^.
3. 3V3  2V2T 11. i\/3  fV6. ^^* ^ "^ ^'^^ .
, r r r 20. 1 + V2.
4.V^V3. I2.i + V3. 21.V7V^.
5. Vl4 + 2 V7. 13. 2  J V3. ^ 3 _^ 3 VsT
6. 3V52V7.' 14. Wm^n\^m^n, 23. 3v^\/5.
7. 2V5 + Ve: 16. a + 3Var+T. 24. 2V3  V6.
8. 3\/5  4V2. 16. V3  1. 26. 2 4 V3.
. EXERCISE 195
L 8. 6.  1. 11. i. 16. 6. 21. 1.
2. 2. 7. V. 12. f 17. 1. 22. 8.
3. 2. 8. 1. 13. ^. }J 5; 23. Vi
4. 8. 9. 9. 14. V. ' (flb)« 24. i.
6. 14. 10. 64.. 15. 18. ^' 2ah 26. 64. /
12. J.
13. ^.
14. J^.
15. 18.
29. J.
30. i.
^^ 16*
26. ^^'. 29. i. 33 ^.
4 ' on 1 O
1 « ^ 20a«
28. 16a. 32. a*. 36. 5.
EXERCISE 106
1. 4*. 2. 0*. 3. 2. 4. 9*. 5. 9.
6. 4*. 7. 7. 8. y*. 9. V.
xlvi • SCHOOL ALGEBRA
EXERCISE 107
1. Vs. 10. 4 + VIE. 20. vr^.
2. V3. 11. 2.88675 +:. 21. i(x  V5^^36).
„ 1 /— 12. .218286 4. ,
'iS^ 13.2V7. S'""^^
4.2+4V3. j,;^ 2^6
6.  2V^.^ jg g^ , 24. V.
6. iv^; ivloB. . 4^' ^ 25.25.
7. 3V^+ V3
16. VIT. 26. f.
'aVk+ViO. 17. 5+2V^. 27. o»x».
8. «  V^^n;. 18. 3 V7 + 2 VlT. 28. lex"  Sxy + »•.
2V3V2 , 19. ^. 29. ^.
33. ^Z^. iy/2. iV^. iV^. V5. y^ 2+%^.
34.  V2. A(3 + V2). 36. 40(3+V3)
35. 2\/3  2V2. 37. 21.5439 +.
41. (xfv^) (ajv^); {x+VE){xVE); (VS+v^) (V^v^);
(VJ + V5)(V^V5); (a;HV3)(x + lV3).
x> + V2.42 ^ ^ 43. ii. 46.ai6
*^ ^4 V2 ' 45. 4, 1.
47. .1732 +. 5.196 +. .5196 +^_
48. ix\a + y/2ax) {x + a \/2ax).
49. (a;+l+\^)(a: + lV2x); (a;» + a« + (M;V^) (a^ + a^oxV^).
51. 3v^ ^ W2. 56. a6\/5*S».
53.  2\^. 55. ?±>^. 57. VT + 2V^.
EXERCISE 109
1. 5V^^. 4 0. 8.  VJ.
2. 6 V^. 5. a V^^  6. 9. 10.
X BV^^ J6.  .(a+3»V^. Jia 7V=T[.
ANSWERS xlvil
it  6V&. 13.  lOVlO. "^ 16. a(a  1)« V^=T.
12. 28V6. 14. (y  x)V^, 16. 3 + V2.
17.  6  &VS. 1». 24.
19. 18 V^  2V^ +dV2  2V3.
20.  2  2 V3. 35. 1  V*^.
21. 3V^+V^V62. 0151^4.2^16^:^1
22. a? 4a; + 7. "^^^ a^+¥
23. &»a». 24f7VlO
24. a;»  2x + 2. "*'' 2
25. a;»  ai + 1. i^aV^n! .
26.  1. ^ . 5 *
27. V^^Ts. 164^:^:
28. .V6. 13
29. 3V5. V70+3 V^^Oi
30. 4aV^=n:. ' 14 *
31.  V6 + 2V5 + 5V::^. 41. 3  V^,
32. l+2V6a»V^. '42. Vs  V^.
34. V^ 43 2V3 + 3V::^.
^* 11 44. 2V:=^  SV^.
14V^^ 45. 43V::^.
'*^* 7 46. 5V:=T + 4V2.
47. (a*i6); (a»*i6«); (a;=fcV:^); (aa;=bi6); (x=fci); (a:*+i)(x«i).
48. X > 2; X > V2.
49. V^^; 1;  1; V^^; V^^; V^^T;  1.
50. 0. 53. 7i  1. 56. 2  2V^=T.
51. 4  6i. 54. QV^, 57. iV^ + 4;
62.3. 55. V^nrll. iy/^^l
16. :fcl.
17. *14.
18. 6.
22. 6.
23. 8rd.X32rd.
24. 6iti.X24r(L
EXERCISE 111
1. *4.
7.
=fc5.
12. «fc3a.
2. ;*:2.
3. *}.
8.
9.
=«=1.
13. . .
4. *2.
5. =fc.jV2'.
6. *f.
10.
11.
=fc a.
" a
11 ±26
14. ±
a
15. * (a + 6).
xlviii SCHOOL ALGEBRA
IZERCI8E 112
1. 2,  12. 11. 1,.  f . 2L * 1. ,5^VZ7
2.2,10. U2. J, f. Jij. 5, V. 4
3.6,1. vJl3.2,6. 23 3_, ^^ l=^2V3i
4. 3,8. ^14. 3,1. 24 \^ ^
6. 1,  }. 15. 3,  . ^' " ^* 31.  1,  3.
6. 2,  f 16.' 3,  1. X 25. J, . 32 ^t V^T.
7.  1, i. 17. 4, ^ f. 26. 5,  f. 33, 6, 1.*
8. 2, V. 18. 1, V. 27. I*y2^34. 4,3*.
9.1,1. 19.2, J. 5*Vi3 35.4,1.
10. 5, ^ }. 20. 1,  V. 6 ' 36. 12,  5*.
EZSBCISE 118
1.2a, 6a. ,^6 36 ,„ 62 6 + 2
• lU. ?r, —IT* ly. — 7Z — I — ^ — ;
2.36,76. 2a' 2a 2 ' 2
20.
3. 2c, 5c. 11 o, 1
4. a6, 6a6. ^ Sb _36^ ""'a+6' a + 6
.36 46 • 2a' a 21 i 1
^* T' "T* 13. 3a,  2. ' a6»' a«6
'*• 3 o 3a c ' aH6
^•a'2S ^^a'r 23.  &, "^.^^
8 ^ P.' ^^ «' i 24. 1, «
c' 7c a ' a+c
1 _6, 17. a*, a 4 6. g
a' a 18. a + 1, a1. ^5. a, ^^y
„« 6+V6»4ac  6  V6»  4ac
2a 2a
EXERCISE 114
1.  1,  7. 5. i,  i. 8. 2,  1 * V^^^
2. 12, 7. 6 A _i ^ *^' *^
3. V, f. • 3a' a* 10, 1, *1, d
4. 3, i 7. *2, =fc2V^=n[, 11. J, J, f.
ANSWERS xlix
12.  1, f . 13. 2,  f . 14. * 2, 7, f.
.  3W^=15 16. 1,3, 4.
^^'^' 6 17. 1, *iV5.
18. 1, i  3.
19. ^l/'^^^ 1V^
f—
^^V£3. 2036 20 + 36
21. 2,  1 * V^^. 30. a  6,  a  26.
22. 4,  2 * 2V^^. ^^ <» + 26, a  26.
24.C, d. 32. ^,« + 6.
25. ±, 5^. 33. a;»  fix + 6 = 0.
^ ^ 36. aj(x  2) = 0.
26. a,a + &^ 37. a^  9a:* + 26x  24  0.
27. 6, 6  1. gg a 6
ac6c * « + &' oH^
^' ' o6ac' 40. 0, =*=4.
IXERCISE 116
1. »1, »4. 2. =1=1, db. 3. l,t, "^^2^^^ > 1=*=V=:3
4. 16, A. 7. 1, A. 10. 1, W. 13. 1, ( })f.
6. 1, f 8. 27,  i. 11. 1, ^. 14. }, yH.
6. 8,  JV. , 9. 1, A. 12. 8,  tIt. 15. 4,  6.
16. =i= 3, =i= JV2. 25. 2,  3, i,  ^
18. 2, *m. ''• '' *' 3
19. 14, ^ 1*. 27 3W5^.
20. 1, «« . , "^ ,
21. 2, 6,  2 * 2V^^. 28. a + V^, a + Vl^
22 2 4 3^V^. 29. 1, *.
" 30. =t JV6, ± 8V^^.
no 1 « "" 3 =*= 3 V5 8 ,,
23. 1,  f, J • 31. _ ^, jvTs.
OA 1 _ B 7*3V2i 32. },  i, 1 * V5.
"^^ ^' *' 2 ■ 33. * 1, * AVliO.
1
SCHOOL ALGEBRA
EXERCISE 116
1. 3, 2. 3. 3*, 12.
2. 1,  i. 4. 6, t*.
9. 2a«, 5.
0» Oj ^fg •
6. 9,  J*.
7. 3,  V*.
8. 4,  f .
11. 6,  A.
12. 0, 5.
10. 2, 1, ^*y~^ . i±v^.
14. 3,
15. f,
16. i,
13*.
¥•
2
17.
18. *a,
19. 2, {.
f *i
2o'.
20. 4.
1. 1,  6
2. 1,  f
3. J,  {
4. i  i
5.*, I
6. * J, ^
EXERCISK 117
_ 2a a
4a a
9. =*= 3, =*= 2V^
10. =t 2, =t i.
11 t1t> — V"'
12. V,  1.
13. a, ^'
14.
16.
lo
a
, 1.
6 — o a f&
EXERCISE 119
1. J,  i 4. 0, ^ 4.
2. d= 2, * JV^. 6. 9, V,
3. 1, 16. 6. 3,2. ^
11. A^, ¥.
^ „ 3=t3V^^
12. 0, 3, 2
13. 1 Tft.
14. V,  3.
15. 1,  A*.
16. 1,3, 2*lV^:n[0.
17. 15,  M.
18. 4, I. ^
„ 6*V^^23 •
19. 1,3, J
20. 2, 648.
7. 6* 3.
8. a, — "
' a
9 ^ ?«
^' a* b
21. 3,  1,
22. 2, f.
23. i±^, i=i.
a
24. 16, 1.
25. — o, — b.
10. 3, J.
~3«fcV43
26. /^
2a
27. 1,  1,  5,  7.
28. 12,  16*.
ANSWERS
u
29. V3.
30. *4, *
31. (^^
2V22
3
2a
32. a  3, a + 2.
33.  1 + \/2,  3  2 V2.
v^ ^^ "" 2 r 2ir"^ 4
4ac)« 2 r ^T 4
^ ' 35. 17. 36. 29.
«
EXERCISE 120
1. 6, 6.
13.' 20 rd, 26. 45 mi. per hr.
2:^7,8:
14. i rd. 27. 6, 9.
3. 2, 5.
15. 10 rd. ♦ 28. 6, 15.
4. 5.
16. 36 in. ^29. 7, 11.
5. 8,
17. $140. ^^^ 80. 8, 9.
6. 6, 7, 8.
18. 3. 31. 3 in.
7. Length;
,x + 5;
19. 2. 32. 20 rd.
area,
a^ + 5x
20. i, 33. 40 hr., 60 hr.
8. 4 yd. X 9 yd.
21. 30 niin., 45 min. 34. 24 in.
9. 18 ft.
22. 4 hr. 10 min. 35. 5 mi. per hr.
10. 8 rd. X 10 rd.
23. 24. 36. 6' X 12'.
11. 4.14 + rd.
24. 32. . 37. 24' X 12'.
12. 20%.
39. 
40. 
25. 5 mi. per hr. 38. 9 mi. per hr.
•
i * iV2a  1, i =fc i V2a  1.
hi hi
43. 2ae + €* when e is + ; 2ae — f^ when c is — .
EXERCISE 121
1. 7.906 .
i. 7.0711 .
3. 5.88 .
4. 10, 2i.
5. 117.7 +ft.
6. 3.03 +.
7. 4/
8. Mv =*=\/v*64«).
9. A[» * V»«  64 (a  A)].
.0. l^fT?.
11. 7.906— sec.
12. 3.4— sec.; 2 min. 26.6+ sec.
13. 144 ft.
14. 7.14 f sec.
15. 353.556 + ft.
16. 335 ft.
lii
SCHOOL ALGEBRA
EXERCISE 122
1 l^'^^
^' \ 2, 16.
. • 10,3.
1 2,  28.
*• 11,16.
4,i.
I 2, ft.
1 1, A
3, V.
• ll,tt.
, f3,7.
• li4.
9.
10.
II.
12.
17 I'*'*
1 1,2.
U, 1.
I ±6.
1*4.
1 1, 10.
lf, I
l,V.
l3,V.
18 P'^
**• 12,3.
14.
IS.
16.
U V
f=6.
1*2.
f 1, 11.
16,1.
■ZEBCI8E 133
I 14,* 14.
^' 1*1, =F 4.
3.
4.
5.
*fV3.
^ jV3.
1=^1, d^jV^.
I =*= 5, :*= 6V2.
1 =*= 2, :f 7 V2.
ab 1 sfe »
' V91
27
6.
7.
* 3, =*« f .
* 5, =*= V.
(
I * 8, * 3.
1 =F 5, * 5.
*6,
"• 15: 1:
{
8.
3,
3,
VSI
9.
10.
f1,
(2,
13
8
V5i'
2.
t.
3V2.
12.
13.
14.
15.
16.
3,
2,
fVii
13, V.
18, V.
{3,2.
12, 3.
1
±3.
*3.
* 5, =t iVs.
* 2, ^ J^V5
21. 52.
KZESCISE 1S4
2
3
f9,4.
• l4,».
4,3.
• 13,4.
f 3, 7.
• 17,3.
f4, 5.
• 15,4.
5;
6.
7.
8.
f "1.7, *3.
1 *3, *7.
I *i *!•
1 *i *i
ft, J
lt,t.
1 2,1.
11,2.
til «•
1 : 1; : i 
10.
11.
a+1, a1.
a1,0+1.
2,3.
3,2.
6,2.
12,6.
ANSWERS
liii
a.
17.
18.
19.
20.
21.
I2,i.
*, f.
it.
^ ♦, =*= f
fit.
lt, 
5,3.
3, 5.
22.
23.
24.
25.
26.
o— 26, —2a— 6.
2a + 6, 26  a.
=t 1 =tV2.
=fcl=FV2.
3,  10.
12, 15.
' ± 2, =t f V30.
=tl,:^AV30.
[2,1.
1 1,2.
27.
28.
29.
*6.
«t 3.
[2,1.
11,2.
[5,2.
13,6.
30. j^
[a.
31. 8, 6.
EXERCISE 125
. J3, 1, 2db3V^.
11,3,
2=f3V^=T.
, f3, 1, l=fc V10.
" 11,3, 1=fV^=10.
1 =b3V^^
3.
4.
3, 2,
2,3,
1=f3V^
, ^ 5=fcV159
1, 4, s ;
4,1,
5±V159
5.
6.
7.
8.
9.
10.
1, 1 =fc V^^.
.  1,  1 =t v:r2.
3, 2, f =t i V151 .
1, f , { :^ i v^=n[5T.
 46, 2.
1 15, 3.
±2, =*= V3.
±1,0.
[4, ».
f ^ 3, =t 1.
1 =F 1, =F 3.
EXERCISE 126
1
2
3
4
• I
■1
■ {
•1
1, 3, H9 =*= V69) *.
3,1,§(9^ V69)*.
1, 6,4, 1.
~~ 3, i, — J, — 2.
4, l,i(5=*=V^=^)*.
1, 4, J (5=F V^ni)*.
4, ¥*.
l,t*.
5.
6.
7.
5,  1, i (9 =*= Vioi);;'.
,1, 5, §(9=tVl01)*
3, 4, =fc2V^.
.  4, 3, «*= 2V3.
. . 1=*=V97
6,  4, —
4, 6,
2
V97
8.
=*= V2, =fc 1.
:F 3V5, =^ 1.
lii
SCHOOL ALGEBRA
EXERCISE 122
1 j^'^^
^' \ 2, 16.
. • 10.3.
[2,  26.
*• 11,16.
5.
6
7.
8.
U, i.
1 1,  if.
I 2, ft.
1 1, A.
1  1, 1*.
f  3, 7.
I  f , 4.
ffo.
• Ifo.
2.f
17 P'*
11, f«.
6.
2.
1
2
3
•(
•1
■I
4.
t
5.
4, »fe 14.
1, =F 4.
3, :^ t V3.
1, =F iV3.
l,=bjV52.
:2,=fV^.
: 5, * 6V^.
= 2, =F 7 V2.
' V91
27
EXERCISE 128
*• 1*5, *V.
7 1*8, *3.
*5,
=f3,
8.
13
Vsl
8
3,
V91
9
10
f1.
• l*f.
12.
1*4,
Vsl
2.
V2.
3V^.
I * 3, * t Vio.
■ 1 * 2. * I VlO.
f3. V.
• 18, V.
3.2.
• 12, 3.
• Us.
[ * 5, * i V5.
12
13
14
15
16
21. 52.
KXEBCISE 134
2
3
4
f9,4.
• 14.9.
4. 3.
• 13.4.
I 3, 7.
• 17,3.
4. 5.
• 16,4.
6;
6.
7.
8.
f *7, *3.
1 *3, ="=7.
I *f, *f.
1 *f. *i
If, i.
1  1, f •
j2,l.
ll,2.
9.
10.
11.
12.
[4,3.
13,4.
7, 6.
1  5. 7.
(
13.
14.
f a+l, a1.
lol,a+l.
16.
* f , =^ f .
=^ t. * f •
*2a, >fc3o. ,«
*3o,*2o. "^^
(2,3.
l3,2.
1 6. 2.
I2.6.
1*1.
1*>,
i.
ANSWEBS
liii
17.
18.
19.
20.
21.
I2,i.
I it.
5, 3.
3, 5.
22.
23.
24.
25.
26.
o— 25, —2a— 6.
2a + 6, 2&  a.
rfc 1 =fcV2.
. =fcl:T=V2.
3,  10.
I  2,  15.
:*: 2, =fc f V30.
=*=1,='=AV30.
f2, 1.
11,2.
27.
28.
29.
=b6.
^3.
12.1.
11,2.
(5,2.
\3,6.
30. (*•
[a.
31. 8, 6.
fa
/5.
Vs.
4
EXERCISE 126
 [3, l,2=fc3V^.
' 1 1,3,
2=f3V^.
> 13,1,1
• 11,3,1
3, 1, 1 =*= V 10 .
Vlo.
3.
3, 2,
2,3,
l=b3V^=3
1=f3V^
4.
1,4,
5=fcV159
4,1,
5=fcV159
6.
6.
7.
8.
9.
10.
1, 1 =*= V^^.
.  1,  1 =*= v^.
3. 2, 1 =fa ^ y151 .
ll,f,f=^iv^=^^.
 46, 2.
1 15, 3.
' d= 2, ± V3.
U 1, 0.
[4, ».
f =b 3, =*= 1.
=F 1, ^ 3.
EXERCISE 126
fl,3,i(9=tV69)*.
• 13,1, J
2.
3.
4.
(9^ V69)*.
1,  6,  4,  1.
2.
J (5 =t V11)*.
1 1,6,4,
1 — 3, i, — i,
14,1,1(5=*=' ^
ll,4, i'(5=F V=Tl)*
f4,V*.
1 1, ♦*.
5.
6.
7.
5^  1, J (9 rfc VloT)*.
1, 5, i(9=*=VIoi)*
3, 4, =*x2V3.
 4, 3, «*= 2V3.
, 1 =fc V97
6,  4, —
4,6,
2
V97
8.
=*= V2, =*= 1.
^ 3V5, =^ 1.
liv
SCHOOL ALGEBRA
9.
12
13
9,  1, 4 =*: ViO, 2,3,
1,  9,  4 * ViO, 3, 2, 
VS.
5«fc VST
■■  <
2
*= Vei
{3,0, =fV6.
^" 1  2, 1,  2 * Vg.
f 25, 4, 43 * 30 V2.
U,25,
1*2.
f*4, *\^.
' 1*2, =bfV2.
43=f30V2.
16
'• {It:
8 * 2 VlT*.
8 ^ 2V1T*.
14 3,2,i(5*V73). 1*2, *11.
U,l,i(6=i.V73). "• 1*11, * 2.
 {J.:'
19.
a
2
&
2
1.
2.
1.
2.
3.
KZESCISE 187
8,4. JO.
2,4. ^ to.
_ 1 2, — 6, 2, 6.
*• li,3,l,3.
«,i . 2,2,f,2.
i, i. 12,6,2,2.
6 i'^'^
10,1.
7 H'l 8 l'(2'
'• 10,2. **• l2*
* V3), 0.
s/3, * V 1.
EZEBCI8S 1S8
•*6. . 10,¥*.
*2. *• U, f*.
7 /**.*»•
N2, ^a
i>i _ [6,6.
t,. *• 15,6.
l*i,=FiV2. *•• U,}.
8.
a
3
2a

3
fl 1,1,6,J.
11,4,2,2.
11.
fl.f.
3,2.
10.
12.
13.
2, ■
13,
(6,1
13, J
i.
H
3.
ANSWERS
Iv
14.
l,t,
3,t,
7*Vl9
5
(•4:,.
f2,i,6, f.
^^ 1 3, J,  10,  i
19 2'*
fi,
20.
21.
22.
31.
17.
1,1,
3:feV33
18
3=f\/33
18.
5, 1,
2,6,
18
S^fV^^^
26  3, 36 + 2.
36 + 2, 26  3.
3,  2,  2 =*= V5.
 2, 3, 2^ V5.
16, 9, (¥)*.
.9, 16, (¥)*.
23.
24.
25.
26.
4, =•= } V36.
1, =*=JV35.
B::;:
1 6, 45.
13, i.
1.
6.
27,  1.
1,27.
28.
64, 1.
1,64.
• 12,8.
.{4,3,6,2.
• If, 2, 1,3.
on {27,1.
^^' ll,27.
33.
2, =fc 1, =*= 2V^, =b V^^.
1, ^ 2, =F v^, =F 2V^n:.
34 {3,2,1,3.
^ 12,3,3, 1.
{ * 2a, =fc 26.
I =fc 6, ^ a.
. {3,4.
35.
36.
38.
39.
{*, 7.
 3,  4, 6 =b V43.
. A  3, 6 =F V43.
40.
fl 4
— , _.
a a
vb
41.
42.
i ii, 0.
4, =fc 2, =«= V^=^ =fc V^=l?.
=*= 2, =t 4, =*= V V =*= V=^.
Ivi
SCHOOL ALGEBRA
43.
a + 6,
a + 5,
6(0  h)
a
a(b — a)
1,¥,2,5.
44. 2 5 1 10
— , — , 1 .
a 7a a a
1^ ab ac be
c a
46.
si
+cb) {!i}\ca)
2(a+6c)
{a +hc) (6+ca)
2(a6+c)
+cb) (a+&c)
2(6+ca)
45
47. 1, 2, 3.
L 3, 7.
2. 4, 28^
3. 2i, 31.
4. 5, 13.
5. 3, 8.
6. r X 12'.
7. t X }.
22. t.
26.
27.
28.
29. 6, 1.
33. 15.
36. Man's rate
37. 40 X 60.
40.
EXERCISE 129
8. 40 yd. X 30 yd. ; 20 yd. X 60 yd.
9. 78'X36'.
10. 7rd.Xllrd.
11. 15rd.Xl8rd.
12. 9'X18\
13. 20'X46^\
14. 30 mi. per hr.
23. Played 24 games, won 16.
Man's rate 4 mi. per hr.; stream's, 2 mi.
Man's rate 6 mi. per hr.; stream's, 2 mi.
40 mi. and 60 mi. per hr.
30. 4i, 7J. 31. 14, 3. 32. 110 ft.
34. 21ft., 13 ft. 35. f, f.
5 mi. per hr.; stream's, 3 mi.
38. 12, 16. 39. 10 ft., 12 ft.
J(a + Vo*  46), i(o  Va«  46).
15.
36.
16.
9.
17.
20.
18.
25^, 50ff.
19.
30^.
20.
A.
21.
t.
24. 28.
1. Real; uneq.
2. Real; imeq.
3. Real; eq.
15. Imag.
16. J.
EXERCISE 134
4. Imag.
6. Real; imeq.
7. Real; eq.
18. ±10. 20. 2.
4 21. 3,  4.
^^' ^ 3' 22.  3, 
17. V.
26. >¥,<¥; none, all (except m =  2).
29. (1)9(2)10(3) 16.
9. Real; uneq.
10. Real; uneq.
12. Imag.
23..  1, V.
25. > J, < J.
ANSWERS
EXERCISE 186
4. 3,  f .
8. A, A
^ a' a»
9 ''^°'.
»• 4 .4
Ivii
1. Sum =s  3.
Product = 5.
2. 1, 7.
11. a:*  5x + 6 = 0. 25. 4x» + a» + 4c«6 = 4(m;. '
13. a:* + 6a; + 5 = 0. 26. a:*  4a; + 5 = 0.
14. a;*  5.04X + .2 = 0. 27. a;»  3a;* + a;  3 = 0.
15. a;« + f a; + f = 0. 28.  f.
17. x" + .12x  .016 = 0. 30. 3a;*  10a;  8 = 0.
18. a;*  (ab  a)a;  a«6 = 0. 32. (x  4) (3a; + 2).
20. a;*  2a;  1 = 0. 33. (x + 1 =*= V2).
22. 2a;«  4a; + 1 = 0. 35. (a;  3 * V^^).
23. 2a;«  2a; 4 1 = 0. 37. 3(a;  i =fc J V^^.
38. t; S;¥; t; V; W; *; f; «•
39. t, J, ¥, f, y, ^, *,  f,  tt;
*"• a' a' a« ' V «' ' «* aV «' A
 ;  V6»4ac; p; g; P»  2g; Vp*4g;
pV^^:i^; p(p* 35); 2 Ivj^TTi^; P"^^"^ .
41. u^ + ix\'^='0. a;»Wa; + W = 0.
43. p =  2; roots = 17, 0.
EXERCISE 136
4. 3:1.
7. 1:50a;.
11.
24:5.
5. a;— l:x+l.
a* + 6*
8. a»: 1.
9. 1:4.
12.
4:3.
^•a*6«
10. 193 + ft. 
13.
7:2.
1. 5:1.
2. 9:2.
o. ol St
14. 3o26: 4a36. 16. 2: 15, 1 : 35, 3: 14. 18. 5175 lb.
15. 3: 1;2: 1. 17. 328,500 Uvea. . 19. 6, 7t
Iviu
SCHOOL ALGEBRA
1. =fc 6aH>. 7. Zy.
2. * 21. 8. f
3. * (a«a:«). ^ a
3«+4
9.
EXERCISE 187
12. (o 1)«.
13. ^^
4. ±;
a1
10. ?5.
o(a + 1)
16. 2,  f.
16. 5,  4.
17. 7,V.
32. 4 ft. 9* in.
18. :« 3,  2,
19. 4.
20. 7 games.
21. 5.
22. 2ft.;3ft.9in.
29. 480, 720.
3x + 5
6. =*=4V3.
6. 66c. 11. .016.
30. 300, 400, 600.
33. $27,180) $12,080; $24,160, $9,060, $6,040.
34. 636t yd. 37. 1,666} lb. 39. 640^ ounces.
36. 26 + sq. mi. 38. 2,666,656 +mi.; 40. 46 lots.
36. 10 + mi. 2.7 + %. 41. 3200 shingles.
42. Hi yd. 43. 20%.
EXERCISE 188
1. 0,  4.
2.0,1, f.
3. 0, 6, 20.
4. 0, 6.
6. 3a, — 4a.
6. 0, 6, f
7. 3,1.
8. V.
9. 3a>.
10.  3,  4.
01
IL
^"^'a+1
a + 1, 1.
24. 4, 10.
26. 6, 11.
26. 13, 19.
27. 4, 8.
31. A, $6200;
B, $8600.
32. 93,146 Uves.
33. 2466 + ft.
34. 649  mi.
36.?^.
2. 31.
3.  26,  81.
4. V,  13.
6. i, 0, 3}.
7.  7.2,  37.8.
8. 88.
9. 164.
10. 189.
11. 148f.
EXERCISE 140
12. 673J.
13. 166.
14. 77V3.
54
16. —  30a.
a
16.
n(3n  1)
17. I (15  3p).
18. n(6  n).
19. I (5a;  4y
+ 2ry — rx),
20. 900.
21. 156 strokes.
22. $26,350.
23. 579.6 ft.;
3622.5 ft.
24. 822 ft. ^
25. 600 ft.
ANSWERS lix
EXERCISE 141
1. a = 4; 8286. 8. n = 21; d = 1. 14. a = 7; n =• 6.
2. a=5i;«=209. 9. n = 21; d = 2. 15. a = l; n=16.
3. a = 5; d = 4. jq ^^jg. d= ^. ig. 12.
4. a = ll;d=3. ^^ ^ . ig. ^ = j. 17. g.
5. a=5i;d=2i.
6.af;d=A. 12.« = 4;n = 5. 18.4or9.
7. a = 3i;d=i. 13. a = 8; n = 5. 19. 9 sec.
EXERCISE 142
1. d =  2. 4. d = f . . o« + y g 10. 106i yr.
2. di. 6. Itt. 2a6 'x«2/« 11. 130.
3. d   A. 6. X. 8. 3i ft.
EXERCISE 148
1. 5, 7. 15. 12.
2. 4J, 3. 16. 5, 4J
4.6075,4066. jg. 2! 6. 8, 11 .U.
!' ^n + l) 20. 30; 13; 150; 100.
7. —=^2 21. 819; 70,336,
9. 102<i tenn. 22. 288 ft.
10. 8, 7i . . . . 23. 14°.
11.3,2,7,12. 24.18°.
12. 1, 3, 5, 7. 25. 0, 357, 826, 212 ft.
13. 24 days. 26. 310, 63, 1336, 398 ft.
14. 1, 4, 7 ... . 27. 948tt ft.; 14,476 ft.; 208ec.
EZKRGISE 144
1. 486. 3.  JV. 4 16
2. 192.  «  33if . « " ^>
\
IX
SCHOOL ALGEBRA
5. A.
7.
1 9.
63.
6. 32.
8.
fl*^! 11.
*m.
12. mi^
13. V(3 + V3).
14. 21V^ + 28.
15. 2V 1.
16. 0, 0, .25, .75,
17. 2,097,150.
18. 386,268,750.
19. S1657.69. $3773.37.
20. $2440.74.
1.25.
21. 9,226,406,250 + bu.
23. $10,737,418.24, cost of en
tire shoeing.
EZEBCISE 146
■
1. 2, 728.
2. 5, 425.
3. 45, W^.
. . 65+19V6
5. 4.
6. f .
7. i.
8. J.
9. 5.
10. 5.
11. 5.
12. 6.
13. 5.
14. 5.
15. 5.
16. 6.
/■
EXERCISE 146
.
1. r = J.
6.
r=J.
10. .49.
11. .25.
12. .025.
2. r = .
3. r =  2.
4. r 4.
6.
7.
8.
=t5.
=b42a*a^.
13. 7.
14. r 4"
15. r
G) ■
EXERCISE 147
1. 3.
2. f
9. i(3V2 + 4).
13. A.
3.  V. 5. W.
4. ¥. 6. i.
10. J(3V2+2V3).
16. 3H». 19. Ixfir.
17. Iftff. 20. 1V».
18. 3iWir. 21. Hit.
7. «.
8. 6(2 + V2).
11. a.
22. 2.
23. 180 ft.
2^. 240 ft.
ANSWERS
Ixi
EXERCISE 148
12. 3, 6, 12.
13. 1, 3, 9, 27.
5, 8, 11.
, 15, 8, 1.
15. $15, 30, 60, 120.
16. 2V2 + 3.
24. .263+.
14.
{2,4,8,12.
'• 1 ¥, ¥, », 1.
18. 24J, i.
1. ti =*= t
2. T61 t • • • •
3. 96, ^ 48 . . . .
10. 5, 15, 45.
[7,14,28.
• 163, 21,7.
21. 0.
22. 52i, 0, 2280, 15560 sq. mi. 26. Between years 1990 and 2000^
19.
2, 4, 6, 9.
, 2, i, — t, 9.
25. 7.89 in.
EXERCISE 149
12
13
EXERCISE 160
2. 32a»  SOalhc+SOa^x^  40a2a^ + lOax*  sfi.
Q 1 . Q , 15x«,5x» ,15a:* ,3a* ,a:«
3. l+3x+^ + ^ + .3g +  + gj.
4. 81a^  216xV + 216a^  96a;V + 16y8.
6. xi  10a;* + 40x»  80a;^ + 80a;V  32^5.
6. x"^ + 7a;i + 21a;"t + 35a;" ^ + 35 + 21a:* + 7a;* + x*.
7. ih^tr^ — Aa;*2/"* + }a;*2r* — fa;*2/"i + ia^y — a;*^/*.
8. a;"  fa;V + ¥ar«2/«  iix*y' + A^^V  uiiy".
9. 243a*a;"¥  405a2ar« + 270a*a;~*  90aaH« + 15oia;"4  1.
10. 16a;« + 32a;'^y* + 24a;*2/* + 8a;V + x V
11. S2a^y''i  40xiy^ + 20a;22^i  5a;V + i^^V^  Aa;"V.
12. 64a*x« + 576a^^a;"* + 2160a""4a;"i + 4320o"ta;i f 4860a"Va;l
f 2916a"V^a;V + 729o9a;«.
13. 81a»62  108a26« + 54a%«  126io + aft".
14. a;«  3a;« + 9a;*  13a?» + 18x«  12x + 8.
15. 8  36x + 66x2 _ q^^ ^ 33^4 ^g^^^^i^
16. 16x8 +. 32x^  72x«  136x« + 145x* + 204x8 162x»108x + 81.
17. a8 + 8a7x + 20a«x« + 8a»x» 26a*x* Sa'x^ + 20o»x« 8ax^ + x«.
Ixii SCHOOL ALGEBRA
18.  14,784a«xio. 2I.  lOSa^ftT, 4^
19. ITWyi. 7920aV&*. ^^ *
20. 3003x1 V. ^ 24.  61,236a«x».
lOOlajVyio. ^ 6a^'
25. ^(^l)(^' + l) ^n,r^r,
ll
27 (n + 2) (n + 1) . . . (n  r + 4) ^_,^^,_,
Ir — 1
28.  1320a:». 32. 24,310a^". 36. 548.75873536.
37. 16,016.
31. \l2,640. 35. 8.157 +. 38.  1,293,6000'"^
♦ 29. 1365a<a;". 33. 3.138+.
30. Hi^^, 34. 8.915 +.
EXERCISE 162
1. (« + 2) (a;  2). 8. 2(x + 1) (2x»  4x  3).
2. {x  7) (x 4 4). 9. {x + 1) (3x 1) (x + 2).
3. (a  6) (a* + a6 + 6« + 3). 10. (x1) (x2) (x+3) (2xHl).
4. (a  6) (a  6 + 3). 11. (x + 1) (x4) (3x2) (2x +3).
5. (a  1) (a« + a + 6). 12. (x  1) (x»  x  1).
6. (x + 5) (2x  3). 13. (x  5) (x«  X  5).
7. (x  iy(x + 2) (2x 3). 14. (x5) (x + 6) (x«  x + 3).
EXERCISE 168
1. X + 1; (x + 1) (2x  3) (4x»  8x + 5).
2. 2x  3; (2x  3) (3x + 4) (3x«  2x  6).
3. x1; (x1) (x« + 2xH3) (x»2x + 3).
4. 3(x  1)»; 6(x  1)« (x + 1). ;
5. x(2x + 1); x(2x + 1) (3x»  4x + 5) (x»  5x  2).
6. 3x(3x + 4); 6x«(3x + 4) (x«  x + 1) (x«+xl). .
7. :t + 3; (x + 3) (3x»  2x + 1) (2x»  3x + 2).
8. x»  2x H 3; (x«  2x + 3) (x« + x  2) (x» + 3x  2).
9. X 4 3; 10(x + 3)<x«  3x + 1) (x« + 1).
10. x« + X + 1; (x» + X + 1) (5x«  3x + 1) (2x»  x» + iP  1).
11. 3xy; (Sxy) (x» + x»y + xj^ + 22/») (2x» + x«2/ + xy« + 2/»).
13, :p(3a?  4); x(3a;  4) (x» + 2x«  1) (x*  2x»  2x* + 2x  1).
ANSWERS , _ Ixiii
13. x2; (x2)ia^+x + l) (a;« + 3) (2x« Ha; ■+• 1).
14. 2x(z  1); 6x{x  1)» {x 2) (x + 3) (a;  3) (x + 6).
EXERCISE 164
5. w' + 6a — 1. o«. Q
6. 2a:»5a;3. "^ y 2i/«*
EXERCISE 166
3. 124. 5. 3204. 7. 70.09.
4. 352. 6. 804.5. 8. .0503.
15. 2.704+ 21. 1.730 +
16. .3968+ 22. .0535 +
S:S2l+ 23. (531.3 +yd)»:
19. 2.0033 + 24. (1493. + ft.)».
20. 2.901 + 25. (592.8 + ft.)».
EXEBCISE 166
5. l3a6. 9. 34.
6. a; J. 10. 1.5704 +
7. 2 + 2x  x». 11. a;  3.
8. 14. 12. 4a:«  J.
EXEBCISE 167
1.0. 2. iVB. 3.18,3,27. 4. 26c + c». 5. a=4,6=5:
9. H. C. F. = a+ 6; L. C. M.  (a+ b) (a + ahi  h) (aahib).
10. a*(a  6). 11. {x\ a) (xa) (a;»+ ax+a«).
12. (x+2y^) (xyh i2x+yi),
jg 2(a+6) ^^ x' + l 21. t,*,f.
a'^'l^ (1a^)* 22. 1, 2.
^^' ~T~' 20. A. 23. tt,f.
1 1 25. ^^ ^
1.
a + 2x.
2.
3  a.
3.
a*  a — 2.
1.
15.
2.
91.
9.
.997.
10.
4.217 +
11.
1.817 +
12.
1.775 +
13.
1.542 +
14.
1.953 +
1.
19.
2.
43.
3.
3.08006 +
4.
.9457 +
13.
2mn
14.
2a
o»6«
24.
1
a'^'b{^c a + b + c a\b + c ' q
26, B =* 4; C «  I; D «  15; B «^ 27.
bdv SCHOOL ALGEBRA
; 5 7»t 35. m^, .16.
28. o« + ad 4 &*. ^'
29.6}. 33. m^. 37.?^.
30. x^ ^. VS+llo 2/*
31. X. V6 + 9a 38. X* + 1 + ar»».
40. 3V2 + 8V3  8 4 2V^. 41. 3.121 +.
42. 0. 43. V. Q7.2xiSxiyi2xiyK ^' ^^ ^y " ^'
^ (ca:)'l y^3^ 68.^^ 24 ft. 7i in. ^^' ^
47. Ja^e +. . 70. ^. ^. «!• (^1" •
48. ^±^. 71 2« ^ 27.309J.
(«'**)* 6,5 93 hya
49. 1. V31, V=:T. 72. y/^. •a. + 2
60. 30 geom.; 72 alg. ^^ ^^^ ^^ 94 96.
61. 32 games. 95. 2 + V3,
52. f. " iVlO + iV^,
53. J. 76. yhkri. VS+5+ V^^.
54 9 77. . 96. $580,046 + .
65. x«  2r» + X. 78. i, 97 9^ j
66. VE, vlT, V^. „ V6+V15 98. 0.
, 79. 5 >
57. 1_,^ + ^. 3 99. j^.
m + n 2.1075 +. iqo. ahi^2\a^hi.
59. 28.6 +. 80. 0. loi i
^^ (2a;l) (3x2) «, kqa,^
60. ^ ^^ ^. 81. 586 + da. 102. n« + n + 1.
62. 16 lb. ^2 ^'^+ *^* 103. $750, $300.
63. 124, 61, 15. ^' " ^' 104. J^VIO.
64. 3. ^ " A 106. c.
11 85. wi — n. 6c  1
65. aj +   ^ gg 7,000,000 H. P. 1^^' ahc  a  c
66. 72. 87. a?' 108. J,  }.
109 a: = q<^~Q'^+^ . _ a»  gfe^ + 6c» _ 2ab  c*
110. ox + 61/  C2. 111. A(3V3  V2).
112. 4a:2/(5:j;« + QOa*y» + 126aV + 60x22/» + 5y^h
ANSWERS IXV
113. 7:05J^. 115. 36.55+ H. P. 118. a; =» f.
114. if.^ ifL'^  12. 117. sViV^. 119. t,  A
120. (a  6) (6  c) (c  o). 121. 76 lb. gold; 30 lb. sUyer.
122. (3m2 + 2mp  %n^)x + (4n«  m^)^.
124. 23A mi. 125. 2. 126. J.
127. 11.5174 +. 128. 2x.
129. a:' + 2g« + 2a; + l jgg. (^ + 1) (x  1) (x  2).
X ^X ""■ 1)
131. (a + 6)«, (ol 6)«. 133. 3  2V2.
4 4. 4V5 v/lO+ 2 V5  V5O+IO V5
134. 1
135. 16. 137. xV^. ¥.
1 1
136. 20 mi., 25 mi. 138 ^ = 5^ 2/ == a*
139. a;i + xi + xi + ar2 + . . . .; xi + ix"* + fx"^ + . . . .
140. x«  xy + i/^.
A + (3A + B + C + D)x K3A + 2B + C)x» + (A + Bjx*
1^^ " X (x + 1)»
1^ —u^ 1 >•>! 2x8 + 2x* — 1 1 A7 1 oQQn a
142. 3 = 27 ^^ 2x^ + 2x ' 147. 1.2930 +.
= 161+ 145.x=;2, = «. l^^^
143. ^^ +4 • 146. 288 V^ 1^9. ^*  2xM + 2y.
150. 3V6(x  D* (x + l)i. 151. A, $27; B,' $13.
16(x^ + a^y . 155. H. 158. ^ V^.
152.
(^«*)* . 3a
156. x»'»H2. 159. ~.
154. 15 da. _ , 4
157. i. 161. 0, 2.
,^« V9  VlO3 V2+3V5 163 §^_£E_^.
162. ^^ 2 2 3 5
164. (x» + 2x + 7)(x2+2x3). (2/* +3y + 4) (2/* + 3y2).
Ixvi
SCHOOL ALGEBRA
1.
1
a
BXEBGISB 168
11.
2. 1+V3, 3V3.
3. 2.608+, 4.408I. 12
4. 2,i
6. 2*, «.
 o + b — 6
' a — h a + b
f4, 20.
120, 4.
[2; I.
14, V.
14.
3,1, 9=fcVV5.
1,3,9=fVV6.
13.
r a + b
15.
16.
f2,}.
Ii2.
10.
18.
19.
f8, L
11,8.
2, 8, 2 =fc4V^
8, 2, 2 =f4V^.
5, 5, 11, 11.
4 V7,  4 V7.
f 5, 5, 11
14,4,
11,9.
19,1.
23. ab.
24. 3.
25. 24 ft.
26. 16, 20, 30.
41. r = * f, a = f , V.
42. 40,040x"*.
43. 8a:»  4  &rVx»  1.
y a — b + c
■ ,/ 2 17 U3, =bVZ7.
20. 25, 9,i(225=fcV5 ^589)*.
9,  25, K 225=*= V56689)*.
21. x»2a;+2 = 0:
2a:»2a;+l = 0.
22. J, i.
37. 300 times.
(26)»»  1
27. ^.
28. 2, J.
33. 960.
36. 11  2n, 7  2n.
39.
40.
261
1
59,049
44.  4,365,900 X 2»V6»7.
45. 1; 6561.
46. Sixth, 924x*; seventh, 231a?*.
47. 70.
48. 3x.
49. 3,  2*.
60. 55,350.
62. 3a + l, a2.
^ (J; :i
64. * 2, * V69*.
55. 4.
56.
,.^.
57. 9a;«+6a;190.
68. Row, 6 mi.; stream, f mi.
iV3, ^ 1.
IV3, =F 1.
61. 0, 1.8,  2.4, 5.8.
62. *5V^, 0.
69. I
U, 9.
65. I.
2x
66. 1 * V3, 1,  }.
n
ANSWERS Ixvii
eg """ . 74. 1, J, Ai • • • 78. K " ?• ^
» or 3, f , A . • • • 7 "
69. n42. 7K ., 4.(7 79 y.'S' + "S^.
70.7,4.4,1. 75.41 + %. 79. T ,^
72. *V5; *1. 7« 5' ® 80 *306.
73.3,9,16,21. 77. V. 81. $60.
«, (9, 7, i(25 =b V373); 87. 0, , * 3.
^ t7,9,l(26=pV373). * '' ,
, ./8«< 88. ~» ^ ^ > •
83. Z = V— • V V V
89. 3 mi. an hr.
, V ,/3Z'*32s«
' = 2*r — 12 — ^1 *■ = *•
/8iS __ r / 3Z^«  328' 92. 15 or 14.
r"S"""2*r 12 ' 93. ».
84. M^V3*¥^V^^. 94. =t=2V3.
85. 40.
86. 10 hr.
96. r = 3, 1; series 2, 6, 18 ... and 2, 2, 2 .. .
97. 0, 1,  2. 102. 1, 27. 107. 10,000.
103. 12, 16. 67.
95 1*4' *»•
104 P»3>4,1.
^"^ 13,2,1,4.
109. n = 4; d =  1.
no 13,1.
^^ 11,3. _. f 2, 3, 4. 1. 108. X = i; y = 2.
100 1^'**
^"" I d= 3, ± . 105. * 3. 110. 40 mi. an hr.
irti [2,3,3,2. _ 1 9, 4. J. 1 2, .7+.
1^1 1 1,  2, 0, 0. ^^ 1 4, 9. "^ 1 0,  1.8+.
,3^,3 1^3783.
120. ^ ^^
112. ^^^^'^6V83. ^^"•S'3^»6a
113. r =  i, « = 3i.
114. «*, 0. .
121. 0.
122. 180 ft.
116. 4 mi. an hr. 123. j2 =^V£2,  2 *\2.
l2=F\A=^, 2=F V6.
117.7=^. 124 &c<«^
6V; 124. ^^^_^_^
118. 32, (  f)». 125. I, 10.
119. ijA; 650. 126. \/7 =»= Vs. 127. 1.
Ixviii
SCHOOL ALGEBRA
130.
131.
5, V, ¥, 10.
4, f , V, 6.
2, =*= J.
132. 2.43 — yd. per second greater
134. a = i, ^^; r = 3, 1.
135. d = f .
136. 2n(2nl)....(n + l)^_^3^,^
^
138. 64o»  48a62.
12,1,1 =tV"
139.
2.
140.
j2n _ yin
li,2, 1:^ V^r2.
142. V = Y
141. 3.5 + in.; 31.8 + in.
5ER
Tp' 23.7 + mi. per hr.
1. logs 9 = 2.
logs 27 = 3.
log4 64 = 4.
2. log2 32 = 5 logs A
3. 1.
9. V64 = 4.
EXEBCISB 169
log4 A =  2. logio 3^ =  1
logs 4 =  2. logio .01 =  2.
logs A: =  4. Ibgio .001 =  3.
 5 log2 f i^ =  7 log4 8 = t logs 16 = t
V 1024 = 4.
^4096 = 8.
1. 2.
2. 4.
3. 2.
EXERCISE 160
4. 1. 7. 0. • 10.  5.
5. 0. 8. 0. 11. 0.
6. 2. 9. 3. 12. 3.
16. 4, 3, 6, 2, 1, 5, 1, 2, 0.
13. 4
14. 2.
15. 1.
EXERCISE 161
1.
1.5682.
6.
2.2430.
11. 8.8797 
10.
16.
9.8914 
10.
2.
1.9294.
7.
1.5172.
12. 3.7619.
17.
8.6309 
10.
3.
0.7782.
8.
0.6767.
13. 7.3365 
10.
18.
2.3706.
4.
1.9542.
9.
8.9031
 10. 14. 1.8008.
N
19.
0.7490.
6.
2.4771.
10.
0.0086.
15. 0.4774.
EXERCISE 162
20.
3.8911.
1.
43.
3.
236.
5. 8400.
7.
4.09.
2.
770.
4.
3.78.
6. 70.4.
8.
.627.
ANSWERS
box
9. .00803. 11. .00502.
10. .0428. 12. .000126.
17. 283.6.
13. 2.69.
14. 30.9.
18.
15. 7080.
16. 77.7.
.4367.
EXERCISE
1. 105.
2. 34.3.
3. .0755.
4. 207.71.
5. 4.082.
6. .04218.
7. 64.7.
8. .7995.
9. 681.
10.  ^2681.
41. $2514.60.
42. $995,200,000,000.
43. $5716.30.
44. $5985.70.
45. 16.924 ft.
11. 1.427.
12. 2.407.
13. .3016.
14. 1.324.
15. .23317.
16. 4.08.
17. .4287.
18. 12.16.
19. 1.596+.
20. 197.68.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
46. 1.6838 ft.
47. 10.632 ft.
48. 1.4029.
49. .7333+.
50. 216.15.
168
.75183. 31. 9.365.
.2526. 32. .3933.
4.35S? 33. .17556.
1.4876. 34. 22.58.
1.502. 35.  1.162.
.6633. 36. 3.2714.
3.936. 37. 2.483.
.459. 38. .873.
14.44. 39. .35142.
5.624. 40. 1.6167.
51. 14.2+ yr.; 10.24+ yr.
52. 7.6717 ft.
53. 31.671ft.
54: 1759.2 1b.
55. 457.1ft.
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