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I
f,^,\-.C. Xai^\ns.Wi
c^v^
JUNIOR HIGH SCHOOL
MATHEMATICS
BOOK II
BY
JOHN C. STONE, A.M.
HBAD OF THB DBPARTHBNT OF MATHEMATICS, 8TATB
NOBMAL SCHOOL, MONTCLAIB, KBW JBB8BT, AUTHOB
OF THB TBACHIKG OF ABITHMBTIC, CO-AUTHOB
OF THB SOUTHWOBTH-STONB ABITHMETICS,
AND THB STONB-MILUS ABITHMETICS,
ALGBBBAS, AND GBOMBTBIBB
BENJ. H. SANBORN & CO.
CHICAGO NEW YORK BOSTON
1922
GOPTRIGHT, 1919,
Bt BENJ. H. 8ANBOKN & CO.
PREFACE
This series of mathematical textbooks marks a new type of
mathematics as to aims, purposes, and material. Unbiased by
tradition, the author seeks to give the mathematics necessary in
order to interpret the quantitative phases of life, met by the aver-
age intelligent person outside of his specialized vocation.
The material, selected with this aim in view, is naturally based
upon some social issue. Such a selection of material not only
leads to the habit of using the mathematics learned in school, to
interpret the quantitative side of everyday life that is met out of
school, but it will also greatly increase the interest in the subject —
a vital factor in the economy of learning.
It should be noticed, then, that this series especially emphasizes
the interpretative function of mathematics, and seeks to develop
both the power to see and the habit of seeing the quantitative
relationships that necessarily arise in topics of general conversa-
tion and reading.
To do this, the series makes use of concepts and processes usu-
ally classed as arithmetic, algebra, geometry, and trigonometry ;
but it uses only such a part of these subjects as is needed to in-
terpret references met in general reading. It is the needs of the
student, then, that is kept constantly in mind in the selection of
material, and not the development of the subject or traditional
subject matter.
Book II reviews methods of computation and introduces a few
of the most-used "short cuts." The formula is reviewed and its
/ use extended, in order that the student may be able to interpret
its meaning and to evaluate it when met in other work, and that
he may see its advantages and use, in expressing quantitative
relationships in the briefest possible forms.
• • •
ui
iv PREFACE
The simple equation of one unknown quantity is introdooed to
acquaint the student with its meaning and with methods of solv-
ing it, and as a necessary tool in solving problems of proportion
that follow. Ratio and proportion precede a study of simila r
figures, to furnish a means of expressing relations found through
measurement, and to furnish a tool to use when applying the prop-
erties of similar figures to the finding of heights and distances.
The study of similar figures naturally leads to scale drawing
and trigonometric ratios as means of finding heights and distances
without actually measuring them. In Book n, only the tangent
relation is used.
A very complete discussion of the graph as used in representing
quantitative relations is given. Instead of graphs being made
for the occasion, they are taken from leading newspapers, maga-
zines, and other sources, thus leading the student to observe and
interpret them as he meets them in general reading. The func-
tional graph is discussed briefly.
After a thorough discussion of relations expressed by per cent,
the last half of Book 11 discusses business terms, forms, and proc-
esses ; banking ; methods of investing money ; the meaning and
nature of insurance; and the meaning and necessity of taxes.
The problems under these topics are real and are given for the
purpose of helping the student to interpret these important topics
which are of interest to all of us. The unreal, indirect problems,
still found in most courses — given through adherence to tradi-
tion or for mental gymnastics — are carefully eliminated as
contributing in no way to the aims of this series.
Properly used, then, this series of textbooks will give an inter-
pretative power to mathematics not developed by the methods
and the problem material of the present type of textbooks written
for these grades.
John C. Stone.
June, 1919.
CONTENTS
OBA^TBB PAOB
I. REVIEW OF ARITHMETICAL PROCESSES: SHORT
METHODS 1
1. Addition: Whole numbers, fractions, and decimals;
2. Subtraction: Whole numbers, fractions, and decimals;
3. Multiplication : Whole numbers, fractions, and decimals ;
4. Division : Whole numbers, fractions, and decimals.
n. THE FORMULA 17
1. Evaluating formulie; 2. Simplifying literal expressions;
3. Factoring a formula; 4. Formulae derived from other
formuUe.
m. THE EQUATION 24
1. Determining the value of the unknown number ; 2. Prob-
lems solved by equations.
IV. RATIO AND PROPORTION 31
1. The meaning and use of ratio; 2. The meaning and use
of proportion.
V. SIMILAR FIGURES 37
1. Fundamental principles of similar figures; 2. Practical
measurement of distances ; 3. Maps and plans : Drawing to
scale.
VL TRIGONOMETRIC RATIOS 46
1. Tangent relations ; 2. A table of tangents.
V
vi CONTENTS
CHAfTKB PAOB
Vn. GRAPHIC METHODS OF REPRESENTING FACTS . 51
1. General illustrations ; 2. Simple comparisons ; 3. Graphs
showing component parts ; 4. Curve plotting : The broken
line graph; 5. Map presentation of facts; 6. Functional
relations shown by graphs.
VXn. MEASUREMENTS, CONSTRUCTIONS, AND OBSERVA-
TIONS 77
I. Measuring any quantity : Denominate numbers ; 2. A
review of areas ; 3. Constructions and observations ; 4. The
area of a parallelogram ; 5. Constructions and observations ;
6. The area of a triangle ; 7. Constructions and observations ;
8. The area of a trapezoid ; 9. The relation of the circum-
ference of a circle to its diameter ; 10. The area of a circle ;
II. Measuring lumber ; 12. The volume of prisms ; 13. The
volume of cylinders ; 14. The surface of a cylinder ; 15. The
volunae of pyramids and cones ; 16. The measurement of a
sphere.
K. SQUARE ROOT AND THE PYTHAGOREAN THEOREM 103
1. Squaring a two-figured number; 2. Finding the square
root of a number; 3. Some applications of square root;
4. The Pythagorean Theorem.
X. GENERAL DISCUSSION OF PERCENTAGE . .113
1. A review of former work in percentage; 2. Interpreting
and finding per cents of increase and decrease; 3. A new
problem in percentage; 4. Applications of the three prob-
lems of percentage.
XI. BUSINESS TERMS, FORMS, AND PROBLEMS . . 124
1. Bills rendered by the retail merchant; 2. Keeping ac-
counts ; 3. Buying and selling at a discount ; 4. Commercial
or trade discount; 5. Successive discounts; 6. Profit and
loss ; 7. Commission and brokerage ; 8. Borrowing and loan-
ing monoy.
CONTENTS vii
GHAPTBB PAOB
Xn. BANKING 145
1. Deposit slips ; 2. The pass book; 3. Making out a check ;
4. Buying a draft; 5. Borrowing money from a bank;
6. Discounting notes at a bank.
Xm. METHODS OF INVESTING MONEY .... 156
1. Loaning money on bond and mortgage; 2. Investing in
bonds; 3. Savings bank deposits ; 4. The growth from regu-
lar deposits; 5. Building and loan associations; 6. Real
estate investments ; 7, Investing in stocks.
XIV. THE MEANING AND NATURE OF INSURANCE . 177
1. Property insurance ; 2. Personal insurance.
XV. THE MEANING AND NECESSITY OF TAXES . .185
1. How city, county, and state expenses are met ; 2. How the
expenses of the National Government are met.
XVI. SOME THINGS YOU HAVE LEARNED DURING THE
YEAR . .195
1. You have learned to interpret and evaluate a formula;
2. You have learned the meaning of an equation and how to
solve it; 3. Problems solved by use of equations; 4. You
have learned to find distances by scale drawings ; 5. You have
learned to find the height of objects from the length of
shadows they cast; 6. You have learned to find heights and
distances by tangent relations ; 7. You have learned to repre- .
sent data graphically ; 8. You have learned the use of many
business terms and problems; 9. You have learned the im-
portant methods of investment; 10. You have learned to
check your work and to know that your computation is
correct.
Tables 212
Index . . • 213
JUNIOR HIGH SCHOOL
MATHEMATICS
BOOK II
CHAPTER I
REVIEW OF ARITHMETICAL PROCESSES:
SHORT METHODS
You have learned how to compute with whole numbers,
fractions, and decimals, but continued practice is necessary,
in order to develop greater skill; that is, in order to be
more accurate and rapid. Skill in computation, however,
depends not only upon recalling the number facts accurately
and rapidly, but upon seeing relations that will save figures
and even whole processes. In this chapter, then, will be
discussed some "short cuts" in computation. These may
occur chiefly in multiplication and division.
1. ADDITION: WHOLE NUMBERS, FRACTIONS, AND
DECIMALS
In all computation, one should form the habit of going
over the work a second time to see if it is correct. This is
called checking the work.
In addition^ check by adding a second time in reverse order.
1
2 JUNIOR fflGH SCHOOL MATHEMATICS
One method of recording the steps in addition and check-
ing the work is shown in the following example.
Explanation. — The sum of the first column
WORK CHECK ^ 4^ j^ ^ written under " check." The sum of
the next, with the 4 carried, is 43. The sum of
the next, with 4 carried, is 40.
Now, beginning with the highest order and
adding in the opposite direction, the sum is 36 ;
this with the 4 of the next lower sum (43) is 40.
The next sum is 39, which with 4 (from 41) is 43.
And the next is 41. So the sum 4031 is recorded.
TqqT Some prefer the check shown in Book L This is
just a little shorter. Use the method you prefer.
495
41
628
43
786
40
549
875
698
Drill Exercises
Add and check:
1.
2.
3.
4.
5.
79,648
65,468
56,981
67,943
68,991
84,796
96,394
98,156
59,946
64,036
72,387
54,387
86,675
78,897
78^38
86,793
86,731
17,386
27,889
42,392
54,635
72,467
39,756
82,137
67,169
68,357
81,120
79,346
64,938
42,627
53,386
27,351
42,932
58,716
63,128
49,962
84,936
93,742
78,345
96,321
51,195
35,426
64,396
27,564
38,528
16,840
26,734
46,321
64,375
16,784
31,178
52,191
32,875
41,728
36,597
28,385
16,207
42,368
50,732
48,096
Adding Two Numbers Without a Pencil
In adding two numbers of two figures each, without a
pencil, it is easiest to add the tens to one of the numbers and
to that sum add the ones. Thus, in adding 48 and 56, think
98, 104 ; or in adding 27 and 58, think 77, 85.
REVIEW OF ARITHMETICAL PROCESSES 3
Drill Exercises
Add at sight :
1. 42 + 56. 11. 19 + 35. 21. 87 + 56.
2. 27 + 35. 12. 46 -h 18. 22. 95 -f- 48.
3. 38 + 53. 13. 55 + 28. 23. 86 + 93.
4. 41 -f. 29. 14. 31 -f 67. 24. 78 -f- 84.
5. 36 + 29. 15. 52 + 27. 25. 57 + 69.
6. 27 + 56. 16. 48 + 39. 26. 48 + 88.
7. 64 + 17. 17. 63 + 29. 27. 54 + 76.
8. 38 + 46. 18. 57 + 17. 28. 85 + 47.
9. 35 + 49. 19. 82 + 49. 29. 96 + 58.
10. 73 + 19. 20. 98 + 54. 30. 87 + 98.
These exercises are to fix the method. Before this, or any
other form of computation, is of value, it must become a
haUt. So, by using this method whenever the sum of any
two numbers of two figures each is wanted, you will soon
find it easy to add all such numbers without a pencil.
Adding a Number Nearly 100, 1000, etc.
To add 98 to 56, observe that it is 2 less than 100 to add,
hence, the sum is 100 + 56 - 2 or 156 - 2 or 154.
996 + 748 = 1744, for it is 1748 - 4. Why ?
Drill Exercises
At sight add :
1. 998 + 645. 6. 875 + 970. ii. 995 + 846.
2. 990 + 875. 7. 694 + 920. 12. 970 + 645.
3. 995 + 767. 8. 736 + 998. 13. 980 + 763.
4. 990 + 843. 9. 645 + 980. 14. 991 + 750.
5. 996 + 735. 10. 563 + 940. 15. 997 + 645.
JUNIOR fflGH
SCHOOI
. MATHEMATICS
Drill Exercises
Add and
cheek :
1.
2.
3.
«.
s.
38.4
16.48
3.96
74.6
63.48
75.64
24.3
98.6
9.16
8.34
9.15
8.67
6.98
18.46
16.9
16.9
9.875
19.8.
36.64
9.85
7.63
10.48
13.66
17.29
48.76
9.375 6.847 9.268 7.175 9.43
42.86 75.37 54.28 16.97 16.38
4.463 7.289 7.543 6.297 6.783
^ML_ 1^-^ 16.28 42.98 53.9
6. In adding decimals, why are the decimal points written
under each other ?
Drill Exercises
Add and check :
1. i + i+J- 3- l + J + A- «• l+* + ^ir-
2. l+^ + f- ♦• f + A + H- 6- i + f + if.
7. In adding fractions, why are they first changed to a
common denominator ?
8. What are fractions called whose numerators are equal
to or greater than the denominators ?
9. How are improper fractions changed to whole or mixed
numbers ?
Add and check :
10. 11.
12.
13.
14.
9^ 26J
24J
82i
28|
8f 48J
16f
46|
42J
n 54|
42,V
16i|
26|
H m
m\
n
41*
^ i9f
24^
n
86^
REVIEW OF ARITHMETICAL PROCESSES 5
Adding Special Fractions
In all computation, one should be on the alert for com-
binations that will save work. Thus, to add i + f + f 4- f
4- J, one should observe that J + | = 1, and that ^ + | = 1.
Hence, the sum is 2J. And to add | + ^^ one should observe
that since 8 and 5 have no common factors and since each
numerator is 1, the new numerators will be 5 and 3, resj)ec-
tively, and the denominator is 3 x 5. Hence, the sum is ^.
That is, the sum is the sum of the two denominators over
their product.
Drill Exercises
Give at sight
•
1. \ + \.
5. J +
f
9.
J + W-
13. i + f
2- h+h
6- i +
\'
10.
i + i-
14. J + T^r-
3. i + f
•>- \ +
f
11.
i + f
15. T^r + T^.
*• i + i-
«• i +
h
12.
i + i-
16. i + T^r.
17. §+| + J
+ i + |.
21. \ + l
+ i + f + i-
18- i + f + i
+ l + f
22. i+J
+ i + i + f
19- i+l + f
+ i + i-
23. f + i
+i+i+^
»• i + j + i
+ a+i-
2*. 1 + I
+ f + i + i-
2. SUBTRACTION: WHOLE NUMBERS, FRACTIONS, AND
DECIMALS
Subtraction is the inverse of addition. That is, the sum
of two addends, and one of the two addends, are given, and
the other addend is to be found. The given sum is called
the minuend. The given addend is called the subtrahend.
The addend found is called the remainder or difference.
Always check subtraction by adding the result to the subtra*
hend to see if it equals the minuend.
Do not rewrite, but check the result as it stands.
JUNIOR HIGH SCHOOL MATHEMATICS
Drill Exercises
Subtract and check :
1.
65,179
28,396
2.
72,307
19,698
3.
61,302
28,496
4.
29,361
17,975
5.
30,621
13,794
6.
81,106
47,647
7.
71,193
53,465
8.
40,069
28,773
9.
61,110
28,326
10.
52,903
49,628
11.
64,216
28,269
12.
81,726
54,392
13.
42,901
16,284
14.
61,110
49,306
15.
53,306
48,729
16.
35.48
16.527
17.
204.3
165.48
18.
39.026
16.48
19.
42.16
19.865
20.
7.36
6.475
21.
38.465
19.8
22.
42.065
19.78
23.
13.065
9.87
24.
47.081
19.765
25.
52.876
48.9
26. In the subtraction of decimals, why are the decimal
points placed under each other ?
27. In the subtraction of fractions, why must the fractions
be changed to common denominators ?
31. 32.
24^ 48J
Subtract and check :
28. 29.
35i 42^
15i 15i
30.
30i
16|
33. 34.
260| 204^
1925 1461
35.
240f
198f
16i 17|
36. 37.
305| 201|
168^ 128|
REVIEW OF ARITHMETICAL PROCESSES 7
Subtracting Special Fractions
When the numerators are each 1 and the denominators
have no common factor, the fractions may be subtracted at
sight. Thus, i — ^ = ^1 for it is seen that the new numer-
ators will be 7 and 4 respectively and the common denomi-
nator will be 4 X 7, or 28. That is, the result is the differ-
ence of the numerators over their product.
Subtract at sight :
2- 1-f 7. i-4. 12. J- i. 17. I-^V
5. J - f 10. i - f 15. J - ^. 20. i - J^.
3. MULTIPLICATION: WHOLE NUMBERS, FRACTIONS,
AND DECIBfALS
When the multiplier is a whole number, multiplication is a
short form of finding a number equal in value to the sum of
a number of equal addends.
Thus, 5 X f 7 = 17 -(- 17 -(- $7 + $7 + $7 = 135.
5 X .07 = .07 + .07 + .07 + .07 + .07 = .36.
5x| = | + | + J + |-f-| = ^ = 4|.
The 5 in each example given is the multiplier ; f 7, .07,
and ^ are the multiplicaiids ; and $35, .35, and ^ are the
products.
From the meaning of multiplying by a whole number, it
follows that :
1. Uie multiplier nrnst be an abstract number.
2. ITie multiplicand may be either abstract or concrete.
8 JUNIOR fflGH SCHOOL MATHEMATICS
3. The product muMt be a number of the Mime name or kind
as the multiplicand.
From principle 3 above, it follows that when the multi-
plier is a whole number, there will be as many decimal
places in the product as. there are in the multiplicand.
It also follows that a fraction is multiplied by a whole
number by multiplying the numerator and leaving the de-
nominator unchanged.
Check multiplieation by going over the work a second time.
Drill Exercises
1. 306x475. 8. 916x70.9. is. 9 x f .
2. 906 x 66.8. 9. 426 x 90.3. 16. 8 x |.
3. 606 X 39.2. 10. 481 x 7.06. 17. 10 x f .
4. 276 X 3.09. u. 4 X f . la 12 x f .
5. 460 X 62.8. 12. 6 X |. 19. 14 x f .
6. 326 X 84.3. 13. 7 X |. 20. 16 x |.
7. 606 X .628. 14. 6 X f . ' 21. 19 x |.
Multiplication by Fractioiis and by Decimals
Multiplying by a fraction is both a multiplication and a
division. The numerator is the multiplier and the denomi-
nator is the divisor.
Thus, I X 24 = 3 X 24 -^ 4 ; .4 x 5.3 = 4 x 6.3 -«- 10.
To multiply a fraction by a fraction^ take the product of the
nvmierators for the numerator of the product and the product of
the denominators for the denominator of the product.
Work is saved by cancelling factors that occur in both
terms, before multiplying.
REVIEW OF ARITHMETICAL PROCESSES 9
In the product of two decimals^ there are as many diffits at
the right of the decimal point as there are in the total number
at the right of the decimal point in both multiplier and multi-
plicand.
Drill Exercises
Find the products :
«
1. |xi|.
11.
W X If.
21.
1.75 X 36.9.
2- fX^V
12.
|x||.
22.
3.45 X 16.3.
3. |xif.
13.
A X e.
23.
40.3 X 5.65.
4. fxf
14.
il X ||.
24.
16.5 X 2.85.
«• |xi?.
15.
HxM-
25.
19.8 X 40.6.
6. |xf|.
16.
if X ||.
26.
57.3 X 30.3.
7. \xl\.
17.
if X If
27.
42.1 X 64.9.
8- *xi|.
18.
H X IJ.
28.
5.06 X 84.7.
9. Jxif.
19.
ii X 4f .
29.
89.2 X 9.03.
10. |xft.
30.
ifxM.
30.
58.2 X 10.8.
Finding a Per Cent of a Number
Per cent is only another name and notation for hundredths.
Thus, 7 % of 480 = .07 x 480 ; ^ % of 360 = .025 x 360.
Change to decimals :
1. 45%.
2. 82%.
3. 156%.
4. 245%.
5. 300%.
6. 12^%.
7. 141%.
8. 26^%.
9. llf%.
10. 5J%.
n. 200%.
12. 550%.
13. 725%.
14. 1100%.
15* 14.5%.
16. 4.5%.
17. 13.25%.
18. .84%.
19. .16%.
20. 1.25%.
10 JUNIOR HIGH SCHOOL MATHEMATICS
Mnd:
«
21. 28 % of 456. 30. 18^ % of 1600.
22. 14% of 96.8. 31. 2.48% of 1940.
23. 3.5% of 1650. 32. 7.8% of 3680.
24. 4 J % of 846. 33. 340 % of 240.
25. 17.25% of 960. 34. 156% of 390.
26. 8J%ofl280. 35. 285% of 346.
27. 8.2% of 34.8. 36. 178% of 1750.
28. 9.3% of 168.4. 37. 204.5% of 34,200.
29. 10J% of 86.3. 38. 196.5% of 17,500.
Multiplying by Powers of 10
Always multiply by any power of 10, as 10, 100, 1000, etc.,
by annexing zeros to a whole number, or moving the decimal
point in decimals. For either has the effect of moving the
digits to higher orders. Thus,
100x75 = 7500; 1000x845=846,000; 10x887 = 8870;
100x6.84=684; 100xl.756»=176.6; 1000x8.46=8460.
Drill Exercises
Give products at sight :
1. 10x86. 6. 100 X. 048. il. 100 x. 1685.
2. 100x64. 7. 10x1.768. 12. 1000x8.8.
3. 100x7.56. 8. 100 X. 165. 13. 1000 x. 046.
4. 100x8.3, 9. 1000x3.5. 14. 1000 x. 0478.
5. 100x17.365. 10. 1000 X. 48. 15. 1000x8.64.
Multiplying by Multiples of Powers of 10
When toth factors end in zeros, work is saved as follows :
500 X 15,000 = 5 x 16 with five zeros annexed.
REVIEW OF ARITHMETICAL PROCESSES 11
400 X 600 = 240,000 ; 700 x 1300 = 910,000 ; 600 x 1800
= 650,000.
Drill Exercises
At iigkt give the products :
1. 30 X 80. 6. 90 x 800. ii. 60 x 130. 16. 60 x 160.
2. 40 X 300. 7. 80 x 600. 12. 70 x 120. 17. 40 x 160.
3. 60 X 70. 8. 70 X 900. 13. 80 x 160. is. 60 x 160.
4. 60 x 800. 9. 30 X 160. 14. 90 x 700. 19. 80 x 200.
5. 300 X 40. 10. 40 X 160. is. 60 x 900. 20. 90 x 120.
21. Find the product of 2800 x 3600.
WORK
8600
2800 Explanation. — Only 36 and 28 were used in the
288 actual multiplication. When this product (1008) was
WQ found, four zeros were annexed.
10,080,000
Find the products :
22. 170 X 8600.* 25. 340 x 860. 28. 420 x 8200.
23. 420 X 6400. 26. 250 x 9800. 29. 350 x 8160.
24. 160 X 8400. 27. 320 x 8620. 30. 230 x 7620.
Hultiplying by Aliquot Parts of 10 or 100
The aliquot part of a number is a number that is contained
in it an integral number of times. Aliquot parts of 10 and
100 are so important that they should be memorized.
Table of Aliquot Parts
6 = JoflO 60 = J of 100 33J= J of 100
2J = ioflO 26«|ofl00 161= J of 100
3} = I of 10 12 J = J of 100 8^ = tt of 1<>0
12 JUNIOR HIGH SCHOOL MATHEMATICS
Tell the reason for the following :
2^x32 = ^ = 80; 3 J^ x 27 = ^il = 90.
25x42 = ^^^0^=1050; 33J x 38 = ^^= 1366|.
Drill Exercises
Find as above :
1. 5 X 846. 7. 50 X 865. 13. 33J x 248.
2. 2J x 936. 8. 25 X 932. 14. 16f x 765.
3. 3J X 729. 9. 12J X 864. 15. 8J x 896.
4. 5 X 1750. 10. 50 X 753. 16. 33 J x 576.
5. 2^ X 1340. 11. 25 X 875. 17. 16f x 645.
6. 3^ X 1650. 12. 12 J X 932. 18. 8 J x 763.
Multiplying by Special Per Cents
Certain per cents are more easily used when changed to
their fractional equivalents. They are :
50% = i 12J% = i , 16|% = i
25% = J S3^% = J 66f% = f
Drill Exercises
Oive at sight :
1. 50 % of 84. 5. 33 J % of 63. 9. 16f % of 180.
2. 25% of 120. 6. 16|% of 36. lo. 16|% of 240.
3. 12J% of 96. 7, 25% of 128. ii. 33^% of 210.
4. 50 % of 420. 8. 121 % of 168. 12. 50 % of 750.
Making Use of Known Products
The following example shows how to make use of a known
product to save work.
REVIEW OF ARITHMETICAL PROCESSES
13
Find 287 x 375.
WORK
375
287
2625
10500
107626
Explanation. — When 7 x 375, or 2626, is known,
280 X 375 can be found by finding 40 x 2625, for this is
40 X 7 X 375, or 280 x 375.
Find as above :
1. 217 X 624.
2. 328 X 725.
3. 426 X 864.
4. 357 X 796.
5. 549 X 834.
6. 637 X 528.
Drill Exercises
7. 459 X 826.
8. 369 X 768.
9. 567 X 936.
10. 248 X 793.
11. 324 X 842.
12. 728 X 966.
13. 546 X 726.
14. 648 X 584.
15. 427 X 645.
16. 355x846.
17. 155 X 964.
18. 287 X 839.
4. DIVISION: WHOLE NUMBERS, FRACTIONS, AND
DECIMALS
For a discussion of the meanings of division, see Book I.
Work is given here for practice and for short methods.
Chech division by going over the work a seeand time.
Find and check :
x. 10,635-^36.
2. 42,738 -J. 63.
3. 90,684 -J- 97.
4. 35,680 -J- 2.96.
6. 42,340-4-36.1.
Drill Exercises
6. 87.35-^2.46.
7. 96.3 -i- 1.76.
8. 8.361 H-. 197.
9. 19.34 -^- .946.
10. 356.2 -*- 42.5.
n. 42.6 + 17.34.
12. 3.98 -J- .063.
13. 17.02 -5- .098.
14. 64.3-8- .185.
15. 4.63-*- .028.
14
JUNIOR HIGH SCHOOL MATHEHfATIGS
16. I+J.
1ft 5. -a- J.
19- i + f
20. 3%-s-|.
25. Divide 368f by 7.
WORK
24. il^f
7 )368|
62, 4| remainder
Hence, 368f H-7 = 52f.
Divide and check :
26. 739^ -^ 8.
27. 676§-^7.
28. 863|-^8.
29. 927J-5-5.
Explanation. — Divide as in whole
numbers until the remainder is less
than the diyiM)r, then divide the frac-
tion or mixed number as shown here.
30. 648§-f-7.
31. 696|-s-8.
32. 726^^9.
33. 697f-^6.
34. 816|-s-6
35. 597f-f-9.
36. 572J-*-8,
37. 678|-^-7,
Diyiding by Numbers Ending in Zeros
Before dividing, all zeros should be cut off the divisor
and a corresponding change made in the dividend. To
divide 1366.48 by 2600, the work should be :
WORK
.62 17
26PJJ)13^66.48
13
66
62
44
26
188
182
Explanation. — Since the divisor was divided
by 100 by cutting off the two zeros, the dividend
was also divided by 100 by moving the decimal
point two places to the left. This follows the
principle that dividing both dividend and divisor
by the same number does not affect the quotient.
REVIEW OP ARITHMETICAL PROCESSES 15
Drill Ezercises
Divide as above :
1. 4686-^200. 9. 8645-S-4800. 17. 36.84-^400.
2. 6786-1-300. 10, 7280 -H 6400. 18. 9.68+500.
3. 783. 6-*- 400. n. 6950-^6300. 19. 17. 35 -^ 600.
4. 875.8-*.900. 12. 746.8-4-3500. 20. 8.3-h700.
5. 658.7-5-700. 13. 693.7-J-2800. 21. 16.2-^500.
6. 9687 -f- 2000. 14. 564. 8 -f- 3200* 22. 4.26 -^ 800.
7. 8766H-5000. 15. 76.84-*.540. 23. 5.6-^700.
8. 9847-^4000. 16. 70.65-8-610. 24. 6.4-8-800.
Diyiding by Aliquot Parts of 10 or lOa
The use of aliquot parts of 10 and 100 is shown by the
following problems. Since 2 J = J^^ 82 -*- 2^ = ^ x 32 = 12. 8 ;
since 33^ = i^, 48 -h 33^ = ^f^ x 48 « 1.44.
The work can be done without a pencil. Thus,
1348-^25 =4x13.48 = 53.92;
1625 -H 33J = 3 X 16.25 = 48.75 ;
3852 -8- 16| = 6 X 38.52 = 231.12.
Drill Exercises
Withovt a pencil find :
1. 8846-1-25. 8. 8692-^-25. is. 165. 8-8-12 J.
2. 1697-I-88J. 9. 1698-S-38J. 16. 21.68-*.16|.
3. 2468-1-50. 10. 2165 -8- 12 J. 17. 75.16-^33^
4. 2178 -^16|. n. 1168 -8- 16|. is. 124.6-8-25.
s. 4268-8-50. 12. 2046-*-25. 19. 216.4-8-50.
6. 1968-^25. 13. 18.48-8-50. 20. 192.8-8-25.
7. 2146-f.l2f 14. 241.5+25. 21. 92.68-8-25.
16 JUNIOR fflGH SCHCX)L MATHEMATICS
When Both Multiplicatioii and Division Occur
When both multiplication and division occur, common
factors should be removed. Thus, 16 x 345 ^ 32 = 345 -i- 2;
24 X 275 -t- 16 = 8 X 275 -J- 2 ; 328 x 16 -*- 8 = 2 x 328.
Drill Exercises
At sight give :
X. 14x284-5-28. 6. 27x84-5-9. u. 9x135^64.
2. 26x125-5-13. 7. 35x96^7. 12. 8x245-f-56.
3. 24x213-5-8. 8. 7x125^35. 13. 56x72-f-8.
4. 7x456^21. 9. 48x96-5-8. 14. 54x86-5-9.
5. 9x378-5-27. 10. 8x160-1-48. 15. 72x68-t-8.
Miscellaneous Drill
At sight give :
1. 90x120. 13. 48+56. 25. 50% of 96.
2. 70x800. 14. 93+49. 26. 25 % of 72.
3. 80x900. 15. 74 + 38. 27. 33^ % of 54.
4. 5x842. 16. 96 + 37. 28. 50 % of 75.
5. 2Jx968. 17. 58 + 96. 29. 25% of 38.
6. 3Jx735. 18. 87 + 56. 30. 12J%of25.
7. 25x840. 19. 78+47. 3i. 16f%of48.
8. 50x348. 20. 93 + 86. 32. 33| % of 58.
9. 12Jx368. 21. 54+73. 33. 75% of 36.
10. 33^x564. 22. 49+56. 34. 66|%ofl8.
u. 16fx456. 23. 75+67. 35. 12 J % of 86.
12. 8Jx560, 24. 63+84. 36. 33j%of84.
^
CHAPTER II
THE FORMULA
In Book I you saw that the principles of mensuration were
expressed in a kind of shorthand, using letters instead of
words. Thus, instead of saying that *^ the number of square
units in the area of any rectangle is the product of the num-
ber of linear units in its two dimensions," you simply said
A = bh.
This shorthand expression is called a formula. This con-
venient form of expressing mathematical relations is used in
various kinds of industrial and commercial work. You will
find formulae used in science, in trade journals, in books on
mechanics, and in various articles that you will read.
State in words the principles expressed by the following for-
mulce^ used in mensuration in Book I:
7. V^Bh.
8. F=7rr2A.
^ Tr 'rrd^h
4
1. EVALUATING FORBIUUB
To evaluate a formula is to substitute the numbers repre-
sented by the letters and then perform the computation.
Thus, in the formula V = abc^ to find V when a = 5, 6 = 6,
and tf = 8, F= 5 x 6 x 8 = 240.
!• Find A when 6 = 5 and A = 6 in the formula A = 6A.
17
1. ^ = hh.
4. A^\cr.
a. ^-2-
5. J. = irr^.
a. A=^J^±n.
2
6. F = abc.
^ I
18 JUNIOR fflGH SCHOOL MATHEMATICS
2. Find A when A = 6, 6 = 14, and J' = 8 in the formula
A - h(h + y )
A 2 •
3. The area of a circle is expressed by the formula ^=7rr^,
where ir = 3.1416 and r the radius. Find the area of a
circle whose radius is 12 ft.
4. The volume of a right circular cylinder is expressed by
the formula V = in^h. Find the volume of a cylinder whose
height is 12 inches and the radius of whose base is 5 inches.
5. Another formula for the volume of a cylinder is
V= — - — , where d represents the diameter. Find the volume
4
of a cylinder whose height is 12 ft. and whose diameter is
10 ft., first using one of the formulae, and then checking by
using the other.
6. If c represents the circumference of a circle, and d its
diameter, express the formula for c in terms of d.
2. smPLiFmro literal exprbssioits
When mathematical principles or relations are expressed
by letters, the expression is called a literal or an algebraic
expression. It is often necessary to simplify these ex-
pressions.
Addition
Just as 31b.+51b.-h21b.=10 lb., so 3 a -f 5a-h 2a=10a
where a represents any value whatever.
At sight give the sums :
1. 2. 3. 4. 5.
8a 4<? 7m 8a: 7y
5a 2e 2m 2x 2y
la 6(? 4m 9a; 3y
6a 8(? 2m 4a; by
THE FORMULA 19
6. Just as 3a + 5a = 8a, so 3x66 + 6x65 = 8x66 =
520 ; and 7 X 428 + 3 X 428 = 10 x 428 = 4280.
At sight give the results :
7. 8x248+2x248. 12. 17x350 + 13x850.
8. 6x785+4x785. 13. 18x620 + 12x620.
9. 7x8.1416 + 3x3.1416. 14. 84x37.5+16x37.5.
10. 4x896+16x896. 15. 48x7.26 + 52x7.26.
11. 12x565+8x665. 16. 85x9.87+66x9.87.
Subtraction
Just as 8 lb. — 3 lb. = 6 lb., so 8a — 3a = 5a, where a
represents any value whatever.
At sight give the differences :
1. 2. 3. 4. 5.
9a 11^ 18a; 16y 12c
2a 66 5x 9y Ic
■ ' ■ ■ ' ■ « ■ ■■
6. Just as 13 a - 8 a = 5 a, SO 13 X 880 - 8 X 380 =: 5 X 380 =
1900.
At sight give the results :
7. 17 X 275 - 7 X 275. 12. 42 x 982 - 32 x 982.
8. 27x820-22x320. 13. 13x640-8x640.
8. 16 X 480 - 12 X 480. 14. 26 x 820 - 21 x 820.
10. 48x750-45x750. 15. 19x720-16x720.
11. 26 X 975 - 16 X 975. 16. 98 x 380 - 88 x 380.
Multiplication
The expression aa or a x a is simplified by writing it a^,
read **a square." Likewise, bbb or b x b x b is written b%
read " b cube " ; and cccc is written c*, read " c to the fourth
power."
2 a X 3 a is simplified to 6 a^ ; 2 a x 4 i, to 8 a&.
20 JUNIOR HIGH SCHOOL MATHEMATICS
Simplify at sight :
1. 35x45. 5. 85x45. 9. 2axSa x. &CU
2. 6ax3a. 6. IcxSc. 10. 4ax2ax^a.
3. 7yx8y. 7. 9dx7d. ii. 25x45x10 6.
4. 9cx4e. 8. 8r X 6r. 12. 5<? x 2<? x 8<?.
13. 3 a X 2 5 is simplified to 6 ab. Simplify 5 5 x 6 <?.
Simplify 3rx6«; 5a x 6t; 7m x 6n.
At sight give the simplest forms of:
14. 5ax75. 17. 8ax7<?. 20. 7dxSe.
15. 65x7<?. 18. 75x6rf. 21. 6cx86*
16. 9d X Qe. 19. 9a x 6d. 22. 9e x Sg.
Division
Since division is the inverse of multiplication, 6 a^ -f- 8 a ss
2a; 8a5H-2a = 45; 16^6-5- 8 = 2de.
At sight give :
1. 8a5-i-4 5. 6. 54 52-!- 6 5. ii. ^2abc-i-6c.
2. dac-i-Sa. 7. 72 ac H- 9 a. 12. 32 acd -s- 8 cd.
3. 16a^-h2a. 8. 48c2^6c?. 13. 45 5cci -s- 6 ic.
4. 21e?d-5-7<?. 9. 56c^-i-7c. 14. 54a;y2-!-6a;y.
5. iSxy-^ex. 10. 63d8-.9rf. 15. 48a25-f-6a2.
16. This work is just what you have done in division of
arithmetical numbers. That is, it is merely canceling like
factors from both dividend and divisor. Thus, 16 tt -f- 2 tt = 8 ;
48 TT -J- 16 TT = 3.
THE FORMULA 21
At sight give:
17. 8 X 9 -h 2 X 9. 23. 49 X 365 -h 7 X 366.
18. 16x7-4-4x7. 24. 72x296-8-9x296.
19. 20 X 17 -4- 5 X 17. 25. 45 X 19 -h 5 X 19.
20. 48 X 87 ^ 8 X 37. 26. 36 x 24 h- 4 x 24. .
21. 54 X 350 -f- 9 X 350. 27. 56 x 52 ^ 8 x 52.
22. 63 X 480 -^ 7 x 480. 28. 64 x 87 ^ 8 x 87.
3. FACTORING A FORMULA
Just as in 5 a -h 3 a the unlike factors 5 and 3 are added,
so in 5 a 4- 5 J the unlike factors a and b may be added. That
is, 5a-h56 = 5(a-f- 6). Thus, if a = 4 and 6 = 6, 4 and 6
may be added before multiplying ; thus making but one
multiplication instead of two.
When 5 a -I- 5 J is changed into 5(a -|- 6), the expression is
said to be factored, for it is changed to the product of two
factors.
■
Itepresent as two factors :
1. 3a + 36. 5. 7a + 7d. 9. Sab + Scd.
2. 7b + 7c. 6. lOx-^lOt/. 10. 5aJ2-f6c2.
3. 5c + 5d. 7. 9a -1-96. ll. 6a6-|-6d.
4. 8a + Sc. 8. 2a2 + 262. 12. ixg + ^ab.
Give the value of:
13. 7 X 35 -h 7 X 65. 16. 8 x 17 -f 8 x 33.
14. 9 X 25 -h 9 X 75. 17. 9 x 21 -|- 9 x 29.
15. 8 X 36 4- 8 X 64. 18. 7 x 83 + 7 x 17.
22 JUNIOR HIGH SCHOOL MATHEMATICS
4. FORMUUB DERIVED FROM OTHER FORMULS
When certain fundamental formulae are known, others
needed may be derived from these. Examples to show how
this is done are*given here.
1. You know that the area of a rectangle is represented
by A = bh, where A represents the number of square units
in the area, and b and A represent the number of linear units
in the base and height, respectively. From this we know-
that J.-4-6 = A orJ.^A=6. Instead of the division sign,
these are usually written
A = 4 and 6 = 4-
b h
This result is evident from the meaning of division. That
is, the product of two factors divided by either gives the
other.
2. From the formula c = wd^ give a formula for d in
terms of ir and c,
3. From the formula F= Bh^ give the value of B in
terms of of Fand A. Of A, in terms of J^and B.
4. From the formula derived in problem 2, find the
diameter of a circle whose circumference is 150 ft.
5. From the first formula derived in problem 3, find the
area of the base of a prism whose volume is 600 cu. in. and
whose height is 20 in. Find the height of a prism whose
volume is 500 cu. in. and the area of whose base is 25 sq. in.
6. From the formula <? = 2 ttt*, give the value of r in
terms of <? and 2 tt.
7. From the formula -4 = ~, it is evident that 2A^hh.
From A^\ h(b -f J')? g^^e the value of 2 A.
THE FORMULA 23
8. From the formula ^ = — , derive a formula for 6, and
one for A.
9. When the area of a triangle is 64 sq. in. and the base
16 in., what is the altitude?
10. In the formula V^ ahc^ give a in terms of FJ 6, and c.
11. In the formula A^\ h(b + 6'), give h in terms of the
other letters.
12. How high must a trapezoid be whose bases are 8 in.
and 12 in., respectively, if the area is 60 sq. in. ?
13. To hold 16 tons of coal when filled to a depth of 6 ft.,
how many square feet must there be in the floor of the bin ?
(1 cu. ft. of coal = 65 lbs.)
14. To hold 16 tons of coal, a bin 8 ft. by 10 ft. will have
to be filled to what depth?
15. The volume of a cylinder is expressed by the formula,
T^= wr^h. To what depth will 6 gal. of milk fill a milk can
14 in. in diameter ?
16. A circular running track J mi. (660 ft.) around has
a diameter of how many feet ?
17. To what depth must a box 80 in. by 42 in, be filled to
hold 6 bu. ? (1 bu. = 2160.42 cu. in.)
18. A garden 120 ft. long must be how wide to contain
the same area as a garden 86 ft. by 96 ft. ?
19. A triangle with a base of 24 in, must have what alti-
tude to contain 192 sq. in. ?
CHAPTER III
THE EQUATION
An equation is a statement that two expressions are equal,
or that they have the same value. The formulae which you
have studied were equations. So are such expressions as
2 X 12 = 4 X 6and3w = 15.
In reducing formulae to other forms, and in solving many
problems that arise in mathematics, use is made of the
equation.
In solving any problem, we are seeking an unknown value.
To solve a problem by use of an equation, the unknown
value is expressed by some letter and the relation of the
known to the unknown is expressed by an equation.
Thus, in the problem, " What number added to 16 gives a
sum of 21 ? " the relation may be expressed
w + 16 = 21, an equation.
It is evident that w = 6, for 6 is the only number which
added to 16 gives 21.
The expression on each side of the sign of equality is a
member of the equation.
Such problems as these are solved as easily without the
use of the equation, but the illustration is given merely to
furnish a simple example of how an equation may be used
in the solution of a problem.
24
THE EQUATION 25
1. DETERMINING THE VALUE OF THE UNKNOWN
NUMBER
To find a value of the unknown number that makes both
sides equal, or satisfies the equation, is to solve the equation.
By inspection you can solve such equations as a: -f 2 = 6, for
this merely asks, " What number added to 2 equals 6 ? " and
you know that it is 4, for 4 + 2 = 6.
At sight solve :
1. a: -h 3 = 9. 6. 7 -h a; = 12. 11. w + 7 = 14.
2. a: + 6 = 10. 7. 8 -fa: = 14. 12. 8 + «=22.
3. a: 4- 5 = 20. 8. 11 + a: = 20. 13. 9 + r = 17.
4. a:+7 = 18. 9. 12 + a: =30. 14. f + 6 = 15.
5. a? + 9 = 14. 10. 15 + a; = 40. 15. 3 + w = 20.
Equations Requiring Subtraction
In the above equations, you found the value of the un-
known by subtraction. Thus, in a; + 3 = 9, to find what
number added to 3 equals 9, you subtracted 3 from 9.
In more complicated equations that arise, use is made of
an evident truth, called an axiom. This is stated thus,
If equals are subtracted from equals^ the remainders are
equal.
This is too evident to be questioned, but an illustration is
of interest. An equation is really an expression of the
balance of values and may thus be com-
pared to the balance used in weighing.
Each member of the equation corre-
sponds to the weights in the two pans.
If the weights in either pan are
changed, there must be a corresponding change in the
other.
26 JUNIOR HIGH SCHOOL MATHEMATICS
Thus, in the equation a: + 8 = 12, if each member repre-
sents weights in the pans of the balance, and 8 is taken from
one pan, it must be taken from the other also.
Hence, from a? + 8 = 12
we have a; » 12 — 8 ; (8 being subtracted from
each member)
hence, a: = 4, the solution.
The solution is checked by substituting the value found
and seeing if it satisfies the equation. Thus, 4 -f 8 =«: 12 sat-
isfies the equation, and the solution is correct.
Solve :
1. 71-1-4 = 9. 6. 84-2^=15.2. 11. 2J-f-a;*=7J.
2. a;+6 = 16. 7. 2.5+a;=:9.76. 12. 3J-ha^=9f.
3. a; + 12 = 38. 8. ic+| = 2J. 13. 1.5-|-rr=16.26.
4. 3x=2x-hl. 9. a:+2.8 = 7.63. 14. 8.4 + a;=9.8.
5. 6ir = 4a;+9. \o. y+1.09 = 6.81. 15. 7.8+a;»16.
Equations Requiring Division
The solution of a problem may require such an equation
as 3ic+5 = 17. Here we see by subtraction that 3a; =12.
The question here is, '' What number multiplied by 3 equals
12 ?" From the meaning of division we see the answer is
4. But, using the balance again to illustrate the two mem-
bers of an equation, we see that if one member is divided
the other member must also be divided by the same number.
This is expressed as an axiom by saying.
If equal numbers are divided by equal numbers (not zeros\
the quotients are equal.
Thus, if 8 a: = 12,
then a; = 12 -^ 3 = 4, by dividing each member by 8.
THE EQUATION 27
Solve:
1. 3ir = 16. 6. 3a: + 7 = 25. U. 5rr+7=c42.
2. 7a: = 42. 7. 5a;4-2 = 32. 12. 6a; + 8 = 60.
3. 9a; = 36. 8. 7a: + 6 = 40. 13. 7a: +10 = 66.
4. 12a: = 72. 9. 8a: + 2 = l8. 14. 8a: 4- 9 =66.
5. 8a: = 56. lo. 9a: + 7 = 70. 15. 7a: + 15 = 36.
Equations Requiring Addition
The solution of a problem may require such an equation
as a: — 5 = 20. The question here is, " From what number
may 5 be subtracted and leave 20?" The answer is evi-
dently 26. But since the expression a: — 5 is 5 less than a:,
5 would have to be added to it to get x. That is, a:— 5 + 5 =a:.
Using the balance again as an illustration, 5 was added to
each member without affecting the equation. Stated as an
axiom,
Jf equals are added to equals, the sums are equal.
Thus, if a: - 5 = 20,
then a: = 20 + 5, by adding 5 to each member.
Solve :
1. a:- 3 = 8. 6. a:- 48 = 125. 11. 8a: -9 = 6.
2. a:- 7 = 15. 7. a: -96 = 84. 12. 4a:- 10 = 18.
3. a:- 3 = 2. 8. a:- 8.4 =14.2. 13. 5a:- 8 = 17.
4. a:- 4 = 7. 9. a: -3.6 = 9.6. 14. 3a: -90 = 90.
5. a;- 6 = 10. 10. a; -1.8 = 2.1. 15. 4a; -60 =100.
Equations Solved by Multiplication
The solution of a problem may require such an equation
X
as - = 7. The question here is, " One-third of what number
is 7 ? " The answer is evidently 21. While you could have
28
JUNIOR HIGH SCHOOL MATHEMATICS
answered this simple question by trial or inspection, it is
usually solved by multiplying both members by 3. The
authority for this, expressed as an axiom, is.
If equals are multiplied by efpjuxls^ the products are equal.
Thus, if
then
Solve :
5 = 7
3 ''
X = 3 X 7, by multiplying both members by 3.
1 £
2
X
'•3
X
'•6
= 8.
s9.
= 7.
4. ? = 12.
•■I
= 9.
Solve :
D* -— = 0«0»
7. f = 2.4.
4
8. ■— = AmOm
8
9. 1 = 3.4.
7
10. 1 = 1.2.
«7
X
11. | = 3|,
X 2
12. - = —
4 3
13. ? = ^.
7 6
X 2
8 3
15. -5- = ?.
10 6
Miscellaneous Exercises for Drill
1.
a: + 3.6 = 8.
7.
X 2
7"3-
13.
i=^i-
2.
a;-6 = 12.
8.
5a:-3 = 42.
14.
a:- 4^ = 3.
3.
2a: + 6 = 28.
9.
a: +3.1 = 9.6.
15.
6x-7 = 47.
4.
4a;- 1 = 23.
10.
3a;+7 = 34.
16.
a; + f = 7}.
5.
X +4| = 7.
11.
a; -3.5= 1.9.
17.
5x4-6 = 46.
6.
|a;=9.
12.
4^ + 3 = 11.8.
18.
8a;-3J = l|.
THE EQUATION 29
2. PROBLEMS SOLVED BY EQUATIONS
You have met no problems in your earlier mathematics
that could have been solved more easily by algebra. And,
at present, any problem to which you can apply algebra is
more in the nature of a puzzle than a problem that meets a
real need. To understand some of the work that follows,
however, you will need a simple knowledge of equations such
as you have gained from studying this chapter. A few
problems follow, to give practice in expressing a relation in
the form of an equation, and not to meet any real need in
life.
1. A line 20 ft. long is to be divided into two parts so
that one part is 2 ft. mor^ than twice the other. Find the
two lengths.
While this problem may be solved by arithmetic, an alge-
braic solution is simpler, as shown below.
SOLUTION
Let X = the number of feet in the shorter part ; then 2 x + 2 = the
longer part, for this is 2 more than twice a:, or the smaller number.
Then a: + 2a: + 2, or 3 a: + 2 = 20.
Hence, 3 a: = 20 — 2 = 18, subtracting 2 from each member.
Then a: = 18 -r- 3 = 6, dividing each member by 3.
And 2a:-f-2 = 2x6 + 2 = 14, substituting 6 for x in the expression
for the longer part.
2. Find the lengths into which a rod 16 in. long
must be cut so that one piece will be 4 in. longer than
the other.
3. In a class of 32 pupils, there are 4 more girls than
boys. How many of each are there ?
4. If it takes 240 ft. of fencing to inclose a rectangular
garden 20 ft. longer than it is wide, how long and how
wide is it ?
30 JUNIOR fflGH SCHOOL MATHEMATICS
5. The perimeter (distance around) of an isosceles tri-
angle is 66 in. Each of the two equal sides is 4 in. longer
than the base. Find the length of the base and of each of
the equal sides.
6. Find two consecutive numbers whose sum is 117.
Suggestion. — Let x = the smaller of the two numbers.
Then x + 1 = the other, for, being consecntive, it must be 1 larger
than the other.
Then x + a: + lor2a: + l = 117. Now solve the equation.
7. Find three consecutive numbers whose sum is 128.
8. Find two consecutive odd numbers whose sum is 56.
Suggestion. — Let x be one and a: + 2 the other. Why ?
9. Find two consecutive even numbers whose sum is 98.
10. Two boys together sold a total of 98 papers. One boy
sold 12 more than the other. ^Find how many each boy sold.
11. A woman paid 90 ^ for a pound of coffee and a pound
of tea. She paid twice as much for the tea as for the coffee.
How much did she pay for each ?
12. A rectangular lot is twice as long as it is wide. The
distance around it is 90 rd. Find its dimensions.
13. Paul and Henry together have 38 marbles. Paul has
10 more than Henry. How many has each ?
14. A rectangular lot 40 ft. longer than it is wide is 400
ft. around. Find its dimensions.
15. A boy's salary doubled each year for 3 years. The
third year it amounted to f 20 per week. What was it the
first and second years ?
16. When 60 is added to a certain number it gives a num-
ber four times the given number. What is the number ?
17. When pears cost twice as much as apples, a boy
bought 10 of each for 60 ^. Find the price of each.
CHAPTER IV
RATIO AND PROPORTION
1. THE MBAKING AlTD USE OF RATIO
We compare numbers in two ways, either by subtraction
or by division. Thus, we say of a number that it is so
much larger or so many times as large as another. Thus, 9
is 6 larger than 8, or it is 3 times as large as 8.
When numbers of the same kind are compared by division,
the relation is often called a ratio. Thus, the ratio of f 12
to 18 is 4 ; of 8 ft. to 8 ft., | ; of 5 in. to 2 in., 2J; of 8.8
in. to 4.5 in., .844+; the ratio of the circumference of a
circle to its diameter is tt; etc. The ratio, then, of one
number to a like number is the quotient found by dividing
the first by the second. That is, tbe quotient is the ratio of
the dividend to the divisor.
Since the quotient of any number divided by a like
number, as feet by feet, or dollars by dollars, is abstract, we
see that a ratio is always abstract.
1. What is the ratio of 8 ft. to 6 ft.?
Always give a ratio in its simplest terms. Thus, the
above ratio is J.
2. Give the ratio of 12.50 to 17.60; of $8.50 to
of 6 ft. to 4J ft. ; of 9 mi. to 6 mi.
Give the ratio of:
3. 8 in. to 12 in. 5. 27 gal. to 18 gal.
4. 9 ft. to 15 ft. 6. $54 to $45.
31
32 JUNIOR fflGH SCHOOL MATHEMATICS
7. 300 mi. to 60 mi. 11. f; 38.40 to $17.
8. 75 yd. to 60 yd. 12. * 96.50 to $120.
9. 80 rd. to 140 rd. 13. $8.75 to $16.30.
10. 700 ft. to 175 ft. 14. 9.3 ft. to 16.2 ft.
15. A certain rectangular garden is 60 ft. wide and 80 ft.
long. What is the ratio of its width to its length ?
16. When a boy sells 48 of 52 papers received, what is
the ratio of those sold to those received ?
In the past, ratio has often been written with a colon be-
tween the two numbers. Thus, the ratio of $7 to $12 was
written $7 : $12 and read "As $7 is to $12." While this
form is but little used, the expression "as 3 is to 4" and
like expressions are still used. Thus, to say that the dimen-
sions of a rectangle are " as 3 is to 4 " means that the width
is I of the length, or that the length is IJ times the width.
17. The dimensions of a garden are as 2 is to 3. If the
garden is 60 ft. wide, how long is it ? If it is 80 ft. longi
how wide is it ?
18. If the dimensions of a page in a book are 7^ in. and
5 J in., what is the ratio of the length to the width ? If a
page is 10 inches long, how wide must it be to have the
same ratio ?
19. Construct any two line-segments that are in the ratio
of 3 to 4 ; of 4 to 5.
20. If a man who earns $ 25 per week is saving $ 10 of it,
he is saving what ratio of his earnings ?
2. THE MEANING AND USE OF PROPORTION
The equation expressing the equality between two ratios
is called a proportion. For example, —^ = zr^-r' ^^ ^ propor-
RATIO AND PROPORTION 33
tion. It is read " $3 is to 18 as 6 ft. is to 16 ft.," or "the
ratio of $ 3 to $ 8 equals the ratio of 6 ft. to 16 ft."
The above proportion was formerly written
: : 6 f t. : 16 ft.
But this form is rapidly going out of use. The ratios are
considered fractions, and the equality of two ratios is but an
equation and follows all the principles of any equation.
Formerly, the subject of proportion was much used in
solving many of the problems of everyday life. Now, it is
rarely ever used. For example, such a problem as "If 5
acres of potatoes yield 800 bu., how many bushels will 3
acres yield at the same rate?" was solved by proportion.
The statement was
a:: 800:: 3:6
That is, the relation or ratio of the yield is the same as
the ratio of the areas producing the yield.
This was then solved by the principle that
The product of the extremes (the end terms) is eqvial to the
product of the means (the second and third terms).
This gave the equation 5 a; = 2400, from which x = 480, by
dividing each member by 5.
Even when proportion is now used for such problems, the
proportion is stated as an equation as follows :
X ^3
800 5*
3 X 800
Hence, x = , by multiplying both members by 800.
X = 480.
Such problems are not usually solved by proportion.
They- are usually solved by "unitary analysis" or by the
"ratio method." That is, one says, "If 5 acres yield 800 bu..
^ I
34 JUNIOR fflGH SCHOOL IJ^ATHEMATICS
1 acre will yield 160 bu. and 3 acres will yield 8 x 160 bu,,
or 480 bu." or ''The 8-acre field will yield f as much, henoe,
f X 800 bu., or 480 bu."
The use of proportion is now confined largely to a method of
stating certain geometric and scientific relations or principles.
Proportionality of Areas
1. Compare the areas of two rectangles, one whose
dimensions are 6 in. and 10 in., and the other whose dimen-
sions are 8 in. and 15 in.
SOLUTION
Area of first _ 6 x 10 _ 1
Area of second 8 x 15 2*
Therefore their areas are to each other as 1 is to 2.
2. Compare two rectangles whose dimensions are 10 in.
and 12 in., and 15 in. and 18 in., respectively.
3. Compare the area ^ of a rectangle whose dimensions
are a and 5, with the area A^ of a rectangle whose dimensions
are a' and V ,
A iih
4. The proportion ^77 = -777 » found in problem 8, is
A^ a'6'
stated in words as a principle often used in mathematics as
follows :
The aretM of two rectangles are to each other 09 the product
of their dimensione.
5. Find the relation of two rectangles having equal bases.
Suggestion. — Let a and b be the dimensions of one, and c and b of
the other, for the base b is the same in both.
A a
6. The proportion — = -, found in problem 5, is stated
-A t?
as a principle as follows :
77ie areas of two rectangles having equal bases are to each
other as their altitudes.
RATIO AND PROPORTION 36
7. Find the relation of two rectangles having equal alti-
tudes, and state the result as a principle.
6. Compare two parallelograms having unlike dimen-
sions, as you compared rectangles in problem 3.
State the proportion as a principle.
9. Compare a triangle whose base is 10 ft. and altitude
8 ft., with one whose base is 12 ft. and whose altitude is
12 ft.
10. Compare a triangle whose dimensions are a and 6,
with one whose dimensions are a! and V.
A t
11. State the relation ---3= ----, which you found in prob-
lem 10, as a principle. (See problem 4.)
12. Compare two triangles, each with a base of 10 in., one
having an altitude of 7 in., and the other an altitude of 9 in.
13. Compare two triangles with equal bases 5, and un-
equal altitudes a and ^, and state the relation as a principle.
(See problem 6.)
14. Compare the area of a square whose sides are each
6 in., with one whose sides are each 7 in.
15. Compare the areas of two squares whose sides are
respectively a and a' units.
16. The relation -jj^^^'^ expressed by a principle as
follows :
The areas of two squares are to each other as the squares of
their sides.
17. Compare the area of a circle whose radius is 5 in.,
with one whose radius is 8 in.
First circle __ ir 25 sq. in. _ 25
Second circle ir 64 sq. in. 64 '
36 JUXIOR fflGH SCHOOL MATHE3^DITICS
18. Show that the areas of two circles are to each other as
the squares of their radii.
19. If one circle has a radius twice that of another, bow
do their areas compare ?
20. If one square has a side three times that of anot^Iier,
how do their areas compare ?
21. If the len^h of a rectangle Ls made twice as great
and its wddth three times as great, how many times i>^ its
area increased ?
22. John's garden is twice as long as Frank's but only
three fourths as wide. John's garden is how manj- times as
large as Frank's ?
23. The carrying capacity of a water pipe depends upon
the area of a cross-section of the pipe. A pipe with twice
its diameter w^iU discharge how many times as much water ?
24. Compare the carrying capacity of a |-inch pipe with
that of a 2-inch pipe.
25. If the linoleum for a square floor costs WO, how much
will the same grade of linoleum cost for a square floor but
half as long, not considering waste ?
CHAPTER V
SIMILAR FIGURES
Any two figures having the same shape are called similar
figures. Thus, two squares, two equilateral triangles, or
two circles are similar, for they have the same shape.
1. FUNDAMENTAL PRINCIPLES OF SIMILAR FIGURES
1. These two polygons are similar. Compare the length
of FG- with that of AB by the use of your ruler or com-
passes.
2. Compare all the corresponding sides, as 6r-H* with jB(7,
^7 with CD, 77 with BE, and TFwith EA.
3. In the following similar triangles, compare corre-
sponding sides and the altitudes that are drawn.
If your measurements were carefully made, you found the
ratio of corresponding sides in problem 1 to be 2. In prob-
lem 3 the ratios were each 1\.
37
38 JUNIOR HIGH SCHOOL MATHEMATICS
And in general,
The corresponding lines in similar figures are proportional.
4. Draw any triangle ABC. Now bisecting each side,
construct a second triangle with these half sides, making' a
triangle similar to the first. Measure the corresponding
angles of the two triangles.
5. If carefully drawn and carefully measured, the cor-
responding angles in problem 4 were found to be equal.
And in general,
ITie corresponding angles of similar triangles are eqiud.
6. Upon any two unequal bases, as AB and D^, con-
struct triangles with base angles 40° and 60°, respectively.
bdI$2L
Find the ratio of the corresponding sides. If caref uUy drawn,
the triangles are similar and the corresponding sides have
the same ratio. And in general,
If the angles of one triangle are respeetivelt/ equal to those
of another, the triangles are similar.
7. Construct any triangle on
cardboard. Then cut off the three
comers and place them as in the
figure. It is seen that the three
angles form a straight angle. And in general,
ITie sum of the three angles of any triangle is equal to 180
degrees.
SIMILAR FIGURES
39
a. If two angles of a triangle are 40° and 60°, respec-
tively, what is the third angle ?
9. It is thus seen that if two angles of a triangle are
known, the third one can be found. Hence, the principle
following problem 6 can be stated,
If two angles of any triangle are equal to two angles of an-,
other ^ the two triangles are similar.
10. In right triangle ABG^ if a line is drawn cutting off
a right triangle ADE^ the triangle ADE is similar to tri-
angle ABO^ for they each have a right angle and angle A
is common to both. If AD = 10 ft., AB=25 ft., and
DE ■= 8 ft., what is the length of CB ?
S. PRACTICAL MEASUREBfBNT OF DISTAlfCBS
The proportion between the corresponding sides of similar
triangles gives a practical method of finding distances where
direct measurement is impossible. A few such applications
are shown in the following problems.
1. Since the time of the ancient Greeks, the heights of
objaots have been found from the lengths of their shadows.
40
JUNIOR fflGH SCHOOL MATHEMATICS
Thns, in the figure, the height of the two objects, BC and EF', theii
shadows, AB and DE ; and the sun's rays passing over their tops, A C
and DF, form two Himilar triangles.
F
\Eji.
Since DE, EF, and AB can be found by measurement, BC, the un-
known length, can be found from the proportion. Suppose DE = 24 ft.,
EF = 20 ft., and AB = 15 ft., by letting BC = x, we have the proportion
_x ^20
15 24'
20 X 15
Hence,
X =
24
, by multiplying both members by 15.
2. When a tree casts a shadow 48 ft. long, a vertica-l staff
6 ft. high casts a shadow 8 ft. long. Find the height of the tree.
3. A telephone pole casts a shadow 42 ft. long when a
fence post 5 ft. high casts a shadow 6 ft. long. How high
is the telephone pole ? ^
4. A boy used the fol-
lowing method to find the
height of a tree : he used
a right triangle whose sides
AB and BC were equal.
He found a spot in the
tree, at i>, just as high as
his eyes. Then he walked
back, sighting along AB
to 2>, until he could see
SIMILAR FIGURES 41
M in line with AC, He then measured the distance Alt.
If AD is 60 ft., how high is the tree if i> is a point 5 ft.
from tlie ground ?
s Using the length of shadows, lind the heights of trees,
telephone poles, water towers, or other heights, then check
the results by use of a right triangle as in problem 4.
SuGOEgTiOM. — In making the right triangle, more accurate reeultd
nil! be obtained if the equal sides are at least 2 or 2\ ft. long.
6. The distance AB across a stream may be found by
measuring along the shore at right angles to AB to some
point f, then measuring at right angles
to AC to some point I>. Now sighting *
from DtoB, mark point E where the line
of sight crosses AC. AB corresponds
to DC and AE to EC. Write the
proportion by which AB may he found.
7. In the figure of problem 6, if DO = 150 ft., CE = 100
ft., and EA = 200 ft., find AB.
8. An easily devised instrument, as shown in the illus-
tration, can be used to find the distance from a given point
to some inaccessible point. AC is a
staff at the end of which is a movable
frame ECD in which EC and CD are
joined at right angles. With CD
pointing to the object B, and by noting
a point F in the ground toward which
CE points, the distance AB can be determined, for triangles
ABO and ACF are similar. Find AB when AC = 5 ft.
and AF = 2 ft.
42
JUNIOR HIGH SCHOOL MATHEMATICS
9. Find the distance from AtoBhj taking the following
measarementB : BP and AP are measuied and are 240 ft.
and 500 ft., respectively. Then
from P, P2> and P(7 are laid
off, respectively, 24 ft. and 60
ft. (each ^ of the BP and AB)^
making triangle POD similar
to triangle PAB. If 2>(7 = 38
ft., how far is it from A to B?
10. In problem 9, if JPB and P (7 have been laid off equal
to J of BP and AP and BO had been found to be 38 ft.,
what would the distance from AtoB have been ?
11. A woodsman measures the height of a tree H)H/J^t5^
to the first limb very approximately as follows : "^^ ^^
Walking back from the tree and holding his axe- ^^"^^
handle perpendicularly at arm's length, he finds a
position from which the axe-handle
just covers the height he wishes to
find. By measuriug the distance to
the tree, he computes the height.
When standing 25 ft. from the
tree, what is the height covered by a 30 in. axe-handle held
27 in. from his eye ?
^ 12. Draw any right triangle. From any
point 2> draw a parallel to AB by erecting a
perpendicular to AC. Carefully measure AB
and BO and find the ratio of AB and
BO. Then measure B£J and JEO and
find the ratio of UB to JSO.
If accurately drawn, measured, and
computed, you will find the ratios to
B be the same. And in general.
SIMILAR FIGURES
43
A line parallel to one side of a triangle divides the other two
sides proportionally/.
13. Wishing to measure the distance across a pond from
AtoB^ some boys made use of the
principle found in problem 12.
They measured to in line with
AB. From B and O perpendiculars
to AC were drawn, and points JE
and D on these perpendiculars, and
in line with A^ were marked. If
AH = 400 ft., HB = 300 ft., and
^(7 = 200 ft., what was AB ?
14. Find heights and distances that you can measure by
the principles of similar triangles.
3. MAPS AND PLANS: DRAWINO TO SCALE
Maps and plans are figures similar to the figures which
they represent. Thus, a map of a state is a drawing similar
In shape to the figure formed
by the state itself. The draw-
ing in the margin represents
an architect's floor plan of a
house. To understand maps
and plans requires a knowl-
edge of the meaning of draw-
ing to scale, for all such figures
are thus drawn. The maps
of any geography usually give
on the map the scale to which
it was drawn. Thus, a map in
which 200 miles are represented by 1 inch is said to be drawn
to scale 200 miles to 1 inch j or " Scale 1 in. « 200 mi."
44 JUNIOR fflGH SCHOOL MATHEMATICS
1. Using a map in your geography and the scale to
which it was drawn, find the distance in a straight line from
New York to Chicago. From New York to San Francisco.
From Chicago to New Orleans.
2. A map of Illinois, scale 1 in. = 200 mi., is 1| in. long.
How long is the state ?
3. On a map, scale 1 in. = 240 mi., it is 3^^^^ in. from
Chicago to Denver. How far is it from Chicago to Denver ?
4. If the floor plan shown on page 43 is drawn to scale
1 in. = 16 ft., find the dimensions of the living room ; of the
dining room ; of the porch.
5. Draw a plan of the floor of yonr room to scale 1 in.
= 10 ft.
6. When the plan of a room 20 ft. by 30 ft. is 5 in. by
7.5 in., what is the scale?
7. Draw the plan of a garden 48 ft. by 120 ft. to scale
1 in. = 24 ft.
8. ** Scale ^ '' means that the dimensions of the plan are
each \ of those of the thing represented. Draw to scale 4
the plan of a rectangle 12 in. by 16 in.
9. Draw to scale ^V the plan of a table top 4 ft. by 7 ft.
10. The distance to an inaccessible object may be found
by drawing a plan to scale. Thus, to ^
find the distance from ^ to ^, measure /I
oS AC perpendicular to AJS. To make j/^ \
a map or plan, lay ofif any length to ^y {^
represent AC and from the ends of the ^^^^
line-segment thus taken construct angles y
equal to the given angles. ^ ^
Supposing that .4(7=800 ft., AACB^W, and ABAC
= 90*^, construct a map to scale 1 in. = 100 ft. and compute
the distance from .4 to B^
SIMILAR FIGURES 45
11. To find the height of
the church spire as shown
here, a line AB, 80 ft. long,
toward the foot of the spire
was taken. From A and B
the angles of elevation of
the top were tnken. /LDAQ
= 50° and ^BBC = 80°.
Make a diagram to scale 1 in.
= 20 ft. and find the height
of the spire. j*
12. The following is the floor plan of a cottage. From
dimensions marked, find
the scale to whicli it was
drawn. Check by using
other dimensions.
13. From the scale
found, find the dimensions
of the porch. Of the
bathroom. Of the entire
floor plan.
14. If possible, bring
to class some architect's
real plans for a house and
interpret them.
CHAPTER \T
TRIGONOMETRIC RATIOS
Ix Chapter V yon saw some of the uses of the prop-
erties of similar triangles in deter-
mining heights and distances. In this
chapter a further use of ratios will
be shown. It will be shown how to
find the height BC oi the figure rep-
resented in the margin bj knowing
the distance from ^ to ^ and the
angle of eleyation at Ai that is, the
angle formed by AB and A C.
1. TAHGBHT RELATIOHS
1. Draw any right triangle ABC^ right angled at J9, with
angle A = 30°. Measure
BC and AB carefuUy c
and find the ratio of BO
to AB. (The longer you
can take AB the more
accurate you will be
likely to get the ratio.)
2. Make other tri-
angles, using different
lengths for AB^ as 10 in.,
15 in., or 20 in., but keeping angle A equal to 30 degrees,
and find the ratio of BC to AB.
46
TRIGONOMETRIC RATIOS 47
If accurately constructed and . computed, you found the
same ratio in each case, and to the nearest hundredth it
was .58.
This follows from the fact that all the triangles you con-
structed were similar, and that the ratio of similar sides of
similar triangles is constant; that is, the ratio is always
the same.
The ratio of the perpendicular BO to the base AB is
BO
called the tangent of angle A. It is written tan Z.A^ — — .
In the problem given, it was found that tan 30*^ = .68.
The tangent is but one of the six possible ratios, called
trigonometric ratios, in any right triangle. Being the one
used in finding heights, it is the only one defined here. The
others are called sine A^ cosine A^ secant A^ cosecant il, and
cotangent A. Their meaning and use will be taken up in
Book III of this course.
3. In the figure of problem 1, if AB = 75 ft., find BO.
80X.UTI0N
Let BC = X,
Then — = .58, for the ratio is the tangent of 30°, which is .58.
75
Hence, x = 75 x .58 = 43.6, the number of feet.
4. In the same figure, if AB =^ 120 ft.,
find BO.
5. Find the height X of the flag pole AB
when the angle of elevation at a point 30 ft.
from the foot of the pole is 70®, having
given that tan 70° = 2.75.
6. If the angle of elevation to the top of
a church spire at a point 50 ft. from the
foot of the spire is 60% having given that
tan 50° = 1.19, find the height of the spire.
48
JUNIOR mCH SCHOOL MATHEMATICS
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71
2.9«> 1
89
57.29
18
^12
36
.73
:a
1.37
72
3.«>S
<«
1. The distance from an observer
pole is 48 ft. The angle of elevation
of the top of the pole at the point of
observation is 52°. Find the height
of the pole.
2. Two boys measure the height of
their kite as follows : when the kite
is directly over one of the bovs the
boy holding the kite string is 700 ft.
away, and the string makes an angle
of 40^ T*ith tlie horizontal. Find the
height of the kite.
to the foot of a flag
TRIGONOMETRIC RATIOS
49
3« To find the distance AJB across a small lake, -A (7 was
run at right angles to AB. AC
measured 750 ft., and the angle
^OA measured 54°. Find the dis-
tance from A to -B.
4. A tower known to be 260 ft.
high forms an angle of elevation of
25° from the point of observation.
How far away is the observer ?
In this problem
?— = tan 25° = .47 or
47
To solve it, write it
X
X
100
0-
^100
260 47 '
Knowing that since one acute angle is 25° the other must be 65", we
could have solved the problem from the equation :
-^ = tan 65° = 2.14.
260
Suggestions. — Solve in both ways. The fact that the two answers
do not quite agree comes from using a table true to the second decimal
place only. Had tan 25° = .4663 and tan 65° = 2.1445 been used, the
answers would have more nearly agreed.
5. A mariner finds that the angle of elevation of a light
from a lighthouse known to be 80 ft. above the level of the
ship, is 8°. How far away is the lighthouse ? {^This is like
problem 4.)
6. When a vertical rod 6.3
ft. high casts a shadow 9 ft.
loBg, what is the elevation of
the sun ?
SOLUTION
tan^ =— = .7,
9
In the tables, tan 35° = .7.
Hence, angle A ^ 35°.
50 JUNIOR fflGH SCHOOL MATHEMATICS
7. When a man 6 ft. tall casts a shadow 10 ft. long,
what is the elevation of the sun ?
8. The angle of elevation of an aeroplane from the
observer was 70®. The aeroplane was directly over a point
1200 ft. away. Find the height of the aeroplane.
9. When the legs of a right triangle are 10 in. and 22.5
in., respectively, find the number of degrees in each of the
two acute angles.
10. When the elevation of the sun is but 20°, how long a
t»hadow will a boy 5^ ft. tall cast ? (See problem 4.)
11. From an aeroplane an observer notes that the angle
of depression of the enemy trench is 46% and that his eleva-
tion is 6000 ft. Find the distance from a point on level
ground directly below the aeroplane to the trench.
Note. — The angle of depression is the same as the angle of elevation
from the trench to the aeroplane.
12. From the point of observation 38 ft. above the water
on a transport, the periscope of a submarine is noted at an
angle of depression of 15^. How far away is the submarine ?
CHAPTER VII
GRAPHIC METHODS OF REPRESENTING FACTS
In order to present quantitative facts so as to make the
relations stand out clear and effective, writers in the news-
papers and magazines, officials in making reports, adver-
tisers, lecturers, and others attempt to picture the relations
to the eye. These pictured forms of presenting facts are
called graphic methods of presentation.
The graphic method has not become a fixed form of
presentation yet, as forms of computation have. The graphs
are nearly as varied in form as the number of persons using
them. They are roughly classified, however, under three
general heads : straight lines or bars ; broken lines ; and
circles.
The following graph is one of the best forms of the bar
graph.
THE NUMBER OF MEAT-PRODUOINO ANIMALS IN 1916. THESE INCLUDE
CATTLE, SHEEP, AND PIGS.
MILLIONS
20 40 60 80 100 120 140 160
UNITED STATES ^^^^^^^^^^^^^^^^^^^^^^^^
170.000.000 pi^^MlfMBI^MMfBBH^iHiMB^iBM
■
RUSSIA ^^^^^^^^^^^^^^^^^^
147.000.000 ^^^^^^^^^^^"^^^^^"
■
ARGENTINA L^^^^^ g-^J^I--
115.000.000 ^^^^^^^^^^^^
HH
^
BRAZIL 1^^^^^^^^
60,000.000 ^^^^^^^^
GREAT BRITAIN ^^H^^^^^B
58.000.000 ^^^^^^^^
FRANCE ^^^^^^
40.000.000 ^^^^^^
61
52
JUNIOR HIGH SCHOOL MATHEMATICS
1. GENERAL ILLUSTRATIONS
1. The following graph is taken from a school report of
the Stamford, Connecticut, schools, showing the number of
pupils per drinking fountain. It is a splendid graph for
this purpose. Thus, it is seen at a glance that the conditions
at three of the schools are "very good." Of these three,
the conditions in the West Stamford Building are the best,
having but about 56 pupils per fountain.
NUMBER OF PUPILS PER DRINKING FOUNTAIN
VERY GOOD
10 to 30 40 50 «0 70 ao 90 lOOIIOI tOlSOl 401 SOI tOITOISOlSOZOO
W. STAMFORD NEW BLDG.
GLENBROOK
WIU.ARD
ROGERS OLD BLDG.
HEW BLDG.
WATERSIDE
W. STAMFORD OLD BLDG.
FRANKLIN ST.
WOODLAND AVE.
WALL ST.
8PRINGDALE
HART '
ELM
CENTER
COVE
GOOD
FAIR I POOR
VERY POOR
2. From the above chart, tell approximately the relation
of the number per drinking fountain in the Hart and Elm
Schools when compared with the number in the West Stam-
ford Building.
3. From the chart, tell approximately how many per cent
more per fountain in the Hart and Elm Schools than in
Woodland Avenue School.
Suggestion. — Since the Woodland Avenue School is about 100, and
the Hart School about 120, it is 20 ^ more.
GRAPHIC METHODS OF REPRESENTING PACTS 53
4. The percentage ia about how much less in the Willard
School than in the Franklin Street School ?
5. The following graph, taken from The Literary Digest
o£ March 16, 1918, showing how food prices increased dur-
ing the War, is a typical chart for picturing changing
relations. It could have been improved by placing the
numbera at the right of the chart aa well as at the left. The
numbera do not indicate any particular price, but ahow what
per cent they were of the price at the beginning of 1914.
6. Obaerving that the graph starts at 100, 120 shows a
20 fe increase ; 150 shows a 50 % increase ; 180, an 80 %
increase. About what was the per cent of general increase
in Canada at the end of 1915 ? ■ At the end of 1916 ? At
the end of 1917 ?
7. What was the per cent of general increase in Germany
at the end of 1915 ? At the middle of 1916?
S. What was the per cent of general increase in the
Unit«d States at the end of 1915 ? At the end of 1916 ?
At the end of 1917?
9. What was the per cent of increase in England at the
end of 1915 ? At the end of 1916 / At the end of 1917 ?
54
JUNIOR HIGH SCHOOL MATHEMATICS
10. The following graph is taken from an article on "' The
Trend in Food Prices " by R. T. Bye, published in The
Annals of the American Academy of Political and Social
Science^ November, 1917. The heavy line marked 100 is
the average price for 1916. How much lower was sirloin
steak in March, 1915, than the average price of 1916 ? Sliced
ham ? Pork chops ?
/^/^ 1 /^/^ 1 /^/r
^^
"^'7riv;7SV^'^/^'-J^«^/^'>^''-'^''-'^'/7/^^j77i»>/73^.^r*Y7ar*/7iVZiT.'7r.
l4»
/■
^i....
AM
1
•
•
,.'">
T7
>-^
r
/
U-
J^
:3?
^
^
*•
••.
^
^
\^
<^
7^
1
Wf**
^S
^^
P
■ '•.
V
'"'
-•'
40
S0
^a
^^
•— XV/«/<i J/k«i -
S/4€«Jf/0m
/
^^^
ij^_
^^^
mmmm
1^^^
^^^
w^Kmt
•
Jn 1 1 1 1 ,
11. Compare the price of sliced ham in May, 1915, with
the price in July, 1917.
12. Compare the price of pork chops in March, 1915, with
the price in July, 1917.
13. Compare the price of sirloin steak in July, 1916, with
the average price of 1916.
*
14. Compare the price of sirloin steak in July, 1917, with
the average price of 1916
15. Make other interesting comparisons as your teacher
may direct.
GRAPHIC METHODS OF REPRESENTING FACTS 55
16. The following graph, taken from The Scientific Monthly,
May, 1918, ehows the variatioD in lossea from hog cholera
from 1894 to 1917, iuclusive.
Does the graph indicate a control of the dieease ?
17. During the drive for the Fourth Liberty Loan in the
fall of 1918, The New York Times published the following
graph showing the relation of our national income per capita
to that of our allies. Compare our per capita income with
that of the other three countries here represented.
UNITED STATES « 372.0
O.E*T.mT.,N ,.,...
FRANCE (182.3
ITALY »lll.l
AVERAse r
ATIONAL INCOME PER
CAPITA
56
JUNIOR mCH SCHOOL MATHEMATICS
IS. The following chart, taken from 7%e Country Gentle-
man of June 9, 1917, shows the world's production of six
leading crops. From the chart, the production of rye is
what per cent of the production of com? Compare the
production of potatoea wiUi that of wheat.
POTATOES. .^^P-aooaOOO BtXSHELB
CMiiS 4i3*O.OOO.OOOSOSaEL3
VBESJT^SSzio
aoR/f aaie.ooQooosa3H£Lo
■y^ 2.4^.00QOOOSUi3SELS
19, The following graph of our exports, on account of the
perpendicular lines and the data written both at the top
and at the right of the bars, is a better type for showing
data, for the comparisons can be more easily made. It ia
taken from !%« WorMft Work of December, 1914.
i
GRAPHIC METHODS OF REPRESENTING FACTS 57
Compare the cotton exports with the foodstuffs. Food-
stuffs with iron and steel. Mineral oil with iron and steel.
20. The exports of coal were about what per cent of those
of iron and steel ?
21. The iron and steel exports were about what per cent
of the cotton exports ?
22. By the use of your ruler or with compasses, compare
the exports of cotton and foodstuffs with all the rest com-
bined.
23. Make other comparisons as your teacher may direct.
24. Formerly, the circle was much used to show the rela-
tion of parts to the whole. Such relations are now more
often shown by a bar, shaded to represent different parts.
The following graph, taken from The Country Grentleman of
June 9, 1917, shows a type of circular graphs.
CORN
WHEAT
25. About what per cent of the world's production of
corn does the United States produce ?
26. About what per cent of the world's wheat crop is
produced by the United States ? By Russia ? By France *^
27. From the graph, rank in order the five leading wheat-
producing countries.
58 JUNIOR HIGH SCHOOL MATHEMATICS
28. The bar is coming to he more used than the circle in
representing the component parts of the whole. It is mora
easily read and more easily made. Thus, a graph like the
following by G, B. Roorbach in The Annals of the American
Academy of Political and Social Science, November, 1917, ia
more often used in careful discussions. How is the number
of cattle, sheep, and hogs distributed in the United States ?
29. Which country has the greatest per cent of sheep?
The smallest per cent ?
30. Which country produces the greatest per cent of
h(^8 ? Which one, except Uruguay, the least ?
31. About what per cent of our food animals are cattle ?
Sheep ? Hogs ?
32. What per cent of Uruguay's animals are cattle ?
Sheep ?
33. Show in per cents the distribution of h(^, sheep, and
cattle in France.
GRAPHIC METHODS OF REPRESENTING FACTS 59
34. The graph shown here was published in newspapers
and magazines during the War to show the property de-
stroyed by Germany in France and Belgium during the first
two years of the War.
}700 MULIONS
Baildings and Indottrial Machinery
■■■B S68Q MUiONS
Agricultural Buildings. Tools, Baw Materials and Livestock, Etc.
^^1 S300 MIUIONS
Railway Property and Bridges
> FRANCE
S1688 MHUONS
TOTAL DESTRUCTION y
■■■^^^■l ;iOOO MHUONS
Buildings and Industrial Machinery
■■■I^B S780 MIILIONS
Agricultural Buildings. Tools, Raw Materials and LiTestock, Etc. \ BELGIUM
j^H $275 MILUONS
Railway Property and Bridges
■■^^■■■■■■■■■H 12055 MILLIONS
TOTAL PROPERTY DESTRUCTION J
a. Using a pair of compasses, tell what per cent of the
total destruction in France was buildings and . industrial
machinery ; tools, raw material, live stock, etc. ; railroad
property.
J. In the same way, make similar comparison of the
destruction in Belgium.
c. Compare the total destruction of property in Belgium
with that of France.
d. Compare Belgium's loss of buildings and industrial
machinery with that of France.
35. Make a collection of graphs from newspapers, maga-
zines, reports, and other sources. Paste them upon card-
board and leave with your teacher to illustrate the various
types. Study them carefully as to whether or not they
present accurately and definitely the quantitative relations.
60
JUNIOR mCH SCHOOL AL\THEMATICS
2. sniPLE OOMPikRISOSS
In the f ollowin;^: problems, use the horizontal bar to show
comparisons. Place the aetoal figures from which the
graphs were constructed at the left end of the bars. The
following from Swift «k Company's Year Book of 1918 is a
gorxl example of a very common t\'pe of the horizontal bar
graph.
19M
14«.4«732«
1»15
3S3.S33.059
191ft
444,440.400
I»I7
4II«473^029
Beef Products
Pork Products
1914
921.9 13.029
1919
1,1 00,1 90.489
1919
1.499.532.294
1917
V499,479.444
The compariwm could have l)een more easUy made had
jierpendicular lines divided each of these bars into sections
repres<?nting 25,000 x>ounds each, as in problem 19, page 66.
1. From the graph sho^vn here, compare the exports of
beef prcxluced in 1915, 1916, and 1917 with those of 1914.
That is, tell about how many times as great. Use a pair of
compasses in making the comparisons.
2. Compare the same exports by telling how many per
cent more one is than the other.
GRAPHIC METHODS OF REPRESENTING FACTS 61
3. From the data, compute the per cents and see how
nearly they agree with your estimates from the graphs.
4. It was estimated that the per capita consumption of
sugar in 1916 in the following six countries was : France,
89.01 lb.; England, 89.69 lb.; United States, 83.83 lb.;
Russia, 29.26 lb.; Italy, 10.45 lb.; and Belgium, 42.79 lb.
Make a horizontal bar graph showing comparisons.
If you represent 20 lb. by 1 in., how long will the bar be
for France ? For England ? For the United States ?
5. Show by graphs the relative importance of beans as a
food crop, from the productions of the following countries
before the War : India, 125 million bushels ; Italy, 23 mil-
lion bushels ; Japan, 21 million bushels ; Austria-Hungary,
19 million bushels ; Russia, 12 million bushels ; and the
United States, 11^- million bushels.
Suggestion. — Select some length to represent a given number of
bushels. Thus, if you select 1 inch to represent 25 million bushels, then
compute what length must be used for each.
6. Always try to make your graph show clearly the rela-
tions that you are trying to picture. Thus, to picture the
following, draw perpendicular lines 1 inch apart, and let
each 1-inch space represent 2 bushels, and see if your graphs
are more easily interpreted than the others you have drawn.
The per capita consumption of potatoes in four countries
is: United States, 2.6 bu.; France, 7.7 bu.; England, 8.3
bu. ; Belgium, 9.4 bu.
Suggestion. — Use a graph similar to that on page 51.
7. Show graphically the relations of our four Liberty
Loan Bond sales of 1917 and 1918. They were : First Loan,
* 3,035,226,850 ; Second Loan, $4,617,532,300 ; Third Loan,
• 4,176,516,850 ; Fourth Loan, $ 6,989,047,000.
62 JUNIOB mCH SCHOOL IfATHEliATIGS
a. The following gnpb shows the per capita conwimptioii
of oleomaigariiie for varioos cc»aiitries« as given in Hke Jn^
depemdtmt of November 17^ 1917. Di^w a graj^ of the
flame, raling perpendicular lines 1 inch apart and let each
^aoe represent 10 lb.
C*^?TA
OC^kiMat '
«3
MORWAY
33H
HOLLAND
TO
GREAT BRfTAM
SK
UMTTED STATES
«f
9u The per capita consomption of meat by Tarions coun-
tries was approximately as follows : in 19<3i Australia, 263
lb. ; in 1902, New Zeakind, 212 lb. ; in 1909, United States,
171 lb.; in 1910, Canada, 137 lb.; in 1906, Great Britain,
119 lb.; in 19W, France, 79 lb.; in 1902, Belgium, 70 lb. ;
in 1899, Russia, 50 lb. At the left of the bars give both
the name of the country and the year. Draw perpendicular
bars about \ in. apart to represent 10 lb. Mark these at the
top of the graph, 0, 10, 20, 30, etc.
Try other plans of graphing the same data and see which
you prefer. For example, draw such a graph as the one
shown in problem 8, letting 50 lb., 60 lb., or any convenient
number be represented by 1 'n.
10. The average yearly production of wheat for three
years (1911—1913) for several countries, given in million
bushels, was as follows: Russia, 727; United States, 705;
Italy, 191; Australia, 89 ; Great Britain. 61 : Belgium, 15.
Show the relations by graphs.
11. In 1913 the production of cane sugar in short tons
C2000 lb.) was as follows for the seven countries leading in
the production : Cuba, 2,909,000 ; India, 2,534,000 ; Java,
GRAPHIC METHODS OF REPRESENTING FACTS 63
1,591,000; Hawaii, 612,000 ; Porto Rico, 398,000 ; Australia,
897,000 ; United States, 300,000. Show the comparison by
graphs.
12. The following is the per cent of games won by the
various clubs of the National League in 1918. Chicago, 65. 1 % ;
ITew York, 57.3 % ; Cincinnati, 53.1 % ; Pittsburg, 52 % ;
Brooklyn, 45.2%; Philadelphia, 44.7%; Boston, 42.7%;
St. Louis, 89.5 %. Show the relations graphically.
13. Our production of grain in 1918 was as follows: wheat,
918,920,000 bu. ; corn, 2,717,775,000 bu. ; oats, 1,535,297,000
bu. ; barley, 236,505,000 bu. ; and rye, 76,687,000 bu. Show
the relations graphically.
14. The price of hogs, live weight, per 100 lb. varied as
follows during a five year period : in 1914, $ 7.45 ; in 1915,
$6.57; in 1916, f6.32; in 1917, $9.16; in 1918, $15.26.
Taking the years in order, show graphically the variation in
price.
15. The average price per acre of cultivated land in 1918
in six states was as follows : Maine, $35 ; Massachusetts, $68 ;
lUinois, $139; Indiana, $106; Iowa, $156; California, $110.
Arrange in regular order from highest to lowest and repre-
sent graphically.
16. Take any data your teacher may direct and draw
neat graphs, showing comparisons, until you are able to make
and interpret graphs easily.
3. GRAPHS SHOWIITG COMPONENT PARTS
The relation of a part to the whole is usually shown either
by the circular graph or by a shaded bar graph. Of the
two methods, the bar graph is more easily made and more
64
JUNIOR HIGH SCHOOL MATHEMATICS
easily read. Thus, the relation of the production to the
importation of sugar in the United States yearly may be
shown by either of the following methods :
THOUSAND TONS
300 733 3306
76.29b
THOUSAND TONS
6.9%^^300
16.9%ral733
The 6.9% is the cane sugar; the 16.9%, is the beet
sugar; and the 76.2% is the importations.
The bars are often made perpendicular as is shown on the
margin. It makes but little difference and
depends largely upon what statements the
maker of the graph wishes to record in the
graphs and the ease with which it may be
read.
1. To make a circular graph, divide the
given whole by 360 to find what each degree
must represent. Thus, in the graph shown
above, 360° must represent 4339 thousand.
Hence, each degree must represent about 12.1
thonsand. Hence, to represent 300 thousand
will require as many degrees as 300 will con-
ain 12.1, or 24.8. In like manner, find how
najij degrees must represent 733 thousand
and 3306 thousand. Before drawing the
graph, see if the sum of all three arcs is 360°. Why ? Now
with a protractor draw a graph like the one shown here.
76.2^
3306
GRAPHIC METHODS OF REPRESENTING FACTS 65
2. While no one making a graph to show more clearly
and vividly a set of facts
will fail to place the data
from which it was made
where they can be clearly
seen and read, let us suppose
that such a graph as the
one in the margin should
be made to represent the
disposition of a family in-
come of $ 1500. By use of
yonr protractor, find the ap-
proximate amount allowed
for each item.
3. In the graph given here, taken from 7%e World'a Work,
December, 1914, the 13 % furnishes sufficient data from
which to find the amounts. Com-
pute, both by per cent and by meas-
uring the arc with a protractor, the
exports of the United States, and see
if the results agree with the figures
given here.
You will find that neither method
gives exactly 2428 million. But the
graph pictures to the eye approximate
relations. The graph, then, is accu-
rate enough to convey the facts to the general reader.
4. In 1917 we consumed about 180 pounds of meat per
capita, distributed as follows : pork, 92 pounds ; beef, 82
pounds ; mutton, 6 pounds. Draw a circular graph showing
the relation of each to the whole.
Suggestion. — How many degrees must be taken to represent 1
pound? How many to represent 92 pounds? ij2 ponnds? 6 pounds 1
66
JUNIOR HIGH SCHOOL MATHEMATICS
UVC CATTLE
127.49
31 6
UNITED STATES
(Check your answers before drawing the graph; that is, see if the sum
is 360°.)
5. Swift & Company's Year Book of 1918 uses the fol-
lowing bar graph to show the relative increase in the price
of cattle and of dressed beef from
1915 to 1917, based upon data from
The Chicago Daily Droveri Journal,
The graph shows what the relation is
between the gains in prices of cattle
and of dressed beef. The figures show what relations ?
6. The following graph was published in The New York
Times^ October 13, 1918, in promoting the Fourth Liberty
Loan.
a. Compare our ^^^■^■■^■■^^^^^^■■•sise.a
national debt in
1918 with our na-
tional wealth.
6. Compare our
national debt in
1918 with our pre-
war national debt,
c. Compare our
national debt per
capita with that of
our three allies.
d. Compare our national wealth per capita with that of
our three allies.
e. Compare Great Britain's national debt with her na-
tional wealth.
/. Compare the national debt of France with her national
wealth.
g. Compare Italy's debt with her wealth.
f 1913.0
$77.5
GREAT BRITAIN
% tes.o
FRANCE
SS55.5
1$ 138.8
76.1
ITALY
f 1812.8
NATIONAL WEALTH
Per Capita.
NATIONAL DEBT
Per Capita.
[
] PRE WAR NATIONAL
DEBT Per. CAPITA.
GRAPHIC METHODS OF REPRESENTING FACTS 67
7. Show by a bar graph the relations shown by the cir-
cular graph in problem 4.
8. We produce yearly an average of 705 million bushels
of wheat, and export 116 million bushels of it. Show the
relation by a bar graph. Show the same by a circular
graph. Which method of presentation do you prefer ?
Why?
9. Great Britain consumes 282 million bushels of wheat
yearly and has to import 221 million bushels of it. Show
the relation by a bar graph.
10. France raises normally 324 million bushels of wheat
and exports 65 million bushels of it. Show this by a per-
pendicular bar graph.
11. Cuba produces 2,909,000 tons of sugar yearly and
exports 2,738,000 tons of it. Show the relation by any type
of graph you wish.
12. In a recent year 86 % of the world's total export of
meat was supplied by five countries : Argentina and Uruguay
together, 36 % ; United States, 31 % ; and Australia and
New Zealand together 19 %. Show by a bar graph the re-
lation of each to the whole.
13. Make a bar graph showing the relation of the boys
and the girls to the entire enrollment of your room. Of the
entire school.
14. Make a bar graph showing the relation of boys and
of girls to the entire number studying second-year Junior
High School Mathematics. -
15. By circular graphs show the same relations shown by
the graphs of problems 13 and 14.
68
JUNIOR mOH SCHOOL MATHEMATICS
4. CURVE PLorrnrG: the brokef uhe graph
Information is shown graphically in many different ways.
The method nsed depends largely upon the user, and not
upon the facts presented. Yet, in a general way, the broken
line, sometimes called " curve plotting," is more often used
than other methods to show the variation through which
quantities pass. Thus, to show the variation in the price of
a commodity over a fixed period, a broken line graph is more
often used than other forms of graphs.
In using the broken line or
** curve," the zero line should be
shown or the chart should show an
irregular line at the bottom, thus
showing '*an incomplete chart."
Thus, from a first glance at the
chart in the margin, taken from
IJhe Country Gentleman of January
6, 1918, the impression is that calves
sell for about half as much in March
and in June as in January and Sep-
tember. But the curve begins at
$8.60, and so, when properly inter-
preted, the price has not made such
a change as a chance observation
would lead one to conclude.
1. By use of a zero line, as in the figure on page 69, one
does not draw the wrong conclusion that would be possible
with the graph shown above. Thus, one sees at a glance
that the value of the exports of 1870 was but about one
third of those of 1900, or that our exports in 1900 had
nearly doubled since 1870. Compare the exports of 1880
with those of 1860. Those of 1890 with those of 1840.
"
HI
^^
^^"^
""■
—
i
7
r
1
T
^ -
1
f
\
4W -
I
J
r—
\,
^
/
\
1
s
\
1
t»-
I
\
Mt
f
>.
fjo -
—
/
(
\
>
*"*•
« re^mt
V
1* AMt
JMur
oo-
__
_
The Selung Price of
Gai^ves in 1917.
GRAPHIC METHODS OF REPRESENTING FACTS 69
Those of 1880 with those of 1910. This graph is one of
the most approved types of curve graphs.
MILLIONS^
OF 2
DOLLARS^
3400
^
r-
CM
00
r».
Oi
o
^
CM
."
00
CM
m
(O
CM
m
<o
00
't CM
— O
CM CO
CM CO
1830*40 *50 *60 *70 *80 *90 *00 '10
The Valub of our Exports from 1890 to 1910.
2. Show the relations given in the graph of problem 1 by
horizontal bar graphs, placing the year and the exports at
the left end of each bar.
3. The average prices received by farmers for wheat
during the first ten months of 1917 were: Jan., $1.50;
Feb., 11.66 ; Mar., $ 1.64 ; Apr., $1.80 ; May, 12.46 ;
June, $2.49; July, $2.20; Aug., $2.29; Sept., $2.10;
Oct., $ 2.00. Show by a curve graph the variations in price.
4. The average September price received by the farmer
for hogs, per 100 lb. live weight, ranged for eight years as
follows: 1910, $8.27; 1911, $6.53; 1912, $7.47; 1913,
$7.68; 1914, $8.11; 1915, $6.79; 1916, $9.22; 1917,
$15.69. Show the variations by a curve graph. Show
them by horizontal bar graphs.
70 JUNIOR mOH SCHOOL MATHEMATICS
5. The following graph is taken from The Amtals of the
American Academy of Political and Social Science of No-
vember, 1917.
Study the graph and
answer the following
questions and simitar
ones that your teacher
may ask.
Note. — The figures do
not show an J particular price,
but show what per cent they
were of the average price of
1916, ahown by the heavy
line marked 100.
a. In May, 1917, the
price of flour had in-
creased how much over
the average price of
1916?
b. It. was how much less in September, 1915, than the
average price of 1916 ?
e. Sugar was how much higher in May, 1917, than at the
end o£ March, 1916 ?
d. Sugar was how much lower at the end of September,
1915, than the average price of 1916 ?
c. Sugar was how much higher the middle of July, 1916,
t^an the average price for the year ?
6. From the reports of the Board of Education of your
city, make a chart showing the variation in the total enroll-
ment in your schools for a period of years.
7. Show by a similar chart the variation of enrollment of
sixth grade pupils for a number of years.
GRAPHIC METHODS OF REPRESENTING FACTS 71
5. MAP PRESENTATION OF FACTS
Maps marked or shaded in various ways f onn a very con-
Tenient way of presenting information. The following ia a
very common form,
Eacb Dot RsPBEasNTS a Pakh Tractob in Una.
1. What general section of the country uses more farm
tractors?
2. What state uses the least number per acre ?
3. What part of New York state uses the greater number?
4. In what general section is the use distributed more
equally ?
5. Compare the use in Indiana and Illinois.
6. Compare the number used in Wisconsin with th«
number in Missouri.
7. In what five states is the greatest use made of farm
tractors?
72 JUNIOR HIGH SCHOOL MATHEMATICS
8. The following map, taken from The Literary ZHgest
of May 18, 1918, shows area in which food might be grown.
The shaded areas show the amount of cultivated land in
each state.
Name four states in wliich most of the available land ia
under cultivation.
9. In what stjite is the Hinallewt proportion of the land
under cultivation ?
10. About what per cent of Washington is under culti-
vation ?
11. About what per cent of Minnesota is under culti-
vation '!
12. About what per cent of the New England states is
under cultivation ? About what per cent of the Southern
states?
13. Which has the greater per cent of it« area under cul-
tivation, North Dakota or South Dakota ?
GRAPHIC METHODS OF REPRESENTING FACTS 73
14. Compare the area under eultivation in Nevada with
that in Utah.
15. Compare the area under cultivation in Montana with
that in Colorado. With that in Oregon.
16. Compare the cultivated land in Oklahoma with that
of Texas.
17. Compare the cultivated land in Georgia with that of
each of the other Southern states.
18. Compare the cultivated land of PeiniHylvania with
that of Oliio. With that of California.
19. Try to find otlier maps of this nature in newspapers
and magazines, and bring to class for sucli pnibleras as those
given liere.
20. The following map, repi-oduced hy special permission
from Tlie Natiimal Geoyraphic Mayaziiie, February, 1917,
shows the foreign stock in the population of the United
States. That is, tlie foreign-bom and the uhiidren of at
least one foreign-born parent.
IBS FooBioH Stock ih oua Fofdi.atioh
74
JUNIOR fflGH SCHOOL MATHEMATICS
21. Name the states having a population of less than half
of native stock.
22. Name the states whose population is made up of more
than one-third of foreign stock.
23. What state has more than 99 out of every 100 of
native stock ?
24. What general section of the country has the larg'est
per cent of native stock ? What section the least per cent ?
f.'2^00
$1.76
6. FUNCTIOKAL RELATIONS SHOWN BT ORAPHS
When one number depends upon another, as when the
cost depends upon the amount purchased, one is said to be
a function of the
other. Thus, the •228
amount bought is
a function of the
cost; the dimen-
sions of a rec-
tangle are func-
tions of its area;
the diameter of a
circle is a function
of its circumfer-
ence. The graph
in the margin is a
price graph of gas-
oline when the
price is 25 ff per
gallon. Since the cost of 1 gal. is 25 ^, we make a dot above
1 and opposite 26 ^ ; then above 2 and opposite 50 ^ ; etc.
By finding several such points and connecting them, the
$1.90
$1.25
•1.00
2S^
2 3 4 9 6 7 6
._J
GRAPHIC METHODS OF REPRESENTINQ PACTS 75
ta.oo
•2.BO
92.00
SI. so
S1.00
BO^
graph is seen to be a straight line. Hence, but two points
need to be found in order to determine such a graph.
In a graph of this kind, two principal lines at right angles
are used to represent the numbers. These are called the
axes of the graph. The horizontal one is called the x-axis
and the perpendicular one the y-axis. These axes, however,
are often given special names, ^
as axis (rf gallons and axis
of coat.
By properly numbering
the axes, the diagonal of a
square may always be used
in a price graph. Thus, in
the graph in the margin,
when articles are selling at
50 f! each, numbering the di-
vision on the y-Bixis 50, 100,
150, 200, etc., and those on the x-axis 1, 2, 3, 4, etc., the
cost of any number of articles may be found.
1. Draw a price graph of gasoline at 26 fi. From it give
the cost of 5 gal. ; 10 gal. ; 8 gal.
2. Draw a price graph of cloth at 45 ^ per yard. From
it give the cost of 5 yd. Of 7 yd. Of 12 yd.
3. Draw a price graph of potatoes at $2 per bushel.
From it give the cost of 4 bu. Of 8 bu. Of 12 bu.
4. Draw a wage graph of wages at 40 fl per hour. From
it give the wages for 8 hr. For 12 hr. For 9 hr.
5. Draw an interest graph showing the interest at 6 %.
From it give the interest of $800. Of $500. Of $1200.
Of $1500.
/
/
/
/
/
-
/
■
I
> :
i A
5 6
76 JUNIOR fflGH SCHOOL MATHEMATICS
6. Where the employer is paying several rates for help,
as 25 ^, 80 ^ 40 ^, and 45 ^ per hour, the graphs could all be
shown upon the same chart as follows : Since the wages for
no time is nothing, all graphs start with zero. But one more
point is needed for each graph. For example, at 25 ^, the
wages for 8 hr. are $2.00. Hence, mark a point above 8
and opposite $2.00 and draw a straight line through this
point and zero. To make the graph for the 30 ^ wage, find
the wage, say for 5 hr. This is $1.50. Hence, draw a
straight line through the point directly above 5 and oppo-
site $1.50. Show how to draw the other two. From the
graph, give the wages for 6 hours at each rate.
7. Make a wage graph to show the wages at 50 ^, 60 ^,
and 75^ per hour. From it, give the wages at each rate for
5 hr. For 6 hr. For 8 hr.
8. Let every other division on the ar-axis be called 1, 2,
3, etc., so as to read wages for half hours, and make a wage
graph at 28^ per hour. Give the wages for 4 J hr. For
7J hr.
9. Draw a price graph from which the cost to ^ of a
pound may be read when the price is 32 ^ per pound.
10. Draw a graph from which the cost of butter can be
read from the cost of 1 oz. to the cost of 4 lb., when butter
is 48 ^ per pound.
CHAPTER VIII
MEASUREMENTS, CONSTRUCTIONS, AND
OBSERVATIONS
This chapter is a review of various problems in measure-
ments that you have had in the other grades and an exten-
sion of the subject to include new areas and volumes, as well
as a new process called square root.
1. MEASURING . ANY QUANTITY: DENOBflNATE NUMBERS
The numerical measure of any quantity is the number of
times it will contain some standard unit of measure.
A denominate number is a number of standard units of
measure, as 6 feet, 6 pounds, 8 bushels, etc. When a num-
ber consists of two or more related units, as 3 bu. 2 pk. ;
5 ft. 8 in. ; 3 gal. 2 qt. ; etc., it is called a compound de-
nominate number.
Compound denominate numbers are changed to single
units by the laws of arithmetic which you already know.
1. Reduce 8 ft. 9 in. to inches.
SOLUTION The work may be short- 8 — 9
8ft. = 8xl2in. = 96 1iL ®^^^ ^^ ^™g abstract 12
96 in. + 9 in. = 105 in. ^^^mbers and computing as 96
in the right-hand margin. _9
105
2. How many quarts in 16 gal. 3 qt. ?
3. How many pecks in 64 bu. 3 pk. ?
4. How many feet in 18 yd. 2 ft. ?
77
1
78 JUNIOR fflGH SCHOOL MATHEMATICS
5. How many minutes in 15 hr. 48 min. ?
6. How many seconds in 38 min. 16 sec. ?
7. Reduce 12 qt. 1 pt. to pints.
Reduce :
8. 12 A. 96 sq. rd. to square rods.
9. 14 mi. 96 rd. to rods.
10. 5 mi. 1230 yd. to yards.
11. 16 lb. 11 oz. to ounces.
12. 5 sq. ft. 84 sq. in. to square inches.
13. 6 cu. yd. 16 cu. ft. to cubic feet.
14. 284 in. to feet and inches.
SOLUTION
284 in. ~ 12 in. = 23 times and 8 in. nndivided.
Hence, 284 in. = 23 ft. 8 in.
Explanation. — The quotient shows the relation of the dividend to
the divisor. Hence, the 23 and a remainder of 8 in. shows that the
dividend is 8 in. more than 23 times the divisor (12 in.). But 12 in. is
a foot. Hence, the quotient shows that 284 in. is 23 ft. and 8 in.
15. Change 18 qt. to gallons and quarts.
Reduce :
16. 125 pt. to quarts and pints.
17. 196 in. to feet and inches.
18. 340 in. to yards and inches. ^
19. 426 sq. rd. to acres and square rods.
20. 324 oz. to pounds and ounces.
21. 175 pk. to bushels and pecks.
22. 185 ft. to yards and feet.
23. 342 min. to hours and minutes.
33. ^^ hr.
37.
fft.
34. f A.
38.
fib.
35. ^^T.
39.
fgal
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 79
24. Reduce ^ bu. to lower units.
SOLUTION
} bu. = } X 4 pk. = 3 J pk. Note. — This does not differ
\ pk. s I X 8 qt = 4 qt. from the reduction of a whole
Hence, } bu. = 3 pk. 4 qt number to a lower unit.
Reduce to lower units :
25. I ft. 29. I lb.
26. I yd. 30. If A.
27. I gal. 31. \^ bu.
41. What part of an hour is 82 min. 40 sec. ?
solution
1 hr. = 60 X 60 sec. = 3600 sec.
32 min. 40 sec. = 32 x 60 sec. + 40 sec. = 1060 sec.
1960 sec. -*- 3600 sec. = J J JJ = 4f ©^ .544+.
42. What part of an hour is 12 min. 30 sec. ?
43. What part of a yard is 2 ft. 4 in. ?
44. What part of a gallon is 3 qt. 1 pt. ?
45. What part of a mile is 960 ft. ?
46. What part of a mile is 720 yd. ? .
47. Reduce 2 pk. 6 qt. to a decimal part of a bushel.
48. Reduce 9 hr. 48 min. to a decimal part of a day.
49. Reduce 1 ft. 10 in. to a decimal part of a yard.
50. What decimal part of a square foot is 96 sq. in. ?
51. If a coat rack is to contain 8 hooks placed at equal
distances apart, find the distance if it is 63 inches from one
end hook to the other.
Suggestion. — By use of a diagram, show that 63 inches is to be
divided into 7 equal spaces, and hence, that the divisor is 7 instead of 8.
80
JUNIOR HIGH SCHOOL MATHEMATICS
52. If 12 plants are to be set at equal distances apart, in
a row measuring 16 ft. 6 in. from the first plant to the last,
how far apart must they be set ?
53. If posts for a fence are set 10 ft. apart, how many
will be needed for a fence row measuring 210 ft. from one
end post to the other ?
54. How many posts set 8 feet apart are needed for a
grape arbor 56 feet long? How many crosspieces will be
needed if there are 3 between each pair of posts ?
2. A REVIEW OF AREAS
The area of a surface is the measure of it when the unit
of measure is the surface of some square whose side is some
linear unit.
1. Show from this figure that the number of square units
in the surface of any rectangle is the product of the number
of linear units in its length and
in its width. To do this, show :
(1) that a strip across the length
and 1 unit wide contains as many
square units as there are linear
units in the length ; (2) that
there are as many such strips each
1 unit wide as there are linear
units in the width ; and (3) that the area of one strip mul-
tiplied by the number of strips gives the entire area.
2. Let A represent the number of units in the area of a
rectangle I units long and w units wide, and write the
formula showing the relations.
3. From the formula A = Iw^ what does I equal in terms
of A and wl win terms of A and I ?
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 81
A
4. Express in words the meaning of the formula, 1 = — .
V
5. What will a sidewalk 4 ft. wide and 120 ft. long cost
at 18 ^ per square foot ?
6. What will the linoleum cost for a kitchen floor 12 ft.
by 15 ft. at $ 1.75 per square yard ?
7. How many square feet of sodding are required for a
lot 80 ft. by 220 ft., deducting for a building 34 ft. by 36
ft, a walk 4 ft. by 24 ft., and 350 sq. ft. for shrubbery beds?
8. If a garden plot 42 ft. by 76 ft. is surrounded by a
sod border 4 ft. wide, how much is left for cultivation ?
How many square feet in the sod border ?
9. When a boy has mowed a strip 10 ft. wide about a
rectangular lawn 70 ft. by 185 ft., what per cent of the
lawn has he yet to mow ?
10. Find the cost of building a sidewalk in your town or
city, and compute the cost of five pieces of sidewalk in the
neighborhood, in order to get an idea of the cost of any
piece of sidewalk that you see.
11. Find the cost of lathing and plastering a room, then find
the cost to plaster a room the size of your schoolroom, and thus
get an idea of the cost of plastering any surface you see.
12. Find the cost of paving streets, and compute the cost
of paving a piece of street one block long in your neighbor-
hood, and thus be able to have some notion of the cost of
other pieces of paving that you see.
13. Find the cost of flooring, the amount to be added to the
surface covered to allow for " tongue and groove," and com-
pute the cost of flooring a room the size of your schoolroom,
and thus be aUe to estimate the cost of flooring other rooms.
82
JUNIOR HIGH SCHOOL MATHEMATICS
3. CONSTRUCTIONS AND OBSERVATIONS
By carefully constructing figures and measuring certain
parts, many useful facts may be discovered. This phase of
mathematics is sometimes called constructive or observa-
tional geqmetry.
1. Carefully construct three
or more rectangles of different
dimensions. By the use of a
pair of compasses compare tlie
diagonals, AC smd BD. If care-
fully drawn and measured, you
found that they were equal.
And in general,
• ITie diagonals of any rectangle are equal.
2. In the rectangles you have drawn, measure with your
compasses the two parts into which each diagonal is divided
by the other. If carefully done, you found them equal.
And in general,
The diagonals of a rectangle bisect each other,
3. Carefully construct three or more squares and draw
their diagonals. By use of your protractor measure the
angles which the diagonals make. If carefully done, you
found them all right angles. And in general.
The diagonals of a square bisect each other at right angles,
4. Draw 'a rectangle and cut along the diagonals into
four triangles. Place those triangles that seem alike upon
each other, and thus compare them.
Figures that coincide throughout are congment. If care-
fully done, you found two pairs of congruent triangles.
And in general,
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 83
The dicigonals of any rectangle divide the rectangle into two
pairs of congruent triangles.
5. Compare the four triangles into which the diagonals of
a square divide the square. State your conclusions.
4. THE AREA OF A PARALLELOGRAM
Carefully construct a parallelogram ABCD of any size or
shape. From point A at the extremity of the base erect a
perpendicular cutting CD in
E. Cut off triangle T thus
formed, and place it in the
position of S. What is the
shape of the figure thus
formed? What are its di-
mensions? Compare these
with the dimensions of the parallelogram. If your con-
structions and observations were carefully made, you found
that the figure formed from the parallelogram was a rec-
tangle having the same base and altitude as the parallelo-
gram. And in general,
A parallelogram is equal to a rectangle having the same
dimensions.
Hence, A = wL
1. What is the area of a parallelogram whose base is 30
ft. and whose width is 12 ft. ?
2. See if you can find surfaces in the form of a parallelo*
gram, and, if so, measure them ; that is, find their areas.
You will not find such surfaces common. A knowledge
of the measure of its surface is useful on account of other
surfaces being transformed into parallelograms in order to
discover how to measure them.
M rrSVj^ HIGH 5CH«>:.L l.L\THEMATICS
!• Carrf^iH J c-*:r.*trTn a pazalLel-ii^iam. By oTudng alcMAg
a di^e'^^t^ diTidt tLe paiaHelogram isto two triangles.
iy^tuyskn: xc^ Vmo trUkZ^JA br i^ac-ing one upon the other.
If e*r«rf.*^ly 'i>f:^ev v-vn V^iilA them CTC-c^nient- And in
S?*rr*^raL,
Z* CoiL^tract a paralUrlogTam which shaLl Lare two ad-
yu^titit Mfb^ aiid the in-
clad^l an^le e^^ioal respec-
tively to two ^%'en line-
fie^fieriU and a given angle.
Exfi^ASATiOiT. — S:ij4i09e that
If aiid A' ai»r the giv^-n luM^-seg- J^
i»^frft» and Jf tli^ giv^n angle.
Tb^ st^^ In tbe cr>b.«tni/rtioiis are ^
an folUjwn:
fu CoriiitnBct angle iBvl 2> equal
to angle JT, a« given in Book L
i&« Mark r>ff by tue of compa«es
^/:f = J/,and^I/ = .V.
<?« With center at B and radios eqoal to X, draw an arc ; then with
CMiUiT at D and radiuii equal to J/, draw an arc catting the first arc and
mark the intersection C.
d. Draw BC and //C, and A BCD is the required paraUelogram.
3. If two forces are exerted in different directions upon
the same object at A^ they have the same effect as a single
force called their resultant.
If the directions and mag-
nitudes of the two forces are
represented by line-segments
AB and AD^ the direction
and magnitude of the re- ^
M
L
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 85
sultant will be represented by line-segment AC^ diagonal of
the parallelogram AB CD, Construct a parallelogram to some
scale and by measurement find the resultant of two forces,
one of 100 lb. and the other 200 lb., acting at an angle of 60°.
4. Two forces acting at an angle of 45°, one of 40 lb. and
the other of 60 lb. , are equivalent to a single force of how
many pounds ?
5. Draw any parallelogram and its diagonals. Compare
the segments into which each diagonal divides the other.
State your conclusion. Compare your conclusion with that
made from a similar observation with rectangles.
6. Compare the four triangles into which the diagonals of
a parallelogram divide it. State your conclusion. Com-
pare your conclusion with that made from a similar obser-
vation with rectangles.
7. Take two pairs of strips of cardboard or light wood
and join them with tacks so as to form pivots by which
the form of the frame may be
changed. Have the opposite
sides exactly equal, as AB—DO
and AD =iBC, Move about so
as to form different shapes.
What is the name of the figure,
whatever angle the sides make
with each other ? This illustrates the fact that,
If the opposite sides of a quadrilateral are equal^ the figure is
a parallelogram. £>. c
8. The figure shows the picture
of an instrument called a parallel
ruler, used for drawing parallel
lines. Study it and show why, if AB is held in a rigid position,
all lines ruled along D (7 as it is raised or lowered will be parallel.
86
JUNIOR fflGH SCHOOL MATHEMATICS
6. THE AREA OF A TRIANGLE
1. It was seen under the study of parallelograms that a
diagonal of a parallelogram divides the parallelogram into
two congruent triangles. From this fact, show how to find
the area of a triangle.
2. Interpret the formula A = ^bh as a rule for finding
the area of a triangle.
3. What is the area of a triangle whose altitude is 12 in.
and whose base is 8 in. ?
4. In measuring some triangular area, as a triangular
plot of ground, what two measurements are necessary ?
5. Draw upon the blackboard some triangle whose sides
are several inches, say from 15 in. to 30 in. Now, by three
different pairs of measures, find the area. That is, take
each side in order as base. This will serve as a check upon
the accuracy of your measurements and computation, for all
results should be the same.
6. The irregular figure in
the margin can be measured
by dividing it up into triangles.
If AO^ZO in., DI!:= 20 in.,,
and BF= 12 in., find the area
of ABOD.
7. Make irregular figures
upon the blackboard and, by
making proper measurements,
find the areas.
8. Draw the figure of problem 6 to a scale, making the
figure several times as large as this one. Now draw diagonal
BD and drop perpendiculars upon it and find the area.
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 87
7. CONSTRUCTIONS AND OBSERVATIONS
1. A triangle may be constructed with sides equal to three
given line-segments a, 5, and c.
The following is the order in
which the construction is made :
a. Draw a straight Une and lay off
AB = c.
h. With center at A and with
radius equal to h, draw an arc.
c. Then with center at B and with
a radius equal to a draw an arc cut- ^
ting the first arc, calling the point of c
intersection C.
d. Draw A C and BC, and ABCis the triangle required.
2. Construct a triangle whose sides are 2 in., 3 in., and
8 J in., respectively.
3. Construct an isosceles triangle whose equal sides are
each 4 in. and whose base is 3 in.
4. Construct an isosceles triangle whose equal sides are
each equal to some line-segment a:, which you have chosen,
and whose base is some other line-segment y.
5. Construct an equilateral triangle each of whose sides
is equal to some chosen line-segment.
6. Nail three strips of wood together so as to form a tri-
angle, using but one nail at each joint.
Is this frame rigid, or can it be changed
into various shapes by exerting pressure
upon it ?
This illustrates the fact that,
The form of a triangle is fully determined by its sides. That
is^ aU triangles whose corresponding sides are equal are con-
gruent.
88
JUNIOR HIGH SCHOOL MATHEMATICS
7. Why is a roof sufficientlj braced when a board is
nailed across each pair of rafters ? . , - ■
8. Why is a long span of a ,^-=^l/l/l\l\
bridge in which the tmss is made
with queen posts and diagonal rods,
as shown in the drawing, sufficiently supported ?
9. Draw an isosceles triangle and
its altitude. Cut out the two tri-
angles and compare them by placing
one upon the other. Wliat is your
conclusion ?
Several observations may be made.
Thus:
I. The altitude of an isosceles
triangle divides the triangle into two
congruent right triangles,
II. The altitude bisects the base and also the vertical angle.
III. ITie base angles of an isosceles triangle are equal.
10. The drawing is that of a plumb level, used for level-
ing before the modem spirit level was
invented. From a pivot at (7 a plumb
line is hung. Mia the middle point of
base A£. Show how to use the plumb
level in determining whether a con-
struction is level or not.
11. Cut from cardboard any triangle and cut off the cor-
ners and place them so that the three angles form a single
angle as on page 38. What is the size of the angle formed
by all three angles of the triangle ?
By trying this experiment with any triangle, you will find
that,
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 89
The sum of the three angles of any triangle is equal to 180
degrees.
12. How many degrees in each angle of an equilateral
tiiangle ? (The angles are all equal.)
13. If the two base angles of an isosceles triangle are
each 66% what is the size of the vertical angle ?
14. In a right triangle, if one acute angle is double the
other, what is the size of each ?
8. THE AREA OF A TRAPEZOID
1. Take two congruent trapezoids and place them as in
the figure. What kind
^ DC -
of figure do they form ? ^
From this, state a rule
for finding the area of
a trapezoid.
2. State in words
the relations expressed
by J. = -i — - — 2 J when A represents the area of a trapezoid.
h its altitude, and b and 6' its two bases.
3. Find the area of a
trapezoid whose bases are ^
4 in. and 7 in. respectively,
and whose altitude is 6 in.
4. This diagram is
that of an irregular field.
AH =20 rd., Ha^^b
rd., (7D = 15 rd., ^^ =
24 rd., aO^ 20 rd., BE
= 24 rd. Find the num-
ber of acres in it.
90 JUNIOR HIGH SCHOOL MATHEMATICS
5. Find the area when ^jy= 30 rd., Ha = 70 rd^ QD = 20
rd., IIF:=50 rd., (3^(7=40 rd., and BR:= 45 rd.
9. THE RELATION OF THE CIRCUMFERENCE OF A
CIRCLE TO ITS DIAMETER
1. Measure tihe circumference and diameter of several
large circular objects, as dining-room tables and large wheels,
or describe circles on large pieces of cardboard and cut' them
out for measurement. Divide the circumference by the
diameter in each case. What relation do you find ?
If you could have been exact enough in all your measure-
ments, you would have found every quotient to be 3.1416.
And in general,
Circumference = 3.1416 x diameter ;
or, (7= 7rd
where the Greek letter tt (pi) represents 3.1416, or the re-
lation of the circumference to the diameter, and d the
diameter.
2. Find the circumference of a circle whose diameter is
12 ft.
3. Find the diameter of a circle that has a circumference
of 200 feet. .
.4. By tying a string to a stake fixed at a point that was
to be the center of a circular running track, and walking
about this center so as to keep the string taut, some boys
laid out a ^-mile (660 ft.) track. Find how long a string
they needed besides the amount used up in tying.
5. How far does a 32-inch automobile wheel carry the
automobile forward each revolution ?
6. How much farther per revolution does a 86-inch
wheel carry a car than a 34-inch wheel would carry it ?
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 91
Note. — The relation found in problem 6 can be expressed as a ratio
as follows : 36ir--34w = 2ir; 2 ir -i- 3^ w = ^f. Hence, the larger wheel
would carry the car ^y farther each revo^'ition.
7. How much farther per revolution would a 32-inch
wheel carry a car than a 80-inch wheel would carry it ?
8. The readings of a speedometer of an automobile are
controlled by the number of revolutions made by the wheels.
If a speedometer is made for a car having a 32-inch wheel,
and a 33-inch wheel is used, what correction must be made
in the readings in order to know the actual speed or dis-
tance traveled ?
Suggestion. — A study of problems 6 and 7 will enable yon to answer
this.
9. Since O = ird^ every increase in the length of the
diameter gives an increase 3.1416 times as great in the cir-
cumference. When increasing the diameter of any circle,
large or small, 10 inches, how much is the circumference
increased ?
10. On a running track having parallel sides and semi-
circular ends, two boys run, the outer boy being 3 ft.
farther from the inner curb than the other. How much
farther does he run each lap than the other boy does ?
11. If one circle has a diameter 16 ft. longer than an-
other, its circumference is how many feet longer ?
12. If one circle has a circumference 15 ft. longer than
another, its diameter is how much longer ? Its
radius is how much longer ?
13. In these two concentric czrcies^ if the outer
one has a circumference 10 inches greater than
the other, how far apart are the two circumferences ? Does
the size of the circles affect the answer ?
92 JUNIOR HIGH SCHOOL MATHEMATICS
14. The following question is often given to catch one :
"If the earth were a smooth and perfect sphere 8000 mi.
in diameter and banded at the equator with a tight fitting
iron band into which a piece 12 in. long could be inserted,
by how much would the insertion make the band stand out
from the surface, if the space were distributed evenly around
the earth ? " The person asked usually says, " It would not
loosen the band perceptibly." Study problems 12 and 13.
then see if you cannot answer this question correctly.
10. THE AREA OF A CIRCLE
1. Describe a circle upon cardboard, taking a i*adius of
from 4 in. to 6 in., in order to have a circle large enough
to use easily.
Erect two perpendicular diameters.
Bisect one of the four right angles formed.
Bisect one of these two angles thus formed.
Using the arc intercepted by the angles thus formed, with
your compasses divide the circumference into sixteen parts.
Now cut the circle
into sixteen equal sec-
tors as in figure A and
rearrange as in figure B.
Of the figures studied,
what does B most re-
semble ?
We infer from the experiment you have made that,
The area of a circle is the same as that of a parallelogravi
whose hose is half the drcmnfereiice^ and whose altitude is iht
radius*
Expressed as a formula,
A^\cr.
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 93
It is not necessary to measure both e and r, for if r is
known, c may be found. That is, c = 2 irr^ hence, \c^ trr.
Substituting, the formula becomes
2. Find A when r is 12 ft.
3. Find the area of a circular flower bed 12 feet in
diameter.
4. A 20-foot basin for a founttiin is surrounded by a
5-foot cement wfilk. Find the area of the walk.
5. Compare the area of a 10-foot circle with that of a
15-foot circle.
6. The water delivered by two pipes from the same main
(since both have the same pressure) varies with the areas of
the cross sections of the pipes. An inch pipe will deliver
how many times as much water in a given time as a |-inch
pipe ? Asa |-inch pipe ?
11. MEASURING LUBfBER
In measuring lumber the unit of measure is a board foot.
This is the equivalent of a board 1 foot square and 1 inch
thick, except in measuring lumber less than 1 inch thick,
in which case the thickness is not considered, but each
square foot makes a board foot.
Lumber is usually sold by the thousand board feet. Thus,
a quotation of "$42 per M" means 142 per 1000 board
feet.
Note. — By lumbermen the term " foot " is used instead of " board
foot " when no confusion in the meaning could arise from such use.
1. How many board feet in a piece of lumber 12 in. wide
14 ft. long, and 1 in. thick ? That is, to how many boards
1 foot square and 1 inch thick is it equivalent ?
94 JUNIOR HIGH SCHOOL MATHEMATICS
2. How much lumber in a board 6 in. wide, 12 ft. long,
and 1 in. thick ?
3. How much lumber is there in a piece 6 in. wide, 18 f t»
long, and 2 in. thick ?
Suggestion. — There is just twice as much lumber in the piece as
there would be if it were but 1 in. tliick.
4. A beam 8 in. wide, 3 in. thick, and 18 ft. long contains
how much lumber ?
5. How much lumber in a piece 16 ft. long and 4 in.
square ?
6. A driveway to a barn is 12 ft. wide, 24 ft. long, and
made of lumber 2 in. thick. Find the cost at $ 35 per M.
7. A board walk 4 ft. wide and 65 ft. long is made of
lumber 2 in. thick, nailed crosswise to three pieces, each
3 in. thick and 6 in. wide, running lengthwise. Find the
entire cost of the lumber at $ 38 per M.
8. Hardwood flooring is called 3-inch flooring when made
from lumber three inches wide when it came from the saw
mill. In making it, there is a waste
of f of an inch in planing and in
cutting the " tongue and groove."
Then how wide a strip of floor is
covered by each 3-inch board ?
9. Compare the | inch lost in making with the 2J inch
strip actually covered by each board.
10. The result of problem 9 may be interpreted as mean-
ing that ^ more lumber is needed than there is floor area to
be covered. How much lumber will be needed to floor
300 sq. ft. with 3-inch flooring ? To floor 450 sq. ft. ? To
floor 1500 sq. ft. ?
11. At $75 per M, find the cost of the 3-inch flooring
needed for a room 15 ft. by 24 ft.
'^mimm^
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 95
U. Measure from crack to crack in the floor of your
Bchoolroom and find what width of flooring was used,
SuoaicsTioN. ^If the liistaDce is 2^ inches, 3'iiich flooring was used ;
if 3^ inches, 4-inch flooring was used. That is, add } inch to the width
[rom crack to crack.
13. How much must be added to the area of the floor in
your room to give the amount of lumber needed to floor it ?
12. THE TOLVHB OF PRISHS
You have learned that any solid whose bases are in
parallel planes, and whose sides are rectangles, is a right
prism, and that the prism is named from the shape of its
base.
In a rectangular priBin the bases, tlien, must be rectangles.
In a rectangular prism the three edges that meet at any
comer are caUed its dlmeosiong.
1. If each division re-
presents a foot, what are
the dimensions of the rec-
tangular prism represented
here? ..__,,
2. The figure represent*
the prism as having been cut into unit cubes. How many
are here represented ?
3. If we think of this prism a« cut into two layers, each
3 units wide, 5 units long, and I unit thick, how many unit
cubes in each layer ? In both layers ?
«. How many cubic inches in a rectangular prism 3 in.
wide, 4 in. long, and 5 in. high ?
5. Call (t, h, and c the dimensions of a rectangular prism
whose volume is V, and express the relation of F" to a, 6,
and by a formula.
96 JUNIOR fflGH SCHOOL MATHE^L\TICS
6. State in words the trath expressed by the formula
V^abe.
7. Find how many cubic feet of space in a coal bin 8 ft
wide, 10 ft. long, and 6 ft. deep. At 35 eu. ft. per ton,
how many tons of coal will it contain ?
8. What is the area of each of the six faces of a 1-inch
cube ? Of a 1-foot cube /
9. How many square units in the base of the prism rep-
resented in problem 1 ? How does this number compare
with the number of cubes in each of the two layers ?
10. Tf there are 20 sq. ft. in the base of a rectangular
prism, how many cubic feet in each layer 1 ft. thick ?
11. How many cubic feet in a rectangular prism whose
altitude is 8 ft. and whose base contains 15 sq. ft. ?
12. Let V^ volume, B= area of base, and h = height of
a rectangular prism. Write the formula sho^ving the rela-
tion oi Vto B and A.
13. Give in words the fact expressed by the formula
r=5A.
14. A rectangular watering trough 8 ft. long, 2 J ft. wide,
and 20 in. deep will hold how many gallons ? (231 cu. in.=
1 gal. ; 1 cu. ft. = 7.48 gal., approximately.)
15. A farmer has a bin 10 ft. wide and 12 ft. long filled
with wheat to an average depth of 6 ft. How many bushels
has he, allowing .8 bu. per cubic foot ?
It has been found by mathematics that the number of cubic
units in any prism is the product of the number of square units
in the base and the number of linear units in the height. That
is^ V^Bh is not only true of a rectangular prism^ but it is true
of all prisrttS,
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 97
16. A concrete retaining wall 3 ft. wide at the bottom
and 18 in. wide at the top is 5 ft. high and 60 ft. long.
How many cubie feet of concrete are in it ?
17. When water is flowing at the rate of 80 ft. per minute
through a drainage ditch 20 in. wide at the bottom and 30
in. wide at the top, and at a depth of 12 in., how many cubic
feet are being discharged per day ?
13. THE VOLUME OF CTLHTOERS
A cylinder may be divided as shown in the flgure and
formed into a solid closely resembling a prism, from which
we infer a fact proved later in mathematics that, just as in
t prism.
The number of cubie
units in the volume of a
eylinder is the product
of the number of square
units in the bate and the
vMviber of linear units in
the altitude.
That is, V=Bh.
X. A cylindrical pail 12 in. in diameter and 14 in. deep
will hold how many gallons ?
2. A hot-water tank 4 ft. long and 14 in. in diameter will
hold how many gallons ?
3. Find the capacity of a cylindrical gasoline tank 3 ft.
in diameter and 8 ft. long.
4. A hollow cylindrical iron pillar whose outer diameter
is 6 in., whose inner diameter is 4 in., and whose length is
10 ft., contains how many cubic inches of iron ?
98 JUNIOR HIGH SCHOOL MATHEMATICS
5. How much water can flow in one hour, through a
water pipe 2 in. in diameter, when flowing at the rate of 80
ft. per minute ?
6. A cylindrical silo 16 ft. in diameter and 28 ft. high
will hold how many tons of silage, allowing 60 cu. ft. per
ton?
14. THE SURFACE OF A CYLINDER
The surface of a right circular cylinder consists of two
circles in parallel planes, called the bases, and a curved sur-
face called the lateral surface.
1. With a strip of paper just
as wide as the height of some
right circular cylinder, roll it
about the cylinder, as shown in
the figure, using just enough to cover the lateral area. Now,
unrolling it, describe the shape and the dimensions of the paper
used. State a rule for finding the lateral area of a cylinder.
2. When S represents the lateral area of a right circular
cylinder whose diameter is d and whose height is A, give' in
words the relations expressed by the formula,
S = 7rdh»
3. Find the lateral surface of a cylinder 10 ft. long and
18 in. in diameter.
4. If a room is heated by the steam passing through 6
pipes each 14 ft. long and 2 in. in diameter (outer diameter),
how many square feet of radiation are there ?
5. If any room in your school is heated by cylindrical
pipes, me^asure them and find the amount of radiation in
the room.
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 99
15. THE YOLUME OF PYRAMIDS AND CONES
A pyramid is a solid bounded by any
kind of polygon as base, and by triangles
meeting at a point,
called its vertex.
By taking a pyra-
mid and a prism
having equal altitudes and equal bases,
and using the pyramid as a measure, and
filling the prism, as in the figure, it will
be seen that a pyramid is but one third as
large as a prism of the same dimensions.
The figure in the margin is
a right circular cone. The base is a circle. The
lateral surface tapers uniformly to a point called
the vertex. The straight line joining the vertex
with the center of the base is the altitude. The
distance from the vertex to any point in the cir-
cumference of the base is the slant height.
The same kind of experiment shows
the same relation between a cone and a
cylinder as between a pyramid and a prism.
1. Give in words the relation expressed
when V is the volume of a pyramid or
cone, B the area of the base, and h the
altitude.
2. Jf the base of a pyramid contains 20 sq. ft. and its
altitude is 9 ft., how many cubic feet in its volume ?
3. Find the volume of a cone the diameter of whose base
is 12 ft. and whose height is 7 J^ ft.
100 JUNIOR fllOII SCHOOL MATHRMATICS
4. The base of a pyramid is 4 ft. m|ii!ire and its altitude is
6 ft. Find its volume.
5. A pile of sand in the form of a cone 20 ft. across the
base and 9 ft. high contains how many loads (cu. yd.)?
6. A conical pile of potatoes 8 ft. across the base and 4 ft.
high contains how many bushels, allowing .8 bu. per cul»ie
foot?
7. A conical pile of grain 12 ft. across the base and G ft.
high contains how many bushels ?
8. In one comer of a bin, a pile of wheat forms ^ of a cone
the radius of whose base is 5 ft. and wliose height is 3 ft-
How many bushels in the pile ?
16. THE HEASUREHENT OF A SPHERE
A sphere is a solid bounded by a curved surface all points
of which are equally distant from the center. A straight
line from the center to the surface is the r&dlus, and a line
through the center terminating in the surface is the diameter.
A plane through the center divides a sphere into two hemi-
spberea. The Aat surface of a hemisphere is called a great
circle of the sphere.
1. If the surface of a
hemisphere be wound by
a hard waxed cord, and
that of its great circle by
the same cord, it will be
found that it takes just
twice as much cord to wind tlie hemisphere as the great
circle. What is your conclusion ?
a. Where S=i surface of a sphere whose diameter is d,
what relation is expressed by the formula
5=47rr3?
MEASUREMENTS, CONSTRUCTIONS, OBSERVATIONS 101
3. Find the surface of a sphere 10 in. in diameter. One
16 ft. in diameter.
4. The radius of the earth is approximately 4000 mi.
How many square miles in its surface ?
5. By drawing a number of planes through the center
of a sphere, it may be divided into a number of solids re-
sembling what solids that you
have studied ? If the bases of
these solids were planes, they
would be pyramids. From this
we infer what is proved in later
courses of mathematics that,
The volume of a sphere is the same as that of a pyramid
whose base is the surface of the sphere and whose altitude is its
radius.
That is, r=J/Sr.
6. What is the volume of a sphere 10 ft. in diameter ?
7. What is the volume of a sphere whose radius is 8 in. ?
Since the surface may be found from the radius, the
volume depends upon the radius only. From problem 5,
r=JXASr, butAS=47rA
Substituting, 1^=1 Trr^, the formula to remember.
8. By the formula, find the volume of a sphere whose
radius is 5 in.
9. If steel weighs 490 lb. to the cubic foot, find the
weight of a steel ball 6 in. in diameter.
10. A bowl in the form of a hemisphere 12 in. in diameter
will hold how many gallons ?
102 JUNIOR HIGH SCHOOL MATHEMATICS
11. A cylindrical haystack 8 ft. high and 14 ft. in diameter
is surmounted (" topped ") with a hemisphere. Find how
many tons of hay in the stack, allowing 512 cu. ft. to the ton.
Suggestion. — This is made up of a cylinder 8 ft. high and 14 ft. in
diameter, and a hemisphere 14 ft. in diameter.
12. If a cubic foot of iron weighs 450 lb., find the weight
of a hollow spherical shell 1 in. thick, with an outer diameter
of 10 in.
13. Compare the volume of a sphere 4 in. in diameter
with the volume of one 8 in. in diameter.
14. Compare the volume of a sphere 5 in. in diameter
with that of one 15 in. in diameter.
Oh%erve from 'problems IS and IJf that the ratio of the voluniie%
18 equal to the cubes of the ratio of the diameters.
15. When one sphere has a diameter 4 times as great as that
of another sphere, its volume is how many times as great ?
16. An orange 4 in. in diameter is how many times as
large as one 3 in. in diameter ?
17. If you knew the weight of a 2 in. steel ball, how could
you find, without weighing, the weight of one 10 in. in
diameter ?
CHAPTER IX
SQUARE ROOT AND THE PYTHAGOREAN THEOl^EM
If you know the sum of two numbers and one of them, by
mbtraction you can find the other. Or if you know the
product of two numbers and one of them, by division you
can find the other. But if you know that a number is the
product of two eqtial numbers j to find them requires a process
called square root. The product is called the square of one
of the equal numbers. Thus, the square of 7 is 49, and
the square root of 49 is 7. These are written 7^ = 49, and
V49=7. They are read <'7 squared is 49," and "the
square root of 49 is 7."
1. SQUARING A TWQ-FIGURSD NUMBER
Subtraction and division are inverse processes depending
upon the direct processes of addition and multiplication.
In the same way, extracting the square root of a number is
an inverse process depending upon the process of squaring a
number. The general process of squaring, a number may be
seen by analyzing the work of squaring some number as 47.
WORK
47 Explanation. — It will be seen
A^ by following the work in the
order in which it is done that the
first step 18 7 X 7, the next 7 x 40,
^880 = "^^ X4 Q + 40^ the next 40 x 7, and the last
2209 = 72 + 2 X 7 X 40 + 402 40 x 40.
103
104 JUNIOR HIGH SCHOOL MATHEMATICS
This being the work in squaring any two-figured number,
it is seen that,
The %quare of any two-figured number is the square of (yrvei
digit 'plus twice the product of the ones hy the value represented
by the tens^ digit^ plus the square of the value represented by the
tens* digit.
Thus, 752=52+2 x6x 704-702=25 + 700+4900 = 5625.
By this method square :
1. 63. 5. 38. 9. 93. 13. 76.
2. 72. 6. 96. 10. 84. 14. 53.
3. 85. 7. 57. u. 43. 15. 87.
4. 47, 8. 35. 12. 91. 16. 89.
2. FINDING THE SQUARE ROOT OF A NUMBER
Not many of the problems that you will jneet in the
ordinary walks of life require the process of square root, but
the subject is needed in mathematics and science that you
may study later and hence it is treated briefly here.
To get the process, first study the following squares to
get the relation of the number of root figures corresponding
to the number in the square.
Number of Ffgures in Roots and Powers Compared
12 = 1. 102 ^ 100. 1002 = 10,000.
92 = 81. 992 = 9801. 9992 = 998,001.
From the above powers and their roots, it appears that,
TTie number of periods of two figures each, beginning at onei
place, int,o which a whole number can be divided, equals the
mrniber of figures in the square root.
SQUARE ROOT AND PYTHAGOREAN THEOREM 105
Extracting Square Root
The process is shown by the following example :
Example. — Find the square root of 2809.
WORK
28'09'(63
52=25
100
103
309
309
Find:
1. V784.
Explanation. — It is seen that there are two
root figures. The first must be 5, for 50^ = 2500 and
602 -- 3600, and 2809 lies between the two. Then,
of the three addends that make 2809, 2500 or 50^ is
known. Subtracting 2500, 309 remains. This must
be the sum of the other two addends, the larger of
which is 2 X 50 X the ones* <iigit. Hence, 309 -f- 100
gives approximately the ones' digit, or 3.
Adding 3 to 100 gives 103, which multiplied by 3
gives 309, the two remaining addends being thus
found by one multiplication.
4. V3136. 7. V6329. 10. V7569.
2. V3364. 5. V6889. 8. V4489. 11. V2916.
3. V8464. 6. V2704. 9. V9801. 12. V9409.
The process is the same for larger numbers, as shown in
the following:
13. Find the square root of 2,137,444.
PROCESS
Find the square root of:
2'18'74'44'(1462
12 = 1
2
113
24
96
74
28
17
286
1716
292
58 44
29!
22
58 44
14. 283,024.
15. 299,209.
16. 404,496.
17. 556,516.
18. 755,161.
19. 6,017,209.
20. 529,984.
21. 484,416.
22. 638,401.
23. 725,904.
24. 294,849.
25. 1,739,761.
106
JUNIOR fflGH SCHOOL MATHEMATICS
26. Square the following decimals: .5, .35, .245.
Observe that the square of a decimal has twice as manj/ deci-
mat places as the number squared.
The process of finding the square root of a decimal is
shown in the following :
27. Find the square root of .734.
Explanation. — Since the square root
of .734 can be obtained only approxi-
mately, we plan to find it to three decimal
places. Hence, zeros are added until three
full periods of decimal figures are formed.
Since the square of tenths is hundredths,
to get the first root figure we take the first
two figures at the right of the decimal
point, or .73, the root of which is .8, nearly.
Twice .8, or 1.6, is taken as the first di-
visor. Each new root figure is determined
by division as in the case of integers. The inexactness of the r6ot is
expressed by + or — after the last root figure computed.
28. Study the process of extracting the square root of
.501 to three decimal places and explain the steps.
.'50'10'00'[^7072
.72=. 49
1.4
PKOCESa
.'73'40'00' .856+
.82=
.64
1.6
. 09 40
1.65
. 08 26
1.70
. 01 15 00
1.706
. 01 02 36
00 12 64
0110
00 98 49
1.407^
t"00 11 51
Find the square root of:
29. .5625. 32. .783.
30. .9216. 33. .89.
31. 42.225. 34, 19.467.
Find the square root to two decimal places :
41. 2. 43. 5.
42. 8. 44. 7.
35. 824.9.
36. .64.
37. .064.
38. 1982.4.
39. 225.9009.
40. .8.
Tial places :
45. 10.
46. 18.
' 47. 24.
48. 39.
SQUARE ROOT AND PYTHAGOREAN THEOREM' 107
Methods of Using the Table
Two methods of using the table given on page 108 to find
approximate roots of numbers larger than 100 are shown as
follows :
Find the square root of 7235.
WORK
85.44003
84.85281
.58722
M
293610
176166
.2065270
84.85281
85.0583870
Explanation. — By the tables, y/7S = 8.544003.
Hence, V7306 = 85.4400il Also V72 = 8.485281.
Hence, V7206 = 84.85281. The difference is .68722,
which is caused by a difference of 100 between the
numbers 7300 and 7200. But the given number
7235 is but 35 larger than 7200. Hence, .35 of the
difference between the roots is added to the root of
7200. This is but a close approximation.
SECOND METHOD
7235 - 7225 10
Explanation. — The given number
7235 lies between two numbers, 7225
yogg 7225 171 •^^"'* and 7396, whose roots are known to be
Root = 85.0584
85 and 86, respectively. The difference
between 7235 and 7225 is ^^ or .0584
of the difference between the two numbers, 7396 and 7225. Hence, the
root is approximately that much more than 85, the root of 7225. Ob-
serve that the results by the two methods agree to three decimal places.
-By the table ^Jind the square roots to nearest hundredth:
1. 4623. 7. 938. 13. 76.2. 19. .3846^
2. 5781.
3. 8746.
4. 1925.
5. ll78.
6. 3462.
8. 722.
9. 634.
10. 816.
11. 738.
12. 972.
14. 84.6.
15. 46.7.
16. 75.6.
17. 35.2.
18. 28.7.
20. .5763.
21. .936.
22. .847.
23. .76.
24. .5.
108
JUNIOR HIGH SCHOOL MATHEMATICS
TABLE OF SQUARES AND SQUARE ROOTS
NrVKKK
' S«rA«
SdrASS Boot
E
. SgCABB
SqicAmm Root
1
1
IMOOOO
* M
3601
7.141428
2
4
1.414213
52
2704
7.211102
3
9
1.73206O
L 53
1 54
2809
7.28D109
4
16
1 2.000000
2916
5
1 ®
2.236068 j
55
1
' 36
3025
7.416198
6
' 36
' 2.449489
3136
7.483314
7
1 49
1 2.645751
57
58
3249
7JMy834
8
64
2.838427
3364
7.615773
9
81
3.000000
69
3481
7.681145
10
100
3.161B77
60
3600
7.745966
11
121
3.316634
61
3721
7.810249
12
144
3.464101
62
3844
7.874007
13
169
3.606551
63
3969
7.937253
14
198
3.741657
64
4096
8.000000
15
2^
1
3.872983
65
4235
8.0G22S7
16
' 256
44W000O
66
4356
8.124038
17
289
4.123105
67
4489
8.185352
18
324
4.212640
68
4634
8.246211
19
361
Y.OOOOiKV
69
4761
8.306623
20
400
4.472136
70
4900
8.366600
21
441
4.582575
71
5041
8.426149
22
484
4.690415
72
5184
8.486281
23
529
4.795831
73
5329
8.544003
24
576
4.898979
74
5476
8.602325
25
625
5.000000
75
6625
8.660264
26
676
5.099019
76
5776
8.71779T
27
729
5.196152
77
5929
8.774964
28
784
5.291502
78
6084
8.a3i7eo
29
841
5.385164
79
6241
o.oooX«l4
30
900
5.477225
80
6400
8.944271
31
961
5.567764
81
6561
9.000000
32
1024
5.656854
83
6724
9.055385
33
1089
5.744662
83
6889
9.110438
34
1156
5.830951
84
7056
9.166161
35
1225
5.916079
85
7225
9.219644
36
1296
6.000000
86
7396
9.273618
37
1369
6.082762
87
7569
9.327379
38
1444
6.164414
88
7744
9.380631
39
1521
6.244998
89
7921
9.433981
40
1600
6.324555
90
8100
9.486833
41
1681
6.403124
91
8281
9.539392
42
1764
6.480740
92
8464
9.591663
43
1849
6.557438
93
8649
9.643650
44
1936
6.663249
94
8836
9.695359
45
2025
6.708303
95
1
9025
9.746794
46
2116
6.782330 ;
96
97
9216
9.797969
47
220fl
6.855654
9409
9*48867
48
2304
6.928303
98
9604
9.899494
49
2401
7.000000
99
9801
9.949874
60
2600
7.071067
100
10000
10.000000
SQUARE ROOT AND PYTHAGOREAN THEOREM 109
3. SOME APPLICATIONS OF SQUARE ROOT
Some of the indirect problems of mensuration require
square root. Thus, the area of a square whose side is 47 in.
is 47 X 47, or 2209 sq. in. But the number of inches in the
side of a square containing 2209 sq. in. is V2209 or 47.
Likewise, the area of a circle whose radius is 15 in. is 15 x
15 X 3.1416, or 706.86 sq. in. ; but the number of inches in
the radius of a circle having an area of 706.86 sq. in. is
V706.86 -h 3.1416, or V225, which is 16.
1. Find the side of a square containing 5329 sq. in.
2. Find the dimensions of a rectangle twice as long as it
is wide that contains 8978 sq. in.
Suggestion. — The rectangle will make two squares each containing
4489 sq. in. Show this by a diagram.
3. Find the diameter of a circular plot that has the same
area as a rectangular one 18 ft. by 24 ft.
4. What must be the diameter of a circle having three
times the area of one 10 ft. in diameter ?
5. What must be the side of a square having twice the
area of one whose sides are each 16 ft. ?
6. A water main 15 in. in diameter is to be replaced by
one having three times the carrying capacity. What must
the diameter of the new pipes be ?
Suggestion. — To have three times the carrying capacity, the area of
a cross section must be three times as large as the area of the smaller.
7. Frank has a garden 38 ft. by 96 ft. If he replaces
this by a square garden of the same area, what must be its
side? ,
8. How much less fencing is needed to inclose a square
garden as large as a rectangular one 120 ft. by 800 ft.?
110
JUNIOR mOH SCHOOL MATHEMATICS
4. TH£ PTTHAGOREAN THEOREM
A right triangle is a triangle of which one angle is a right
angle. The side opposite the right angle
is called the hypotenuse, and the other
sides are called the legs. By drawing a
right triangle whose legs are 3 in. and
4 in., respectively, it will be seen that the
hypotenuse is just 5 in., and that the area
of the square on the hypotenuse equals
the sum of the areas of the squares on the
two legs.
This important truth was proved by Pythagoras, about
500 B.C., to be true of any right triangle. That is.
The square on the hypotenuse of any right triangle is equal
to the sum of the squares on the other two sides.
Note. — Carpenters make use of this fact in laying out the foundation
of a building when they want the walls at right angles to each other.
Starting at one comer, a line 8 ft. long is taken in one direction along
which the foundation is to be laid. Starting from the same comer, an-
other line 6 ft. long is fastened to the end of the first line and moved
about until a 10-ft. rod will just reach the outer extremities of the two
lines.
The truth of the Pythagorean theorem may be seen by
drawing, or cutting from cardboard, figures like the
following : ^c
A. B
SQUARE ROOT AND PYTHAGOREAN THEOREM 111
Let ABC be the right triangle. The square on the hy-
potenuse AC i^ equal to the four triangles, 1, 2, 3, and 4,
and the small square, 5. Now put 1 and 2 in the position
of the figure at the right, and the figure is equal to a square
on AB and one on CB.
1. One leg of a triangle iy 48 ft. and the other is 36 ft.
What is the hypotenuse ?
2. The hypotenuse of a right triangle is 86 ft. and one
leg is 51 ft. What is the other leg ?
3. One leg of a right triangle is 76 ft. and the hypotenuse
is 95 ft. What is the other leg ?
4. What is the diagonal (distance between the opposite
corners) of a rectangle 92 ft. long and 69 ft. wide ?
5. How long is the diagonal of a 30-foot square ?
6. What is the length of the longest straight line that
can be drawn on a sheet of paper 16 in. by 20 in. ?
7. How far is a place 12 mi. east of you from one 18 mi.
north of you ?
8. What is the distance between the opposite corners of
a field 200 rd. long and half as wide ?
9. If a window is 18 ft. from the ground, how long must
a ladder be to reach to the window if the foot of the ladder
is placed 6 ft. out from the building ?
10. In decorating a room two ribbons are stretched, con-
necting the opposite corners. If the room is 30 ft. wide and
40 ft. long, how many yards of ribbon does it take ?
11. A baseball diamond is 90 ft. square. How long is
the throw from first to third base ?
12. A derrick is 48 ft. high, and is supported by three
steel cables, each reaching from the top of the derrick to a
stake in the ground 45 ft. from the foot of the derrick. How
/
112 JUNIOR HIGH SCHOOL MATHEMATICS
much steel cable does it take, allowing 10 ft. for fastening
all three cables ?
13. For reaming round holes .5 of an inch in diameter, a
square reamer is used. Find the dimensions of the reamer.
14. If the base of an isosceles triangle (a triangle having
two equal sides) is 12 in. and its altitude is 10 in., find the
length of its equal sides.
Suggestion. — The altitude of an isosceles triangle divides the tri-
angle into two equal right triangles.
15. If the equal sides of an isosceles triangle are each 15
in. and the base 12 in., what is the altitude ?
16. Find the area of an isosceles triangle whose base is
14 in. and whose equal sides are each 12 in.
17. Find the area of an equilateral triangle each of whose
sides is 20 in.
CHAPTER X
GENERAL DISCUSSION OF PERCENTAGE
f ou have become familiar with the use of per cent to ex-
press the relations between two numbers, and have seen that
it is but another name and notation for hundredths. Thus,
8% = T*(F = -08; 2J% = .026;
.28 = ^ = 28% ; .035 = ^%
Problems whose relations are expressed in per cent are
sometimes called percentage problems.
1. A REVIEW OF FORMER WORK IN PERCENTAGE
In the mathematics that you have had, you had but two
kinds of problems involving the use of per cent. They
were :
(1) To find a per cent of some number ; and
(2) To find what per cent one number is of another.
These two problems are by far the most common in all
ordinary uses of mathematics. As you have seen, the first
is but an application of the multiplication of decimals after
the per cent has been expressed as a decimal ; and the second
is but an application of the division of decimals, with the
exception of expressing the quotient as a per cent.
113
114 JUNIOR HIGH SCHOOL MATHEMATICS
Drill Exercises
Change to decimals :
1. 45%. 6. 16%. 11. 15.4%. 16. 4^%.
2. 63%. 7. 16^%. 12. 3.6%. 17. 6J%.
3. 7%. 8. 4%. 13, .8%. 18. 8|%.
4. 9%. 9. 4J%. 14. 125.4%. 19. 9J%.
5. 188%. 10. 240%. 15. 245.6%. 20. 200%.
Find :
21. 24% of 650. 27. 125% of 980. 33. 2.4% of 780.
22. 38% of 98.5. 28. 240 % of 650. 34. 24.6% of 540.
23. 6.4% of 760. 29. 350% of 920. 35. 2.46% of 360.
24. 5.2% of 84.5. 30. 175% of 685. 36. 17^ % of 560.
25. 21.5% of 780. 31. 16^ % of 840. 37. 37J% of 560.
26. 9.5% of 866. 32. 4f % of 720. 38. 200% of 765.
Change to per cent :
39.
.35.
43. .735.
47. 1.35.
51.
.015.
40.
.48.
44. .864.
48. 2.48.
52.
.008.
41.
.09.
45. .025.
49. 2.9.
53.
.0085.
42.
.095.
46. .258.
50. 3.2.
54.
.1225.
Find what per cent :
55. 48 is of 85. 58. 46.8 is of 39.6. 61. 9 is of 4.48,
56. 95 is of 148. 59. 175 is of 84.5. 62. 1.8 is of .96.
57. 7.3 is of 16.6. 60. 28.7 is of 36. 63. 7.2 is of 4.35.
GENERAL DISCUSSION OF PERCENTAGE 115
2. INTERPRETING AND FINDING PER CENTS OF
INCREASE OR DECREASE
In general reading we constantly meet references to in-
creases or decreases in production, consumption, prices, wages,
and in various other things in which we are interested, all
given in terms of per cent. To read intelligently such
articles we must be able to interpret such references and to
find such relations for ourselves.
1. In a recent news item it was stated that from 1914 to
1918 the cost of food had advanced 64% and of clothing
66%. What does this mean? Food that cost |1 in 1914
would cost how much in 1918 at this rate of increase ?
Clothing that cost $20 in 1914 would cost how much in
1918 ?
2. The lAterary Digest^ Sept. 7, 1918, says that an unmis-
takable evidence of thrift among wage-earners is shown by
the fact that the membership in Building and Loan Associa-
tions has increased 52 % since the beginning of the War in
1914. What does this mean? To every 100 members in
1914 there were how many in 1918 ?
3. The same article (problem 2) states that the increase
in the amount of deposits during the 15 years preceding
1918 was 205%. What does this mean? To every 1 100
deposited fifteen years before, how many dollars were de-
posited in 1918 ?
4. The same article (problem 2) says that while the
membership increased 52 % during the four years preceding
1918, the amount of the deposits increased but 30 %. From
this, are the average individual deposits larger or smaller
than those four years ago ?
116 JUNIOR HIGH SCHOOL MATHEMATICS
5. The Bureau of Immigration reported that our immi*
^ration had lost 78% from January, 1917, to January, 1918,
and that it had lost S6% since 1913. What does this mean?
To every 100 immig^nts landing in 1913, how many landed
in 1918 ? To every 100 landing in 1917, how many landed
in 1918 ?
6. Our potato crop averages but 90 bushels per acre,
while that of France averages 135 bushels, and that of Great
Britain averages 124 bushels. The production of potatoes
in each of these countries is an increase of what per cent
over the average amount that we produce per acre ?
7. A magazine article in June, 1918, says that the fact
that the price of imports into and exports from the United
States has increased from 50 % to 100 % within a year shows
the general advances in prices to be world-wide. What
does an increase of 50 % to 100 % mean?
8. Olive oil was imported from Italy at an average price
of $ 1.25 per gallon in 1914, and of $ 3.05 per gallon in 1918.
Find the per cent of increase in price.
9. Flax was imported at $290 per ton in 1914, and at
f 1188 per ton in 1918. Find the per cent of increase.
Linen being made from flax, what would such an increase
indicate as to the cost of all linen articles ?
10. We exported upland cotton at an average price of
8.5 ^ per pound in 1915, and at an average price of 81.7 ^
per pound in 1918. Find the per cent of increase.
11. A magazine article of August, 1918, says, " That sub-
marine warfare has still a long way to go to stop or even
check our trade with the rest of the world, is shown by the
following report : "
GENERAL DISCUSSION OF PERCENTAGE
117
1915
1916
1917
Exports
Imports
92,500,041,944
1,516,474,600
♦ 3,867,115,373
1,962,033,212
9 5,718,000,000
2,342,000,000
Find the rate of increase each year over the preceding,
both in exports and-in imports.
12. During the War there was a marked decrease in the
importation of many articles of food usually classed among
the luxuries. There were but 9,000,000 pounds of cheese
imported in 1918 as against 15,000,000 pounds in 1917, and
64,000,000 pounds in 1914 ; the import of currants in 1918
was but 5,000,000 pounds as against 25,000,000 pounds in
1916, and 32,000,000 pounds in 1914 ; and the importation
of dates dropped from 34,000,000 pounds in 1914 to 6,000,000
pounds in 1918. Find the per cent of decrease of each
article from one date to the next.
3. A K£W PROBLEM IN PERCENTAGE
Since a per cent expresses the relation of one number to
another, if one number and its relation to the other, expressed
in per cent, are known, the other number may be found.
Thus, if 35 % of some number is known to be 70, the number
is evidently 200. For 35 % of 200 is 70. This, then, is the
inverse of the first kind of problems and is called the indirect
problem of percentage. This type of problem has fewer
practical applications than the two kinds already studied.
However, it has a use ; hence, the way to solve it will be
shown here. The general problem is,
To find all of a nvmher when a certain per cent of it is known.
This problem is easily recognized,* for the per cent named
in the problem refers to a number not given in the problem,
118 JUNIOR HIGH SCHOOL MATHEMATICS
but to the one to be found, instead of referring to the
number given, as in the first type of percentage problem
studied.
1. An article costing $23.10 will have to sell for what
price in order to give the dealer a gross profit of 40 % of the
selling price ?
Analysis of the Problem. — Our experience tells us that when a
dealer sells an article at a profit, he is getting back, in the selling price,
both the cost and the profit. That is, the selling price equals the cost
plus the profit. But since 100 % of anything is all of it, if 40 9& of the
selling price is profit, the remaining 60 ^ of it must represent the cost
Hence, we have the following relation :
60 9^0 of the selling price = J 23.10.
This means .6 x (an unknown factor) = f 23.10. That is, the product of
two factors and one of the factors are known. The problem is to find
the other. From the meaning of division the solution is
123.10^.6 = f 38.50.
2. An article costing f 52.§0 must sell at what price to
give a profit of 30 % of the selling price ?
Explanation. — Since 30 9^ of the selling price
.7)j^62.50 was profit, 70 9fc of it must be the cost, or < 62.60.
$76 Then % 52.50 -;- .70 must give the selling price.
3. At what price must goods be marked in order that
20 % of the marked price may be deducted and leave a sell-
ing price of 1 7.60 ? (80 % of marked price = % 7.60.)
4. At what price must goods be marked if f 5.60 is to be
received for them after deducting 30 % of the marked price ?
5. A merchant having mislaid the cost price of some
goods marked at f9.75 remembers that they were marked
to sell at 30% above the cost. Analyze the problem and
show how he can find what the goods cost him.
GENERAL DISCUSSION OF PERCENTAGE 119
6. In making white flour, 72 % of the wheat is used.
How many bushels (60 lb.) of wheat will it take to make
12 bbl. (196 lb. per bbl.) of wheat flour?
7. In making whole wheat flour, 85% of the wheat is
used. How many bushels of wheat will it take to make
12 bbl. of whole wheat flour ?
8. In making graham flour, 96 % of the wheat is used.
Find how many bushels of wheat are needed to make 12 bbl.
of graham flour.
9. Experience shows that cattle lose 44 % of their live
weight in dressing. What must be the live weight of
cattle that dress 616 lb.? That dress 846 lb.? That dress
580 lb.?
10. When small undressed fish that will lose 45 % in
dressing are selling at 18 ^ per pound, a slice of a larger fish
having no waste is selling at 24 ^ per pound. Which will cost
less and how much less when 6 lb. of dressed fish are needed ?
11. Hogs lose 20 % in dressing. How large a hog will it
take to dress 192 lb.?
12. About 20 % of the dressed weight of a hog goes into
lard. A farmer got a 48-pound pail of lard from one hog.
How large must it have been ?
13. In canning berries a woman found she got an average
of but 40 % as much canned fruit as she used of raw fruit.
How many berries must she buy for 48 quarts of canned fruit ?
14. If a dealer sells you an article for $24 which he tells
you is 20 % less than the usual price, find the usual price.
15. During a special sale, a firm sold all of its athletic
goods for 10 % less than the regular price. How much
would you have to pay at regular price for what you could
buy for $14.86 during the special sale?
120 JUNIOR HIGH SCHOOL MATHEMATICS
Drill Exercises
1. 115.2 is 20 % more than what number ?
2. 76.8 is 20% less than what number?
3. $10.92 is 40 % more than what sum ?
4. $ 4.68 is 40 fo less than what sum ?
5. A gain of $4.50 is 20% of what an article cost.
Find the cost.
6. A gain of $7.50 is 15% of what goods sold for.
Find the selling price.
7. When selling goods for $10.08, 20 % of the cost is
gained. Find the cost.
8. When goods costing $8.75 are to sell at a profit of
30 % of the selling price, find the selling price.
9. When chickens lose 25 % in dressing, what size, live
weight, will dress IJ pounds?
10. A dressed hog weighs about 80 % of its live weight.
What live weight will dress 280 pounds ?
4. APPLICATIONS OF THE THREE PROBLEMS OF
PERCENTAGE
Most of the relations expressed about the quantitative
side of life are expressed in terms of per cent. In our daily
reading we see increases or decreases of all kinds referred to
in per cent. To enable you to interpret such references is
the purpose of this list of problems.
1. It is estimated that a man having a family of five and
receiving a salary of $3500 per year should portion his
several expenses as follows : food, 25 % ; rent, 20 % ; cloth-
ing, 22 % ; operating expenses, 15 % ; and use the balance
for savings, charity, and recreation. Find how much this
would allow for each item.
GENERAL DISCUSSION OF PERCENTAGE
121
2. It is estimated that a family of four, with an income of
11500 per year, should divide it as follows : food, 85 % ;
rent, 20 % ; operating expenses, 15 % ; clothing, 18 % ; and
use the balance for savings, charity, and recreation. Find
how much this would allow for each item.
3. A study made in 1907 showed that untrained girls
were earning a maximum of $8.75 per week, while the
trained girls of the same age were earning $20.25. Find
what per cent more the trained girls were earning.
4. In 1909 it was found that at the age of thirty the
average salary of a group of men taken at random, all of
whom had received a grammar school education, but no
further training, was % 1258, while the average yearly wage
of a number of illiterate workers at the same age was but
$500. The first group received how many per cent more ?
5. Statistics show that at the end of 1917 the average
increase in price of food since the end of 1915 had been 63 %.
If so, $100 at the end of 1917 would buy as much as what
sum in 1915 ? It would take how much in 1915 to buy as
much as $ 100 would buy in 1917 ?
6. The increase in price of six important commodities
is shown in the following table. Compute the per cent of
increase in the price of each commodity.
Obop
Unit
1916
1917
Wheat
bushel
• 1.071
• 2.289
Com
bushel
.794
1.966
Barley
bushel
.593
1.145
Rye
bushel
.834
1.781
Potatoes
bushel
.954
1.708
Cotton
pound
.126
.243
122 JUNIOR fflGH SCHOOL MATHEMATICS
7. In 1916 we produced but 640,000,000 bushels of
wheat. This was a decrease of what per cent of a five-year
average of 728,000,000 bushels ?
8. In 1915 we produced 1,026,000,000 bushels of wheat
This was an increase of what per cent over a five-year
average of 728,000,000 bushels ?
9. What is meant by saying that the production of a
certain article is 226 % of its former production ?
10. What is meant by saying that the production of an
article has increased 225 % over its former production ?
11. If the factory output of, a certain article is now 226 ^
of its former output of $ 726,000 yearly, what is it now ?
12. If the factory output of a certain article is now 226 ^
more than its former output of $ 726,000 yearly, what is it
now ?
13. If in 1918 we decreased our average consumption of
wheat, which was 680,000,000 bushels yearly, by 27 %, what
was the consumption of that year ?
14. Before the World War, France used 380,000,000
bushels of wheat annually, 86 % of which she produced. If
her production was cut 40 % during the War, how much
did she have to import annually to give the same consump-
tion ?
15. A news item says, " Montana, Idaho, Wyoming, and
Oregon produced 86,256,000 pounds of wool in 1916, which
is about 30 % of the total production of the entire United
States." From these data find the total production of that
year.
16. With an increase in the acreage under cultivation
and the consequent restricting of pasturing acreage, the
number of sheep raised in Texas decreased from 4,260,000
in 1890 to 1,600,000 in 1915. Find the per cent of decrease.
GENERAL DISCUSSION OF PERCENTAGE 123
17. Vermont at one time was a very large sheep-produc-
ing state, but the number decreased from 1,682,000 in 1840
to 47,000 in 1916. The number in 1915 was what per cent
of the number in 1840 ?
18. Rye being hardier than wheat and succeeding in
poorer soils, the Department of Agriculture in 1917 recom-
mended an acreage of 5,131,000. " If planted, this will be
an increase of 22 % over our ten-year average," says a news
item. From these data find our ten-year average.
19. Our acreage of beans, an especially important food in
war time, was 84 % niore in 1917 than in 1916. If you
knew the acreage in 1916, how could you find the acreage of
1917? If you knew the acreage of 1917, how could you
find the acreage of 1916 ?
20. If a merchant pays $24.50 for an article and marks
it 80 as to give a discount of $5.50 from the marked price
and still make 20 % of the marked price, find the price at
which he marked it. The discount was what per cent of
the marked price ?
21. If fish lose 40 % in dressing, what is the cost per
pound of the dressed fish when undressed fish are 18 ^ per
pound ?
22. Which is cheaper and how much : live chickens at
25 ^ per pound, or dressed ones at 85 ^, if the loss in dressing
is 80 % ?
23. What per cent of his sales is the ice man making if
he sells ice at 60 fl per 100 pounds, for which he pays, includ-
ing the cost of delivery, $7.50 per ton, the loss through
melting being 15 % of each ton ?
CHAPTER XI
BUSINESS TERMS, FOBMS, AND PROBLEMS
In Book I you studied certain common business terms and
forms that you meet in the ordinary walks of life. These
will be reviewed and extended so that you will be able to
interpret references to them, which you will meet more and
more in general reading and in conversation.
1. BILLS RENDERED BT THE RETAIL MERCHANT
Bills are statements rendered to a purchaser showing the
date and price of purchases made, credits if any, and the
final amount due. To keep one's credit with a store, all
bills should be settled promptly. That is, within a few
days of the time they were rendered.
1. If you bought goods, paying $3 at the time of purchase,
and returned goods costing $ 1.25, what is the total credit
allowed on the bill ?
2. If your mother buys 6^ yd. of cloth at 95 ^ per yard,
5 yd. of lining at 65 ^ per yard, a house dress at $ 3.85, and
a waist at $2.98, find the total amount of the bill.
3. If, in problem 2, the dress is returned, what credit item
will the bill show when the bill is rendered, the goods
having been charged to her account ?
4. Check the following bill, that is, see if there is any
error in the computation :
124
BUSINESS TERMS, FORMS, AND PROBLEMS 125
1918
Sept.
Itsms
Total
GlIAROEB
Total
Grkdits
Balahos
4224
4906
1128
926
3
6
8
13
14
17
1 dress
4i yd. satin 93.50
^ yd. satin 2.75
3 yd. percale .59
1 skirt
2 yd. lining .59
1 waist
i yd. silk 2.50
1 pr. gloves
1 skirt retd.
12J yd. braid .20
Ibag
} doz. buttons 1.25
9 15.75
12.38
1.77
922.98
29.90
3.98
7.16
7.11
93.98
3107
2986
1732
4102
4396
1.18
2.98
1.25
1.75
4510
3946
4098
2.50
3.98
.63
■
971.13
9 67.15
5. Rule paper and make out a bill for the following,
heading it with the name of some store in your city and
naming Mrs. Richard Roe as buyer : 8 pr. of hose at 69 ^ ;
2 skirts at $4.25; 6 yd. satin at $8.98 ; 4 yd. lining at 59^ ;
1| yd. belting at 82^; 6^ yd. silk at $3.25 ; 1 suit case at
19.75 ; 4| yd. satin at $3.25; credit 1 skirt returned, $4.25 ;
and $10 cash payment at time of purchase.
6. Why should one keep a receipted bill, that is, a bill
that has been paid? Sometimes, when payment has been
made by a check on some bank, the top part of the bill
showing the name of the purchaser and the amount of the
bill is torn off and that part only is returned with the pay-
ment, the part shown in problem 4 being kept by the pur-
chaser. In such cases, no receipt of the paid bill is returned.
See if you can find out from some older person why no
receipt is necessary.
126
JUNIOR HIGH SCHOOL MATHEMATICS
7. Bring to class paid bills. Tell of whom the goods
were purchasted, by whom, and when payment was made.
Check them to see if any error was made in computation.
2. KEEPING ACCOUNTS
Careful persons in all walks of life keep some sort of
account 'of their business dealings. An account is a record
of value received and of value delivered. These accounts
are kept in various forms, depending upon the needs of those
keeping them. A few forms are shown here.
Personal Cash Accounts
One should early form the habit of keeping a careful
record of all money received and when and for what it was
expended. The following simple form is a very common
type.
Personal Cash Book
191»
Received
Paid
Jan.
6
Balance on hand
4
60
7
Received allowance
2
00
t
Paid for book
1
25
8
Paid for lunch
30
8
Received for errand
50
9
Received for shoveling snow
75
10
Paid for R.R. ticket
1
20
12
Deposited in savings bank
4
00
12
13
Balance
Balance on hand
1
10
7
1
85
7
85
Jan.
10
The balance item is entered in the smaller column, so that
each side will "total" the same. It is the excess of the
total amount received over the total amount paid out.
BUSINESS TERMS, FORMS, ANI> PROBLEMS 127
1. Rule paper like the form shown here and make out a
^cash account" showing the balance at the end of each week.
Receipts : March 5, 1919 (Monday), balance on hand, $3.20 ;
March 6, allowance, $1; March 8, errands, 60^; March 9,
payment for bicycle, $15; March 10, errands, 90^; March 18,
allowance, $1 ; March 15, balance on bicycle, $10; errands,
40^; March 17, for delivering papers for the week, $2.75;
March 20, allowance, $1; March 2l, for 8 hens, $14.30;
March .24, errands, 70 ^ ; March 24, delivering papers for
the week, $2.75. Paid : March 8, carfare, 20^ ; March 9,
lunch, 30 ^, carfare, 10 ^ ; March 14, lunch, 35 ^ magazine,
20^; March 16, carfare, 20^ ; March 17, deposited in sav-
ings bank, $ 9 ; March 22, bought 4 hens, $ 7.10 ; March 23,
feed for hens, $1.50 ; March 24, deposited in savings bank,
$5. Show the balance at each week-end and the balance
on hand Monday, March 26.
2. Arrange the following as a cash account balanced at
the end of each week. It begins Apr. 1, 1919 (Tuesday),
hence, balance it Apr. 5, 12, 19, and 26, leaving the last
items, beginning Apr. 28, unbalanced : 1. Cash on hand,
$4.50, spent 15 ^ carfare, 30 ^ for lunch ; 2. Allowance, $ 1,
from errands, 40 ^ ; 3. Bought magazine, 20 ^, had bicycle
repaired, $ 1.20 ; 5. Deposited $ 3 in savings bank, received
$3.25 for delivering papers; 7. Allowance, $1, carfare,
20 |zf ; 8. From errands, 60^, for lunch, 30 ^, carfare, 10 ^ ;
9. Movie ticket, 20^, carfare, 10^; 11. Bought catcher's
glove, 85 ^, carfare, 10 ^ ; 12. Received the week's wages
for delivering papers, $3.25, deposited $3 in savings bank ;
14. Allowance, $1, sold old catcher's glove, 35^ ; 16. Errands,
50 ^ carfare, 15^ ; 17. Bought magazine, 20^, fruit, 10^ ;
19. Week's wages for delivering papers, $3.25, deposited,
13; 21. Allowance, $1, for carrying packages, 65^; 22. For
128 JUNIOR HIGH SCHOOL MATHEMATICS
raking lawn, 60 ^ ; 23. Carfare, 10 ^ ; 26. From errands,
60^; 26. Week's wages for delivering papers, $3.25, de-
posited, $ 4 ; 28. Allowance, $ 1 ; 29. Earned 60 ^ deliver-
ing packages, spent 30 ^ for lunch ; 30. Movie ticket, 20 ^
carfare, 10 ^, fruit, 16 ^.
3. Keep a cash account of your own receipts and expendi-
tures for a month, and bring it to class one month from the
time you study this.
Household Accounts
A monthly household account should show a record of all
incomes for the month, as salary and other items, and of all
expenses, as items for rent, food, operating expenses (fuel,
lights, wages, etc.), clothing, and higher life (amusements,
travel, church, charity, education, etc.). The usual form is
that of the personal cash account which you have studied.
The items in the monthly account are taken from "totals"
shown in the daily and weekly accounts.
1. Rule paper and balance the following, supplying dates
Receipts: balance of cash on hand, $48.60; salary, $226
miscellaneous, % 30.40. Paid out : rent, % 60 ; food, $ 48.76
clothing, f62.80; operating expenses, $14.30 ; higher lif<5,
$28.30 ; savings bank deposit, $30.
2. Balance the following : Cash on hand, $28.30 ; monthly
allowance for expenses, $ 160. Paid out : food, $ 45 ; cloth-
ing, $38.60; operating expenses, $16.80; rent, $85; highet
life, $6.80 ; health, $3.76.
3. Balance the following : Cash on hand, $82.75; monthly
allowance, $126. Paid out: rent, $80; food, $42.60;
clothing, $34.20; operating expenses, $10.30 ; higher life,
$4.26.
BUSINESS TERMS, FORMS, AND PROBLEMS 129
4. In the following weekly account find : (a) the total of
each item for the week ; (5) the total expense for each day ;
and (jo) the total expense for the week.
Weekly Household Expense Account
MON.
Tirss.
W»D.
Thvbs.
FBI.
Sat.
Sfn.
Total
Food
Groceries
$2.10
3 .90
$1.20
Meat
.70
8 .30
.50
$ .40
9 .30
1.30
Milk
.16
.16
.16
.32
.16
.32
Clothing
4.80
6.30
Operating expenses
Fuel
4.60
light
1.20
Telephone
.40
Laondry
1.80
Higher life
Amusements
.60
1.00
Papers and magazines
.02
.02
.02
.02
.07
.04
$ .07
Church
1.00
Charity
.50
Health
2.50
Total
Ledger Accounts
In business, more elaborate accounts are kept. There are
two terms that the business man employs that were not used
in the simple accounts that you have studied. These two
terms are debit and credit. A record of deMU is a record of
debts or of value received ; a record of credits is a record of
value delivered. Among the most common accounts in
business are personal accounts; cash accounts; merchandise
accounts ; and expense accounts.
A merchant's personal account shows the amount owed
to or owed by the person whose name appears at the head of
the account.
130
JUNIOR HIGH SCHOOL MATHEMATICS
The following shows the form of ledger account between
L. Harris & Sons (merchants) and J. S. Lee, a customer.
Dr,
J.
S. Lee,
56 Elm
St.
Cr.
1919
1919
■»
Jan.
3
Mdse.
28
40
Jan.
6
Cash
25
00
8
«
16
70
15
«
15
00
28
i<
19
64
80
90
Feb.
1
Balance
24
64
90
90
Feb.
1
Balance
-^
90
10
Cash
30
00
6
Mdse.
17
80
20
«
20
00
15
it
26
30
Mar.
1
Balance
19
00
69
00
^
69
00
1. Check the personal account shown above, that is, see if
the computation is correct.
2. Who bought the merchandise shown by the items of
the account ? Of whom were they bought ? Who kept the
account ?
3. What does the balance on Feb. 1 show ? On March 1 ?
Had there been a balance on the "Dr." side, what would
it have shown ? In such an account is there likelv to be a
balance on that side ? Why ?
4. Had the merchant loaned Mr. Lee $15, upon which
side would the item have been written ? Why ?
5. Had Mr. Lee sold the merchant some produce, or
rendered him some service, upon which side of the account
would it have been placed ? Why ?
6. Pretending that you are a merchant and that some
student is your customer, make up and balance an account,
and explain the meaning of each item.
BUSINESS TERMS, FORMS, AND PROBLEMS 131
A merchant's Cash account is a record of debits and credits
of cash, the balance showing the cash on hand. It is as if the
merchant was keeping an account with his own cash box.
Hence, "Cash" is debtor of all that is put into it, and
credited with all that is taken out.
Dr,
Cash
Cr.
I9i»
1919
May
1
Balance
986
30
May
7
Mdse.
1250
00
6
Mdse.
680
50
10
u
750
00
7
<(
950
00
10
Office furniture
150
00
15
J. Morris
250
00
15
Wages
200
00
16
Balance
80
80
15
Balance
516
80
2866
2866
80
May
616
7. Do the " mdse." items of May 6 and 7 show a record
of merchandise bought or sold ? Why ?
8. What does the " J. Morris " item of May 16 show ?
9. What do the merchandise items on the credit side
show?
10. What is the meaning of the item " office furniture "
of May 10 ?
11. Check the account and see if any mistakes haye been
made in the computation.
The Merchandise account is a record of the cost of goods
bought (debits) and of the receipts from goods sold
(credits). The inventory item on the credit side shows the
value of the goods on hand. The balance shows whether
there was a loss or gain.
132
JUNIOR HIGH SCHOOL MATHEMATICS
Dr.
Merchandise
O.
1919
1919
June
1
On hand
840
00
June
5
Cash
576
80
10
Brown & Co.
960
40
■
7
Note
240
00
12
K L. Smith
796
30
18
S. C. Hart
160
00
18
Cash
536
80
24
Cash
1080
00
30
1
Balance
On hand
448
60
10
30
30
Inventory
1525
30
3582
3582
10
July
1525
12. Check the merchandise account shown here.
13. What items show the amount of goods bought ? Of
goods sold ? Of goods unsold ?
14. What does the balance of $448.60 on the debit side
show ? What would it have shown had it been on the credit
side ?
15. What does the " note " item of June 7 show ? The
" S. C. Hart " item of June 18th ?
16. What do the " cash " items on the credit side show ?
The " cash " item on the debit side ?
The Expense account shows the cost of doing business.
To the account, rent, fuel, lights, postage, salaries, etc., are
debited, and unused coal, postage, etc., are credited in find-
ing the balance, which is the net cost of doing business.
Farm Accounts
Successful farmers often keep an account with each crop or
kind of stock raised, as account mth wheat ; account with com;
account with hog%^ etc. More often they are like the follow-
ing form :
BUSINESS TERMS, FORMS, AND PROBLEMS 133
Account with Wheat, 40 Aches
Cost
RSTITBKB
1918
Oct.
Plowing and seeding
Seed
170
225
00
00
June
Cutting
70
00
Aug. .
Threshing
Interest on land investment
820 bu.@ 92.20
60 tons straw @ $4.50
40
240
00
00
1804
225
00
00
17. Find the net profit per acre from the wheat.
Rule forms for a merchant's personal account with his mis-
tomers and balance thefollomng:
18. Roberts & Sons in account with R. L. Jones. Sales :
Sept. 3, furniture, $ 386 ; Sept. 5, rugs, $ 175 ; table, % 48 ;
Sept. 15, refrigerator, $ 38, range, $ 48. Credits : Sept. 3,
cash, $ 250 ; Sept. 10, cash, $ 150.
19. Cuthbertson Bros, in account with W. A. Miller.
Sales : Aug. 1, groceries, $5.60 ; Aug. 4, clothing, $7.80 ;
Aug. 15, shoes, $6.25 ; Aug. 20, groceries, $3.85 ; Aiig. 25,
hardware, $5.60. Credit;S : Aug. 5, cash, $10; Aug. 15,
services, $3.50 ; Aug. 28, cash, $15.
20. Willey & Son in account with A. P. Smith. Sales :
Nov. 3, lumber, $18.60 ; nails, $1.20 ; paints, $3.50. Credits :
Nov. 7, nails returned, $.50; cement returned, $1.20; paint
returned, $.75 ; Nov. 20, cash, $18.
21. West & Son in account with E. R. Harris. Sales:
$86.30; $94.30; $68.70; $42.30; $86.90; $75.80. Credits:
By cash, $150 ; $75 ; $40 ; by returned goods, $16.50 ; $7.60.
(Supply dates.)
134 JUNIOR fflGH SCHOOL MATHEMATICS
22. A. Sellers & Co. in account with R. G. Lyons. Sales :
1168.80; 190.30; 184.70. Credits by returned goods,
service, and cash: $6.30; $4.80; $126; 186. (Supply
dates.)
3. BUYING AND SELLING AT A DISCOUNT
You, no doubt, have heard some one say that he bought or
sold some article at a discount, A discount is a deduction
from some former price. Thus, goods out of season or for
cash often sell at a discount from a former or regular price.
1. If you should buy a bicycle listed at $ 35 at a discount
of 20 %, what would it cost you ?
2. If a merchant gives a 5 % discount on all cash pur-
chases, what is the yearly saving to a family that spends
f 850 per year, regular price, at that store ?
3. When a grocer advertises 10 ^ package goods for 9 ^,
what per cent of discount is he giving ?
4. When 25 ^ packages are sold at 22 ^, what is the per
cent of discount ?
5. When $ 30 suits are selling at $ 25, what per cent of
discount is allowed ?
6. When $ 25 suits are selling at a discount of 10 %,
how much will they cost ?
7. A f 2500 automobile used for demonstration purposes
was offered at a discount of 15 %. At this discount, how
much will it cost ? ,
8. A dealer advertised that he had an $ 1800 automobile
that had run less than 600 miles, which he would sell at
f 1500. This was a discount of what per cent from the
regular price ?
BUSINESS TERMS, FORMS, AND PROBLEMS 135
9. A dealer during a " special sale " offers the following
discounts :
A 10 % discount on all $ 350 parlor sets ;
A 20 % discount on all $ 150 bedroom suites ;
A 15 % discount on all $ 175 dining-room sets.
How much will a customer have to pay for each ?
10. At a. ^^ special sale " a dealer offered the following
prices : All $ 85 suits, $ 80 ;
All 9 50 suits, $ 40 ;
AU$25 suits, $16.50
Upon which class was the rate of discount the greatest?
Can you give any reason for allowing different rates ?
11. Find an advertisement of a ^^ special sale " and reckon
the rate of discounts allowed. Can you give a reason for
the different discounts ?
12. I bought an article for $21.60. The dealer told me
that this was 25 % less than his former price. From this,
show how to find the former price.
13. At a special '' cash sale " I got a discount of 20 %
from regular prices. At this ^ sale- price " I bought the
furniture for a new house at a total cost of $ 938.40. How
much was saved over the former price ?
14. At a ^^ 25 <fo discount sale," I bought goods costing
me $70.95. What would they have cost me at the regular
prices ?
15. Make up problems in discount, giving what you con-
sider a reasonable discount from former prices, giving as a
reason that the goods were not in season, sold for cash, or
any other reason for which you think a discount might be
given.
136
JUNIOR HIGH SCHOOL MATHEMATICS
4. COMMERCIAL OR TRADE DISCOUNT
The wholesale merchant who supplies the retail merchant
with goods often has expensive catalogues of his goods, and
in these catalogues he has a printed price called a list price
from which he allows a discount to the dealers. The price
that the goods cost the dealer after the discount has been
deducted is called the net price. Since the discount is given
by the wholesaler to the dealer handling his kind of goods,
it is called a trade or commercial discount.
1. Athletic goods listed at #3.75 were sold to dealers at
a discount of 30 %. Find the net price.
2. When golf shoes listed at $7.50 sold at $6.46 net,
what per cent of discount was allowed ?
From the follovnng data^ find the net price :
List Price
Tbadb
DiBCOUNT
List Price
Tkadb
Discount
3.
4.
5.
6.
7.
» 38.50
42.80
65.20
86.50
48.20
30 9fc
20 9^,
25 <^
15 9fc
20 9b
8.
9.
10.
11.
12.
$ 96.80
142.50
84.30
72.40
64.70
lO^f)
^0<fo
20 <jlh
15 9{?
From the following data^find the rate of discount :
List Price
Nbt Price
List Price
Net Price
13.
912.50
910.00
18.
913.50
912.15
14.
16.80
15.12
19.
16.70
13.36
15.
26.25
21.00
20.
38.70
25.80
16.
42.80
29.96
21.
74.30
44.58
17.
36.50
25.55
22.
46.80
31.20
BUSINESS TERMS, FORMS, AND PROBLEMS 137
23. Check the following bill :
A. Q. 8PAUL0INQ & CO.
Athletic Goods
Chicaoo, III. Aug, 4, 1919
Sold to
Morgan & Dale
Dixon, 111.
Terms : Net SO days
3
Drivers
»4.50
$13
50
4
Mid irons
3.75
15
00
2
Putters
3.25
6
35
50
00
Less 25 ^
8
75
• 26
25
5. SUCCESSIVE DISCOUNTS
Usually the list price of goods remains the same for long
periods, but as the market changes, new discounts are made.
When the market price decreases, it is usual for a new dis-
count to be given and applied to the previous net price.
Thus, if goods have sold for $15 less 20 % and the market
goes lower, a further discount of, say, 10 % may be given on
the former net price of $12. This is quoted as $15 less
20 % and 10 %.
1. How much will a dealer have to pay a wholesaler for
goods listed at $ 84.50 less 20% and 10 % ?
WORK
5)$ 84.50
16.90
10)67.60
6.76
$60.84
Explanation. — A discount of 20 9^, or J, is f 16.90,
leaving 967.60. A further discount of 10 %, or ^, of
this is 96.76, leaving 960.84, the net price.
138
JUNIOR HIGH SCHOOL MATHEMATICS
2. Find the net price of goods listed at $ 86.40 less 15 ^
and 12^ %
WORK
$86.40
^
48200
69120
8)73.4400
9.18
Explanation. — Since 15^ iB not an aliqnot
part of 100 ^y the first discount is deducted by find-
ing 85 ^ of • 86.40, for if the discount is 15 %, 85 ^
of the list price remains. Since 12^ ^ is \^ the last
part is found as in problem 1, leaving a net price of
1 64.26.
$64.26
3. Find the net price of goods listed at % 94.50 less 33| %
and 10 %.
From the following data^ find the net price :
4.
List Pricic
Discounts
List Pbick
Discounts
927.80
25 ^,10 9b
9.
9 06.30
3319^,10^
5.
36.50
20 <fo, 10 <j/h
10.
120.50
15 9^,10 96
6.
54.90
SSi<fh, 20 <jlh
11.
148.60
15 9>, 5 9()
7.
65.20
20^,5^
12.
365.70
20^,15^
8.
66.40
20 <jh, 121 9fe
13.
448.60
25 <f>, 15 ^
14. Check the following hill :
Boston, Mass. Oct IS, 1919.
SPENCER AND BROWN
SlLVBRWARB, ChINA, AND CUT GlASS
Sold to S. L, Scott & Son
Burlington, Vt,
Terms: 60 da., 2 ^ 10 da.
6 doz. plates
8 doz. dishes
3 tea sets
Less 33} <fo
Less 10 ^
93.20
6.80
4.25
19
54
12
20
40
75
5
86
28
35
78
57
76
51
81
BUSINESS TERMS, FORMS, AND PROBLEMS 189
15. What will the bill cost S. L. Scott & Son if paid
before Oct. 28 ?
6. PROFIT AND LOSS
There are certain terms used in buying and selling that
should be understood by all, for they are met in general
reading and in conversation. These have to do with the
profit or loss to one who sells goods.
The prime or net cost of an article is the amount actually
paid for it. When transportation charges, insurance, com-
mission for buying, etc., are added, the result is the gross
cost. The selling price is what the dealer actually receives
for the goods. The difference between the selling price and
the gross cost is the gross gain, or gross profit. When all
the expenses of selling, as salaries, traveling expenses, and
all other costs of doing business, are deducted from the gross
profit, the result is the net profit. In case the selling price
is less than the gross cost, or if the cost of doing business is
greater than the gross gain, there is a loss.
There is no uniform agreement among business men as to
what should be used as the basis in finding the per cent of
loss or gain. Some reckon it on the prime cost^ some upon
the gross cost^ and some upon the selling priee. No confusion
arises, however, if the basis upon which it is reckoned is
stated. But to say that a man made a profit of 26 % is
meaningless unless the basis is stated, as ^ 25 % of the prime
cost," '' 26 % of the gross cost," or " 26 % of the sales."
To ask what per cent a boy makes when buying Saturday
Bveving Posts at 3 ^ and selling them at 6 ^ is indefinite.
He makes 66} % of the cost or 40 % of the selling price.
1. A boy sold brushes at $3, for which he paid $2.10.
What per cent of the cost did he make? What per cent
of the selling price did he make ?
140 JUNIOR fflGH SCHOOL MATHEMATICS
2. A dealer bought shoes at $4.50 per pair. The cost of
buying, delivery, etc., was 15 ^ per pair. The cost of selling
averaged 35 ^ per pair. If the shoes sold for $ 6.50, the net
profit was what per cent of the prime cost ? Of the gross
cost ? Of the selling price ?
3. A retail grocer's sales for the year amounted to
$86,324.50. The gross profits were $18,991.39. The
entire cost of doing business was $13,811.92. His gross
profit was what per cent of the sales ? His net profit was
what per cent of the sales ? The cost of doing business was
what per cent of the sales ?
4. If a wholesale grocer's sales for a year amounted to
$350,680 and he makes a gross profit of 12 % of the sales,
and his expense of selling is 5.2 % of the sales, find the gross
profit, the expense of selling, and the net profit.
5. If a dealer in hardware gets an invoice listed at
$387.50, less 33J % and 10 %, and sells it at a gross gain of
35 % of the net cost, how much does he get for it ? If the
cost of doing business is 22 % of the sales, find the net profit.
6. The "Profit and Loss Statement" of three departments
of business one year showed the following : Clothing de-
partment, sales, $ 94,500 ; gross gain, 32 % of the sales.
Shoe department, sales, $ 26,400 ; gross gain, 24 % of the
sales. Men's furnishings department, sales, $ 19,680 ; gross
gain, 35 % of the sales. If the total cost of doing business
averaged 20 % of the sales in each department, find the net
gain in'each.
7. If a merchant's sales for the year are $ 83,450, with a
gross gain of 23J^ % of the sales, what is his net profit if
clerk hire is $ 8640, and the other expenses, $9364.50 ? The
net profit is what per cent of the sales ?
BUSINESS TERMS, FORMS, AND PROBLEMS 141
8. It is estimated that the average gross profit of the
retail grocer is 21 % of his sales and that the gross profit of
the wholesaler, of whom he buys, is 12 % of the wholesale
price. In a city spending f 986,500 yearly for groceries,
find the gross profit that goes to the retailer, and to the
wholesaler.
9. An importer bought green coffee at 15 ^, and sold it
roasted at 24 ^ per pound. If it lost 15 % of its weight in
roasting, and the cost of selling was 2 J ^ per pound, the net
profit was what per cent of the selling price ?
10. If a grocer pays $1.25 per basket for peaches and
sells them at f 1.75 per basket, after losing 10 % of them by
decay, he is making a gross profit of what per cent of the
sales ?
7. COMMISSIOlf AND BROKERAGE
One who buys or sells for others is often paid a per cent
of the amount bought or sold. This fee is called his com-
mission or brokerage. The one buying or selling is called
a broker or a commission merchant. The general distinction
between the two depends upon whether the agent actually
handles the goods or not. If the one selling the goods actu-
ally handles them, he is usually called a commission mer-
chant. If he merely arranges for the purchase or the sale,
he is called a broker.
1. If a real estate agent sells a house for $9500 and re-
ceives 2| % of the sales as his fee, find the amount of the fee.
2. If a commission merchant sells f 2500 worth of produce
on a 5 % commission, find the amount of his commission.
3. At 5 % find the commission of a shipment of 300 cases
of eggs, 30 doz. per case, when sold at 43^ per dozen.
142 JUNIOR fflGH SCHOOL MATHEMATICS
4. If a broker sells $236,600 worth of goods at 2J%, find
the brokerage.
5. A truck gardener shipped his commission merchant
vegetables which sold for 1238.60. After deducting 5% for
commission and $12.60 for freight and drayage, how much
should the merchant remit to the gardener ?
6. If an agent is selling goods on a 20 % commission, how
much per month will he make net from sales averaging
$1260 per month, after deducting $96 per month for ex-
penses ?
7. An agent bought hogs for a shipper on a 2% com-
mission, averaging $18,600 worth per month, at an average
expense of $86. At this rate find his net earnings per year.
8. A boy sold aluminum ware on a 30 % commission one
summer vacation. His average sales per week were $184.50
for 9 weeks, at a total expense of $185. Find his net
earnings.
9. A salesman in a large store got a fixed salary of $1800
and 6% of the sales that he made. If his sales amounted
to $17,600 per year, find his total income.
10. If you know agents who work on commission, get
data and information of interest and make a report of it
to the class.
8. BORROWING AND LOANING MONET
You often hear people speak of borrowing or loaning
money, and of receiving or paying interest. Interest is
money paid for the use of money or paid for an accommoda-
tion on an unpaid debt. It is reckoned as a certain per cent
of the debt, called the principal, for a year's use of it, even
though the interest is collected every half year or more often.
The interest paid varies from 6% to 7%. Sometimes the
BUSINESS TERMS, FORMS, AND PROBLEMS 143
rate is even less than 5%, but seldom more than 7%. The
rate of interest to be paid, unless paid in advance, is stated
in the promissory note held by the debtor. This is a signed
promise by the person borrowing the money or getting the
accommodation that he will pay a certain sum of money at a
specified time to the party holding the note.
USUAL FORM OF PROMISSORY NOTE
f^^' ^ New York Cl^/' S, / 9/-^.
U/^ ^rncvCt/L af ter rlnh cX promise to pay to
\Jznij^ 6r r?tdU<rv\^' or order
^/jHh^ J^^^vW" Dollars
for value received, interest at (^/hi^ ,,
1. How much interest will be due on the above note at
the end of six months ?
2. Who holds the note ? How much money will he
receive when the note is due ?
3. If your father loans $2000 at 6%, how much interest
will he get each year ? If the interest is payable every
half year (semiannually), how much will he get in each
payment ?
4. If a man borrows $750 at 5%, how much interest will
he have to pay each year ?
5. A man bought a house for $12,500. Find the interest
on the investment at 5^%; that is, this amount loaned at
5 J % would earn how much per year ?
144
JUNIOR fflGH SCHOOL MATHEMATICS
Find the yearly interest of:
6. $350 loaned at 6^.
7. $940 loaned at 5%.
8. $1150 loaned at 5^%.
9. $1580 loaned at 6%.
10. $2450 loaned at 5%.
11. $7500 loaned at 5%.
12. $9600 loaned at 4}%.
13. $13^00 loaned at 5J^0.
14. $16,250 loaned at 5%.
15. $18,600 loaned at ^%.
CHAPTER XII
BANKING
You have seen banks in your city, been in them, no doubt,
and perhaps you have money deposited in some savings
bank. The savings bank will be discussed in the next
chapter. The banks discussed here are commercial banks.
They are institutions where money is deposited for safe
keeping and paid out on the order of the depositor, and
where drafts may be bought to send in payment of debt
instead of sending the actual money. Banks also collect
debts by means of drafts, and loan money on personal or
other security. The modem bank is
essential to the commercial life of the
country and a great convenience to all
of us.
1. DEPOSIT SLIPS
When making a deposit with a bank
you will fill out a deposit slip like the
one in the margin and hand it, with the
money and checks, to the teller, who
checks each item. This serves as a
sort of receipt of the transaction; for,
in the future, if any questions arise as
to the deposit, the slip in your own
handwriting and checked by the teller is proof of the trans-
action.
145
DEPOSITED TO THE ACCOUNT OF
•N THB
First national Bank
of detroit. michigan
^^^^""^
DOLUin 1
dun
foif .,
y
4e
Mil*
Z6
»r
/S^
c*
M
7
S9
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/J
fe
10TM.
210
r
146
JUNIOR HIGH SCHOOL MATHEMATICS
Write out deposit slips for the follovring :
1. R. N. Doty deposited with the Farmers' Bank of Co*
lumbus, Ohio, on Apr. 3, 1919 : $ 150 in gold, $ 380 in bills,
and the following cheeks, $740, $36.80, $210.40, and $98.
2. C. L. Henry deposited in the Merchants' Bank of
Kansajs City, Mo., on Aug. 10, 1920 : $285 in bills, $200 in
gold, $56.50 in silver, and the following checks, $980,
$ 178.50, $209.30, $16.80, and $18.32.
3. Make out a form of deposit slip for a " make-believe "
bank of your school, as " The Students' Bank of the Detroit
High School." Write out a deposit slip in which you are
supposed to have deposited $10 in silver, $25 in gold, $45
in biUs, and checks for $7.50, $8.35, and $16.80.
2. THE PASS BOOK
When making your first deposit with a bank, you will be
given a pass book in which your deposit is entered to your
credit, and in which future deposits will be entered. This
pass book is left at the bank every month or so to be bal-
anced. The following shows a page from such a book,
showing the amount of each deposit and the total amount of
the "vouchers returned." These vouchers are the returned
checks that have been paid.
June i, 1919
balance
316
11
5, 1919
deposit
145
00
7, 1919
SJfi
00
9, 1919
237
00
18, 1919
150
22
SO, 1919
Total credits
186
51
1374
84
Vouchers refd as per list
876
52
July 1, 1919 balance
498
32
Rvle forms like the above and balance the follow^
iTig hank accounts :
BANKING 147
The "vouchers ret'd as per list" item was the 21.04
sum of all checks paid. This was found, perhaps, ' 20.67
by an adding machine. The ''list," shown in the 2.16
margin, is returned with the balanced book and 42.11
canceled checks. 33.50
17.28
38.88
60.66
1. May 1, balance, $387.42. Deposited: May 6, 1.44
$340; May 16, $636.70; May 26, 1763.40. 55.6I
Vouchers returned, $1634.87. 41.17
2. Junel, balance, $496.34. Deposits : June 1, 66.26
$842.60; June 10, $346.93; June 21, $684.76; 29.80
June 30, $ 963.70. Vouchers returned, $ 2384.39. 40.00
116.00
3. Sept. 1, balance, $ 398.46. Deposits : Sept. 3^ 07 71
$348.90; Sept. 6, $196.30; Sept. 7, $206.30; 2I 74
Sept. 17, $687.66; Sept. 26, $498.76. Vouchers gQ^gj
returned, $ 1698. 32. 4^' gg
4. Dec. 1, balance, $286.70. Deposits : Dec. 2, 68.60
$263.40; Dec. 10, $366.80; Dec. 13, $296.87; 6.46
Dec. 28, $164.30; Dec. 30, $463.70. Vouchers 29.00
returned, $ 1364.86. 6.46
5. Oct. 1, balance, $489.40. Deposits: Oct. 5, ^0.84
$246.38; Oct. 7, $ 178.26 ; Oct. 10, $316.42; Oct. ^'^^
16, $190.78; Oct. 24, $206.42. Vouchers re- §76 62*
turned, $1148.63.
6. Check the form at the head of this list. First see if
the sum of vouchers returned is $876.62; then see if the
pass book is properly balanced.
7. If you have a school bank, balance actual accounts. If
not, make up and balance " make-believe " accounts.
148 JUNIOR HIGH SCHOOL MATHEMATICS
3. MAKING OUT A CHECK
When you have a deposit with the bank you will be given
a check book for writing out orders on the bank to pay out
any of the money which you have on deposit. The usual
form is shown below :
No. ^W Q.
Rkhrrtond. Va..Jsi&dM 191^
f Imtt^ra' National Vattb
Pay to the order n f/B^et^/:f^*i^^!^S f/P. ¥o
C^^j/TQ
^^LAA^^
J
1. Who is signing this order? Who, then, has money
on deposit in the Planters' National Bank ?
Before the bank will pay this money, Mr. Smith must
indorse the check by writing his name, as it appears in the
check, across the back of it.
The words " the order of " make the check negotiable. ,
That is, Mr. Smith, by indorsement, may transfer it to some
other person for collection instead of collecting it himself
from the Planters' National Bank.
2. Suppose that R. L. Brown has a deposit in the Mer-
chants' Bank of Indianapolis, Ind., and wishes this bank to
pay $38.40 from the deposit to C. R. Reed. Write the
proper form and show the indorsement.
3. Suppose that you have a deposit of f 800 in the Stu-
dents' Commercial Bank of your school* Write out a check
to J. L. Hayes & Co. for $36.20 which you owe them.
BANKING 149
4. Each check hook has a form for entering frequent
balances, new deposits, the amounts drawn, and for what
purpose. The following is a common form :
s. Find the amonnt of one's credit when balance brought
forward is $296.80, and $31.28 and $92.24 had been de-
posited, and the checks drawn were $126.40, $84.70, and
$46.28.
Find the balance of credit from the following data :
6. Brought forward, $196.87; deposits, $34.96, $78.27,
and $63.98. Checks drawn, $84.87, $27.68, and $64.96,
7. Brought forward, $208.76; deposits, $84.26, $154.37,
$75.80. Checks drawn, $126.75, $98,37, and $42.96.
8. Brought forward, $138.28; deposits, $103.42, and
$116.28. Checks drawn, $93.48, $86.42, and $74.39.
1. BUTIITG A DRAFT
If one wishes to pay a debt or send money to some one
in another city, he may buy a draft of any bank and send
that instead of the actual money. A draft is a written order
from one bank to another bank to pay a specified sum to a
third party. It is like a check, then, except that it is an
order issued by a bank rather than by an individual.
150 JUNIOR HIGH SCHOOL MATHEMATICS
USUAL FORM OF BANK DRAFT
Burlington, I//, (LX^^. /9/f
Pay to the order nf ^^.:^2^l& $ 2^^^^
To The National City Bank o /£:>
New York -^^
^
Cashier
1. At what bank is this draft bought ?
2. Upon what bank is the order drawn ?
3. Of what bank is Mr. Porter cashier ?
4. Who is to receive the money ?
5. What is the purpose of the words " to the order of " ?
6. How can Haynes & Co. transfer this to some other
party for collection ?
7. Suppose that this draft was not bought by the party
named in the draft (E. B. Haynes & Co.) but by J. C.
Smith who wished to remit this sum to E. B. Haynes & Co,
He could have had it made out to himself and then in-
dorsed it over to E. B. Haynes & Co. by writing across the
back, *'Pay to E. B. Haynes & Co.," and then signing
''J. C. Smith." Of the two methods, which wouid you
think the better ?
8. Suppose that E. L. Holmes wishes to remit to L. Harris
& Bros. $360, buying a draft of the First National Bank of
Lansing, Mich., issued upon the Bankers' Trust Co. of New
York. Fill out the two forms discussed above and show
the indorsement of each.
BANKING 151
9. Suppose that you wished to send $85 by draft to
A. G. Spaulding & Co., Chicago, for athletic goods. Where
would you get the draft, who would sign it, to whom would
you have it made out ?
5. BORROWIKG MONEY FROM A BANK
A bank's chief income is interest from the money that it
loans. A large part of the money in any bank, which it
loans, is that of its depositors. It is for the use of this
money that banks can afiford to take care of the money of
their depositors and pay it out for them as they order, with-
out making any charge for this service.
Banks usually loan their money to be paid " on demand "
or for short periods, usually 30 da., 60 da., or 90 da. The
interest on these time notes is paid in advance and is called
bank discount, in distinction from simple interest^ which is
paid when the note is paid, or at fixed times.
USUAL FORM OF TIME NOTE
Chicago, ///., Cfb^ /^,JSU^
Jd^^^^ after date la promise to pay to the
order of the ^t«te Vattk of (El^fraQQ
^'wuuuy^jCut^iA.i^^^ ^!^^^r^ Dollars
Payable at the State Bank
of Chicago
Value received
152 JUNIOR HIGH SCHOOL MATHEMATICS
No interest is named in the note, for it has been paid by
Mr. Morgan at the time of the loan. If the bank's rate is
6%, it charged Mr. Morgan $3 interest (bank discount) at
the time of the loan. This was taken from the f 300 and
$297 was given Mr. Morgan or credited to his account.
The 1297 is called the proceeds of the note.
Find the bank discount at 6% cind the proceeds of:
1. $500 for 30 da. 7. $1500 for 90 da.
2. $750 for 60 da. 8. $1650 for 45 da.
3. $980 for 30 da. 9. $1860 for 30 da.
4. $765 for 90 da. 10. $1780 for 90 da.
5. $1250 for 30 da. 11. $1560 for 70 da.
6. $1575 for 60 da. 12. $1350 for 20 da.
For short periods (less than one year) 30 days are con-
sidered an interest month or .^ of a year. Hence, at 6 %,
the interest is 1 % for each 60 days. Thus, the interest of
$1350 at 6% for 60 days can be seen at sight to be $13.50.
For 30 days it would be half as much, or $6.75.
At sight give the discount atQ^o of:
13. $1200 for 60 da. 19. $1600 for 30 da.
14. $1950 for 60 da. 20. $1450 for 30 da.
15. $2480 for 60 da. 21. $1200 for 90 da.
16. $1375 for 60 da. 22. $1600 for 90 da. '
17. $1800 for 30 da. 23. $2400 for 90 da.
18. $2400 for 30 da. 24. $3600 for 90 da.
BANKING 153
USUAL FORM OF A DEMAND NOTE
New YorkQf^J/Zt9Jj^
Uny di/mo/H^fM value received,^jjL^promw
to pay to the order nf nrt/uiii/^ $mM^
AudbLiAy-h^4/uu/^ gjuiJsL ^^s^ ri^Uat
with interest at ny^ , at
The Market Exchange Bank
of New York
If the sum is large, or if a man's financial standing is not
high, a bank will demand some security. This security will
he one of two kinds. Either the note will be made out to
the order of some one of high financial standing, who will
indorse it and thus become responsible for the payment, or
the borrower will put some security, called collateral, worth
more than the face of the note, in the care of the bank to
secure payment. This collateral will be sold by the bank to
pay themselves if the note is not paid when due.
Often demand notes are given for large sums overnight or
for a very few days.
Find the interest on the following demand notes of :
25. $6000 for 5 da. at 4 %. 28. $18,000 for 3 da. at 4%.
26. $10,000 for 6 da. at 4 J %. 29. $36,000 for 8 da. at 5%.
27. $12,000 for 1 da. at 6 %. 30. $50,000 for 5 da. at 3 %.
154 JUNIOR HIGH SCHOOL MATHEMATICS
6. DISCOUNTING NOTES AT A BANK
If one has a note and needs money before the note is due,
he can discount it at a bank and get the money at once, the
bank charging interest on the maturity value of" the note
for the time the note has yet to run. Or if banks themselves
need more money, they may rediscount notes which they
hold, at a Federal Reserve Bank.
Thus, if you have a note dated Apr. 4, 1919, for 11200,
interest 5%, to run 6 months, it is due Oct. 4, 1919, and
worth $1230 at that time. If you wish the money on this,
Aug. 20, 1919, that is 45 days before it is due, a bank will
buy the note, charging you interest at their regular rate
on $1230 for 45 days. At 6% this is $9.23, and you will
receive $1230-19.23 or $1220.77, called the proceeds.
That is, the solution is :
Face of note $1200
Int. for 6 mo. at 5 % 30
Maturity value $1280
Discount of $1280 at 6% for 45 da. 9.28
Proceeds $1220.77
1. Find the proceeds of a note of $900, to run 8 mo. at
6%, discounted at 6 %, 60 days before it is due.
2. If a merchant takes a 90-day note without interest for
$1200 for goods, and discounts it at 6% 20 days after date,
how much will he get for it ?
Suggestion. — Since the note does not bear interest, the maturity
value is but $ 1200. Being discounted 20 days after date, it has 70 days
to run.
3. How much will a bank whicli charges 6 % interest pay
you for a note of $750 to run 6 mo., bearing interest of 5^o^
if discounted 40 days before it is due ?
BANKING 155
4. Find the proceeds of a note of $1600, dated May 5,
1919, to run 6 mo. at 6 %, if discounted on Sept. 20, 1919, at
6%.
5. Find the proceeds of a note of %1850, dated July 10,
1919, to run one year at 6J^%, if discounted on May 16, 1920,
at 6^.
6. A note of $ 2400, dated Aug. 20, 1919, to run 8 mo, at
5 %, was discounted at 6 % on March 10, 1920. Find the
proceeds.
7. A note of $ 8600, dated Sept. 6, 1919, to run 4 mo.
without interest, was discounted on Oct. 10, 1919, at 6 %.
Find the proceeds. r
8. How much will a bank which charges 6 % interest pay
you for a note of $ 1650, dated Nov. 16, 1919, to run 6 mo.
at 5 J %, if discounted on Feb. 10, 1920?
/ 4.
CHAPTER XIII
METHODS OF INVESTING MONEY
Knowledge of investments is a very fundamental part of
one's education. At your age such knowledge is valuable
to you in enabling you to understand and appreciate much
that you read and the conversation that you hear in the
home concerning investments. Later, when you have earn-
ings to invest, such knowledge may prevent your being
persuaded by the representatives of some "get-rich-quick"
scheme to invest your money in some hazardous undertaking,
and to enable you to invest more judiciously.
1. LOANING MONEY ON BOND AND MORTGAGE
The real standard by which the rate of income on an in-
vestment is measured is the rate at which money can be
loaned on a note secured by a mortgage. For a number of
years this has ranged from 6% to 6% of the investment
(principal) per year. When an investment pays less than
this, it has a low rate of income; when it pays more, it lias a
high rate of income
You have seen in Chapter XII that when money is loaned,
the one receiving the loan gives his " promise to pay " or a
promissory note together with some satisfactory security
that the money will be repaid when due. The note is some-
times called a bond and the security given is sometimes a
mortgage, whicli is an agreement that in case the one giving
the note fails to pay the note or interest when due, certain
156
METHODS OF INVESTING MONEY 157
real estate or other property belonging to him may be sold to
pay it. The mortgage becomes void when the money is paid.
Thus, you hear one say that he has given a mortgage on
certain property, which means that he has given his note
secured by a mortgage. When one loans money on a note
given to run for a long period of years and secured by a
mortgage, he speaks of the transaction as loaning on " bond
and mortgage." Under such a contract the interest is
usually paid semiannually, or annually, the note and
mortgage, however, running for several years.
1. At 5 % what is the yearly interest on a note of $2500 ?
2. At 6% what is the semiannual interest on a note of
13500?
3. A man bought a home for $12,000, paying $7000 cash
and giving a 6% mortgage on the home for the balance,
interest payable semiannually. How much interest must he
pay each half-year ?
4. A man bought a farm for $18,000, paying half cash
and giving a 5 J % mortgage on the farm for the rest, interest
payable annually. How much interest must he pay each
year ?
5. Mr. Taylor bought a house of Mr. Barnes for $12,500,
paying $6,500 cash and giving him a 6% mortgage on the
property for the rest, interest yearly.
(a) What is the face of the note ?
(6) Who gives the note and who holds it ?
(c) Who pays the interest ? How much and when ?
(d) What security has Mr. Barnes that he will get the
interest when due, and the face of the note when due ?
(e) Why would Mr. Barnes refuse to take a note for the
whole value of the property secured by a mortgage on this
property alone ?
158 JUNIOR HIGH SCHOOT. MATHEMATICS
FtTid the yearly interest on:
6. $9500 at 5%. 11. $1250 at 5^%.
7. $11,500 at 5^%. 12. $1375 at 5%.
8. $10,250 at 5%. 13. $2125 at 6%.
9. $8760 at 6%. 14. $1125 at 5J%.
10. $4500 at 6%. 15. $1450 at 6%.
Find the semiannual interest on:
16. $9000 at 5%. 19. $16,250 at 6%.
17. $12,000 at 5^ %. 20. $10,500 at 6%.
18. $15,500 at 5%. 21. $11,125 at 5%.
2. INVESTING IN BONDS
A bond is an agreement under seal to pay a certain sum of
money at a stipulated time, with interest at a specified rate,
issued by governments, municipalities, or corporations. In
buying a bond, one should consider: (1) The safety of the
principal; (2) The rate of interest paid; (3) The readiness
with which it may be sold if he needs the money; and
(4) The stability of its market value.
Government Bonds
You are all familiar with the Liberty Loan Bonds sold by
the United States Government to help meet its expenses of
carrying on the great World War. These were issued in
denominations from $50 to $100,000. The Third and
Fourth Liberty Loan Bonds paid 4^ % interest. In buying
one of those bonds, you were merely loaning your money to
your government and the bond which you held was the
Government's promise to pay you the face of the bond at
some specified time and to pay you a certain rate of interest
METHODS OF INVESTING MONEY 159
every half-year, until the bond was due. Over 20,000,000
people bought bonds in the Fourth Liberty Loan of over
^6,000,000,000. This was an average of a bond for nearly
every family in the United States, and an averse of nearly
$G0 for each i>erson.
States, too, issue bonds for internal improvements, as
building roads, canals, bridges, schools, eto. These usually
[lay 3J %, 4 %, or 4J %.
Government and state bonds are paid by a tax levied
upon the people, and on account of this are considered the
safest kind of investment. Hence, they find a ready sale
at a rather low rate of interest.
There are two general types of bonds : the coupon bonds
and the registered bonds. The coupon bonds have small
coupons attached, which are certificates representing the
interest due each period. As the interest becomes due,
these may be cut off and deposited with a bank for collec-
tion. Thus, a $1000 5% bond to run 20 years, interest
payable semiannually, would have forty coupons attached,
similar to the following : v
A registered bond is registered in the name of the owner
by the corporation issuing it, and a check is malted to the
ir as the interest falls due.
160 JUNIOR HIGH SCHOOL MATHEMATICS
1. If you own a flOO Liberty Bond paying 4^% semi-
annually, how much is each coupon worth when 'due ?
Where can you get the money on the coupon ?
2. What is the semiannual interest, at 4| %, on a $500
Liberty Bond? On one for $10,000? One for $50,000?
3. What is the yearly interest on a New York State 4 J %
bond of $5000 ? On one for $10,000 ?
4. How many $1000 bonds bearing 4J% interest will
give an annual income of $900 ? Of $1890 ?
5. The Third and Fourth Liberty Loan Bonds bearing
4f % interest amounted to $10,782,980,000. What interest
must the Government pay yearly on these two issues ?
Municipal Bonds
Municipal bonds are those issued by cities, counties, and
other political divisions of the state, and paid by special
taxation. They generally run from twenty to fifty years
and pay 4%, 4 J %, or 5%. Occasionally the rate of interest
is higher.
1. What is the difference in income between a government
4J% bond for $10,000, and a municipal bond of the same
size for 5 % ? Which would you consider the safer in-
vestment ?
2. If one holds a twenty-year 4^ % municipal bond of
$5000 from the time it was issued until it matures, what is
the total amount of interest that will be received ?
3. If a city issues $1,000,000 worth of 4^% bonds to
build new schools, what yearly interest must it pay on the
issue ?
METHODS OF INVESTING MONEY 161
Railroad Bonds
Railroad bonds, as the name implies, are those issued by
railroads. The security back of them is a mortgage on the
company's property, as roadbeds, stations, terminals, equip-
ment, etc. The safety of the security lies in the value of
the property and the company's earning capacity. As these
vary, the market value varies more than in the class of bonds
already discussed.
Public Utility and Industrial Bonds
Public utility bonds are those issued by electric light, gas
and power, street railway, and similar companies. Industrial
bonds are those issued by manufacturing concerns, oil, coal,
and steel companies, etc. These two classes of bonds, like
the railroad bonds, are secured generally by a mortgage of
the properties. As these depend upon trade conditions,
their market value varies with the general state of the
industry.
Yield or Investment Returns on Bonds
The par value of a bond is the face value or the sum
named to be paid at maturity. The market value is the
sum it can be bought or sold for in open market. The in-
terest, as you have seen, is reckoned upon the par value.
But the yield or investment return depends upon the price
at which it was bought. Hence, if you pay "par" for a
5 % bond, you get 5 % on your money ; if you pay le%% than
par, you get more than 5 ^ on your money ; and if you pay
more than par, you get le%% than 5 % on your money.
Thus, if you pay $950 for a ten year $1000 bond paying
5 % interest, you get $50 per year, and at the end of the ten
years you get par value or $1000, thus making $50 heMes
162 JUNIOR HIGH SCHOOL MATHEMATICS
interest. This being an average of $5 per year, you have
really made $55 per year on an investment of $960, or
about 5.79 %. While if you paid $1050 for it, you paid $60
more than you get back at maturity, so there is a loss of $5
per year, leaving a net income of but $46 per year upon an
investment of $1050, or about 4.29%.
Note. — Since 9 50 at the end of ten years is not the same as 9 5 per
year for ten years when interest is considered, this gives but an approxi-
mate yield or return. Any bond broker will give his customers the exact
yield on bonds offered for sale.
1. Find the yield or investment return on a 5% bond
for $500, to run for 5 years, when bought for $475.
2. Find the yield on the same bond if bought for $510.
3. If a 4J % $1000 bond, due in 10 years, is selling for
$980, what is the yield?
4. When a $1000 bond, due in 8 years, and paying 6%
interest is selling at $1020, what yield is that on the in-
vestment ?
5. Find the yield on a $1000, 3 year, 5% bond, when sell-
ing at $970.
The Market Quotations
In the market reports found in the daily newspapers you
will see such quotations as " N. Y. City 4^'s, May '67 98|,"
*'Un. Pac. 6's 103|," etc. The first means that bonds
issued by the city of New York, paying 4 J- % interest, and
due in 1967, are selling for 98| % of their par or face value ;
that is, for 1^ % below par. The second means that bonas
issued by the Union Pacific R.R., and bearing 6% interest,
are selling for 103|% of their par value; that is, for 8-|%
above par. The first of these is sometimes said to be selling
at a discount, and the other at a premium.
METHODS OF INVESTING MONEY 168
1. Find the cost of $4600 worth of bonds (par value)
when selling at 102.
SOLUTION
$4500
^ r^p Explanation. — The quotation means that the bonds
* are selling at 102 ^b, or 1.02 of their par value. Hence,
^0 ^0 for 1.02 X f 4500.
4600
$4690.00
2. Find the cost of a $6000 bond when selling at 98.
3. Suppose that a man bought eight $1000 bonds from
the foUowing quotation: "N. Y. Tel. 4i's 89J." Find
how much they would cost him and how much interest he
would receive each year.
Bonds are usually bought and sold through an agent
called a bond broker. His fee is usually ^ of one per cent
(i%) ^^ ^^® ^^^® value of the bonds bought and sold. Thus,
the fee for buying or selling a $1000 bond is $1.25. This
fee is called brokerage.
4. Find the cost, including brokerage, of six $1000
bonds selling at 98.
Suggestion. — The total cost is d8J <fo of the par value.
5. How much will a man receive for five $1000 bonds
sold through a broker at 98 ?
Suggestion. — After paying brokerage, he will receive but 97} ^ of
the par value.
Find the cost including brokerage of:
6. $6000at88J. 8. $9000 at lOlf
1. $7000at96|. 9. $6600 at 96f
164 JUNIOR HIGH SCHOOL MATHEMATICS
Find the amount, received for the following^ if sold through
a broker:
10. $5000 sold at 98. 12. $8500 sold at 89J.
11. $7500 sold at 101. 13. $9600 sold at 87|.
The price of a bond usually includes the statement '^ and
accrued interest." This means that the buyer pays the
interest that the bond has earned since the last coupon was
due.
14. Find the cost of a $1000 bond bearing 5% interest,
payable Jan. 1 and July 1, quoted at 98J, including broker-
age, bought Apr. 1.
SOLUTION
$1000 at 98 J =$985.
Int. from Jan. 1 to Apr. 1 @ 5% = 12.50
Brokerage at ^ % = 1.25
Total cost = $998. 75
Note. — On July 1 the owner of the bond would cash his ^ 25 coupon
reimbursing himself for the $ 12.50 paid as accrued interest, leaving him
$ 12.50 as the interest from Apr. 1 to July 1.
15. Find the total cost, including accrued interest and
brokerage, of a $5000 bond paying 4J % interest, payable
Sept. 1 and March 1, if bought on Jan. 1 at 99^.
Why Market Prices Change
You have seen that the interest earned on a " bond and
mortgage " is the standard by which income returns on
investments are regulated. Bonds are usually issued to run
from twenty to fifty years. During that time money rates
may change and this will cause a change in the price of the
bond. Thus, if money on " bond and mortgage " is worth
6%, one would not pay par value for a bond paying but
METHODS OF INVESTING MONEY 165
4 % . On the other hand, if money is worth but 4 %, a bond
paying 6 % would be worth more than par.
Another element that enters into the price of a bond is
the security back of it. During such a long period the
value of the property may change and thus affect the price
of the bond. In general, the four leading factors that regu-
late the price of bonds are :
i. The security hack of the hands.
2. The rate of interest the hond is payiwf compared with
the general interest rates of money,
3. The length of time the hond has to run
4' The confidence of the huying puhlic in the stability and
general earning power of the corporation issuing the honds,
1. Would you expect a bond on a corporation heavily in
debt and earning but little to sell above par or below par?
2. When general interest rates are 5 %, would you expect
a 6 % bond on a prosperous corporation to seU for more or
less than par ? ,
3. When general interest rates are but 6%, could you
afford to pay 102 for a 6 % bond with good security if due in
2 years ?
4. If the bond described in problem 3 had 5 years to run,
could you afford to buy it at that price ?
5. Describe a bond that you feel would not be worth par,
6. Describe one that you feel sure would sell for more
than par.
3. SAVINGS BANK DEPOSITS
The investments with which you may be more acquainted
are savings bank deposits. A savings bank is an institution
for receiving and investing savings. Usually one cannot
buy a bond or loan his money 'ju "bond and mortgage"
166 JUNIOR HIGH SCHOOL MATHEMATICS
unless be has $ 100 or more on hand. But he may start a
sayings bank account with $1. The accumulated deposits
of a large number of depositors allows the bank an oppor-
tunity to invest these in bonds or loan them on mortgages
at the usual rates of 5% ot 6%^ thus enabling them to pay
the depositors 8^ % or 4 %.
When the interest is due a depositor at a savings bank» it
is not sent to him, but it is added to his account and thus
begins to draw interest. When interest due is added to the
principal and thus draws interest, the principal is said to be
drawing compound interest, or the interest is said to be com-
pounded. If the interest is added every six months, as in
most savings banks, it is compounded semiannually ; if it is
added once a year, it is compounded annually.
1. If you deposit $ 50 on Jan. 2 in a bank, adding the in-
terest on Jan. 1 and July 1 each year, how much will be
added July 1, at 4 % ? How much will then draw interest
until Jan. 1 ? How much interest will then be added ?
2. If $600 is deposited on Jan. 2, 1920; in a bank paying
4 % on Jan. 1 and July 1 each year, to how much will the
principal and interest amount on July 1, 1924 ?
The amounts are reckoned more quickly by a table like
the one on the following page.
Bi/ the tables find the amount of:
3. $ 600 at 3 J % for 20 yr. 8. $1200 at 3 % for 6 yr.
4. $800 at 4% for 15 yr. 9. $1500 at S^% for 10 yre
6. $ 300 at 31 % for 20 yr. 10. $ 250 at 3 % for 20 yr.
6. $ 260 at 4 % for 10 yr. 11. $ 750 at 4 % for 10 yr.
7. $ 876 at 4 % for 15 yr. 12, $ 900 at 3 % for 15 y r.
METHODS OP INVESTING MONEY
187
COMPOUND INTEREST TABLE
{The amount of one dollar principal)
Tbam
2%
2i%
3%
3i%
4%
6%
6%
Ybabs
1
1.0200
1.0250
1.0300
1.0350
1.0400
1.0500
1.0600
1
2
1.0404
1.0506
1.0609
1.0712
1.0816
1.1025
1.1236
2
8
1.0612
1.0769
1.0927
1.1087
1.1248
1.1576
1.1910
8
4
1.0824
1.1038
1.1255
1.1475
1.1699
1.2155
1.2625
4
5
1.1041
1.1314
1.1593
1.1877
1.2167
1.2763
1.3382
6
6
1.1262
1.1597
1.1941
1.2293
1.2653
1.3401
1.4185
6
7
1.1487
1.1887
1.2299
1.2723
1.3159
1.4071
1.5036
7
8
1.1717
1.2184
1.2668
1.3168
1.3686
1.4775
1.5938
8
9
1.1951
1.2489
1.3048
1.3629
1.4233
1.5513
1.6895
9
10
1.2190
1.2801
1.3439
1.4106
1.4802
1.6289
1.7908
10
11
1.2434
1.3121
1.3842
1.4600
1.5395
1.7103
1.8983
11
12
1.2682
1.3449
1.4258
1.6111
1.6010
1.7969
2.0122
12
18
1.2936
1.3785
1.4685
1.5639
1.6651
1.8857
2.1329
18
14
1.3195
1.4130
1.6126
1.6187
1.7319
1.9800
2.2609
14
16
1.3459
1.4483
1.5580
1.6754
1.8009
2.0789
2.3966
16
16
1.3727
1.4845
1.6047
1.7340
1.8729
2.1829
2.5404
16
17
1.4002
1.5216
1.6529
1.7949
1.9479
2.2920
2.6928
17
18
1.4283
1.5597
1.7024
1.8575
2.0258
2.4066
2.8543
18
19
1.4568
1.5987
1.7535
1.9225
2.1069
2.5269
3.0256
19
20
1.4860
1.6386
1.8061
1.9898
2.1911
2.6533
3.2071
20
21
1.5156
1.6796
1.8603
2.0594
2.2788
2.7860
3.3996
21
22
1.5461
1.7216
1.9161
2.1315
2.3700
2.9253
3.6035
22
28
1.5770
1.7646
1.9736
2.2055
2.4647
3.0715
3.8198
28
24
1.6076
1.8087
2.0328
2.2835
2.5633
3.2251
4.0489
24
26
1.6405
1.8539
2.0938
2.3628
2.6658
3.3864
4.2919
25
4. TH£ GROWTH FROM REGULAR DEPOSITS
Of more interest to many are the amounts to which regular
deposits will grow in a fixed time at compound interest.
These are easily calculated by use of the following table :
168
JUNIOR HIGH SCHOOL MATHEMATICS
TABLE SHOWING AMOUNT ACCUMULATED AT END OF A
PERIOD OF YEARS BY PAYING $1 AT BEGINNING OF EACH
YEAR IN THE PERIOD
Year
2
Per Ceni
2i
' Per Cbni
3
^ Per Cbni
3i 4 5 6
'PerCbnt Per Cent Per Cent Per Cent
Ybab
1
1.020
1.025
1.030
1.035
1.040
1.050
1.060
1
2
2.060
2.076
2.091
2.106
2.122
2.152
2.184
2
3
3.122
3.153
3.184
3.215
3.246
3.310
3.375
3
4
4.204
4.256
4.309
4.362
4.416
4.526
4.736
4
5
5.308
5.388
5.468
5.550
5.633
5.802
5.975
6
6
6.434
6.547
6.662
6.779
6.898
7.142
7.394
6
7
7.583
7.736
7.892
8.052
8.214
8.549
8.897
7
8
8.755
8.955
9.159
9.368
9.583
10.027
10.491
8
9
9.950
10.203
10.464
10.731
11.006
11.578
12.181
9
10
11.169
11.483
11.808
12.142
12.486
13.207
13.972
10
11
12.412
12.796
13.192
13.602
14.026
14.917
15.870
11
12
13.680
14.140
14.618
15.113
15.627
16.713
17.882
12
13
14.974
15.519
16.086
16.677
17.292
18.599
20.015
13
14
16.293
16.932
17.599
18.296
19.024
20.579
22.276
14
15
17.639
18.380
19.157
19.971
20.825
22.657
24.673
15
16
19.012
19.865
20.762
21.705
22.698
24.840
27.213
16
17
20.412
21.386
22.414
23.500
24.645
27.132
29.906
17
18
21.841
22.946
24.117
25.357
26.671
29.539
32.760
18
19
23.297
24.545
25.870
27.280
28.778
32.066
35.786
19
20
24.783
26.183
27.676
29.269
30.969
34.719
38.993
20
21
26.299
27.863
29.537
31.329
33.248
37.505
42.392
21
22
27.845
29.584
31.453
33.460
35.618
40.430
45.996
22
23
29.422
31.349
33.426
35.667
38.083
43.502
49.816
23
24
31.030
33.158
35.459
37.950
40.646
46.727
53.865
24
26
32.671
35.012
37.553
40.313
43.312
50.113
58.156
25
26
34.344
36.912
39.710
42.759
46.084
53.669
62.706
26
27
36.051
38.860
41.931
45.291
48.968
57.403
67.528
27
28
37.792
40.856
44.219
47.911
51.966
61.323
72.640
28
29
39.568
42.903
46.575
50.623
55.085
65.439
78.058
29
30
41.379
45.000
49.003
53.429
58.328
69.761
83.802
30
31
43.227
47.150
51.503
56.334
61.701
74.299
89.890
31
32
45.112
49.354
54.078
59.341
65.210
79.064
96.343
32
33
47.034
51.613
56.730
62.453
68.858
84.067
103.184
33
34
48.994
53.928
59.462
65.674
72.652
89.320
110.435
34
35
50.994
56.301
62.276
69.008
76.598
94.836
118.121
35
36
53.034
58.734
65.174
72.458
80.702
100.628
126.268
36
37
55.115
61.227
68.159
76.029
84.970
106.710
134.904
37
38
57.237
63.783
71.234
79.725
89.409
113.095
144.058
38
39
59.402
66.403
74.401
83.550
94.026
119.800
153.762
39
40
61.610
69.088
77.663
87.510
98,827
126.840
164.048
40
METHODS OF INVESTING MONEY
169
1. If one can deposit $ 100 yearly in a savings bank pay-
ing 4 % yearly, how much will he have to his credit at the
end of 5 yr. ? Of 10 yr. ? Of 15 yr. ? Of 20 yr. ?
2. If for 10 yr. a man can deposit $400 per year in a
bank paying 4%, how much will he have to his credit at
the end of the 10th year ?
3. If a man from the age of 30 to the age of 60 makes a
regular yearly deposit of 1300 in a savings bank paying
3^ %, how much will he have in it by the end of that time ?
To salaried people and others who can invest small sums
monthly, a saving based upon monthly deposits is more
interesting. The following table based upon 5 % is interest-
ing. While savings banks seldom pay more than 4 %, after
one's savings have grown to a few hundred dollars, he can
easily make them earn 5%. This table then is based upon
the supposition that all interest is reinvested as it falls due.
APPROXIMATE GROWTH AND INVESTMENT RETURN BY
MONTHLY PAYMENTS OF $10 — COMPUTED ON 6% BASIS
COMPOUND INTEREST
YSAB
Capital at
Interest at
Year
Capital at
Intbbebt at
End of Year
End of Year
End of Year
End of Year
1
123.27
6.16
16
2931.14
146.55
a
252.78
12.63
17
3202.80
160.14
3
388.85
19.44
18
3488.22
174.41
4
531.81
26.59
19
3788.08
189.40
S
682.01
34.10
20
4103.13
205.15
6
839.80
41.99
21
4434.12
221.70
7
1005.59
50.27
22
4781.87
239.09
8
1179.77
58.98
23
5147.22
257.36
9
1362.77
68.13
24
5531.07
276.55
10
1555.03
77.75
26
5934.35
296.71
U
1757.03
87.85
26
6358.05
317.90
12
1969.25
98.46
27
6803.20
340.16
13
2192.21
109.61
28
7270.89
363.54
14
2426.47
121.32
29
7762.25
388.11
15
2672.58
133.62
30
8248.74
413.92
170 JUNIOR HIGH SCHOOL MATHEMATICS
From the table fmd the amowni at b% of:
4. $ 5 per month for 10 yr.
5. $ 5 per month for 25 yr.
6. $ 15 per month for 30 yr.
7. 1 20 per month for 20 yr.
8. $ 25 per month for 20 yr.
9. $ 30 per month for 25 yr.
10. $ 100 per month for 30 yr.
11. 1 200 per month for 20 yr.
5. BUILDING KSJ} LOAN ASSOCIATIONS
Many wage-earners and salaried people build homes
through building and loan associations. These associations
offer a safe form of monthly investment at a higher rate of
interest than that paid by savings banks.
Such associations usually issue stock in f 100 shares.
These shares are said to mature when the monthly pay-
ments, called dues, together with the earnings (largely
interest on their loans), are equal to $100. From the table on
page 169, we see that if the net interest earned is 6 %» it
will take $ 1 monthly for about 7 years to mature a share of
$100.
If one borrows money from a building and loan associa-
tion, he takes enough stock to cover the loan. His monthly
payment must cover his monthly interest on the loan and
pay the dues on the stock. When the stock matures, it
cancels the indebtedness. The loan, of course, is secured by
a mortgage on the property.
1. If one should borrow f 3000 to build a house, how
many $ 100 shares would he have to take out ? At 50 ^ per
share, what would his monthly dues be ?
METHODS OF INVESTING MONEY 171
2. What would the monthly interest at 6 % be on the
loan of problem 1 ? What payment would have to be made
monthly to the association ?
3. If the earnings amount to 5 %, find by the table, page
169, about how long it would take to pay for the home.
Suggestion. — The table gives the accumulated amount of J 10.
A monthly payment of 50 ^ per share will give 9 16, which will amount
to i more than 910. Since 1} x 91969.25 is a little less than 93000
and 1} X 92192.21 is a little more than 93000, it will take between 12
and 13 years for the stock to mature.
4. tiad the dues been $ 1 per share, what would the total
monthly payment have been ?
5. At $ 1 per share, about how long would it take for the
stock to mature at 5 % ?
6. If you have a building and loan association in your
city, let a committee from the class find the amount of dues
charged, the dividend it usually makes, the rate of interest
charged for loans, the general time required for stock to
mature, and any other points of interest, and make a report.
6. REAL ESTATE INVESTMENTS
In buying real estate, as houses, buildings, or farms to
rent, one must consider not only the interest on the invest-
ment, but the taxes, repairs, insurance, and probable de-
preciation of value, in order to determine whether the in-
vestment pays.
1. If one pays $10,000 for a house, what monthly rent
must he get in order to make 6 % net on the investment,
estimating that the expenses will average yearly : taxes,
$175 ; insurance, $18.50 ; repairs and general upkeep, $75 ;
and general depreciation in value, $ 150.
172 JUNIOR HIGH SCHOOL MATHEMATICS
2. A man has a house which he can sell for $ 7600. He
could loan this at 6 % on a note and mortgage. If he keeps
it, he can rent it at $60 per month. The general yearly
expenses are: taxes, $115; insurance, $13.60; repairs,
$60. Not considering any increase or decrease in value,
which will pay better and how much ?
3. A man is ojffered a house which rents for $66 per
month in exchange for six $ 1000 4 J % city bonds. If the
general upkeep, taxes, etc., on the house will average $ 20O
per year, what would you advise him to do ?
4. In case the money returns on bonds and real estate are
the same, which would you consider the better investment ?
Note. — In answering such a question, there are many factors to be
considered. Without knowing all these, no positive answer could be
given. In general, bonds are more readily converted into cash in case
one needs for other purposes the money invested.
7. INVESTING IN STOCKS
You have noticed from advertisements that most of the
things you use are produced by some company or corporation.
Thus, you see the advertisements of Swift & Co. ; The Quaker
Oats Co. ; The National Biscuit Co. ; Colgate & Co. ; Cadil-
lac Motor Car Co. ; etc. These companies consist of a num-
ber of individuals united by the consent of the state, and
empowered by the state to transact a certain form of business.
The list of powers, rights, and duties of each are stated in
writing in an instrument called their charter.
Stock is a name given the capital with which they do busi-
ness. This capital stock is divided into shares, usually $100
each, called the par value, but they may be of any size.
Thus, if a company has a capital of ♦1,500,000 divided into
ilOO shares, there will be 15,000 of them. Any one may
METHODS OF INVESTING MONEY 173
become a part owner of the company by buying one or more
of these shares. If one owned 15 of these 15,000 shares, he
would own one one-thousandth of the business and be en-
titled to that part of its earnings.
The owner of one or more shares of stock in a company is
called a stockholder in the company. As evidence of owner-
ship, each stockholder receives a stock certificate showing the
number of shares he owns and the par value of each.
The earnings of a corporation that are divided among its
stockholders are called the dividends. They are distributed
as a per cent of the par value of the stock. Thus, a 12 %
dividend gives the holder $ 12 for each f 100 share that he owns.
1. If the capital of a corporation is $500,000 and divided
into $100 shares, how many shares will there be? The holder
of 100 of these shares would own what part of the business ?
2. If a man owns 100 of the 1000 shares in a company, he
owns what part of the business? If $8000 in earnings
(dividends) are distributed, how much will he get ?
3. If a company with a $100,000 capital divides it into
$50 shares, how many shares will there be ? For each share
held, one will own what part of the business ?
4. If a company with a capital of $500,000 distributes
$75,000 in dividends, what per cent of the capital is this?
5. If one owns twenty $100 shares in a company and an
8 % dividend is declared, how much wiU he get ?
6. If one owns twenty $50 shares and a 10 % dividend is
declared, how much will he get?
7. When a company with a capital of $1,000,000 declares
a 6% dividend, how much will the whole dividend be?
How much will a man get who owns fifteen $100 shares ?
8. When a man gets $ 180 in dividends from fifteen $ 100
shares of stock, what rate of dividend has been declared ?
174
JUNIOR HIGH SCHOOL MATHEMATICS
The Market Value of Stock
The market value of stock is the price at which it can be
bought or sold in open market. A number of factors affect
the market price of stock, chief among which are : (1) the
real or prospective earning power of the corporation ; and
(2) the confidence of the buying public, or the lack of it, in
the general stability of the enterprise. When the real or
prospective earnings are small, the price is low ; when large,
the price is high.
As these two factors change with cost of labor and mate-
rial, and the public demands for the company's products, and
for numerous other reasons, the price of stock varies greatly.
For that reason there is much speculation in stocks. By
speculation is meant buying in expectation of a rise in price,
or selling in expectation of lower prices, with the intention
of buying back. In other words, speculation is dealing in
uncertainties. A stock investment is always more specula-
tive than a bond investment, owing to the fluctuation of
the market value.
STOCKS AND BONDS COMPARED AS INVESTMENTS
Stooks
1. The dividends depend upon
the earning power of the corpora-
tion.
2. The dividends are not due
until they have been declared by
the board of directors.
3. Subject to sudden fluctuations
in value.
Bonds
1. The interest is a fixed rate.
2. The interest is paid at regular
fixed periods.
3. Only slight fluctuations in
value.
METHODS OF INVESTING MONEY 175
Stock quotations, like bond quotations, are a per cent of
the par value. Thus, a quotation of '* U. S. Rubber 70 J "
means that United States Rubber Co. stock is selling for
70^% of its par value, or §70.25 for each $100 share.
Through a broker this would cost a buyer $70.26 + $.125 or
$70,376 per share and net a seller $70.25 - $.125 or $70,125
per share. (The brokerage for buying and for selling is ^ %
of the par value.)
1. Find the cost of twenty $ 100 shares of U. S. Steel
when quoted at 107J, no brokerage. Find the cost of the
same if bought through a broker.
a. How much will a man receive net from a sale of fifty
$100 shares of Studebaker Co. stock when quoted at 70 J,
sold through a broker at J % brokerage ?
3. How much is made on forty $100 shares of Western
Union Telegraph Co. stock when bought at 87J^ and sold at
92|, after paying brokerage for both buying and selling?
4. When stock selling for 80| is paying a 6% dividend,
what per cent of the investment is it earning? (The
question is, '' What per cent of $80.50 is $6 ? " Why ?)
5. When stock selling for $120 is paying an 8% dividend,
what per cent does the investment yield ?
Investing in Preferred Stock
The stocks already discussed, in which the stockholder
becomes a part owner of the corporation by his investment,
has a vote in the control of the business, and shares its
profits, are called common stocks.
There is a growing tendency among industrial corporations
to obtain capital requirements through the issue of a type of
stock called preferred stock. This differs from the common
stock in that the holder has no vote in the control of the
176 JUNIOR HIGH SCHOOL MATHEMATICS
corporation, and does pot share in the earnings except to the
extent of the dividends guaranteed in the certificate. Pre-
ferred stock usually guarantees, in the certificate, a dividend •
of 7%. It is sometimes but 6%. Since preferred stock is
not in the form of a note or bond against the company,
secured by a mortgage on the property of the corporation,
and having a date at which it matures, it is less safe as an
investment and hence has to pay a higher rate in order to
find buyers.
!• What is the yearly income from 40 shares of f 100
each of 7 % preferred stock ?
, 2. Find the difference in income between 110,000 invested
in 7 % preferred stock at par, and the same amount invested
in its 4 J % bonds at par.
3. When "Liggett & Myers pf." is quoted at 107 J, what
per cent of returns is the investor getting if the guaranteed
dividend is 7 % ?
4. A certain preferred stock paying a 6% dividend is
selling for 89 ; find the rate of income on the investment.
5. The total capital of a certain corporation is $2,700,000,
of which $200,000 is 7% preferred stock. One year the
gross earnings were $296,465. The total expenses were
$132,465. The net profits were all distributed as dividends.
How much went to the holders of the preferred stock and
what per cent of dividend did the holders of the common
stock receive ? In this case, which would you expect to sell
higher in the market ?
6. When common stock selling at 198 is paying a 12^
dividend and preferred stock selling at 105 is paying a 7%
dividend, which is giving the better return on the invest-
ment?
CHAPTER XIV
THE MEANING AND NATURE OF INSURANCE
Tou have heard people speak of carrying insurance on
their property or on their health or life. Insurance is an
agreement by an insurance company, for a consideration
called a premium, to compensate the insured party for actual
losses or damages arising from certain stipulated causes.
The agreement or contract is called the policy. The sum of
money specified in the policy to be paid in case of loss is
called the face of the policy. There are two general classes
of insurance: property insurance, ^ndi personal insurance.
1. PROPERTY INSURANCE
Property insurance, as its name implies, is insurance having
to do with damage or loss of property of any kind. There
are numerous kinds of such insurance, among the most
common being fire, tornado, lightning, burglary, live stock,
marine, plate glass, steam boiler, transit, automobile, etc.
The principles of all are similar. The contract between the,
insured and insurer is always called the policy, and shows
the conditions upon which the insurer agrees to indemnify
for losses. As fire insurance is perhaps the most common of
all these forms, it is discussed as representative of all.
Fire insurance is an agreement to compensate or indemnify
the insured against actual losses arising from accidental fires.
The "loss by fire" includes any damage resulting from
chemicals or water used in extinguishing the fire. Fire
caused by lightning is usually included under "accidental
fires."
177
178 JUNIOR HIGH SCHOOL MATHEMATICS
The rate of premium varies with conditions and is usually
stated as a specified sum on each $ 100 in the face of the
policy. The periods of fire insurance are usually one or
three years.
1. For which do you think an insurance company would
charge the larger premium : a $ 6000 policy on a house in a
town with no fire protection, or the same amount on a house
in a city having good protection ?
2. For which would a company charge more in the same
city : a policy on a modern fireproof building, or one of
the same size on a wooden building with shingle roof ?
3. Which do you think should pay the higher rate of
premium on his policy : the owner of a home on his home,
or the owner of a public garage on his garage ?
4. Which would cost more, to insure a building in a
neighborhood of brick or stone buildings, or the same build-
ing if it were surrounded by frame buildings ?
The premium rate upon a building depends upon : (i) the
location; (^) the nature of the construction; (e?) the use made
of the building; and (^) the construction and use of the adjoin-
ing buildings.
5. Some towns are divided into zones and the rates vary
in each zone according to their distance from a fire depart-
ment station. If you can, visit some local agent and get
from him the rates where you live.
6. Rates are usually quoted as so many cents or dollars
and cents per $ 100. 24 ^ per $ 100 is what rate per cent ?
7. If the rate on a 3-year policy in a certain city is $ 1.62
per $ 100, what is the rate per cent ?
8. Show two ways of finding the premium on a 3-year
policy of i 6000 when the rate is 96 ^ per $ 100,
THE MEANING AND NATURE OF INSURANCE 179
9. For 16 years a man has kept his house insured at
$ 6000 by taking out 1-year policies at 48 ^ per 1 100. He
could have taken out 3-year policies for 2^ times the rate on
a 1-year policy. How much could he have saved in taking
out the longer policies ?
10. Mr. Reed pays $1.70 per 1 100 on a $3000 policy on
a summer cottage in the country, and but 86 jf per $ 100 on
a $ 5000 policy on his home in town. How much does he
pay on each, and how do you account for the great difference
in the rate ?
11. A man has his property insured for f 2000 in one
company and $ 3000 in another. In case of a f 4000 loss,
how much will he collect from each ?
12. On account of the fire protection in most cities, a
total loss of property by fire is unusual. For this reason a
man often carries insurance to protect but partially the full
value. Find the premium at 24^ per $100 on a policy
covering but 80 % of property valued at $ 12,000. In case
of total loss, how much would the policyholder receive?
How much in case of a loss of $ 6000 ? A loss of $ 500 ?
13. In some states, if a man agrees, by accepting a certain
clause in the policy, to carry a certain amount of insurance
and fails to do so, he can collect (in case of loss) but such a
part of it as the face of the policy bears to the amount
agreed upon. Under such a contract, if a man agrees to
carry $ 5000 and carries but $ 3000, what part of any loss up
to $ 3000 can he collect ?
14. Some policies contain an 80 % coinsurance clause,
which is an agreement to carry 80 % of the value of the
property. How much insurance would a man have to carry
under such a contract if his property is worth $ 12,000 ?
180 JUNIOR HIGH SCHOOL MATHEMATICS
15. A man having property worth f 10,000 insures it for
f 6000. If there is an 80 % coinsurance clause in the con-
tract, how much does he thus agree to carry ?
16. If the man, problem 16, agrees to carry $8000 and
carries but % 6000, what part of a loss up to $ 6000 can he
collect ? (See problem 13.)
17. In 1916 the total premiums received by fire insurance
companies were $419,361,346 and the total losses paid were
% 221,701,359. The losses were what per cent of the
premiums ? The expenses were % 157,728,585. These were
what per cent of the premiums ?
18. In 1916 the policies written amounted to approximately
$53,000,000,000. The losses paid were what per cent of this?
19. Can you bring to class a canceled fire insurance
policy for study and discussion ?
2. PERSONAL INSURANCE
Personal insurance is that form of insurance in which the
insurance company agrees to pay a certain sum of money in
case of accident to the insured, or in case of his sickness or
death. These are called accident^ healthy and life insurance.
Most common among these is life insurance.
There are four general forms of life insurance policies:
(1) ordinary life; (2) limited life; (3) endowment; and
(4) term inmirance.
In the ordinary life policy, the premiums are paid, usually
annually or semiannually, during the life of the insured,
and the insurance company agrees to pay a fixed sum to the
heirs of the insured, or to some other party designated in
the policy, at his death.
The person named to receive this sum is called the
beneficiary.
THE MEANING AND NATURE OF INSURANCE 181
In the limited life policy, the premiums are paid for
a fixed number of years, after which the policy is called
paid up ; but the face of the policy is not paid the beneficiary
until the death of the insured.
In the endowment policy, the premiums are paid for a
fixed number of years, as ten, fifteen, or twenty, and the
face of the policy is paid the insured at the end of the
period.
Note. — In both the limited life and the endowment, the face of the
policy is paid the beneficiary in case of death before the end of the
period during which premiums are payable.
In term insurance, the premiums are paid for a fixed period
and the face of the policy is paid the beneficiary in case of
death during this period. At the end of the period the con-
tract ceases.
1. Would you expect an ordinary life policy or a twenty-
payment life to cost more ? Give a reason.
2. Would you expect a twenty-year endowment or a
twenty-payment policy to cost more ? Give a reason.
3. Arrange the four kinds of policies in order of what
you consider the rate of premium.
THE TABLE SHOWS THE PREMIUM CHARGED BY A LEADING
LIFE INSURANCE COMPANY FOR A $1000 POLICY. THE
PROBLEMS THAT FOLLOW ARE BASED UPON THESE RATES.
Aqm or Inburidd
Ordinary Lifb
20-Patment Life
20-Yr. Endowment
20
9 18.01
• 27.82
» 47.67
25
20.14
30.12
48.15
30
22.85
32.87
48.83
35
26.35 .
36.22
49.85
40
30.94
40.38
51.48
45
37.08
45.73
54.22
50
45.45
52.87
58.81
•
182 JUNIOR HIGH SCHOOL MATHEMATICS
4. How much per year will a $ 5000 ordinary life policy
cost a man who insures at the age of 25? How much a
year will it cost him if he insures at the age of 40 ?
5. Find how much a $ 5000 policy of each of the three
types will cost a man taking hisurance at the age of 35.
6. Suppose that a man 30 years of age, taking out a
$ 15,000 policy, dies after making the 15th payment. His
beneficiary would get $ 15,000 under any of the three policies
named above. Show how much he would have paid out in
each.
7. If a man of 30 takes out a 20-year endowment policy
of $10,000 and lives 20 years, he will receive the face of
the policy. How much less is this than the amount of the
premiums if placed in a savings bank paying 4 % ?
8. If a man of 30 takes out an ordinary life policy of
$10,000 and dies in 20 years (after making 20 payments),
would his beneficiary get more or less than the amount of
the premiums if placed in a savings bank paying 4 % ?
9. Make the same kind of comparison as in problem 8,
supposing that he died in 10 years.
10. Suppose a man of 50 should take a 20-payment life
policy of $ 20,000 and die at the end of 20 years. Compare
what the beneficiary would receive with the amount of the
premiums placed in a savings bank paying 4 % .
11. Make the same kind of comparison as in problem 10,
supposing the man to have been but 20 years of age when
taking out the insurance and dying in 20 years.
12. Make up and solve other problems using the data
from these tables.
THE MEANING AND NATURE OF INSURANCE 183
The Tbree Elements that Make ap the Premium
The annual premium paid by the insured is made up of
three items : (1) mortality cost ; (2) reserve ; and (3) ex-
pense loading.
The mortality cost is the amount reckoned as necessary to
collect each year to pay the death claims of that year. This
is determined by "mortality tables" compiled from long
experience, showing the deaths expected each year out of a
certain number of any age.
The reserve element is the amount from each premium
necessary to amount to the face of the policy in a given
time. It is a sort of savings bank account of the insured
with the company, bearing 3 % or 3 J % compound interest.
It may be withdrawn at any time by surrendering the policy
and thus terminating the contract, and is thus called the
cash surrender value of the policy.
The expense loading is the amount estimated as necessary
to meet the expenses of the management of the company.
It is usually about one-fifth or one-sixth of the total premium.
1. A man 40 years of age taking out a $10,000 ordinary
life policy at the rates given on page 181 may surrender it
at any time after 2 years and get the reserve or " cash sur-
render value." At the end of 20 years, this cash value is
$3834.70. This is how much less than he has paid out?
Note. — All rates refer to the table given on page 181.
2. If a man of 30 takes out an ordinary life policy of
$1000, he may surrender it in 15 years and receive $276.02.
This is how much more or less than he has paid out ?
3. A man insuring for $1000 at 25 on the 20-payment
plan may surrender it for $504.58 after having made the
last payment. Compare this with the amount paid out.
184 JUNIOR HIGH SCHOOL MATHEMATICS
4. A man insuring for f 10,000 at 30 on the 20-year en-
dowment plan may surrender his policy in 16 years and
receive $6748.50. Compare this with what he has paid out.
5. The cash surrender value, at the end of 10 years, of a
$5000 20-payment policy taken by a man of 25 is $1044.75.
Compare this with the amount of the premiums at 4 % com-
pound interest.
CHAPTER XV
THE MEANING AND NECESSITY OF TAXES
Taxes are the money raised in some form to meet the ex-
penses of government. These are raised in various ways to
meet the expenses of the various units of government.
Towns and cities must raise money to meet the expenses
of fire and police protection, of building and maintaining
schools and other public buildings, to pay its officers, etc.
Townships and counties must meet the expenses of build-
ing roads and bridges, maintaining public institutions, pay-
ing certain salaries, certain courts, charities, etc.
The state has many salaried officials to pay, and helps build
the roads of the state. It also keeps up certain state in-
stitutions, as prisons, schools, and asylums, all of which
demands the expenditure of large sums of money.
The United States Government also requires large sums of
money to meet its expenses. Among these are the salaries
of its officials ; the maintenance of its army and navy ;
interest on its national debt; and the pension of disabled
soldiers. The total government expense of 1916 was about
8725,000,000; during our first year in the great World War
it rose to over $18,000,000,000.
1. HOW CITY, COUNTY, AND STATE EXPENSES
ARE BEET
Most of the expenses of towns, cities, counties, and states
are met by a tax levied by the proper officers upon the
property of the town, city, county, or state. The property
185
186
JUNIOR HIGH SCHOOL MATHEMATICS
is divided into two classes for taxation : (1) real estate, re-
garded as immovable property, as lands and buildings, mines,
railroads, etc.; and (2) personal property, including all
movable property, as money, stocks, bonds, furniture, live
stock, etc.
Assaaa^MTS, elected or appointed, estimate the value of the
property to be taxed. This is called • the assessed valuation
of t^ property. From the total assessed valuation and the
tax to be raised, the tax rate is determined. This tax rate
is stated in various ways* In some states it is a certain
number of mills (tenths of a eent^ on the dollar ; in others,
it is a certain number of dollars per $100 or per $1000.
In some states, it is stated as a rate per cent.
Thus, a rate of 12^ mills on the dollar is $1.25 per $100,
$12.50 per $.1000, or 1\%.
A FORM OF TAX BILL
■ - - ■
Rates 92,02
tb18 biul mxttkt bm bitvurnsd wbkn you fat ythtb taxes
Mr* John Doe
No, 56 N. Walnut St,
Page 123 Line 39
Map 3 Block C Lot No. SS
Rbal
ESXATV
7200
PrntSOlTAZi
PsomRTT
900
Total
vaij7a.tiov
8100
AND CouKTY Tax
65
64
SoUChOCj
Tax
35
08
Toww
Tax
7290
Poll
Tax
16^82^
1. If the assessed valuation of the property of a village
is. $&,500^0QO and $»'!,500 is to be raised,, the tair » how
many mills cm the daUar? How inany dolkirs per $10d?
What per cent ?
THE MEANING AND NECESSITY OF TAXES 187
2. If the school tax of a town is 4 mills on the dollar, how
mnch school tax must a man pay whose property is assessed
at 112,000 ?
3. When the town tax is 8^ mills on the dollar, how
much is the tax on property assessed at $9500?
4. When one's total tax is $1.98 per $100, what will he
have to pay on property assessed at $17,256 ?
5. A man's total tax at $2.02 on $100 was $168.62.
From this find at what value his property was assessed ?
6. If taxes increase from $1.65 to $2.17 per $100, how
much will it increase one's tax whose property is valued at
«18,500?
7. Who is paying the highest rate, one who pays lOJ
mills on the dollar, $1.01 per $100, or 1.1 % ?
Give the rate per $100:
8.
9.
10.
11.
Ambbssd
Valuation
Tax to bb
Raisxd
AsSEfiSBD
Valuation
Tax to bb
Raised
94,800,000
16,500,000
51,000,000
89,000,000
936,000
288,750
750,000
763,000
12.
13.
14.
15.
9245,000,000
356,000,000
758,000,000
986,000,000
91,260,000
1,850,000
3,762,000
10,248,000
Q-ive the tax on:
16.
17.
18.
19.
Absbssbd
Valuation
Tax Ratb
assbbsbd
Valuation
Tax Ratb
9 12,500
9,750
10,500
13,750
4 J mills on 9 1
9} mlllR on 9 1
9 1.65 per 9 100
9 1.08 per 9 100
20.
21.
22.
23.
96,780
11,250
17,750
16,350
9 12.25 per 9 1000
917.60 per 9 1000
9 2.02 per 9 100
188 JUNIOR HIGH SCHOOL MATHEMATICS
24. Find from the assessor of your city the assessed value
of the property and the tax to be raised, and compute the
tax rate.
2. HOW TH£ £XP£NS£S OF THE NATIONAL GOVERNMENT
ARE MET
The people are not taxed directly upon the property they
own, to support the National Government, as they are to
support state, county, and local governments. The ex-
penses are met chiefly by : (1) tariffs, duties, cr customs,
which are levied upon goods imported from other countries ;
(2) internal revenue, which is levied upon things made in
this country, as alcoholic beverages and tobacco products ;
and (3) an income tax, levied upon the incomes of indi-
viduals and corporations.
Tariffs, Duties, or Customs
Some imported goods are not subject to duty. Such
goods are said to be on the free list. The duties are of two
kinds : (1) ad valorem duty, which is a per cent of the in-
voice price of goods at the place of purchase ; and (2) spe-
cific duty, which is a certain amount per unit, as pound, ton,
bushel, barrel, yard, etc. Some goods are subject to one
duty and some to both.
The customs revenue is collected at custom-houses situ-
ated at the various ports of entry.
The tarijBf rates are frequently changed by Congress. For
example, in 1913 the first income tax law was passed and
the tariff rates were lowered that year.
1. The duty on watch and clock movements is 30 % ad
valorem. Find the duty on a watch movement costing
18.50 in Europe.
THE MEANING AND NECESSITY OF TAXES 189
2. The duty on drugs and medicines in pills, capsules,
tablets, etc., is 25% ad valorem. Find the duty on an in-
voice valued at $16,628 in Europe.
3. The duty on olive oil in bottles and cans is 30 ^ per
gallon. Find the duty that an importer must pay on
15,000 gallons.
4. The duty on automobiles valued at more than $2000 is
45%. Find the duty on an automobile valued au 4 5600.
5. The duty on blankets and flannels is 30%. Find the
duty on an invoice of $35,500 worth of flannels.
6. The duty on wool is 8%. In 1917 we imported
372,372,218 pounds, valued at $131,137,170. How much
revenue did the government get from this one item ?
7. In 1917 we imported $93,704,230 worth of manufac-
tured copper ware at a duty of 20%. Find the revenue
from copper.
8. Under the law of 1909, known as the Payne-Aldrich
Tariff Law, the duty on wool was 30 % ad valorem plus a
specific duty of 24| ^ per pound. Under the law of 1913,
known as the Underwood-Simmons Tariff Law, the specific
duty was dropped and the ad valorem duty lowered to
8%. In 1917 we imported 372,372,218 pounds, valued at
$ 131,137,170. Find the decrease in revenue by the law of
1918 from that of 1909.
Internal Revenue
Before the World War we obtained nearly one half of the
money needed to support the National Government from
revenues on tobacco, spirits, and fermented liquors. In
a recent year this internal revenue amounted to about
$888,000,000.
190
JUNIOR fflGH SCHOOL MATHEMATICS
1. During 1917 there were 60,729,509 barrels of fermented
liquors (beer, ale, etc.) taxed at $3 per barrel. Find the
amount of revenue from this source.
2. During the same year 7,390,183,170 cigars weighing
more than 3 lb. per 1000 were taxed $4 per 1000. Find
this tax.
3. At $2.05 per 1000, find the tax on 21,066,196,672
cigarettes sold in 1917.
4. At 13^ per pound, find the tax on 417,235,928 pounds
of chewing and smoking tobacco sold in 1917.
5. The following table shows the internal revenue receipts
for 5 years. Make a bar graph for each item, showing the
relative amounts each year.
. Year
Spirits
Tobacco
FsnmNTBO Liquors
1913
9108,879,342
• 76,789,424
• 66,266,989
1014
169,098,177
79,986,639 ,
67,081,512
1915
144,619,699
79,957,373
79,328,946
1916
158,682,439
88,063,947
88,771,103
1917
186,563,054
102,230,205
61,532,065
6. Make a bar graph of the internal revenues received in
1917, showing the relative amount received from each of
three sources.
Revenue and Expenditures of the Post Office Department
The revenue from the sale of postage stamps is seldom
listed under the internal revenues, for the income practically
balances the expenses of the department. Thus, in 1917 the
revenue of the department was $329,726,116 and the ex-
penses were $319,838,718.
1. In 1917 the compensation paid to postmasters was
$31,890,860. This was what per cent of the total expense
THE MEANING AND NECESSITY OF TAXES 191
of $8199888,718? By inspection, estimate the lesolt before
solving and see how nearly correct you estimate.
2. The cost for transportation of mail in 1917 was
$111,522,255. This was what per cent of the total expense?
Estimate the result before solving.
3. In 1917 there were 48,388 rural delivery carriers, and
the daily mileage was 1,112,556 miles. Find the average
mileage per carrier.
4. The cost of rural delivery in 1917 was 152,420,000.
That is what per cent of the total cost ?
5. The cost of rural delivery increased from $41,859,422
in 1912 to $52,420,000 in 1917. Find the average increase
per year.
6. From 1907 to 1917 the expenditures, of the post ofi&ce
department increased from $190,288,288 to $319,888,718.
Find the average yearly increase.
7. In 1917 the total cost of rural delivery was $52,420,000
and the total mileage was 1,112,556. Find the average
yearly cost per mile for rural delivery.
8. There were 55,413 post offices in 1917, of which 10,381
were presidential appointment offices. This was what per
cent of the whole ?
The Income Tax
The income tax, as its nScme implies, is a tax upon incomes.
This is a new form of raising money to support the govern-
ment, having first been made a law in 1913. The rate has
changed several times to meet new demands upon the
government. The income tax is upon individuals and cor-
porations. Since the beginning of this form of taxation
the personal or individual tax has been divided into a normal
tax and an additional tax or surtax.
192
JUNIOR fflGH SCHOOL MATHEMATICS
The normal tax upon the incomes of 1918 was 12 % of the
net income in excess of $2000 in case of a married person,
and $1000 in case of an unmarried person, except upon the
first $4000 of such excess, upon which the rate was but 6 %.
This law, which was passed in the early part of 1919, pro-
vided that the norvud tax for each calendar year after 1918
should be 8 % of the net income (less the same exemptions
as above), except upon the first $4000 of such excess, upon
which the rate was to be 4 %.
The surtax provided in the law of 1919 was as follows :
Pbb Ckkt
From
To
Per Cent
From
To
1%
$ 5,000
$ 6,000
28%
$ 58,000
% 60,000
2%
6,000
8,000
29%
60,000
62,000
3%
8,000
10,000
30%
62,000
64,000
4%
10,000
12,000
31%
64,000
66,000
5%
12,000
14,000
32%
66,000
68,000
6%
14,000
16,000
33%
68,000
70,000
7%
16,000
18,000
34%
70,000
72,000
8%
18,000
20,000
36%
72,000
74,000
9%
20,000
22,000
36%
74.000
76.000
10%
22,000
24,000
f37%
76,000
78,000
11%
24,000
26,000
38%
78,000
80,000
12%
26,000
28,000
39%
80,000
82,000
13%
28,000
30,000
40%
82,000
84,000
14%
30,000
32,000
41%
84,000
86,000
15%
32,000
34,000
42%
86,000
88,000
16%
34,000
36,000
43%
88,000
90,000
17%
36,000
38,000
44%
90,000
92,000
18%
38,000
40,000
46%
92,000
94,000
19%
40,000
42,000
46%
94,000
96,000
20%
42,000
44.000
47%
96,000
98,000
21%
44,000
46,000
48%
98,000
100,000
22%,
46,000
48,000
62%
100,000
150,000
23%
48,000
50,000
56%
150,000
200,000
24%
50,000
52,000
60%
200,000
300,000
26%
52,000
54,000
63%
300,000
500,000
26%
54,000
56,000
M%
500,000
1,000,000
27%
56,000
58,000
05%
1
over
1,000,000
THE MEANING AND NECESSITY OF TAXES 193
How to Compute an Individual Income Tax
(For incomes of 1918)
Single Person : Net Income |7600
Income $7500
Exemption 1000
Subject to normal tax .... 6500
Tax on first ^4000 of excess @ 6 9^0 240.00
Excess over $4000; 92500® 12 9() 300.00
Surtax: Net income $7500
Not taxable 5000
Subject to surtax $2500
From $5000 to $6000; $1900 @ 19b $10.00
From $6000 to $7500; $1500 @ 2 9«> 30.00
Total tax $580.00
Head of Family — 3 Dependent Children : Income $16,000
Note. — Aside from the $2000 exemption, the head of a family is
allowed $200 for each dependent child under 18 years of age.
Income $15,000
Specific exemption, $2000; plus al-
lowance for children, $ 600
Total allowance 2600
Amount subject to normal tax . . $ 12,400
Noimal tax, 6 9^ on first $4000 of excess of credits .... $240.00
Normal tax of 12 9^ on taxable income over $ 4000 ; $ 8400 . 1008.00
V«rtox : jjg^ income $ 15000
Not taxable 5000
Subject to Surtax $10000
Subject to surtax as follows :
From $5000 to $6000— $1000 @ 1 9^ $10.00
From $6000 to $8000— $2000 @ 2 9b 40.00
From $8000 to $10000— $2000® 3 9^ 60.00
From $10000 to $12000 — $2000® 49b 80.00
From$1200bto$14000 — $2000® 59b 100.00
From $14000 to $15000 — $1000 ® 6 9fe 60.00
Taxable income . . $10000
Total tax $1598.00
194 JUNIOR fflGH SCHOOL MATHEMATICS
Since the income tax rate changes frequently to meet the
government's need of revenue, but few problems are given
based upon the rates given here.
If interested in the subject, get the income tax rate at the
time you study this and solve problems as your teacher may
direct. The tax rate can be obtained at any bank.
1. A single man's income for 1918 was $6000. Find the
income tax he had to pay in 1919. The head of a family
would have had what tax upon the same income ?
2. What was the income tax in 1919 for a single person
whose income for 1918 was $10,000? A married person
with 4 dependent children would have had what income tax
upon the same income ?
3. Find the income tax for the head cf a family having
2 dependent children, if his income for 1918 was $8000.
CHAPTER XVI
SOME THINGS YOU HAVE LEARNED DURING
THE TEAR
This chapter is a brief review of some of the new phases
of the course learned early ii^ the year. It may be used as
a final review, or to supplement the topics as they are studied,
or for both* purposes.
1. Ton HAVE LEARNED TO INTERPRET AND
EVALUATE A FORMULA
You learned to use letters for numbers in expressing a
mathematical relation. These relations were called formula
and you found that they were merely shorthand rules of
computation.
To evaluate a formula^ you learned to substitute the numeri-
cal valu£ of the letters and perform the computation.
1. Evaluate A = lw when 1=12 and w = 8. What prin-
ciple of mensuration is expressed by this formula ?
hh
2. Evaluate A = ~ when J = 16 and h = 12. Inter-
2
pret this formula as a principle in mensuration.
3. Evaluate (7=:27rr when r = 20. What principle of
mensuration is expressed by this formula ?
4. Evaluate A::sirr^ when r=s24. What principle of
mensuration is expressed by this formula ?
195
196 JUNIOR HIGH SCHOOL MATHEMATICS
5. Evaluate V=:lwh when /=20, w=15^ and A = 12,
and tell what relation is expressed by the formula.
6. Evaluate V^Bh when jB = 40 and A = 12, and tell
what relation is expressed by the formula.
Hit
7. Evaluate and interpret 1^= — when 5 = 60 and
A = 15. ^
8. Evaluate and interpret V =^ irr^h when r = 6 and
A = 15.
9. Evaluate and interpret S=4 irr^ when r = 8. (The
letters refer to a sphere.)
10. Evaluate V=^ f irr^ when r = 6. Interpret the formula
when the letters refer to a sphere.
11. What area is expressed by -A = J A(J -H ^') ? Evaluate
the formula when A = 8, 5 = 12, and b' =10.
12. The area of a triangle in terms of its sides is repre-
sented by the formula :
A = V/SXaS- «)(/»- bXS-
where A = area, S = half of the sum of the three sides, and
a, ft, and c are the length of the three sides. From the
formula state a rule for finding the area of a triangle when
its sides are known.
13. By use of the formula given in problem 12, find the
area of a triangle whose sides are 20 in., 30 in., and 34 in.,
respectively.
14. How many acres in a triangular field whose sides are
24 rd., 30 rd., and 36 rd., respectively?
15. Evaluate d(^a — 6) when d = 10, a = 12, and 6=5.
16. Find the value of (a; + y)^ when a; = 10 and y = 12.
17. Find the value of (r -f- 10) -j- (7 — r) when r = 6.
18. If aS= a(t - J), find S when <p= 40 and t = 8.
SOME THINGS YOU HAVE LEARNED 197
19. Bring to class any formulae you find in general read-
ing or in other subjects that you are studying.
2. YOU HAVE LEARNED THE MEANING OF AN
EQUATION AND HOW TO SOLVE IT
You have learned that an equation expresses the fact that
one value' equals or balances another ; and that, just as in a
scale pan, if any change is made on one side of the equation,
the same change must be made on the other side.
You have learned, too, that one side of the equation con-
tains an unknown value and that to solve the equation is to
find a value for the unknown number that satisfies the
equation.
Solve by inspection :
1. a: + 3 = 7.
6.
2a: =10.
11.
2a:-5
2. 4 + 2? = 9.
7.
3a; = 18.
12.
^a; = 6.
3. a;-fT = 12.
8.
2a: + l=9.
13.
i^ = 6.
4. aj — 3 = 5.
9.
3a: + 2=:ll.
14.
\x=8.
5. ic-7 = 10.
10.
2a;-3 = 7.
15.
i^ = 4.
16. Give the four axioms used in the solution of equations.
Solve and state the axiom used :
17. w + 6 = 10. 21. 3a: -10 = 20. 25. |w + 2 = 5.
18. Sn=:2n-{'S. 22. 6a: + 15 = 75. 26. 8 a: - 10 = 70.
19. 3/1-8 = 10. 23. 3:c-7 = 29. 27. fa: + 8 = 18.
20. 5iC4-2 = 27. 24. Jw-3 = 1. 28. fa:- 7 = 3.
3. PROBLEMS SOLVED BY USE OF EQUATIONS
You have learned to express a word-statement in the form
of an equation and then to solve it. Further practice is
here given.
198 JUNIOR fflGH SCHOOL MATHEMATICS
1. A rectangular garden 60 ft. long contains 2400 sq. ft.
Find its width.
Let X = the number of feet in the width.
Then QOx = 2400 ;
and X = 40, the number of feet in the width.
2. How wide a strip 40- rd. long will contain 3 acres ?
3. Find the height of a triangle containing 68 sq. in.
when the base is 12 in.
4. A rectangular prism 20 in. high contains 480 cu. in.
How many square inches in the base ?
5. Find the depth of a bin 8 ft. by 10 ft. that will con-
tain 400 cu. ft.
6. There are 45 sq. ft. in the base of a pyramid contain-
ing 90 cu. ft. What is the altitude ?
7. 20% of a certain number is 98. What is the number?
8. If 25 % of a number is 120, what is the number ?
9. James sold 75 % of his pigeons and had 12 left. How
many had he at first ?
10. A merchant had forgotten the cost of an article, but
remembered that he had marked it 25 % above cost. If it
was marked $80, what did it cost him ?
U. At 4^ %, the interest on John's Liberty Bond is $4.25
each half year. He has a bond of what size ?
12. In a Junior High School class in mathematics, there
were 7 more boys than girls. In all there were 85. How
many boys and how many girls in the class ?
13. In another class of 86, there were twice as many boys
as girls. How many of each ?
14. One day Donald sold half as many papers as Ralph.
Together they sold 90. How many did each sell ?
SOME THINGS YOU HAVE LEARNED 199
16. It i» 240 ft. around a rectangle 3 times as long a» it
18 wide. Find the dimensions of the rectangle.
IS. James caught 4 more than twice as many fish as
Robert. Together they caught 34. How many did each
catch?
17. Ralph and his sister raised vegetables to sell. They
agreed that Ralph should do the heavy work and have twice
as much of the money received as his sister. The total sales
were $96. How shall they divide the money ?
IS. To make ke cream, Mary was going to use twice as
much milk as cream. How much of each in 4»5 qt. of the
mixture ?
19. During a thrift-stamp campaign, Frank sold twice as
many stamps as James did, and Ralph sold as many as both.
Together they sold 240. How many were sold by each ?
ao. Together John and Ralph have 65 marbles. John
has & more than Ralph. How many has each ?
21. Ralph and Donald take care of Mr. Brown's lawn
and garden for |T5 for the summer. They agree that
Ralph should have 1 J times as much of the money as Donald.
How much must each receive ?
4. Ton HAYX TJIABireP lO IlED mnAB€MS> BT
SCALE BRAWIB68
1. Draw to scale 1 in. = 4 ft. a floor plan of a room 24 ft.
by Si ft. and by measurement find the diagonal of the room.
2. If the base of a triangle is 100 ft. and the base angles
SSf and 60^, reapeetively, draw to scale 1 in. == 20 ft. a tri-
tam^ and detearmine the other two sides by measuiring the
plan you have drawn.
200 JUNIOR HIGH SCHOOL MATHEMATICS
3. Some boys wished to know the length of a small pond.
They drove stakes at each end of the pond and found a
point back from the pond from which they could measure
to each stake. From this point it was 400 ft. to one stake
and 500 ft. to the other, and the angle made by the two
lines was 80*^. Make a drawing to scale 1 in. = 40 ft. and
find the length of the pond.
4. Some boys found the distance to a tree on the opposite
side of a river from them by running a straight line 400 ft.
long between two points, A and B, and noting the angles
that the line of sight from each point made with line AB.
If these angles were 60° and 80°, respectively, find how far
the tree was from each point, by drawing a similar triangle
to any scale.
5. A boy standing 80 ft. from the foot of a tree found
that the angle of elevation to the top of the tree was 60°.
By any scale you wish to use, find the height of the tree.
6. When a staff 8 ft. tall casts a shadow 10 ft. long, make
a drawing to a scale and by the use of your protractor find
the elevation of the sun.
7. From an observation balloon at an altitude of 6000 ft.,
the observer notes . the enemy trenches are at an angle of
depression of 20°. How far are the trenches from a point
on the ground directly below the balloon ?
Note. — The angle of depression is the angle made with the horizontal
and is equal to the angle of elevation from the trenches to the balloon.
8. From a point 200 ft. above the surface of the water,
the angle of depression of a boat is 16°. How far away is
the boat ?
SOME THINGS YOU HAVE LEARNED 201
5. TOU HAVE LEARNED TO FIND THE HEIGHT
OF OBJECTS FROM THE LENGTH OF
THE SHADOWS THEY CAST
You learned from a study of similar triangles that at any
given time of day, the ratio of the shadow of an object to its
height is constant, and hence any two such ratios form a
proportion.
1. When a boy 5 ft. tall casts a shadow 8 ft. long, how
high is a church tower that casts a shadow 240 ft. long ?
FIRST SOLUTION Since the ratios are equal, they form a
Let X = height of tower, proportion. — — is the ratio of the height
T ft *
Then =-• 5
240 8* of the tower to its shadow, and - is the
o
5
and X = 240 x - = 150. ratio of the height of the boy to his
shadow.
SECOND SOLUTION Since the boy's height is - of his shadow,
- X 240 = 160. the height of the tower is but 5 of its
shadow.
2. When a staff 10 ft. high casts a shadow 8 ft. long,
how tall is a tree that casts a shadow 120 ft. long ?
3. Some boys found the distance across a stream by find-
ing that a pole 20 ft. tall cast a shadow to the opposite bank
when a rod 4 ft. tall cast a shadow 7 ft. long. Find the
width of the stream.
4. When a flag pole known to be just 100 ft. tall casts a
shadow 460 ft. long, how long a shadow will a boy 5 ft. tall
cast?
5. An anchored observation balloon casts a shadow 1200
ft. from a point on the ground directly below it at the same
time that a rod 5 ft. high casts a shadow 4 ft. long. How
high is the balloon ?
202 JUNIOR HIGH SCHOOL MATHEMATICS
6. Measure heights in the vicinity of the school by use
of the shadows that they cast.
6. TOU HAVE LEARNED TO FIND HEIGHTS AND
DISTANCES BY TANGENT RELATIONS
You have learned that the tangent of an acute angle of a
right triangle is the ratio of the side opposite the angle to
the side adjacent to the angle.
1. From a point 100 ft. from the foot of a tree, the angle
of elevation of the top of the tree is 50°. What is the height
of the tree ?
2. The base angles of an isosceles triangle are each 65°
and the base is 30 ft. Find the altitude and area of the
triangle.
Suggestion. — The altitude divides the triangle into two congruent
right triangles.
3. When the angle of elevation from the top of a tele-
phone pole is 40° at a point on level ground 60 ft. from the
foot of the pole, what is its height ?
4. An observer notes that the angle of elevation of an
aeroplane is 60° when a second observer 2000 ft. away notes
that he is directly below it. Find the height of the aeroplane.
5. When a flag pole 60 ft. high casts a shadow 80 ft.
long, what is the elevation of the sun ?
Suggestion* — The tangent of the angle is .75, To what angle does
that most neariy correspond ?
6. When the two legs of a right triangle are 60 in. and
60 in., respectively, what are the angles of the triangle ?
7. When a balloonist whose altitude is 8000 ft. notes an
enemy gun at an angle of depression of 25°, how far is the
gun from a point on level ground directly below the bal«
loonist ?
SOME TfflNQS YOU HAVE LEARNED 203
8. From a ship, the angle of elevation of a light from a
lighthouse known to be 80 ft. above the level of the ship is
8^. How far away is the lighthouse ?
9. When the sun is 50° above the horizon, a church spire
casts a shadow 66 ft. long. How high is the church spire ?
7. TOU HAVB L£ARN£D TO REPRESBNT DATA
GRAPHICALLY
1. The average price received by the producer of butter
over a range of six years was as follows : 1913, 28.4 ^ ; 1914,
29.2 ff; 1916,28.7^; 1916, 28.3 jzf ; 1917, 34.0 )?( ; 1918,
43.1 ^. Show .these relations by a bar graph. Also show
the variation in price by a broken line or curve graph.
2. As in problem 1, show in both ways the variation in
the price of eggs through a six-year period from the follow-
ing data : 1913, 26.8 ^ ; 1914, 30.7 ^ ; 1915, 31.6 ^ ; 1916,
80.6 ff; 1917,87.7^; 1918,46.8^.
3. Show by graphs as above the variation in the price of
farm land in Illinois during a four-year period when the
prices per acre ranged as follows : 1915, $ 110 ; 1916, $115 ;
1917, % 120 ; 1918, % 132.
4. The following is the approximate population of the
eight largest cities in the world. Show the relations by a
bar graph: New York, 5,738,000; London, 4,523,000;
Paris, 2,888,000 ; Tokio, 2,186,000 ; Chicago, 2,076,000 ;
Berlin, 2,070,000 ; Vienna, 2,081,000 ; Petrograd, 1,900,000.
5. The following shows the per cent of our working
population in the various occupations : agriculture, 33.2 % ;
mining, 2.5 % ; manufacturing, 27.9 % ; transportation, 6.9 % ;
trade, 9.5%; public service, 1.2%; professional service,
4,4% ; domestic service, 9.9 % ; clerical occupations, 4.5 %.
Show the relations by a bar graph.
204 JUNIOR fflGH SCHOOL MATHEMATICS
6 The average weekly wages of factory workers for a
live-year period were as follows: 1914, $11.89; 1915, $12.69;
1916, $14.55; 1917, $16.66; 1918, $21.01. Show the
variation by a broken line graph.
7. A man with an annual income of $ 3000 used 75 % of
it for living expenses, saved 20 % of it, and gave 5 % of it
to charities. Show the distribution both by a circular graph
and by a shaded bar graph. Which kind do you prefer
and why ?
8. In February, 1919, the price per pound of the best
quality of sirloin steak varied as follows in the different
sections of the country : San Francisco, 32 ^j Seattle, 36^ ;
Denver, 36 i ; Minneapolis, 28 ^ ; Chicago, 37 ^ ; Pittsburgh,
45^; Philadelphia, 49 ^ ; New York, 43^; New Haven, 50 ^ ;
Boston, 56^; and Portland, Me., 57^. Show graphically
the variation in price.
9. In January, 1919, the average sales of the War Savings
Stamps were 45^ for every person in the United States.
The eight states leading in the sales that month were
Vermont, $1.20 per capita; Montana, $1.05; Utah, 94^
North Carolina, 82^; Idaho, 81^; South Dakota, 75^
Oregon, 72^; and Colorado, 71^. Make a graph by which
these can be compared with the average sales in the United
States and with each other.
8. YOU HAVE LEARNED THE USE OF MANY
BUSINESS TERMS AND PROBLEMS
1. Write out a bill showing the amount due on the follow-
ing purchases by Mrs. S. A. Smith of Howe & Co., Detroit,
Mich.: Apr. 3, 5J yd. gingham at 48^; 6f yd. satin at
$2.18; Apr. 12, 2 skirts at 15.85; \ yd. ruffling at 36^;
Apr. 19, 2 J yd. net at 85^; Apr. 16, 1 skirt returned, $5.85;
Apr. 20, 2 waists at $2.98; 3 pr. hose at 79^.
SOME THINGS YOU HAVE LEARNED 205
2. Make out a bill from A. G. Spaulding & Co. (whole-
salers), to E. L. Brown & Co. (retailers), Aug. 3, for:
5 doz. tennis rackets at $19.50 per dozen; 8 doz. tennis
balls at $3.75 per dozen; 6 pr. athletic stockings at $1.15
per pair; 12 pr. tennis shoes at $2.95 per pair; 6 baseman's
mitts at $4.20 each; and 4 Youth's League masks at $1.60
each. Allow discounts of 30% and 10%.
3. Write out an interest-bearing note covering a loan from
E. R. Young to L. E. Barnes amounting to $950, to run
8 months at 6%, dated the day you study this problem.
Find the interest. Who pays it and when ? Who signs the
note and who holds it ? How much will the holder receive
when the note is due?
4. Tell the ways that the payment, when due, might have
been secured. That is, tell the kinds of security that might
have been demanded or offered.
5. Write out a non-interest-bearing note such as E. L. Rice
would be required to give The First National Bank of Topeka,
Kansas, for a loan of $1200 for 90 days at 6%. How much
interest would the ,bank get and when would it get the
interest ?
6. What is interest paid in advance called ? What is the
amount received by Mr. Rice for his $1200 note called ?
7. Write out the form of a check given by A. M. Smoot
on The Merchants' Bank for $12.75 to E. R. Holmes. Show
how to indorse it and who endorses it. Where can Mr.
Holmes get the money ?
8. If you should want to purchase a draft of $18.75 to
send to John Wanamaker & Co. for goods, tell where you
could get it and show the form in which it should be made
out.
206 JUNIOR HIGH SCHOOL MATHEMATICS
9. If you have had experience in sending away for goods,
tell how you transmitted the equivalent of money without
actually mailing the money.
9. TOU HAVE LEARNBD THE mPORTAITT METHODS
OF XNYESTMENT
1. What is meant by " loaning on bond and mortgage " ?
What rate of interest could you get in your community?
Is this a safe kind of investment ? (Discuss fully.)
a. In your community, what yearly interest would f 2600
loaned on bond and mortgage yield? (To answer, you
must know the rate of interest paid.)
3. What is a railroad bond ? How is the bond secured ?
Are railroad bonds safe investments ?
4. How much interest would the holder of a $5000 bond
receive every half year if the rate is 4J%, payable semi-
annually ?
5. What are the Liberty Loan and Victory Loan Bonds
issued by the United States Government ?
6. Find the semiannual interest on a $1500 Fourth
Liberty Loan Bond paying 4 J %.
7. When stock in some corporation is paying an 8 %
dividend, how much will the holder of ten f 100 shares
receive ?
8. If one buys ten $100 shares of stock when quoted at
115, what will they cost without brokerage ? With broker-
age of ^ % of the par value ?
9. What income from his investment will the holder of
the ten shares (problem 8) receive when a 7 % dividend is
declared? Is this more or less than the interest which the
SOME THINGS YOU HAVE LEARNED 207
coBt of the stock, including brokerage, would have earned
at 6 % ? How much ?
10. If a man pays $ 12,000 for a house and rents it for
$90 per month, is he making more or less than he would
have made by loaning the money at 6%, allowing $150
for taxes, $85 for repairs, and $200 for depreciation in
value ?
11. How many shares of stock could you buy for $ 1000
when quoted at 125, no brokerage ? How much would you
get in dividends if an 8 % dividend was declared ?
12. Which would earn you the more money per year,
stock bought under the conditions of problem 11, or a 6 %
" bond and mortgage " for the $ 1000 ? Which would be
the safer investment ?
13. A man has $ 2000 to invest. He can buy 7 % pre-
ferred stock at par or loan his money on a mortgage at 6 ^.
Tell what you would advise him to do, and why.
14. If a man at the age of 25 is able to save $ 300 per
year and continue this saving until he is 60 (35 years), keep-
ing all interest reinvested at 5 %, find by the tables on page
168 how much he will have saved.
15. How much would the saving found in problem 14
earn yearly if loaned at 6 % ?
16. The saving of $ 1 per week ($ 52 per year) for 20
years will amount to how much when placed in a savings
bank paying 4 % ? (Use the tables on page 168.)
17. If you had $5000 to invest, discuss the ways you
could invest it, the probable returns from each, and the
safety of each investment.
208 JUNIOR fflGH SCHOOL MATHEMATICS
10. YOU HAV£ LEARITED TO CHECK YOUR WORK AND
TO KNOW THAT YOUR COMPUTATION IS CORRECT
Before the solution of a problem is of any value, we must
krww that tlie result is correct. This requires that every
computation be carefully checked. The following exercises
may be used as a final test of your skill in computation, or
as drill-work throughout the term, or for both purposes.
Directions
1. Write your name and the date on your exercise paper and
he ready to begin work at a signal.
2. At a signal from your teacher^ begin work on the exercise
assigned.
8. Check each computation until you know that your results
are correct.
4. Then hand in your work and your teacher will record the
time taken.
Note. — You may use these exercises for private drill when working
for greater speed and accuracy. In that case, keep a record of the time
taken for each exercise as you use them from time to time, then by com-
parison you can see what progress you are making.
Exercise 1
(a) Add 34.6, 9.47, 100.38, 96.475, 87.09, 432.8.
(h) From 300.98 subtract 96.09.
((?) Multiply 396.4 by 7.28.
(d) Divide 2634.916 by 73.2.
Exercise 2
(a) Add 500.4, 67.98, 175.9, 80.96, 8.175, 29.64.
(6) From 409.06 subtract 98.78.
(c) Multiply 93.42 by 6.29.
(d) Divide 3936.812 by 6.83.
SOME THINGS YOU HAVE LEARNED 209
Exercise 3
(a) Add 96.308, 207.96, 18.063, 203.9, 98.45, 72.628.
(6) From 720.06 subtract 196.8.
(J) Multiply 576.3 by 92.6.
(d) Divide 4868.916 by 76.7.
Exercise 4
(a) Add 278.16, 74.382, 97.65, 208.75, 96.84, 78.09.
(J) From 603,98 subtract 390.462.
(c) Multiply 82.46 by 37.8.
((T) Divide 3040.704 by 5.76.
Exercise 5
(a) Add 59.086, 403.97, 175.86, 93.42, 80.76, 176.9.
(h) From 480.93 subtract 198.47.
(c) Multiply 936.4 by 27.8.
(dT) Divide 40,483.68 by 87.4.
Exercise 6
(a) Add 4g.2, 806.96, 87.46, 180.95, 72.96, 204.8.
(J) From 601.28 subtract 97.376.
(c) Multiply 809.6 by 87.6.
(d) Divide 46,868.12 by 81.7.
Exercise 7
(a) Add 64.66, 108.38, 96.4, 308.75, 87.246, 30.49.
(6) From 98.026 subtract 49.86.
(c) Multiply 760.98 by 34.8.
(d) Divide 4926.896 by 62.7.
Exercise 8
(a) Add 304.9, 58.47, 89.42, 890.82, 19.43, 109.46.
(J) From 300.4 subtract 68.298.
(c) Multiply 576.8 by 9.37,
(d) Divide 2807.954 by 2.89.
210 JUNIOR HIGH SCHOOL MATHEMATICS
Exercise 9
(a) Add 93.08,46.93, 75.068, 138.9, 65.886, 78.9.
(6) From 560.92 subtract 178.39.
le) Multiply 76.38 by 94.2.
(d) Divide 1991.672 by 3.76.
Exercise 10
(a) Add 40.68, 97.9, 240.75, 19.284, 63.9, 78.46.
(6) From 906.8 subtract 342.96.
(0 Multiply 893.7 by 4.58.
(d) Divide 3665.436 by 5.82.
Exercise 11
(a) Add 34J, 56f, 18|, 16|, 19^, 24^.
(b) From 342^ subtract 196|.
(c?) Multiply 348 by 48|.
Cd) Divide 5384§ by 7.
Exercise 12
(a) Add 28J, 19f , 14J, 43J, 32^, 14^.
(b) From 1508| subtract 509f
(c) Multiply 386 by 39f
((^) Divide 6039f by 8.
Exercise 13
(a) Add 42J, 46f, 18|, 271, 17^, 46^^.
(6) From 3048f subtract 932f .
((?) Multiply 495 by 58f .
(d) Divide 1730^ by 6.
Exercise 14
(a) Add 20|, 16f, 43 J, 62f, 13^^, 42f
(6) From 2061| subtract 973|.
((?) Multiply 387 by 46|.
((f) Divide 2631f by 8.
SOME TfflNGS YOU HAVE LEARNED 211
Exercise 15
(a) Add 34|, 16f, 21|, 29f, 46J, 58|.
(5) From 1706§ subtract 960J.
(c) Multiply 792 by 46f
(d) Divide 3842f by 7.
Exercise 16
(a) Add 43§, 96|, 45^, 16|, 17^6^^, 21^.
(J) From 2096 jTj subtract 906f .
(c) Multiply 384 by 66|.
(<0 Divide 1763| by 5.
Exercise 17
(a) Add 53§, 46J, 18J, 43|, 14^^, 16§.
(6) From 3106J subtract 1940f.
(c) Multiply 664 by 57|.
(<i) Divide 2063f by 8.
Exercise 18
(a) Add 48^, 19§, 48f , 16|, 47^, 54f
(6) From 1960f subtract 1068f
(c) Multiply 387 by 43J.
(<?) Divide 3046| by 8.
Exercise 19
(a) Add 52j, 48f, 16|, 21f, 12J, 16|.
(6) From 1930^ subtract 398|.
(c) Multiply 347 by 64f
(d) Divide 1296f by 7.
Exercise 20
(a) Add 32f, 16f, 34|, 46,^, 17J, 19f
(J) From 2063f subtract 970|.
(c) Multiply 534 by 78|.
(d) Divide 3576f by 7.
TABLES OF MEASURES
Linear Measure
12 inches (in.) = 1 foot (ft.)
3 feet = 1 yard (yd.)
16J feet = 1 rod (rd.)
320 rods = 1 mile (mi.)
1 mile = 1760 yards = 5280 feet
Square Measure
144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 square feet = 1 square yard (sq. yd.)
272J square feet = 1 square rod (sq. rd.)
160 square rods = 1 acre (A.)
1 square mile (sq. mi.) = 640 acres
1 acre = 43,660 square feet
Cubic Measure
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
128 cubic feet = 1 cord (cd.)
Liquid Measure
2 pints (pt.) = 1 quart (qt.)
4 quarts = 1 gallon (gal.)
1 gallon = 231 cubic inches
Dry Measure
2 pints = 1 quart
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)
1 bushel = 2150.42 cubic inches
Avoirdupois Weight
16 ounces (oz.) = 1 pound (lb.)
2000 pounds = 1 ton (T.)
212
INDEX
Above par, 162.
Accounts, 126-132.
cash, 126.
expense, 132.
farm, 132.
household, 128.
ledger, 129.
merchants, 131.
Adding, 2-18.
algebraic expressions, 18.
decimals, 4.
fractions, 4.
special fractions, 5.
whole numbers, 2.
without a pencil, 2, 3.
Addition, 1.
checking, 1.
Ad valorem duty, 188.
Algebraic expressions, 18.
addition of, 18, 19.
division of, 20.
multiplication of, 19.
subtraction of, 19.
Aliquot parts, 11.
dividing by, 15.
multiplying by, 11.
Angle of elevation, 46.
Area, 80.
of a circle, 92.
of a parallelogram, 83.
of a rectangle, 80.
of a trapezoid, 89.
of a triangle, 86.
Axes, 75.
Axiom, 25.
Balance, 126.
Bank discount, 151.
Banking, 145.
Banks, 145, 165.
commercial, 145.
savings, 165.
Bar graph, 51.
Below par, 162.
Beneficiary, 181.
Bills, 126.
Board measure, 93.
Bonds, 158-161.
Borrowing money, 142.
Broken line graph, 51, 68.
Broker, 141.
Brokerage, 141.
Building and loan associations, 170.
Capital, 172.
Cash accounts, 126.
Checking work, 1.
Checks, 148.
Circles, 92.
Circular graphs, 51.
area of, 92.
Circumference, 90.
relation to diameter, 90.
Collateral, 153.
Commercial banks, 145.
Commercial discount, 136.
Commission, 141.
Compound interest, 166.
Compound numbers, 77. .
Cones, 99.
altitude of, 99.
slant height of, 99.
volume of, 99.
Constructions, 82-87.
of a parallelogram, 84.
of a triangle, 87.
Cost, 136, 139.
Coupons, 169.
Credits, 129.
Curve plotting, 68.
Customs, 188
Debits, 129.
Denominate numbers, 77.
213
214
INDEX
Deposit slips, 145.
Diameter, 90, 100.
of a circle, 90.
of a sphere, 100.
Discounts, 134, 136, 137, 161.
bank, 151.
conunercial, 136.
meaning of, 134.
successive, 137.
Dividends, 173.
Division, 13, 14, 15.
by aliquot parts, 15.
by decimals, 13.
, by fractions, 14.
by whole numbers, 13.
Drafts, 150.
Drawing to scale, 43.
Duty, 188.
ad valorem, 188.
specific, 188.
Equations, 24-27.
solved by addition, 27.
solved by division, 26.
solved by multiplication, 27.
solved by subtraction, 25.
Evaluating formulae, 17.
Expense accounts, 132.
Expense loading, 183.
Factoring formulse, 21.
Farm accounts, 132.
Formulse, 17.
evaluating, 17.
factoring, 21.
Functional graphs, 74.
Graphs, 51-74.
bar, 51, 52, 55, 56.
broken line.. 53, 54, 55.
circular, 51.
of functions, 74.
showing component parts, 63.
Gross profits, 139.
Household accounts, 128.
Hypotenuse, 110.
Income tax, 191.
Insurance, 177, 180.
life, 180.
property, 177,
Interest, 142, 166.
compound, 166.
simple, 142.
Internal revenue, 189.
Inventory, 131.
Investments, 156, 158, 170, 171
in bonds, 158.
in mortgages, 156.
in real estate, 171.
in savings banks, 165.
in stocks, 172.
Keeping accounts, 126.
Ledger accounts, 129.
Literal expressions, 18.
Loaning money, 142, 156.
Lumber measure, 93.
Map presentation of facts, 71.
Maps and plans, 43.
Market value, 161.
Measurement of distance, 39.
Measuring, 77.
Measuring heights, 39-47.
by scale drawing, 43.
by shadows, 39.
by tangents, 47.
Members of an equation, 24.
Merchant's account, 131.
Mortgages, 156.
Multiplication, 7, 8, 10, 11, 12.
by aliquot parts, 11.
by decimals, 9.
by fractions, 8.
by special methods, 12.
by whole numbers, 7.
Negotiable, 148.
Normal tax, 192.
Notes, 151, 153.
Par value, 161, 172.
of bonds, 161.
of stock, 172.
Parallel ruler, 85.
Parallelograms, 83,
area of, 83.
construction of, 84.
Pass book, 146.
Per cent, 9.
INDEX
215
Percentage, 113.
Personal insurance, 177.
Plumb level, 88.
Policy, 178.
Post office revenue, 190.
Preferred stock, 175.
Premium, 177, 183.
elements of, 183.
on insurance, 177.
rates of, 178.
Price graph, 74.
Prime cost, 139.
Principal, 142.
Proceeds, 152.
Profit and loss, 139.
Promissory notes, 143.
Property insurance, 177.
Proportion, 32.
Pyramids, 99.
altitude of, 99.
slant height of, 99.
volume of, 99.
Pythagorean Theorem, 110.
Ratio, 31.
Real estate investments, 171.
Rectangles, 80.
Savings banks, 165.
Security, 153.
Selling at a premium, 162.
Simplifying literal expressions, 18-20.
Slant height, 99.
Solving an equation, 25-27.
Sphere, 100.
surface of, 100.
volume of, 101.
Square root, 103, 105.
Squaring a number, 103.
by formula, 103.
by tables, 105.
Stockholder, 173.
Stocks, 172.
common, 175.
preferred, 175.
Subtraction, 5, 7.
Successive discounts, 137,
Surtax, 192.
Table of squares, 109.
Table of tangents, 48.
Tangent relations, 46.
Tariff, 188.
Tax rates, 185.
Taxes, 185.
on income, 191.
on property, 186.
Trapezoids, 89,
Triangles, 86.
areas of, 86.
construction of, 87.
Unit of measure, 77.
Unknown values, 24.
Vertex, 99.
Volume, 95-101.
of cones, 99.
of cylinders, 97.
of prisms, 95.
of pyramids, 99.
of spheres, 101.
Vouchers, 146.
ANSWERS
JUNIOR HIGH SCHOOL MATHEMATICS
Pag« 2
1.
677,662.
8.
699,412.
8.
700,934.
4.
692,710.
5.
638,607.
Page 4
1.
224.328.
8.
173.661.
8.
230.361.
4.
228.462.
6.
232.823.
1.
Fraotloi]
If
2.
2A.
8.
lii.
4.
li.
5.
2^.
6.
IJ.
10.
40}.
11.
187A.
12.
144^.
18.
112).
14.
174iJ.
Page 6
1.
36,783.
2.
62,609.
8.
32,806.
BOOK 11
4.
11,386.
Page 8
9.
f
6.
16,827.
1. 144,876.
10.
*.
6.
33,469.
2. 61,460.8.
11.
».
7.
^7,728.
8. 19,835.2.
12.
h
8.
11,296.
4. 849.75.
18.
Jf
9.
32,784.
5. 24.288.
X4.
J.
10.
3,275.
6. 27,481.8.
15.
f
11.
35,947.
7. 317.768.
16.
i-
12.
27,334.
8. 64,944.4.
17.
H.
18.
26,617.
9. 38,467.8.
18.
f
14.
11,804.
10. 3,396.86.
19.
*.
15.
4,577.
11. 2}.
20.
i.
16.
18.963.
12. 3}.
21.
64.676.
17.
38.82.
18. 4}.
22.
56.236.
18.
22.646.
14. 4}.
28.
227.696.
19.
22.296.
15. 7}.
24.
47.025.
20.
1.886.
16. 1}.
25.
783.68.
21.
18.666.
17. 7f
26.
1736.19.
22.
22.286.
18. 7}.
27.
2732.29.
28.
3.196.
19. 7}.
28.
428.682.
24.
27.316.
20. 10}.
29.
363.976.
25.
8.476."
21. 7}.
80.
628.56.
28.
20|.
tf
29.
26§.
Page 9
Page 10
80.
13}.
1. A.
21.
127.68.
81.
8i.
2. A.
22.
13.662.
82.
30}.
8. If
28.
67.76.
88.
67f
4. A.
24.
88.07.
84.
67}.
5. A.
25.
166.6.
86.
42}.
6. }.
26.
106.6.
86.
136}.
7. }.
27.
2.8586.
87.
72}.
•. A.
28.
15.6612.
ANSWERS
8.
«.
6.
6.
29. 9.0615.
80. 212.
81. 48.112.
82. 287.04.
88. 816.
84. 608.4.
85. 986.1.
86. 8116.
87. 69,939.
88. 84,387.6.
Page 11
1. 2,400.
2. 12,000.
3,500.
48,000.
12,000.
72,000.
7. 48,000.
8. 6.3,000.
9. 4,500.
10. 6,400.
11. 7,800.
12. 8,400.
18. 12,800.
14. 63,000.
15. 54,000.
16. 8,000.
17. 6,000.
18. 9,000.
19. 16,000.
20. 10,800.
22. 612,000.
28. 2,268,000.
24. 1,344,000.
25. 292,400.
26. 2,450,000.
27. 2,726,400.
28. 3,444,000.
29. 2,886,000.
80. 1,762,600.
Page 12
1. 4,280.
2. 2,340.
8. 2,430.
4. 8,750.
6. 3,350.
6. 5,600.
7. 43,250.
8. 23,300.
9. 10,800.
10. 87,660.
11. 21,875.
12. 11,660.
18. 8,266f.
14. 12,750.
15. 7,466}.
16. 19,200.
17. 10,760.
18. 6858^.
Speolal Per
Cents
1. 42.
2. 30.
8. 12.
4. 210.
5. 21.
6. 6.
7. 32.
8. 21.
9. 30.
10. 40.
11. 70.
12. 375.
Page 13
1. 135,408.
2. 237,800.
8. 368,064.
4. 284,172.
5. 457,866.
6. 336,336.
7. 379,134.
8. 283.392.
9. 530,712.
10. 196,664.
11. 272,808.
12. 703,248.
18. 896,806.
14. 378,432.
15. 275,416.
16. 300,330.
17. 140,420.
18. 240,798.
• 13-14
1. 296.4166.
2. 678.8809.
8. 934.8865.
4. 12,054.064.
5. 1,172.8681.
6. 35.6081.
7. 54.7159.
8. 42.4416.
9. 20.4489.
10. 8.3811.
11. 2.4667.
12. 63.1746.
18. 173.6784.
14. 347.5676.
15. 166.8671.
16. A.
n. «.
18. 2^.
19. lA.
20. li.
21. 2,^.
22. If
28. lA.
24. lA.
26. 92A*
27. 82^.
28. 807H«
29. 185}.
80. 92}.
81. 74}}.
82. 80^.
88. 116^.
84. 136^.
85. 66J}.
M. 71^-
87. 96}.
Page 15
1. 23.425.
2. 22.468.
S. l.vuv.
4. .9781.
5. .941.
6. 4.8485.
7. 1.753.
8. 2.461725.
9. 1.801.
10. 1.8888.
11. 1.1081.
12. .2138.
18. .2477.
14. .1765.
15. .14220.
16. .1158.
17. .0921.
18. .01936.
19. .028916.
20. .01185.
21. .aS24.
22. .005326.
28. .008.
24. .008.
Divldtag t>7
Aliquot Parts
1. 158.84.
2. 50.91.
8. 49^.
4. ld(MK.
ANSWERS
5.
86.26.
6.
78.72. .
7.
171.6.
8.
147.68.
9.
60.04.
10.
173.2.
11.
69.78.
12.
81.84.
13.
.2696.
14.
9.66.
16.
18.264.
16.
1.8008.
17.
2.2548.
18.
4.984.
19.
4.328.
90.
7.712.
21.
3.7072.
Page 1
1.
142.
2.
260.
8.
639.
4.
152.
5.
126.
6,
252.
7.
480.
8.
25.
9.
576.
10.
25.
U.
22}.
12.
35.
18.
504.
14.
516.
16.
612.
MiiceUaneoiu
DriU
1. 10,800.
2. 56,000.
8. 72,000.
4. 4,210.
6. 2,420.
6. 2.450.
7. 21,000.
8. 17,400.
9. 4,600.
10. 18,800.
11. 7,600.
12. 4,666}.
18. 104.
14. 142.
15. 112.
16. 133.
17. 154.
18. 143.
19. 125.
20. 179.
21. 127.
22. 105.
28. 142.
24. 137. .
26. 48.
26. 18.
27. 18.
28. 37f
29. 9J.
80. 3f
81. 8.
82. 19|.
88. 27.
84. 12.
86. 10|.
86. 28.
Pages 17-18
1. 30.
2. 66,
8. 452.3904 sq.
ft.
4. 942.48 cu. in.
6. 942.48 en. in.
6. c = IT (2.
Pages '18-19
1. 21a.
2. 20 c.
8. 15 m.
4. 2Sx.
.6. 17 y.
7. 2480.
8. 7850.
9. 81.416.
10. 17,920.
11. 11,300.
12. 10,500.
18. 18,600.
14. 3,750.
16. 726.
16. 987.
Page 19
1. 7 a.
2. 5 6.
8. 8x.
4. 7y.
6. 5 c.
7. 2,750.
8. 1,600.
9. 1,920.
10. 2,250.
11. 9,750.
12. 9,820.
18. 3,200.
14. 4,100.
16. 2,160.
16. 3,800.
Pages 20-21
Multiplication
1. 12 62.
2. 18 a2.
8. 21 y2.
4. 36 c2.
6. 32 &2.
6. 56 c2.
7. 63 (P.
8. 40 r2.
9. SOaK
10. 40 aS.
11. 80 68.
12. 80 c».
18. 80 6c ; 18 r8 ;
dOat;42mn.
14. 35 a6.
16. 42 6c.
16. 54 de.
17. 56 oc.
18. 42 6(2.
19. 54 0(2.
20. 56 (2c. ,
21. 48 c6.
22. 72 eg.
Division
1. 2 a.
2. 3 c.
8. 8 a.
4. 3(2.
6. Sy.
6. 9 6.
7. 8 c.
8. 8 c.
9. 8c2.
10. 7cP.
11. 7a6.
12. 4 a.
18. 9(2.
14. 9z.
16. 8 6.
17. 4.
18. 4.
19. 4.
20. 6.
21. 6.
82. 9.
28. 7.
24. 8.
ANSWERS
85. 9.
M. 9.
27. 7.
8.
Page 21
1. 3(o+6).
2. 7(6 + c).
8. 6(c + <0.
4. 8(a + c).
6. 7(a + (0-
6. 10(x + y).
7. 9(a+6).
8. 2(a« + 6').
9. 3(a6 + cd).
10. 5(a6»+c»).
U. 6(a6+d).
12. 4(2y + a&).
18. 700.
14. 900.
15. 800.
16. 400.
17. 450.
18. 700.
Pages 22-23
2. d = -.
X
A
4. 47.74 ft.
5. 30 sq. in. ;
20 m.
c
2x"
7. 2A =
A(6 + 61).
8. 6 = ?^;
A
h =
2A
9.
la
6. r =
8 m.
11- * = ir4:,
6 + 61
12. 6 in.
18. 92.3 sq. ft.
14. 5.709 ft.
15. 7.503 ft.
16. 210.08 ft.
17. 8.53 in.
18. 68 ft.
19. 16 in.
Page 25
1. 6.
2. 4.
8. 15.
4. 11.
5. 5.
6. 5.
7. 6.
8. 9.
9. 18.
10. 25.
11. 7.
12. 14.
18. 8.
14. 9.
15. 17.
Page 26
1. 5.
2. 9.
8. 26.
4. 7.
5. 9.
6. 7.2.
7. 7.25.
8. 1}.
9. 4.83.
10. 5.72.
11. 5.
IS. 6|.
18. 14.75.
14. 6.4.
15. 8.7.
Page 27
Di^aton
1. 5.
2. 6.
8. 4.
4. 6.
5. 7.
6. 6.
7. 6.
8. 5.
9. 2.
10. 7. .
11. 7.
12. 7.
18. 8.
14. 7.
15. 3.
Additioii
1. 11.
2. 22.
8. 5.
4. 11.
5. 16.
6. 173.
7. 180.
8. 22.6.
9. 13.1.
10. 3.9.
11. 5.
12. 7.
18. 5.
14. 60.
15. 40.
1. 16.
2. 27.
8. 35.
4. 48.
5. 54.
6. 7.
7. 9.6.
8. 20.
9. 23.8.'
10. 10.8.
11. 71.
12. 2}.
18. 5t.
14. ^.
15. 6.
Mlacellaneomi
1. 4.5.
2. 17.
8. 11.
4. 6.
5. 4|.
6. 12.
7. 4}.
8. 9.
9. 6.5.
10. 9.
11. 5.4.
12. 2.2.
18. 7^.
14. 7).
15. 9.
16. 6}.
17. 8.
Pages 2d-30
2. 6in. ; 10 in.
8. 14 boys ;
18 girls.
4. 50 ft. ; 70 ft,
5. 16 ft. ; 20 ft.
ANSWERS
6. 68 ; 60.
7. 40; 41; 42.
8. 27 ; 20.
0. 48 ; 60.
10. 43 ; 66.
11. 30^ ; 60^.
12. 16 rd.; 30 rd.
18. 14 ; 24.
14. 80 ft.; 120 ft.
15. $6; $10.
16. 20.
17. 2^ ; 4^.
Pages 31-32
6. 1}.
6. If
7. 6.
8. H.
9. f
10. 4.
11. 2.2688+.
12. .804+.
18. .6868+.
14. .674+.
15. }.
le a-
17. 00ft.;63Jft.
18. 1^; 7 m.
20. f
Pages 34-36
2. |.
ft ^
c
7. —
aft
a'b*
9. f.
10. -777-
a' 6'
12. f
18. —
a
^ •
c
14. IJ.
15. ~
19. 4 times.
20. times.
21. 6.
22. If
28. 4.
24. ^ as great.
25. $10.
8
Pages 37-39
8.
80°.
10.
20 ft.
Pages 39-43
1.
12i ft.
2.
36 ft.
8.
36 ft.
4.
66 ft.
6.
AB DC
AE EC
7.
300 ft.
8.
121 ft.
9.
380 ft. .
10.
100 ft.
11.
27 J ft.
18.
266} ft.
Pages 44-45
2. 366} mi.
8. 866 mi.
4. 10' Q'f by 0'
6";
8' 6" by 0' ;
18' 6" by 6'.
6. 1 in. = 4 ft.
7. 2" by 6",
sr 3" by 4".
9. 4" by 7".
10. 672 ft.
11. 120 ft.
12. 1 in. = 20 ft.
18. 4O'by8'0";
8'li" by 6'
3";
40' by 27' 6".
Page 47
4. 60.6 ft.
5. 82.6 ft.
6. 60.6 ft.
Pages 48-50
1. 61.44 ft.
2. 688 ft.
8. 1027.6 ft.
4. 663.10 ft.
5. 671.42 ft.
7. 31°. '
8. 3300 ft.
9. 66°, 24°.
10. 16.277 ft.
11. 6406.4 ft.
12. 140.74 ft.
Pages 52-59
2.
2.2
4.
6.
30 9(>
10 9^;
66 9b.
Hart
times.
About
less.
About
329b;
7. About 83 9fc;
118 9^0.
About 2 <fo ;
229b; 609b.
About 46 9b ;
86 9^; 106 9b.
8
9.
10. About 139b;
16 9^ ; 26 9b.
11. About
64.88 9^0.
12. About 629b.
18. About 69b
increase.
14. About 18 9b
increase.
17. 160 -14 9b of
Gt. Br.;
204.33 9b of
France ;
336.28 9^ of
Italy.
18. 46.67 9fc ;
143.11 9^.
19. 141.86 9^;
143} 9^;
60} 9b.
20. 209b.
21. 40.18 9b.
22. About 46 9b ;
48 9^.
25. 70.749b.
28. 18.42 9b ;
10.81 9b ;
8.479b.
28. Cattle 66
millions ;
Sheep 62
millions ;
Hogs 62 mil-
lions.
29. Australia
87.01 <j^ ;
Grermany
11} 9^.
80. Germany
48.14 9b ;
Australia
.8 9^.
ANSWERS
81. About
82.04^;
60.69 ^ ;
36.47 9b.
82. 22}^;
77J^.
38. Cattle
39.47 ^ ;
Sheep 42. 19(7;
Hogs 18.42 9^.
84. a. — 41}9b;
40.47 9b ;
I7f9b;
6. 48.66 9b
37.96 9b
13.38 9b
c. 122.329b
of France ;
d. 142f 9b of
France.
Pages 60-63
1. About 2J;
2f;2}.
5. 160 9b;
188| 9b ;
166J9b.
8. 268.28 9b
299.28 9b
277.11 9b
4. 1.96 in.; 4.48
in.; 4.19 in.
Pages 64-67
1. 61° 80';
275° 30'.
2. Food 9460;
Rent $300;
O.Ex.928S^;
Clothing
9260;
Misc. $216}.
8. 2481 miinons;
2337.6 mil-
lions. '
6. Beef 61.889b;
609b (graph).
6. a. Debt6.99b
of wealth;
h. Debt of
1918
403.16 9b
of Debt;
c. 15.03 9b of
Gt. Br ;
20.71 9b of
France ;
91.79 9b of
Italy.
d. 112.86 9b
ofGt. Br.;
142.7 9b of
France ;
388. 62 9b
of Italy;
e. Debt
44.319b
of wealth;
/. Debt
40.679b of
wealth;
g. Debt
24.99 9b
of wealth.
Pages 68-70
1. 1880 — about
4.28 times
1860;
1890 -6
times 1840;
1880 - 44.11
9b of 1910.
5. a. 1009b ;
h. 109b ;
c. 119b;
d. 309b;
e. 109b.
Pages 77-80
2. 67 qt.
8. 269 pk.
4. 56 ft.
5. 948 min.
8. 2296 860.
7. 26 pt.
8. 2016 sq. rd.
9. 4676 rd.
10. 10,030 yd.
11. 267 oz.
12. 804 sq. in.
18. 161 cu. ft.
15. 4gal. 2qt.
16. 62 qt. 1 pt.
17. 16 ft. 4 in.
18 9 yd. 16 in.
19. 2 A. 106 sq.
rd.
80. 20 lb. 4 oz.
21. 43 bu. 3 pk.
22. 61 yd. 2 ft.
28. 6 hr. 42 min.
25. 9 in.
26. 2 ft.
27. 3 qt. 1 pt.
28. 10 in.
29. 14 oz.
80. 116 sq. rd.
81. 3 pk. 6 qt.
82. 1 ft. 8 in.
88. 16 min.
84. 100 sq. rd.
85. 6261b.
86. 56 sec.
87. 10 in.
88. 12 oz.
89. 3qt
40. 8 qt. 1 pt.
42. .208 hr.
48. J yd.
44. }gal.
45. ^ mi.
46. .409 mi.
47. .6876 bu.
48. .408 da.
49. .611yd.
50. .6667 sq. ft.
51. 9 in.
52. 18 in.
58. 22 posts.
54. 8; 21.
Pages 80-81
2. A=Iw.
• T A. A
8. I=— ;w=^.
w I
5. $86.40.
6. $36.
7. 16,930 sq. ft.
8. 2312 sq. ft;
880 sq. ft.
9. 63.71%.
8. 2661b.
4. 931b.
Page 86
8. 48 sq.in.
6. 480 sq.in.
Pages 87-88
12. 60°.
18. 50°.
14. 30°; 60^.
8. 27 J sq. in.
4. UJA.
5. 44^ A.
ANSWERS
Fa8«« 90-92
2. 87.0992 ft.
3. 68.661 ft.
4. 106 ft
6. 8.8776 ft. '
6. .6286 ft.
7. A; .6286 ft.
8. Add A of it.
9. 81,416 in.
10. 18.8406 ft.
11. 47.124 ft.
12. 4.7746 ft.
2,8878 ft.
18. 1.5916 in.
14. 1.9099 in.
Pages 92-93
2. 452.3904 sq.
ft.
3. 113.0976 sq.
ft.
4. 892.7 sq. ft.
6. f as laige.
6. 4 times; 1}
times.
Pages 93-95
1. 14 bd. ft. 14
boards.
2. 6 bd. ft.
8. 18bd. ft.
4. 86bd.ft.
5. 21}bd. ft.
6. $20.16.
7. $80.88.
8. 2^ in.
9. }.
10. 400bd. ft.;
OOObd.ft.;
2000 bd. ft
11. $36.
Pages 95-97
1. 5 ft by 3 ft
by 2 ft
8. 30 oubea.
3. 15 cubes ; 80
cubes.
4. 60 cu. in.
b. V=z abc,
7. 480 cu. ft;
13.71 T.
8. 1 sq. in.;
1 sq. ft.
9. 15 sq. units.
10. 20ou. ft
11. 120CU. ft
12. V=Bh,
14. 249.33 gal.
16. 576 bu.
16. 675 cu. ft.
17. 240,000 ou.ft
Pages 97-98
1. 6.8644 gal.
2. 31.9872 gal.
8. 423.0144 gal.
4. 1884.96 cu.
in.
5. 783.36 gal.
6. 112.5949ton8.
Page 98
8. 47.124 sq. ft
4. 43.9824 sq. ft
Pages 99-100
2. 60 cu. ft
8. 282.744 cu. ft
4. 32 cu. ft
5. 34.906 loads.
6. 63.6166 bu.
7. 180.96616 bu.
8. 15.708 bu.
Pages 100-102
8. 314.16 sq. in.;
804.2496 aq.
ft.
4. 201,062,400
sq. mi.
6. 528.6 ou.ft
7. 2144.6666 cu.
in.
8. 523.6 cu. in.
9. 82.07061b.
10. 1.9684 gal.
11. 3.8083 tons.
12. 66.54081b.
18. i as laige.
14. ^ as large.
16. 64 times.
16. 2.37 times.
17. 126 times.
Page 104
1. 3969.
2. 5184.
8. 7226.
4. 2209.
5. 1444.
6. 9216.
7. 3249.
8. 1226.
9. 8649.
10. 7056.
11. 1849.
12. 8281.
13. 6776.
14. 2809.
15. 7569.
16. 7921.
Pages 105-106
1. 28.
2. 68.
8. 92.
4. 56.
5. 83.
6. 52.
7. 78.
8. 67.
9. 99.
10. 87.
11. 54.
12. 97.
14. 532.
15. 647.
16. 636.
17. 746.
18. 869.
19. 2453.
20. 728.
21. 696.
22. 799.
28. 852.
24. 643.
25. 1319.
26. .26; .1226;
.060026.
29. .75.
80. .96.
31. 6.498+.
32. .885-.
38. .943+.
34. 4.412+.
36. 28.721.
86. .o.
37. .263'.
38. 43.959-.
39. 16.03.
40. .894+.
41. 1.414+.
42. 1.732+^..
43. 2.236+..
44. 2.646+..
45. 3.16+..
46. 4.24+.
8
ANSWERS
47. 4.898+.
48. 6.246-.
Page 107
By First Method
1.
2.
8.
4.
5.
6.
7.
8.
9.
10.
11.
12.
18.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
67.99.
76.03.
93.62.
43.87.
38.44.
68.84.
30.62.
26.86.
26.16.
28.66.
27.16.
31.17.
8.73.
9.197.
6.83.
8.69.
6.93.
6.36.
.62.
.76.
.97.
.92.
.87.
.71.
Page 109-
1. 73 sq. in.
2. 134 in. by
67 in.
3. 23.44 ft.
4. 17.32 ft.
6. 22.627 ft.
6. 26.98 in.
7. 60.39 ft.
8. 81.06 ft.
Pages 111-112
1. 60 ft.
8. 68 ft.
8. 67 ft.
4. 116 ft.
5. 42.426 ft.
6. 26.61 in.
7. 21.63 mi.
8. 223.6 rd.
9. 19 ft.
(18.97 ft.)
10. 33} yd.
11. 127.27 ft.
12. 207.386 ft.
18. .3636 in.
14. 11.66 in.
16. 13.76 in.
16. 68.22 sq.in.
17. 173.206
sq.in.
Page 114
1. .46.
2. -63.
8. .07.
4. .09.
5. 1.38
6. .16.
7. .166.
8. .04.
9. .046.
10. 2.4.
11. .164.
12. .036.
18. .008.
14. 1.254.
16. 2.466.
16. .046.
17. .0626.
18. .0876.
19. .096.
20.
2.
68.
187.6%.
21.
166.
68.
166.62 %.
22.
37.48.
Pages 115-U.7
28.
48.64.
84.
4.894.
l:
$1.64;
26.
167.7.
$33.20.
26.
82.176.
2.
16^members.
27.
1226.
8.
$306.
28.
1660.
4.
Smaller.
29.
3220.
6.
14 ; 22.
80.
1198.76.
6.
60%;37J9^.
81.
138.6.
7.
FromlJ
82.
34.2.
times to twice
88.
18.72.
as much.
84.
132.3.
8.
144%.
86.
8.82.
9.
309.66%.
86.
96.6.
10.
272.94 %.
87.
210.
11.
Exports
88.
1630.
64.68 % ;
89.
36^.
"47.86 % ;
40.
48 9{j.
Imports
41.
9%.
28.72%;
42.
9.6%.
1
19.98 %.
48.
73.6%.
12.
Cheese,
44.
86.4%.
a. '17-'18-
46.
2.6%.
40%;'
46.
26.8%.
6. '14-'17-
47.
136%.
76.66 % ;
48.
248%.
Currants,
49.
290%.
c. 1918-80%;
60.
320 %.
d, 1916-
61.
1.6%.
21.88 % ;
62.
.8%.
Dates, c. 'lg-
68.
.85%.
82.36%.
64.
12.25%.
Pages 118-119
66.
66.47%.
66.
64.19%.
8.
$9.60.
67.
44.24 %.
4.
$8.
68.
118.18%.
5.
$7.60.
69.
207.1 %.
6.
64Jbu.
60.
79.72%.
7.
46j?7 bu.
61.
200.89 %.
8.
41tV bu.
ANSWERS
y
9. 110011).;
4.
160.6 9fc
Pages 124-126
Pages 130-134
1608J}lb.;
more.
1.
$4.25.
17.
$32.10.
1036J lb.
6.
$61.36;
2.
$ 16.26.
18.
$295.
10. Larger 43.6/
$163.
3.
$3.86.
19.
$.60.
less.
6.
Wheat
5.
$ 69.44.
20.
$2.85.
11. 2401b.
113.73%;
Pages 127-128
21.
$ 165.20.
12. 2401b.
dressed.
Com
147.61 % ;
1.
$ 20.10 ;
$24.50;
$ 29.65.
$4.30; $4.30.
$6.96; $7.76.
22.
$117.70.
13. l^qt.
Barley
Pages 134-135
14. $30.
15. $16.60.
93.09 <fo ;
Rye
113.66 <fo ;
2.
1.
2.
3.
$28.
$17.60.
10%.
Page 120
Potatoes
Pages 128-129
4.
12%.
1. 96.
79.04 % ;
1.
$79.76.
5.
16J%.
2. 96.
Cotton
2.
$ 32.46.
6.
$22.60.
5. $7.80.
92.86%.
3.
$86.50.
7.
$2126.
4. 7.80.
7.
12.09%.
4.
a. Food
8.
16} %.
6. 22.60.
8.
40.80%.
$8.98;
9.
$315; $120;
6. $60.
9.
2.26 times its
Clothing
$ 148.75.
7. $8.40.
former pro-
.
$11.10;
10.
14f %; 20%;
8. $12.60.
duction.
0. Ex-
34%.
9. 21b.
10.
3.26 times.
penses
12.
Divide by .75.
10. 3601b.
11.
$1,631,260.
$7.90;
13.
$234.60.
12.
$2,366,260.
Higher
14.
$ 94.60.
Pages 120-123
13.
423,400,000
T.ife
1. Food $875;
bu.
$3.26;
Page 136
Rent $ 700 ;
14.
186,200,000
Health
1.
$2.63.
Clothing
bu.
$2.60;
2.
14%.
$770;
15.
287,616,666}
6. Mon.
3.
$26.95.
0. Expenses
lbs.
$9.78;
4.
$ 34.24.
$626;
16.
62.44 %.
Tues.
5.
$ 48.90.
S. C. fl.nd R.
17.
2.79 %.
$2.98 ;
6.
$ 73.63.
$630.
18.
4,206,737.7
Wed.
7.
$ 38.56.
3. Food $626;
A,
$6.88;
8.
$87.12.
Rent $300;
19.
Multiply by
Thurs.
9.
$99.75.
Clothing
1.84; Divide
$.74/ ;
10.
$ 60.58.
$270;
by 1.84.
Fri. $.63 ;
11.
$67.92.
0. Expenses
20.
$37.60;
Sat.
12.
$56-
$226;
14j%.
$11.76;
($64,995).
S. C. and R.
21.
30/.
Sun.
13.
20%.
$180.
22.
Dressed ^/.
$ 1.07 ;
14.
10%.
3. 131J%.
23.
$26.47%.
c. $33.74.
15.
20%.
47
4.80
48
B.24.
P«ge
By First
1
67.99.
ANSWERS
11
Page 160
1. Alternating
.|2.12 and
12.13.
2. $10.62-10.63;
$212.50;
$ 1062.50.
3. $225; $450.
4. 20 bonds; 42.
6. $458,276,650.
Municipal
Bonds
1. $75 per year.
2. $4500.
8. $45,000.
Page 162
1. 6.32^.
2. 4.519^0.
8. 4.796%.
4. 4.657%.
5. 6.185%.
Pages 163-164
2. $4900.
8. $7160; $360.
4. $5887.50.
5. $4893.75.
6. $5317.50,
7. $6720.
8. $9146.25.
9. $6296.88.
10. $4893.75.
11. $7565.63.
12. $7596.88.
18. $8324.38.
15. $6056.25.
Page 165
1. Below par.
2, More than
par.
8. No. Earns
4.90%.
4. Yes. Earns
5.49 %.
Page 166
1. $1; $51;
$1.02.
2. $597.55.
8. $994.90.
4. $1440.72.
5. $596.94.
6. $370.05.
7. $675.34.
8. $1391.16.
9. $2116.90.
10. $451.53.
11. $1110.15.
12. $1402.20.
Pages 169-170
1. $563.30;
$1248.60;
$2082.50;
$3096.90.
2. $4994.40.
8. $8780.70.
4. $777.62.
6. $2967.18.
6. $12,373.11.
7. $8206.26.
8. $10,267.83.
9. $17,803.05.
10. $82,487.40.
11. $82,062.60.
Pages 170-171
1. 30 shares;
$16.
2. $12.50;
$27.50.
4. $42.60.
6. Between 6
and 7 years.
Pages 171-172
1. $85 ($84.88).
2. Rent $91.60
more.
8. House $190
more.
Page 173
1. 5000 shares;
2. ^-y $800.
8. 2000 shares;
4. 15
6. $160.
6. $100.
7. $60,000; $90.
8. 12%.
Page 175
1. $2145;
$2147.60.
2. $3600.
8. $186.
4. 7.46%.
6. 6|%.
Page 176
1. $280.
2. $260.
8. 6.53%.
4. 6.74%.
5. $14,000; 6%;
preferred.
6. Preferred 6%
better.
Pages 178-180
6. 24%.
7. 1.62%.
8. $67.60.
9. $60.
10. Cottage $61;
Home $18.
11. $1600;
$2400.
12. $23.04;
$9600;
$6000;
$500.
18. f.
14. $9600.
15. $8000.
16. f
17. 52.87%;
37.61 %.
18. .42%.
Pages 181-182
4. $100.70;
$ 154.70.
6. $131.75;
$181.10;
$249.26.
6. $6141.26.
$7396.76.
$10,986.76.
7. $6122.16.
8. $2923.68
more.
9. $7146.95
more.
10. Insurance
$ 12,746.62
• less.
11. Insurance
$2,768.85
more.
Pages 183-184
1. $2353.30.
2. $66.73 less.
8. $97.82 less.
. I
12
ANSWERS
4. S. V. 1576
less.
6. Ins. 9835.64
less.
PagM 186-188
1. 15 miles ;
$1.60; li9fc.
8. $48.
8. $80.75.
4. $841.67.
6. $8100.
6. $96.20.
7. 1.19^.
8. $.75.
9. $1.75.
10. $1.47.
11. 85.73^.
12. 51.02^.
18. 51.96/.
14. 49.63j^.
16. $1,039.
16. $56.25.
17. $92.63.
18. $173.25.
19. $148.50.
90. $83.06.
81. $198.
88. $310.63.
88. $330.27.
Pages 188-189
1. $2.55.
8. $4132.
8. $4500.
4. $1575.
6. $10,650.
6. $10,490,-
973.60.
7. $18,740,846.
8. $121,012,-
301.36.
Page 190
1. $182,188,-
527.
8. $29,560,-
732.68.
8. $43,185,-
703.18.
4. $54,240,-
670.64.
Pages 190-191
1. 9.97^,
2. 34.87 9(>.
8. 25.67 mi.
8. 16.39^.
5. $2,112,-
115.60.
6. $12,960,043.
7. $47.12.
8. 18.739fc.
Page 194
1. $370; $250.
8. $950; $734.
8. $482.
Page 195
1. 96.
8. 96.
8. 125.664.
4. 1809.5616.
5. 3600.
6. 480.
7. 300.
8. 1696.464.
9. 804.2496.
10. 523.6.
11. 88.
18. 297.8321.
14. 2.23A.
15. 70.
16. 484.
17. 71.
18. 300.
Page 197
1. 4.
8. 5.
8. 5.
4. 8.
5. 17.
6. 5.
7. 6.
8. 4.
9. 3.
10. 5.
11. 6.
18. 10
18. 18.
14. 12.
16. 20.
17. 4.
18. 8.
19. 6.
80. 5.
81. 10.
88. 10.
88. 12.
84. 16.
86. 4.
86. 10.
87. 25.
1. 15.
Pages 198-199
8. 12 rd.
8. lljin.
4. 24 sq. in.
6. 5 ft.
6. Oft.
7. 490.
8. 480.
9. 48 pigeons.
10. $24.
11. $200.
18. 14 girls ;
21 boys.
18. 12 girls;
24 boys.
14. Ralph 60;
Donald SO.
16. 30 ft. ; 90 ft.
16. Robert 10 ;
James 24.
17. Ralph 64;
sister 32.
18. 1.5 qt. cream;
3 qt. milk.
19. James 40 ;
Frank 80;
Ralph 120.
80. John 35;
Ralph 30.
81. Donald $30 ;
Ralph $45.
Pages 199-200
1. 6 in. by 8 in.;
40 ft.
8. 81 J ft; 91 J ft.
8. About 585 ft.
4. About 540 ft.;
610 ft.
6. 138.4 ft.
6. About SO**.
7. 16,666fft.
8. 740.74 ft.
Page 201
8. 150 ft.
8. 35 ft.
4. 221ft.
6. 1,500 ft.
ANSWERS
13
Pag0s 202-203
1. 119 ft.
2. 32.1 ft; 481.5
Bq. ft.
8. 60.4 ft.
4. 3460 ft.
6. 37°.
k 50°; 40°.
7. 17,021.276 ft.
8. 571fft.
9. 77.35 ft.
Pages 204-205
X. 9 3o.I7f7.
8. $126.88.
8. V 38 ; Xi. it/.
Barnes ;
$988.
Pages 206-207
4. $112.60.
6. $31.87-
$ ol.bo.
7. $80.
8. $1160;
$ 1151.26.
9. $70; 92|^
more.
10. $ 76 less.
11. 8 shares; $64.
18. Stock $4
more ; Bond
and mort-
gage.
18. Stock earns
more but
mortgage
safer.
14. $28,450.80.
16. $1707.048.
16. $1610.39.
Pages 208-211
Exercise 1
o. 760.816.
6. 204.89.
c. 2886.792.
d. 34.68.
Exercise 2
a. 863.066.
b. 310.28.
c. 587.6118.
d. 576.4.
Exercise 3
a. 697.294.
b. 623.26.
c. 53,365.38.
d. 63.48.
Exercise 4
a. 833.872.
b. 213.518.
c. 3116.988.
d. 627.9.
Exercise 5
a. 988.996.
b. 282.46.
c. 26,031.92.
d. 463.2.
Exercise 6
a. 860.31.
6. 663.906.
c. 70,840.
d. 673.6.
Exercise 7
a. 685.916.
6. 48.666.
c. 26,482.104.
d. 93.48.
Exercise 8
a. 972.6.
6. 232.107.
c. 5404.616.
d. 798.6.
Exercise 9
a. 493.263.
b. 382.63.
c. 7194.996.
d. 629.7.
Exercise 10
a. 540.974.
6. 563.84.
c. 4093.146.
d. 629.8.
Exercise 11
a. 170f.
6. 146^.
c. 16,966.
d. 7693^r.
Exercise 12
a. 152|.
6. 998H.
c. 16,362f.
d. 754}i.
Exercise 13
a. 199A--
6. 2116iJ.
c. 29,040.
d. 288,25.
Exercise 14
a. 199i}.
6. 1087ii.
c. 18,043f
d. 328}}.
Exercise 15
a. 208,'';^.
6. 765i}.
c. 36,907i.
d. 648}f.
Exercise 16
a. 241}^.
&. 1189^.
c. 21,811^.
d. 362}}.
Exercise 17
a. 193J.
6. 1165^.
c. 32,524.
d. 267H.
Exercise 18
a. 234,^.
6. 89lij.
c. 16,979}.
d. 380 J.
Exercise 19
a. 168^|.
&. 1631^.
c. 22,406J.
d. 185^.
Exercise 20
a. 167f
6. 1092i}.
c. 42,008.
d. 610j}.
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