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IN MEMORIAM
FLORIAN CAJORl
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PREFACE.
n->n'h'rn^lci
However forcibly an author may be impressed with
the conviction that his work constitutes an important
improvement on all similar efforts which have preceded
it, he is still aware that its favorable reception by the
public depends upon the recognition of its merits by
other minds than his own.
In regard to this book, if the improvements which
I have attempted to incorporate into it are not readily
recognized by the experienced teacher as he peruses it,
they are certainly of so little value as not to be worth
pointing out. I have, then, only to suggest to the
reader to turn to the subjects, say of Ratio and Pro-
portion, and critically peruse the articles as they occur,
including the examples in the application of principles
and the references. If I mistake not, a few pages will
reveal to him many of the iinportant features which
distinguish this work; and, if an experience of nea.rly
thirty years in the school-room justifies me in express-
ing the opinion, he will be surprised that they have not
been hitherto developed.
q-v
PREFACE.
There is only one point to which I will expressly
refer. It will be seen that decimal fractions are the
orflfepring of decimal notation, and not of ^^vidgar"
fractions, and that their notation is early introduced for
the sake of scientific accuracy, as well as the early in-
sertion of problems involving United States currency;
for if there is any concrete quantity which the Amer-
ican child readily understands, it is that involving
dollars and cents.
P. A. TOWNE.
Mobile, Ala., January^ 1866.
0 r^h),
CONTENTS
a
PAGE
Definitions 9
Notation 11
Arabic Notation 11
Numeration 17
Decimals 20
Notation and Numeration 20
Principles of Arabic Notation 25
United States Money — Notation and Numeration 26
Roman Notation and Numeration 30
Addition 31
Subtraction 45
Multiplication 55
Division 69
Short Division 71
Long Division 78
Properties of Integral Numbers 87
Definitions 87
Factoring 91
Greatest Common Divisor 93
Least Common Multiple 99
Fractions 108
Nature of Fractions 108
Notation of Fractions 109
Classification of Fractions 112
Value of a Fraction 113
Propositions in Fractions 113
Pteduction of Fractions 113
(•5)
%
♦
^■yt t-ti'^
6 ' " *■-** ■ COSTHNIS.""
D
Fractions — [Continued.) pagk
Addition of Fractions 119
Subtraction of Fractions 121
Multiplication of Fractions 124
Division of Fractions 130
Reduction of Common Fractions to Decimal Fractions 136
Compound Numbers 151
Definitions 151
English Money 152
French Money ; 152
Troy Weight 153
Avoirdupois Weight 154
Apothecaries Weight 155
Long Measure 155
Cloth Measure 156
Superficial or Square Measure 158
Solid Measure 159
Wine Measure 160
Ale or Beer Measure 161
Dry Measure 161
Time 162
Circular Measure 164
Reduction of Compound Numbers 166
Compound to Concrete 166
Concrete to Compound 166
Denominate Fractions to Compound Numbers 169
Compound Numbers to Denominate Fractions 169
Compound Numbers to Decimal Fractions 173
Denominate Decimal Fractions to Compound Numbers 173
Addition of Compound Numbers 175
Subtraction of Compound Numbers 177
Time between Dates 179
Multiplication of Compound Numbers 181
Division of Compound Numbers 184
Longitude in Time 186
Analysis by Aliquot parts 187
Review in Addition 195
Review in Subtraction 198
Review in IMultiplicatiou and I'ivision 200
CONTENTS. -^ 7
PAGE
Percentage 202
Applications of Percentage 210
Insurance 210
Commission 212
Stock 213
Brokerage 215
Profit and Loss 215
Duties or Customs 220
Interest 222
Problems in Interest 232
Present Worth *.... 235
Bank Discount 236
Promissory Notes 237
Compound Interest 241
Ratio 244
Proportion 246
Rule of Three 250
Partnership 261
Equation of Payments 265
Alligation Medial 266
Alligation Alternate 268
Position 273
Single Position 273
Double Position 276
Involution 281
Evolution 285
Square Root 286
Cube Root 295
Problems 303
Arithmetical Progression 305
Geometrical Progression 311
Permutations, Arrangements, and Combinations 317
Practical Geometry 319
Definitions , 319
Pythagorean Proposition 323
Proposition on the Triangle 326
Mensuration 327
Area of Triangle, I...^ 327
8 CONTENTS.
a
Practical Geometry — [Continued.) page
Area of Triangle, II 328
Area of Quadrilateral with Parallel Sides 328
Area of Trapezium 329
Proposition on Similar Figures 330
The Grindstone Problem 331
Mensuration of Solids 332
Definition 332
Contents of a Cylinder or Prism 333
Contents of a Cone or Pyramid 334
Contents of a Frustrum of a Cone or Pyramid 335
ConCents of a Cistern 335
Proposition on Spheres 335
Surface and Contents of a Sphere 336
Miscellaneous Examples 336
Appendix 351
Table of Multiplication 351
Table of Square Roots 362
Table of Cube Roots 353
Strength of Building Materials 354
Annuities •. 355
French Weight 357
French Linear Measure 357
French Superficial Measure 358
French Solid Measure 358
French Measure of Capacity 358
Table of Foreign Money (fixed by law) 358
Books and Paper 360
ELEMENTARY ARITHMETIC.
DEFINITIONS
1. Science is knowledge reduced to order.
2. Art is the practical application of the principles
of a science.
3. Quantity is a term that is applied to any thing
that can be measured.
4. A Unit is a quantity to which the term one may
be applied. Thus: one horse, one ten, one half. —
(Vide 128.)
5. A Number is a unit, or a collection of units.
6. Figures are characters used to represent any given
number. They include the cipher, naught, or zero, and
nine digits. Thus :
naught
one
two
three
four
five
six
seven
eight
nine
0
1
2
3
4
5
6
7
8
9
The ni7ie digits are called significant figures, to distin-
guish them from the cipher, which has, when written
alone, no value whatever. Its effect when joined to
other figures is explained under Notation.
7. Integral numbers or integers are tvhole numbers.
Thus: seven, twenty, one hundred and six, etc., are
integral numbers.
(i>)
10 DEFINITIONS.
8. A Fractional number or fraction represents one,
or more than one, of the equal parts of a unit. Thus :
one seventh, five twentieths, two thirds, four ninths, etc.,
<ire fractional numbers or fractions. — (Vide 127.)
O. A Decimal fraction represents 07ie, or more than
one, of the parts of a unit which is divided into ten, one
hundred, one thousand, ten thousand, etc., equal parts.
Thus : one tenth, four tenths, thirty-seven hundredths,
forty-five thousandths, etc., are decimal fractions.
10. Mathematics is the science of quaiitity.
11. Arithmetic is that branch of mathematics which
treats of numbers. It is a science and an art.
13. A Problem is a question proposed which requires
a solution.
13. An Operation is the method of solving a problem.
14. An Analysis is an investigation of the several
parts of a problem. An a^ialysis leads to the operation
which obtains the answer to a problem.
15. A Rule is a direction for performing an opera-
tion. A rule is usually derived from an analysis of one
or more problems.
NOTATION. 11
NOTATION.
IG. Notation is the method of representing num-
bers by figures or letters. Arabic notation represents
numbers by figures. Roman notation represents num-
bers by letters.
ARABIC NOTATION.
PROBIiEM I.
17. To represent an integral number between naught
and ten,
Write the necessary figure as in Definition 6.
EXAMPLE.
1. Write in figures, three, seven, one, five, four, eight,
six, nine, two, and naught.
2. How many units does the figure 5 represent ?
3. How many marbles does the figure 9 represent?
4. How many apples have you when they can be
represented by 0? 3? 4? 6? 7? 8? 2?
PROBIiEM II.
18. To represent any integral number between te7i
and nineteen, inclusive.
Place the several figures to the right of the figure 1.
EXAMPLE.
1. Write, by means of figures,
ten eleven twelve thirt'n fonrt'n fift'n sixt'n sevont'ii eiglil'n nitict'n
Ans. 10 11 12 13 14 15 16 17 18 19
12 NOTATION.
The figures at the right hand, or first place, in these
numbers represent units of the first order.
The figure in the second place represents a unit of
THE SECOND ORDER. — (Vide Def. 4.)
PROBI.EM III.
lO. To represent the units of the second order,
between ten and ninety inclusive,
Place the cipher at the right of the several digits.
example.
1. Write, by the aid of figures,
ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.
Ans. 10 20 30 40 50 60 70 80 90
PROBI.EM IV.
SO. To represent any integral number between
twenty and one hundred,
Unite the required units of the first order to those of
the second.
examples.
1. Write, by the aid of figures, the number twenty-
five. Here five units of the first order are required, and
two of the second. Ans. 25.
2. Write on your slate all the integral numbers be-
tween naught and one hundred, by the aid of figures.
PROBLEM V.
21. To represent the units of the third order,
called hundreds.
Place hvo ciphers to the right of the several digits,
thus :
NOTATION. 13
One hundred... 100 Four hundred, ..400 Seven hundred. ..700
Two hundred. ..200 Five hundred. ..500 Eight hundred. ..800
Three hundred. 300 Six hundred 600 Nine hundred. ...900
PROBLEM TI.
22. To represent any integral number between one
hundred and one thousand,
Write down the figure representing the required units
of the third order, and annex to it the figures repre-
senting the required units of the second and first orders.
EXAMPLES.
1. Write, by the aid of figures, one hundred and
twenty-three. Here one unit of the third order, two of
the second, and three of the first, are required.
Ans. 123,
2. Write, by the aid of figures, seven hundred and
eight. Here seven units of the third order, none of the
second, and eight of the first, are required.
Ans, 708.
3. Write on your slate five hundred and forty-nine,
three hundred and forty, six hundred and one, three
hundred and seven, nine hundred and nine, seven
hundred and ten.
4. Write all the integral numbers between three
hundred sixty-five and four hundred.
5. Write all the integral numbers between seven
hundred three and seven hundred fifty.
PROBLEM Til.
23. To represent units of the fourth order,
called THOUSANDS,
J 4 NOTATION.
Place three ciphers to the right of the several digits^
thus:
One thousand 1000 Four thousand. .4000 Seven thousand. .7000
Two thousand '2000 Five thousand..5000 Eight thousand.. 8000
Three thousand. ..3000 Six thousand. ..6000 Nine thousand. ..9000
Units of the fifth order are called tens of thou-
sands.
Units of the sixth order are called hundreds of
'Thousands.
probxem viii.
24. To represent any number of thousands by the
aid of figures,
Write the figures as if they zvere to represent a number
less than one thousand, and then annex three ciphers.
EXAMPLES .
1. Write, by the aid of figures, one hundred and
twenty-three thousand. Ans. 123000.
2. Write seven hundred and eight thousand.
A71S. 708000.
3. Write five hundred and forty-nine thousand, six
hundred and one thousand, nine hundred and ten thou-
sand, five hundred and fifty-five thousand, twenty-one
thousand.
PROBIiEM IX.
25. To represent any integral number less than one
million.
Write the number as if it were required to write the
thousands only, but instead of the three ciphers place that
part of the number less than a thousand.
NOTATION. 15
EXAMPLES.
1. Write one hundred and twenty-three thousand one
hundred and ninety-one. Ans. 123191.
2. Write seven hundred and eight thousand two hun-
dred and five. Ans. 708205.
3. Write nine hundred and nine thousand nine hun-
dred and nine. Ans. 909909.
4. Write four hundred and forty-four thousand four
hundred and forty-four. Ans. 444444.
5. Write twenty-seven thousand and one.
Ans. 27001.
Remark. — Two ciphers are retained because no mention is
made of units of the second or third order.
6. Represent in figures twenty-seven thousand three
hundred and twenty-one, twenty-seven, three hundred
and one, twenty-seven thousand and twenty-one, tw^enty-
seven thousand and five, sixty-seven.
Units of the seventh order are called millions.
Units of the eighth order are called tens of mill-
ions.
Units op the ninth order are called hundreds of
millions.
PROBI.£i!»I X.
26. To represent in figures any number of millions,
Write the number as if no mention were made of mill-
ions and then annex six ciphers.
examples.
1. Write one hundred and twenty-three millions.
Ans. 123 000 000.
2. Write seven hundred and eight millions.
Ans. 708 000 000.
10 NOTATION.
27. In the previous section, the three figures on the
right of the answer to each example form the period op
UNITS.
The three next figures form the period of thousands.
The three next figures form the period of millions.
A few succeeding periods are named as follows :
4. Billions. 7. Quintillions. 10. Octillions.
5. Trillions. 8. Sextillions. 11. Nonillions.
6. Quadrillions. 9. Septillions. 12. Decillions.
Remark. — These names might be continued to any extent.
PROBIiEM XI.
28. To represent in figures any number whatever,
Write each period in the order named, as if it were the
period of units; hut if intermediate periods are not
named, fill their places luifh ciphers.
exajiples.
1. Write four hundred and twenty-three trillions two
hundred and seven billions five hundred and six thou-
sand and one. Ans. 423,207,000,506,001.
Remark. — For convenience the periods may be separated by a
comma.
2. Write in figures twenty-seven; forty-three; one
hundred and fifty-four ; five thousand and ten ; twenty-
six thousand and forty -five; three hundred thousand
and seven ; two millions and five ; forty-seven millions
and thirty-three ; three decillions one nonillion twenty-
seven octillions three quintillions one hundred and
twenty billions and thirty-four.
Last Ans. 3,001,027,000,000,003,000,000,120,000,000,034.
NUiMEllATION. 17
NUMERATION
29. Numeration exhibits the method of reading num-
bers represented by figures.
EXA]\IPLES.
1. Read, in common language, the number 631230-
405078901.
We begin by separating the number into periods of
three figures each, commencing at the right. Thus :
631, 230, 405, 078, 901.
Next, apply, either mentally or in writing, the riames
of each of the jjeriods as given in Notation, beginning at
the right. Thus :
Trill. Bill. Mill. Tlions. Units.
631, 230, 405, 078, 901.
Next, hegin at the left^ and give to the third figure
the name of the unit of the third order, (vide 21,) and
to the second figure the name of the unit of the second
order, (vide 19,) and to the first figure the name of the
digit which it represents, (vide 6.) Finally, apply the
name of the period itself.
Do the same thing to the successive periods toward
the right, omitting the names of places filled by ciphers.
Thus:
Six hundred thirty-one trillions two hundred thirty
billions four hundred and five millions seventy-eight
thousand nine hundred and one.
18 NUMERATIOX.
Remark 1. — The word and is used when required by custom.
Rkmark 2. — It is not customary to apply the name of the period
on the right. Nine hundred and one is the same as nine hundred
and one units.
2. Read the number 63021457823675301742601930.
OPERATION.
Scptil. Sextil. Quin. Quad. Trill. Bill. Mill. Thoiis.
63, 021, 457, 823, 675, 301, 742 601, 930.
Sixty-three septillions twenty-one sextillions, etc.
Remark. — It is observed that the period on the left need not
be full, for a cipher in the vacant place would be of no service. —
(Vide 6.)
3. Read the number 319415012.
Arts. Three hundred nineteen million four hundred
fifteen thousand and twelve. — (Vide 18.)
30. From these examples we have the following
RULE FOR READING NUMBERS.
1. Separate the number into periods of three figures
each, beginning at the right.
2. Apply to the third, second, and first figures of the
period on the left, if it is fidl, the names of the units of
the third, second, and first orders, and afterivard apply
the name of the period, omitting names* in all cases where
ciphers occur.
3. Read each period in the same way, passing to the
right.
4. If the middle figure of a p)eriod is 1, apply the name
which it, in connection ivith the figure on its right, rep-
resents.
NUMEllATION.
19
EXAMPLES.
Read the following numbers
1.
2.
3*
4.
5.
6.
12.
184.
3261.
81765.
987123.
7. 6345555. 14.
9.
10.
11.
12.
13.
3.
10.
530.
4021.
56304.
740357.
15.
16.
17.
18.
19.
20.
1312547.
24340586.
382003125.
4960606544.
51730004163.
664800000532.
6401836. 21. 7562345940051.
22. 30014820016.
23. 372536370001.
24. 8134000500010.
25. 370010050020.
26. 1010101010101.
27. 4040506070809.
28. 3004005006001.
29. 7000600050004.
30. 9000040000700.
31. 4000000130000.
32. Read the number 3014056000451.
33. Read the number 40700369997823.
34. Read the number 100370059431001.
35. Read the number 6001478900462357.
36. Read the number 81705430267891456.
37. Read the number 123456789098765432.
38. Read the number 9876543210123456789.
39. Read the number 10203040506070809011.
40. Read the number 908070605040302010999.
41. Read the number 1002003004005006007008.
42. Read the number 199:^991^91^Q1^g]^9^1.
f
20 DECIMALS.
DECIMALS
NOTATION AND NUMERATION.
SI. The several units in decimals are named TENTH,
HUNDREDTH, THOUSANDTH, TEN-THOUSANDTH, HUNDRED-
THOUSANDTH, MILLIONTH, etc.
32. The relation between integral and decimal units,
and the manner of representing them by figures, are
given below.
One million 1000000.
One hundred thousand 100000.
One ten thousand 10000.
One thousand 1000.
One hundred 100.
One ten 10.
One t One.
.1 One tenth.
.01 One hujidredth.
.001 One thousandth.
.0001 One ten-ihotisandth.
.00001 One hundred-thoiisandlh.
.000001 One millionth.
(1.) The Decimal Point distinguishes the decimal from
the in!9jfa0m^Sy^tsJpJa/e0>emg at the left of all
decimals. The period (.) is commonly employed for this
purpose.
(2.) By one tenth is meant one of the ten equal parts
into which the nnit one is divided.
DECIMALS. 21
(3.) Bj cue hundredth is meant one of the hundred
equal parts into which the unit one is divided.
(4). By one thousandth is meant, etc.
l>ROBL,£M I.
33. To represent by the aid of figures any number
of tenths,
Place the decimal point to the left of the proper digit.
EXAMPLES.
1. Represent five tenths with a figure. Ans. .5
2. Represent two tenths, three tenths, four tenths,
six tenths, seven tenths, eight tenths, nine tenths, each
by the proper digit.
Hemakk. — By two tenths we mean two of the ten equal parts into
which the unit one is divided.
PROBIi£M II.
34. To represent by the aid of figures any number of
hundredths.
(1.) If the number of hundredths is less than ten,
Place a cipher between the decimal point and the proper
digit.
(2.) If the number of hundredths is ten or more than
ten,
Place the decimal point to the left of the given number.
examples'.
1. Represent two hundredths by figures.
Ans. .02
2. Represent three hundredths, four hundredths, etc.,
to nine hundredths by figures. Ans. .03 etc.
22 DECIMALS.
3. Represent ten hundredths by figures.
A71S, .10
4. Represent eleven hundredths, twelve hundredths,
etc., to ninety-nine hundredths. Ans. .11 etc.
Remark. — By tivo hundredths we mean two of the hundred equal
parts into which the unit one is divided.
PROBIiEM III.
35. To represent by the aid of figures any number of
thousandths.
(1.) If the number of thousandths is less than ten,
Place two ciphers between the decimal point and the
proper digit.
(2.) If the number of thousandths is ten or more, and
less than a hundred.
Place a cipher between the decimal point and the given
number.
(3.) If the number of thousandths is one hundred, or
more than one hundred,
Place the decimal point to the left of the given number.
EXAMPLES.
1. Represent three thousandths by figures.
Ans. .003
2. Represent one* thousandth, two thousandths, etc.,
to nine thousandths. Ans. to last, .009
3. Represent ten thousandths by figures.
Ans. .010
4. Represent eleven thousandths, twelve thousandths,
etc., to ninety-nine thousandths. Ans. to last, .099
DECIMALS. 23
5. Represent one hundred thousandths.
Ans, .100
6. Represent six hundred and twenty-one thou-
sandths. Ans. .621
7. Represent three hundred and six thousandths.
Ans. .306
8. Represent one hundred and one thousandths, one
hundred and two thousandths, etc.
Ans. .101 .102 etc.
Remark. — By /our thousandlhs we mean /our of the thousand equal
parts into which the icnit one is divided.
PROBLEM IV.
36. To represent by figures any decimal whatever,
(1.) Write the given number, as in sections 17 to 28.
(2.) WJien necessary, prefix ciphers enough to make the
right-hand figure of the number occupy the place of 1
when representing the given decimal unit. — (Vide 32.)
(3.) To the left place the decimal point.
Remark. — Observe that one figure only on the right of the point
is required to represent tenths^ two figures to represent hundredths,
three figures to represent thousandths, etc. (Vide 32.)
EXAMPLES.
1. Represent by figures one ten-thousandth.
Ans. .0001
2. Represent by figures two ten-thousandtJis.
Ans. .0002
3. Represent twenty-one ten-thousandths.
Ans. .0021
4. Represent three hundred and six ten-thousandths.
Ans. .0306
24 DECIMALS.
5. Represent three thousand and five ten-thousandths.
Ans. .3005
6. Represent one hundred-thousandth.
Ans. .00001
7. Represent six hundred and one hundred-thou-
sandths. Ans. .00601
8. Represent one millionth. Ans. .000001
9. Represent one thousand and five millionths.
Ans. .001005
10. Represent bj figures, four tenths^ twelve hun-
dredths, seven hundredths, five thousandths, thirty-seven
thousandths, one hundred and eleven thousandths, forty-
six ten-thousandths, nine hundred and one ten-thou-
sandths, three hundred and sixty-one hundred-thou-
sandths, ten thousand four hundred and fifty-six
millionths, one ten-millio7ith, twenty-seven ten-millionths,
one hundred-millionth, sixty-five hundred-millionths, one
billionth, three thousand and fifty-seven hillionths.
Ans. to last, .000003057
PR OBI. EM V.
•17. To read any decimal represented by figures,
(1.) Read the figures as if represe7iting an integral
number. — (Vide 29.)
(2.) Apply the name of the decimal unit indicated by
the right-hand figure. — (Vide 36, Remark.)
EXAMPLES.
1. Read .5; .05; .005; .0005; .00005; .000005 in
words.
2. Read .3; .13; .213; .1111; .22222; .999999 in
words.
DECIMALS. 25
3. Read .9; .24; .031; .0461; .00231; .009999 in
words.
Remark. — An integral number and a decimal may be written
together. Thus : 3.7 are three and seven tenths.
4. Read 2.5; 4.05; 8.005; 21.0005 in words.
5. Read 8.1; 2.12; 9.224; 27.1234 in words.
6. Read the following expressions :
3.004 23.005 67.431 5.6789
17.115 48.673 12.6001 27.3004
126.432 12.6432 1.26432 .126432
.1345 1.345 13.45 134.5
38. Principles of Arabic Notation.
I. All numbers are derived from the unit one.
II. Removing any figure one place toward the left
increases its value .ten times. Thus, in the expressions
.001 .01 .1 1. 10. 100.
the value of the figure 1 is increased ten times in each
step of its passage from right to left past the decimal
point.
Remark 1. — Any digit, then, may have a simple or it may have
a local value; it has a simple value when written alone, and a local
value in all other cases.
m, ^ r The simple value of three is 3.
' \ Some local values of three are 30. ; .3 ; .03
Remark 2. — The principal use of the cipher is to give a local
value to the digits. — (Vide 6.)
Remark 3. — It is evident that one tenth of any quantity, as for
instance a dollar, is the same as ten hundredths of the same quan-
tity; that is, .1 is the same as .10; hence.
Placing a cipher to the right of a decimal, does not
change the value of the decimal.
3
26 DECIMALS.
UNITED STATES MONEY.
NOTATION AND NUMERATION.
39. The several units of the currency of the United
States are named the Eagle, Dollar, Dime, Cent, and
Mill. Of these only the dollar, cent, and mill are con-
sidered in arithmetical Notation.
40. The dollar is the primary unit, and figures rep-
resenting dollars are considered as integral numbers.
41. The cent is the one hundredth part of one dollar.
42. The mill is the one thousandth part of one dollar.
43. The sign $, when placed before figures, denotes
that United States money is meant.
Remark. — The gold coins of the United States are the double-
eagle, eagle, half-eagle, quarter-eagle, and dollar.
The silver coins are the dollar, half-dollar, quarter-dollar, dime,
and half-dime. The nickel coin is the three-cent piece.
The copper coins are the two-cent and the one-cent pieces. '
The value of the eagle is ten dollars, and of the dime ten cents.
The eagle weighs 10 pennyweights 18 grains.
PROBIiEM I.
44. To represent by figures any number of dollars,
cents, and mills,
(1.) Place the figures indicating the dollars on the left
of the decimal point.
(2.) Consider the figures indicating the cents as so
many hu7idredths, arid write them as directed by 34.
(3.) Write the figure indicating the mills in the third
place on the right of the decimal point.
(4.) To the whole prefix the sign |.
Remark. — If mills only are to be represented, the places of the
dollars and cents must be filled with ciphers. — (Vide 35, 1.)
DECIMALS. 27
EXAMPLES.
1. Represent one mill by figures. Arts, f 0.001.
2. Represent two mills, three mills, four mills, etc.,
to nine mills. Ans. to last, |0.009.
3. Represent one cent by figures. Ans. §0.01.
4. Represent 2 cents, 3 cents, 4 cents, etc., to 9 cents.
Ans. to last, $0.09.
5. Represent twenty-five cents by figures.
Ans. 10.25.
6. Represent 6 dollars 27 cents. An^. |6.27.
7. Represent 8 dollars 10 cents 4 mills. *
Ans. $8,104.
8. Represent 24 dollars 25 cents 1 mill.
Ans. $24,251.
9. Represent 103 dollars 6 mills. Ans. $103,006.
10. Represent 904 dollars 37 cents.
Ans. $904.37.
11. Represent 1 dollar 20 cents 5 mills.
Ans. $1,205.
12. Represent 4 cents; 5 mills; 13 cents; 65 cents;
5 dollars 14 cents 2 mills; 167 dollars 55 cents 7 mills;
1 dollar 1 mill; 65 dollars 6 mills; 125 dollars.
Ans. to last, $125.00.
13. Represent 4 dollars 3 cents 1 mill; 6000 dollars
1 cent; 1245 dollars 3 mills; 45 dollars; 7 mills; 5 cents;
222 dollars 22 cents 2 mills; 9167 dollars 54 cents 9
mills.
14. What is meant by the expression $1,251 ?
Alls. 1 dollar 25 cents 1 mill.
15. Read in words the following sums of money :
28
DECI3:
[ALS.
13.265
§3.043
$20,072
$0,714
19.118
§6.259
$18,013
$7.14
$0,001
$0,141
$00,162
$71.40
$1,111
$5,001
$11,001
$714.
45. Exercises ix Review.
1. Express by figures twenty-six; one hundred and
one ; six hundred and forty ; seven hundred and fifty-
three ; five hundred and sixty-seven ; three hundred and
eleven; six thousand and four; eight thousand and
ninety.
2. Expre'ss by figures two thousand and twenty-two ;
three million and twenty-two; forty -five million and
twenty-two ; three hundred and eight thousand.
3. Express by figures six billion and five million;
eight trillion and seven thousand; six quadrillion and
one ; three quintillion and sixty-seven.
4. Express by figures four hundred and forty-three
thousand five hundred and twenty-five.
5. Express by figures sixty-eight billion two hundred
and three million five hundred and five thousand six
hundred and forty-five.
6. Express by figures twenty-six Jiundredths; one
hundred and one thousandths; six hundred and forty
millionths; sixty-four hundred-thousandths.
7. Express by figures seven tenths ; seven hundredths ;
seven thousandths ; sixty-five thousandths; one hillmith;
one trillionth; one quadrillionth; one quintilliontli ; one
sextillionth; one septilUonth; three octilllonths; seven
nonillionths; one hundred and eleven deeilliontlis.
8. The number of inches from the Equator to the
DECIMALS. 29
North Pole is three hundred ninety-three million seven
hundred seven thousand nine hundred. What figures
express them?
9. The numher of seconds in a year is thirty-one
million five hundred fifty-six thousand nine hundred
twenty-seven and fifty-seven hundredths. What figures
express them ?
10. The distance from the earth to the moon is two
hundred thirty-eight thousand six hundred and fifty
miles. What figures express this distance ?
11. The distance from the earth to the sun is about
ninety-five million miles. What figures express this
distance ?
12. It is about twenty trillion miles to the nearest
star. Express the distance in figures.
13. Express by the aid of figures two dollars sixteen
cents four mills ; six dollars seven cents ; eight dollars
and one mill; one hundred twenty-five dollars sixty
cents. Ans. to last, §125.60
14. Express by the aid of figures one hundred
twenty-five ; one hundred twenty -five thousandths ; one
dollar twenty -five cents ; twelve dollars fifty cents ;
twelve cents five mills ; one and twenty-five hundredths ;
twelve and five tenths; twelve and fifty hundredths.
15. Read the following expressions: 3071; $3071;
3.071; §3.071; 30.71; §30.71; 307.1; §307.10;
307.10; 307.100.
30 ROMAN NOTATION AND NUMERATION.
ROMAN NOTATION AND NUMERATION
46. The Roman Notation makes use of seven Capi-
tal Letters to represent numbers. They are
I, V, X, L, C, D, M;
and their values are, respectively,
1, 5, 10, 50, 100, 500, 1000.
47. Principles of Roman Notation.
I. The repetition of a letter repeats the value of the
letter. Thus : II are 2, III are 3, XX are 20, XXX are
30, CCC are 300.
II. If a letter is placed hefore another of greater value
than itself, the value of the less is taken from that of the
greater. Thus : IV represent 4, XL represent 40, XC
represent 90.
III. If a letter is placed after another of greater value
than itself, and a letter of greater value does not follow
both of them, the value of the less is added to that of
the greater. Thus : XI represent 11, XIV represent 14,
OX represent 110, CXL represent 140.
48. Examples.
1. Represent by the aid of letters the numbers 1, 2,
3, 4, 5. Ans. I, II, III, IV, V.
2. Represent by the aid of letters the numbers, 6, 7,
8, 9, 10. A71S. VI, VII, VIII, IX, X.
ADDITIOX. 31
3. Represent by the aid of letters the numbers 11,
12, 13, etc., to 50. Ans. XI, XII, XIII, etc., L.
4. Represent 54, 80, 90, 100, 150, 199, 500, 1099,
1865. Ans. to List, MDCCCLXV.
5. Represent 60, 63, 71, 94, 83, 101, 565, 1741.
Ans. to last, MDCCXLI.
6. Represent 1001, 1005, 1008, 1010, 1499.
Ans. to last, MCDXCIX.
7. Represent 1410, 1951, 1673, 1467, 1866.
Ans. to last, MDCCCLXVI.
ADDITION
49. Addition is the operation of finding the sum of
two or more numbers.
50. The sum is a number which contains as many
units as all the numbers taken together. Thus : the sum
of 5 and 3 is 8.
• SIGNS.
51. The sign + is called plus, and signifies that the
numbers between which it is placed are to be added to-
gether. Plus is a Latin word, signifying more.
52. The sign = is called the sign of equality, and
signifies that the lohole expression placed before it is
equal to that placed after it. Thus: 5-f 3=8, is read
five plus three equals eight, and the meaning is that
there are the same number of units in 8 as in 5 and
3 taken together.
32
ADDITION.
53. In the following table the sign + may be read
by the word and, the sign = by the word are. Thus :
5 and 3 are 8.
ADDITIO
N TABLE.
2 + 0=2
3 + 0=3
4 + 0=4
5 + 0=5
2 + 1=3
3 + 1=4
4 + 1=5
5 + 1 = 6
2 + 2=4
3 + 2=5
4 + 2 = 6
5 + 2=7
2 + 3=5
3 + 3=6
4 + 3=7
5 + 3=8
2 + 4=6
3 + 4=7
4+4=8
5 + 4=9
2 + 5=7
3 + 5=8
4 + 5=9
5 + 5 = 10
2 + 6=8
3 + 6=9
4 + 6 = 10
5 + 6 = 11
2 + 7=9
3 + 7 = 10
4 + 7 = 11
5 + 7 = 12
2 + 8 = 10
3 + 8 = 11
4 + 8 = 12
5 + 8 = 13
2 + 9 = 11
3 + 9 = 12
4 + 9 = 13
5 + 9 = 14
6 + 0 = 6
7 + 0=7
8 + 0=8
9 + 0=9
6 + 1=7
7 + 1=8
8 + 1 = 9
9 + 1 = 10
6 + 2=8
7 + 2=9
8 + 2 = 10
9 + 2 = 11
6 + 3 = 9
7 + 3 = 10
8 + 3 = 11
9 + 3 = 12
6 + 4 = 10
7 + 4.= 11
8 + 4 = 12
9 + 4 = 13
6 + 5 = 11
7 + 5 = 12
8 + 5 = 13
9 + 5 = 14
6 + 6 = 12
7 + 6 = 13
8 + 6 = 14
9 + 6 = 15
6 + 7 = 13
7 + 7 = 14
8 + 7 = 15
9 + 7 = 16
6 + 8 = 14
7 + 8 = 15
8 + 8 = 16
9 + 8 = 17
6 + 9 = 15
7 + 9 = 16
8 + 9 = 17
9 + 9 = 18
PROBLEM I.
54. To add any number of figures representing units
of the^rs^ order, (vide 18,)
(1.) Set the figures under each other, and add from the
bottom upward or from the top downward.
(2.) Place the sum under the column, so that the figure
representing units of the first order shall fall exactly/
underneath the figures above.
AJJi
UllUiN.
i5
EXAMPLES.
(1.) (2.)
(3.)
(4.) (5.)
(6.) (7.)
(8.) (9.)
(10.)
4 5
1
6 8
7 5
2 4
5
6 8
0
7 2
2 9
0 3
9
3 2
7
2 9
4 8
0 2
6
7 9
3
5 4
3 1
9 5
7
20 24
11
20 23
16 23
11 14
27
33
11. Add 6, 3, 5, 9, 4, 0, 2. 15. Add 9, 8, 7, 6, 5, 4, 3, 2, 1.
12. Add 8, 3, 9, 0, 9, 9, 9. 16. Add 1, 4, 7, 2, 5, 8, 3, 6, 9.
13. Add 4, 1, 3, 2, 5, 7, 9. 17. Add 3, 2, 5, 7, 9, 1, 4, 6, 8.
14. Add 8, 2, 6, 4, 1, 5, 0. 18. Add 1, 3, 5, 7, 9, 2, 4, 6, 8.
PR OB I. EM II.
55. To add any number of figures representing units
of the second order, (vide 19,)
(1.) aS'^^ the figures under each other, and add as in
54, (1.)
(2.) Place the sum under columns, so that the figure
represeyiting units of the second order shall fall exactly
underneath the digits above.
EXAMPLES
.
(1.)
(2.)
(3.)
(4.) (5.)
(6.)
(7.)
(8.)
40
50
10
90 50
50
80
70
60
80
00
80 40
90
50
30
30
20
70
70 70
80
30
20
70
90
30
60 30
70
40
40
200 240 110 300 190 290 200 160
9. Add 60, 30, 50, 90, 40. 12. Add 10, 20, 90, 80.
10. Add 10, 20, 40, 70, 80. 13. Add 30, 50, 70, 90.
11. Add 60, 50, 40, 30, 20. 14. Add 20, 40, 60, 80.
34
ADDITION.
Remark. — It is evident that figures, all of which represent any
one given order of units, may be added in the same way.
(15.)
400
600
300
700
(16.)
5000
8000
2000
9000
(17.)
10000
00000
70000
30000
(18.) (19.)
900000 6000000
800000 7000000
700000 1000000
600000 9000000
2000 24000 110000 3000000 23000000
In the same manner add —
(20.) (21.) (22.) (23.)
8 70 500 4000
6 20 400 3000
5 70 800 9000
19 160 1700
24. Add 4, 9, 3, 7, 6.
25. Add 5, 8, 4, 3, 0, 7, 1.
26. Add 30, 40, 50, 60.
27. Add 400, 300, 500, 100.
28. Add 9000, 4000, 3000.
29. Add 40000, 30000, 10000.
16000
Ans. 29.
Ans. 28.
Ans. 180.
Ans. 1300.
Ans. 16000.
Ans. 80000.
l»ROBI.EM III.
56. To add any numbers together where the sum of
the corresponding orders of units in all the numbers is
9 or less than 9,
(1.) Write the numbers so that the corresponding orders
of units may stand mider each other.
(2.) Begin at the right, and add each column separately,
placing the smn exactly under the column added.
ADDITION.
35
EXAMPLES.
1. Add together 19, 160, 1700, and 16000.
OPERATION.
19
160
1700
16000
Ans. 9999.
Ans. 8778
17879 Ans.
2. Add 1025, 6712, 1111, 1151.
3. Add 1234, 4321, 2222, 1001.
4. Add 31004, 13121, 22102, 21101, 11210.
Ans. 98538
5. Add 9, 10, 300, 130, 4110, 71100. Ans. 75659
PROBIiEMIV.
57. To add any numbers whatever together.
EXAMPLES.
1. Add together 4578, 3426, and 9875.
OPERATION.
4578
• 3426
9875
Vide 55, Ex.
20 .
. . 19
Vide 55, Ex.
21 .
. . 160
Vide 55, Ex.
22 .
. . 1700
Vide 55, Ex.
23 .
1 .
. 16000
Vide 56, Ex.
. nS79 Ans
36 ADDITION.
A moment's attention shows how the above operation
may be contracted.
The sum of the first column is 19, which is composed
of 9 units of the first order and 1 of the second. Set
down the 9 units under the units of the first order, j^r ^o
and add the 1 unit of the second order to the 342(3
column of units of the same order, making 17, 9875
which is composed of 7 units of the second order, TI^Z^
and 1 of the third. Set down the 7 units of the
second order under that column, and add the 1 unit of
the third order to the column of units of that order,
making 18, which is composed of 8 units of the third
order and 1 of the fourth. Set down the 8 units of the
third order under that column, and add the 1 unit of the
fourth order to the column of units of that order, making
17, which is written down as in 55. Hence,
RULE.
(1.) W^Hte the numbers so as to j^l^toe the figures in the
corresponding orders of units directly under each other,
and draw a line underneath.
(2.) Begin at the right hand, and add each column
separately, setting doivn the right-hand figure of the result
under the column added, and add- the left-hand figure or
figures to the next column on the left.
(3.) Set dozvn the tuhole amount of the last column.
2. Add 234, 589, 613. Ans. 1436.
3. Add 7123, 6054, and 9123. Ans. 22300.
4. Add 70561, 23564, and 34625. Ans. 128750.
5. Add 123456, 654321, 456123. Ans. 1233900.
6. Add 123, 240, 85, 36, and 7. Ans. 491.
ADDITION.
7. Add 1, 7, 43, 76, 65, 15, and 100. Ans. 307.
8. Add 13, 165, 48, 6251, and 19. A71S. 6496.
9. Add 108, 5012, 4103, 60450, and 6.
Ans. 69679.
10. Add 3456, 6543, 4563, 3645, and 5634.
A71S. 23841.
11. Add 31236, 415, 621437, 90053, and 34.
Ans. 743175.
12. Add 31, 280, 4560, 78930, and 672140.
Ans. 755941.
(13.)
(14.)
(15.)
12343247
213673
13021654
6015400
13021654
12343247
13021654
6015400
6015400
213673
12343247
213673
31593974
(16.)
(17.)
12346721305
3126754
8917259679
25678960
763421893
763421893
25678960
8917259679
3126754
12346721305
22056208591
58. To add several decimals together, proceed exactly
as in 57, and then place the decimal p>oint in the sum
directly under the decimal points above. — (Yide 38, Re-
mark 3.)
8
ADDITION.
EXAMPLES.
(1.)
(2.)
(3.)
(4.)
^34
71.23
705.61
314.5
.589
60.54
235.64
21.346
.613
91.23
* 346.25
5.17
1.436
223.00
1287.50
341.016
(5.)
(6.)
(7.)
345.012
785.432 •
987.65
45.78
1234.6
12.1453
121.3
257.87
1.67
87.125
2
12.431
1436.123
599.217
1290.333
2437.5883
8. Add 12.4, 3.47, 27.67, and 86. Ans. 129.54.
9. Add 1.24, 34.7, 2.767, and .86. Ans. 39.567.
10. Add .124, .347, .2767, and 8.6. Ans. 9.3477.
11. Add 57.76, 98.54, 38.72, and 43.65.
Ans. 238.67.
12. Add 5.776, 985.4, 38.72, and 4365.
Ans. 5394.896.
13. Add 577.6, 9.854, 3.872, and .4365.
Ans. 591.7625.
14. Add 4.8, 43.31, 74.019, and 11.204.
Ans. 133.333.
15. Add 29.0029, 3.4476, and 58.123.
Ans. 90.5735.
16. Add twelve and four tenths, three and forty-seven
hundredths, twenty-seven and sixty-seven hundredths,
and eighty-six. Ans. 129.54.
17. Add twenty-nine and twenty-nine ten-thou-
»
ADDITIOX. 39
sandths, three and four thousand four hundred and
seventy-six ten-thousandths, fifty-eight and one hun-
dred and twenty-three thousandths. Ans. 90.5735.
18. Add three and seven tenths, four and five hun-
dredths, one hundred, five thousandths, sixty-seven
millionths, five hundred and three, eight and six ten-
thousandths. Ans. 618.755667.
59. To add United States Money, consider the several
items as decimals, adding as in 58 ; then prefix the sign
% to the sum.— (Vide 44.)
EXAMPLES.
1. Add two dolhirs sixteen cents four mills, six dol-
lars seven cents, eight dollars one mill, one hundred
twenty-five dollars and sixty cents.
OPERATION.
_ $2,164
6.07
8.001 .
125.60
§141.835 Ans.
2. Add §241.075, $45.06, |37.05, §1216.131.
Ans. §1539.316.
3. Add §3124.162, §812.95, §67.12, §2145.75.
A71S. §6149.982.
4. Add §1.132, §56.075, §931.87, §4621.953.
Alls. §5611.03.
5. Add §27.413, §45.084, §607.219, §205.03, §25.25,
and §405.006. Ans. §1315.002.
40 ADDITION.
6. Add $136,255, §10.30, $248.50, $100,125, and
$65.38. Ans. $560.56.
7. Add $2600, $1927.404, $1603.40, $3304.17,
$165.47, and $2600.08. Ans, $12200.524.
8. Add $170, $400.02, $130, $250.10, and $845.22.
Ans. $1795.34.
9 Add $17.15, $23.43, $7.19, $8.37, and $12,315.
Ans. $68,455.
10. Add $6.75, $2.30, $0.92, $0,125, and $0.06.
Ans. $10,155.
11. Add $56.18, $7,375, $280.00, $0,287, $17.00,
and $90,413. A71S. $451,255.
12. Add 241 dollars 7 cents 5 mills, 45 dollars 6 cents,
37 dollars 5 cents, and 1216 dollars 13 cents 1 mill.
Ans. $1539.316.
Remark. — In adding a long column of figures, it is of much
assistance to divide it into several parts at pleasure, add each of
the parts separately, and finally the several partial sums for the
sum total.
(13.)
(14.)
(15.)
45678
76.345
$27,251
12345
18.237
43.026
37425
5.404
126.007
3128- 98576
12.36 -112.346
185.214
8462
1.1
243.671
71351
33.33
453.172-1078.341
81250
45.54
999.999
11111-172174
8.8 - 88.77
471.862
3333
75.464
125.281
7812
21.853
931.452
4512
27.306
813.161
76251- 91908
31.452-156.075
13.20 -3354.955
362658
357.191
$4433.296
ADDITION.
41
(16.)
(17.)
(18.)
43267
143.01
$25.04
14567
26.435
87.05
76543
506.146
125.113
81234
81.237
37.40
30506
67.21 -
- 824.038
103.046
4736-
-250853
1.004
95.062
154
65.042
127.111
58463
121.251
1237.086
81460
67.132
906.07 -
-2742.978
70120
9.25 -
- 263.679
81.023
93126
14.062
3410.192
47615-
-350938
87.643
1.20
82361
100.916
19.02
'
95864
2147.05 -
-2349.671
127.45
3729
432.876
87.40 -
-3726.285
26-
-181980
91.91
487.103
9428
125.125
45.073
32193
37.126
110.029
86159-
-127780
85.437-
- 772.474
$
3145.671-
-3787.876
911551
4209.862
10257.139
PRACTICAL EXAMPLES.
19. A gentleman purchased 234 bushels of corn at
one time, 589 at another, and 613 at another. How
many bushels did he buy in all? — (Vide 57, Ex. 2.)
20. During one year my crop of cotton was sold for
$7123.00 ; the next year it brought $6054, and the year
after I received |9123.00. How much did I receive for
cotton during the three years? — (Vide 57, Ex. 3.)
21. January has 31 days, February 28, March 31,
4
42 ADDITION.
April 30, May 31, June 30, July 31, August 31, Sep-
tember 30, October 31, Noveraber 30, December 31.
How many days in the year? Ans. 365.
22. Washington was born in 1732 and lived 67 years.
In what year did he die? Ans. 1799.
23. From the creation of the world to the flood, there
were 1656 years ; from the flood to the siege of Troy,
1164 years; from the siege of Troy to the building of
Solomon's Temple, 180 years; from the building of the
Temple to the birth of Christ, 1004 years. In what
year of the world did the Christian Era commence ?
Ans. 4004.
24. How many years have intervened from the
creation of the world to the year 1865? Ans. 5869.
25. Homer was born 733 years before the Christian
Era. How many years from the birth of Homer to the
year 1865 ? Ans. 2598.
26. I bought a barrel of flour for |6.78; ten pounds
of raisins for |2.30; seven pounds of sugar for |0.92;
one pound of coWeQ for $0,125, and two oranges for
10.10. What was the whole amount? Ans. |10.225.
27. A collector has bills in his possession of the fol-
lowing amounts: one of §43.75; another of $29.18;
another of $17.63; another of $268.95, and anotlicr of
$718.07. What amount has he to collect?
Ans. $1077.58
28. A man has the following sums of money due him,
viz: $420,197, $105.50, $304,005, $888,455. What is
the amount due him? Ans. $1718.157. .
29. What is the. sum of 429, 21.37, 355.003, 1.07,
and 1.7? Ai^s. 808.143.
ADDITIOX. 43
30. What is the sum of .2, .80, .089, .006, .9000, and
.005? Ans.2.
31. A gentleman bought at one thne 13.25 bushels of
corn; at another, 8.4 bushels; at another, 23.051
bushels ; at another, 6.75 bushels. How many bushels
did he buy in all? Ans. 51.451 bushels.
32. A gentleman owns five farms ; the first is worth
§11500; the second, §3057; the third, §2468; the
fourth, §9462; and the fifth is worth as much as the
four together. What is the value of the five farms ?
Ans. §52974.
33. By the census of 1850, the population of the ten
largest cities of the United States was as follows : New
York, 515547 ; Philadelphia, 340045 ; Baltimore, 169054 ;
Boston, 136881; New Orleans, 116375; Cincinnati,
115436; Brooklyn, 96838; St. Louis, 77860; Albany,
50763; Pittsburg, 46601. What was the population of
all combined? Ans. 1665400.
34. By the census of 1860, the population of the
following cities was ascertained to be — of New York,
805651; Philadelphia, 562529; Brooklyn, 266661;
Baltimore, 212418; Boston, • 177812; New Orleans,
168675; St. Louis, 160773; Cincinnati, 161044; Chi-
cago, 109260; Bufi-alo, 81129; Louisville, 68033 ; New-
ark, 71914; San Francisco, 56802; Washington, 61122;
Providence, 50666; Rochester, 48204; Detroit, 45619;
Milwaukee, 45246; Cleveland, 43417; Charleston,
40578; Troy, 39232 ; New Haven, 39267; Richmond,
37910; Lowell, 36827; Mobile, 29258; Jersey City,
29226; Portland, 26341 ; Cambridge, 26060; Roxbury,
25137; Charlestown, 25063; Worcester, 24960; Utica,
44
ADDITION.
22529; Reading, 23161; Salem, 22252; New Bedford,
22309; Dayton, 20081; Nashville, 16988. How many
inhabitants in all these cities combined? Ans.
35. By the census of 1860, the population of the
several States and Territories was as follows :
Alabama 964201
Arkansas 435450
California 379994
Connecticut 460147
Delaware 112216
Florida 140425
Georgia 1057286
Illinois 1711951
Indiana 1350428
Iowa 674948
Kentucky 1155684
Louisiana 708002
Maine 628279
Maryland 687049
Massachusetts 1231066
Michigan 749113
Minnesota 173855
Mississippi 791305
Missouri 1182012
New Hampshire 326073
New Jersey 672035
New York 3880785
North Carolina 992G22
Ohio 2339502
Oregon 52465
Pennsylvania 2906115
Rhode Island 174620
South Carolina 703708
Tennessee 1109801
Texas 604215
Vermont 315098
Virginia 1596318
Wisconsin 775881
Colorado 34277
Dakotah ,. 4837
District of Columbia... 75080
Kansas 107206
Nebraska 28841
New Mexico 93516
Utah.." 40273
Washington 11594
Nevada 6857
How many inhabitants in the United States in 1860?
Ans. 31445080.
36. How many inhabitants in the six New England
States taken together? Ans. 3135283.
37. How many inhabitants in the States bordering on
the Gulf of Mexico? Ans. 3208148.
38. How many inhabitants in the States watered bj
the Tennessee River? Ans. 4020991.
SUBTRACTIOX.' 45
39. How many inhabitants in the States bounded in
part by the Ohio River? Ans. 8153883.
40. How many inhabitants in the States watered by
the Mississippi River? Ans. 8718889.
41. How many inhabitants in the States and Terri-
tories lying wholly west of the Mississippi River?
Ans. 3830340.
SUBTRACTION
60. Subtraction is the operation of finding the dif-
ference between two numbers.
61. The difference is such a number as added to the
less will give the greater.
SIG-NS.
63. The sign — is called minus, and when placed
between two numbers it signifies that the one on the
right is to be subtracted from that on the left. 3Iinus
is a Latin word, signifying less.
63. The expression 8 — 5=3, is read eight minus five
EQUALS three, and the meaning is, that three is the dif-
ference between eight and five. The expression may
also be read, five from eight are three.
64. The greater of the two numbers is called the
minuend, and the smaller is called the subtrahend. The
result of the subtraction is called the difference, and
oftentimes the reniainde''\
46
SUBTIl ACTION.
SUBTRACTION TABLE.
2 — 2 = 0
3-3 = 0
4 — 4 = 0
5-5 = 0
3-2 = 1
4-3 = 1
5-4=1
6-5 = 1
4 — 2 = 2
5-3 = 2
6—4 = 2
7 — 5 = 2
5-2 = 3
6-3 = 3
7-4 = 3
8-5 = 3
6-2 = 4
7-3 = 4
8 — 4 = 4
9 — 5 = 4
7-2 = 5
8-3 = 5
9-4 = 5
10 - 5 = 5
8 — 2 = 6
9-3 = 6
10 - 4 = 6
11 — 5 = 6
9-2 = 7
10 — 3 = 7
11 _ 4 = 7
12 — 5 = 7
10 — 2 = 8
11 - 3 = 8
12 - 4 = 8
13 - 5 = 8
11 - 2 = 9
12 — 3 = 9
13 — 4 = 9
14 — 5 = 9
6-6 = 0
7-7 = 0
8 — 8 = 0
9-9 = 0
7-6 = 1
8-7 = 1
9-8=1
10 — 9 = 1
8-6 = 2
9 — 7 = 2
10 — 8 = 2
11 - 9 = 2
9-6 = 3
10 - 7 = 3
11 -^ 8 = 3
12 - 9 = 3
10 — 6 = 4
11 - 7 = 4
12 - 8 = 4
13 — 9 = 4
11 - 6 = 5
12 - 7 = 5
13 - 8 = 5
14 - 9 = 5
12 - 6 = 6
13 - 7 = 6
14 - 8 = 6
15 - 9 = 6
13 - 6 = 7
14 - 7 = 7
15 - 8 = 7
16 - 9 = 7
14 - 6 = 8
15 - 7 = 8
16 - 8 = 8
17 - 9 = 8
15 - 6 = 9
16 - 7 = 9
17 - 8 = 9
18 — 9 = 9
65. If tlie same number is added to any hvo niim-
bers, the difference bettveen the resulting numbers is the
same as that between the given numbers. Thus :
The diiFerence between 7 and 2 is 5. If now 6 be
added to both 7 and 2, the difference between the re-
sulting numbers, 13 and 8, is still 5.
PROBI.KM I.
00. To subtract one number from another, Avhen each
figure of the subtrahend is equal to, or less than, the cor-
responding figure of the minuend, counting from the right,
1. Write the less number under the greater, so as to
SUBTRACTION.
47
place the figures representing the corresponding orders of
units directly under each other.
2. Begin at the right, and subtract each figure of the
subtrahend from the figure of the minuend above it; the
results placed under the figures from which they were
obtained Avill express the difference between the two
numbers.
(1.)
From 5346
Take 2145
EXAMPLES.
(2.) (3.)
7890 4567
3450 1023
(4.)
67814
6514
Ans.
3201 4440
3544
61300
5.
From 76503 take 65402.
Ans.
11101.
6.
From 84321 take 62100.
Aiis.
22221.
7.
From 54360 take 21030.
Ans.
33330.
8.
From 74215 take 3115.
Alls.
71100.
9.
From 21036 take 24.
Ajis.
21012.
10,
From 762137 take 1025.
Ans. '
761112.
11.
From 12345 take 2345.
Ans.
10000.
12.
From 54321 take 4321,
Alls.
50000.
13.
From 20037 take 10036.
Ans.
10001.
(14.) (15,)
(16.)
(17.)
From
45.13 78.64
§3.105
§41.043
Take
34.02 67.64
2.104
21.032
Ans. 11.11 11.00
18. From 87.36 take 43.15.
19. From 96.125 take 5.01.
20. From 128.41 take 127.2
21. From |45.16 t:ike |3Lr
§1.001
§20.011
Ans. 44.21
A71S. 91.115.
Ans. 1.21
Ans. §11.01
48 SUBTIIACTION.
22. From $3426.45 take §113.23. Ans. 3313.22.
23. From §4327.871 take §3216.461.
Ans. §1111.410.
24. From §945.375 take §334.275.
Ans. §611.100.
25. From §12.032 take §1.021. Ans. §11.011.
26. From §119.457 take §8.236. Ans. §111.221.
PROBLiEM II.
G7. To find the difference between any two numbers
whatever.
EXAMPLES.
1. From 493 take 287.
OPERATION.
493
287
206 Ans.
Here 7 can not be taken from 3, because 7 is larger
than 3. Mentally add 10 units to the 3 units, and from
the sum 13 units, take 7 units, placing the difference,
6 units, under the figures representing units of the first
order.
Since now the minuend has 'been increased by 10
units, we must increase the subtrahend by the same
number of units to preserve the true difference. — (Vide
65.)
Mentally add 1 unit of the second order (which is the
very same thing as 10 units of the first order) to the 8
units of the second order, and we have 9 units of the
second order. Proceed now as in 66. Hence,
SUBTRACTION. 49
Tt U L E .
1. Write the numhers as in G6.
2. If the unit figure of the minuend is equal to or
greater than the unit figure of the subtrahend, subtract as
in (j6; but if the unit figure of the minuend be less than
that of the subtrahetid, mentally add 10 ^o the upper
figure and subtract the loiuer figure from the sum, placing
the result under the unit column.
3. Add 1 to the next figure of the subtrahend, and
proceed tvith the result precisely as with the figures repre-
senting units of the first order.
4. Do the same thing to the figures representing each
order of U7iits, and the results will express the difference
between the two numbers.
Remark 1, — The work may be verified by adding the remainder to
the subtrahend.
2. From 7804 take 5936.
OPERATION.
7804 min.
5936 sub.
1868
7804 proof.
Remark 2. — The pupil should say at once, 6 fi'om 14 leave 8; 4
from 10 leave 6; 10 from 18 leave 8; 6 from 7 leave 1. Do not say
^from 4 you can^t, 1 to carry, etc.
(3.)
From 5432
Take 1685
(4.)
* 34.57
23.19
(5.)
56.19
24.396
(6.)
125.4
37.236
Alls. 3747
11.38
34.57
31.794
56.19
88.164
Proof 5432
5
125.4
50
SUBTRACTION.
(7.) (8.)
(9.)
From
$125,456 $1243.18
$7256.372
Take
87.25 125.914
199.20
Ans.
$38,206 $1117.266
$7057.172
(10.)
(11.)
From 645.00037
10000.0000
Take .00198
.0111
Ans. 644.99839
9999.9889
12.
From 40000 take 9.
Ans. 39991.
13.
From 123456789 take 87654321.
Ans. 35802468.
14.
From 101010101 take 90909090.
Ans. 10101011.
15.
From 303030303 take 40404040.
A71S. 262626263.
16.
From 234702358 take 54321987.
Ans. 180380371.
17.
From 1000000 take 1.
Ans. 999999.
18.
From 1000000 take .1.
Ans. 999999.9.
19.
From 1000000 take .01.
Ans. 999999.99.
20.
From 34567 take .003.
Ans. 34566.997.
21.
From 1 take .000001.
Ans. .999999.
22.
From 5 take 4.000001.
Ans. .999999.
23.
From 12.4 take 3.756.
Ans. 8.644.
24.
From 100.25 take 75.12.
Ans. 25.13.
25.
From $100.25 take $75.12.
Ans. $25.13.
26.
From $20.05 take $5.50.
Ans. $14.55.
27.
From $90.00 take $70,045.
Ans. $19,955.
28.
From $1000 take $1,111.
Ans. $998,889.
SUBTRACTION. 51
29. From 6 dollars take 5 mills. Ans. $5,995.
30. From 8 dollars take 7 cents. Ans. $7.93.
31. America was discovered in A. D. 1492 by Chris-
topher Columbus. How many years from that event to
A. D. 1865 ? A71S. 373 years.
32. George Washington was born in A. D. 1732, and
died in A. D. 1799. To what age did he live?
Ans. 67.
33. The Declaration of Independence was published
July 4, 1776. How many years have intervened up to
July 4, 1865? A^is. 89.
34. Henry Hudson sailed up the river of his name,
A. D. 1609. How many years from that time to A. D.
1865? Ans.2b6.
35. The Mariners' Compass was invented in A. D.
1302. How many years to A. D. 1865 ? Aiis. 563.
36. What length of time from the birth of Francis
Bacon, A. D. 1561, to A. D. 1865? Ans. 304 years.
37. What length of time from the birth of Shake-
speare, A. D. 1564, to A. D. 1865 ? Ans. 301 years.
38. What length of time from the birth of John
Milton, A. D. 1608, to A. D. 1865? Ans. 257 years.
39. Pliny died in A. D. 17. How many years to A.
D. 1865 ? A71S. 1848.
40. Sir William Herschel was born in A. D. 1738,^
Galileo, A. D. 1564. How many years elapsed from the
birth of the one to that of the other? Ajis. 174.
41. Oliver Cromwell was born A. D. 1599. How
many years from that time to the death of Washington ?
A71S. 200.
42. Patrick Henry was born A. D. 1 736. How many
52 SUBTRACTION.
years from that; time to the publication of the Decla-
ration of Independence ? Aiis. 40.
43. The Revolutionary War began A. D. 1775 ; the
last war with Great Britain, A. D. 1812. How many
years from the beginning of the one to the beginning
of the other? Ans. 37.
44. What was the increase in the population of New
York from A. D. 1850 to A. D. I860?— (Vide 59, Ex.
33 and 34.) Ans. 290104.
45. How many more inhabitants in New York in
1850 than in Philadelphia? Ans. 175502.
46. How many more in 1860 ? A^is. 243122.
47. How many more in Boston than in New Orleans
in 1850? A71S. 20506.
48. How many more in New Orleans than in Cincin-
nati? Ans. 939.
49. How many more in Cincinnati than in Brooklyn ?
Ans. 18598.
50. How many more in Brooklyn than in St. Louis ?
Ans. 18978.
51. How many more in St. Louis than in Albany?
A71S. 27097.
52. How many more in Albany than in Pittsburg ?
A71S. 4162.
53. . How many more in New York than in Pittsburg ?
A71S. 468946.
54. The polar diameter of the earth is 7898.973
miles; the equatorial diameter, 7925.249 miles. How
much greater is the equatorial than the polar diameter ?
Ans. 26.276 miles.
55. The length of a degree of longitude at the equator
SUBTRACTION. 58
is 69.161 miles ; at New York it is 52.536 miles. What
is the difference ? A^is. 16.625 miles.
56. A is worth §6542.37; B is worth $9341.95 ; C
is worth §18425.63. How much are all three together
worth? Ans. §34309.95.
How much is B worth more than A ?
Ans. §2799.58.
How much more is C worth than B?
Ans. §9083.68.
How much more is C worth than A?
Ans. §11883.26.
How much more is C worth than A and B together?
Ans. §2541.31.
57. To stock a farm, the land of which was worth
§22475.96, I bought two horses for §327.80 ; two yoke
of oxen at §175.47 per yoke ; five cows at §27.36 each ;
a pair of mules for §275; and sixty-seven sheep for
§201.45. How much more is the land worth than the
stock? Ans. §21183.97.
How much more were the oxen worth than the horses ?
Ans. §23.14.
How much more were the oxen worth than the cows ?
A71S. §214.14.
Which were worth most, the oxen and horses together,
or the cows, mules, and sheep together, and by how
much? Ans. Oxen and horses, by §65.49.
58. A merchant bought at one time 3476 yards of
cloth; at another, 5426 yards; at another, 4221 yards.
He sells 3210 yards to one person, and 4345 to another.
How many yards has he left? Ans. 5568 yards.
59. A farmer bought of a merchant broadcloth to the
54 SUBTRACTION.
value of $137.50; cotton cloth, §93.45; sugar, |37.63;
molasses, 114.37; coffee, §11.45; flour, §28.13. He pays
the merchant, in corn, §123.65 ; in hay, §47.24; and the
balance in cash. What was the amount of cash paid?
Ans. §151.64.
60. Bought a yoke of oxen for §150 ; a horse for
§237; three cows for §87.45; and sold the whole for
§500. What was my gain? Ans. §25.55.
61. The length of a pendulum which vibrates once a
second, at London, is 39.1393 inches. One ten-millionth
of the meridian distance from the Equator to the North
Pole is 39.37079 inches. What is the difference?
Ans. .23149 inches.
62. The equatorial circumference of the earth is
24897.883 miles, and the circumference on a meridian
is 24855.296 miles. What is the difference ?
Ans. 42.587 miles.
63. What is the difference between 25 dollars 1 cent
4 mills and 6 dollars 17 cents 9 mills ?
A71S. §18.835.
64. What is the difference between 181 dollars 7
cents 9 mills and 140 dollars 9 cents 7 mills?
A71S. §40.982.
65. What is the difference between 9 dollars 5 cents
3 mills and 10 dollars 3 cents 5 mills?
MULTIPLICATION. 65
MULTIPLICATION.
68. Multiplication is the operation of increasing
one number as many times as there are units in another.
69. The number to be increased is called the multi-
plicand.
70. The number indicating how many times the
multiplicand is to be increased is called the multi-
plier.
Tl. The result of the operation is called the product.
SIGNS.
72, The sign X is called sign of multiplication, and
when placed between two numbers, signifies that they
are to be multiplied together. Thus, the expression,
8x5-=40,
is read, eight multiplied hy five equal forty, or eight
times five equal forty, or eight times five are forty,
73. A bar or imrenthesis is used to indicate that
several numbers are to be taken as a single number, thus :
2+6X5--40, or (2-f 6)X5=-40; but 2+6x5=32.
•74. When two or more numbers are multiplied to-
gether to produce a single number, each of the numbers
involved is called a factor of the product. Thus, in
the expression 3X2X5=30, each of the numbers, 3, 2,
5, is a factor of 30.
The product of the factors is a composite number.
56
MULTIPLICATION.
75. It is evident that 3X5 is the same as 5X3. For
3+3+3+3+3 is the same as 5+5+5.
76. Any number multiplied by 0 produces 0; and
any number multiplied by 1 produces the number itself.
Thus, 8X0=0, and 8X1=8.
TABLE OF MULTIPLICATION.
2x0 =
0
3x0
= 0
4x0
= 0
5x0=0
2x1 =
2
3 X 1
= 3
4 X 1
= 4
5x1=5
2x2 =
4
3x2
= 6
4x2
= 8
5 X 2 = 10
2x3 =
6
3x3
= 9
4x3
= 12
5 X 3 = 15
2x4 =
8
3x4
= 12
4x4
= 16
5 X 4 = 20
2x5 =
10
3x5
= 15
4x5
= 20
5 X 5 = 25
2x6 =
12
3x6
= 18
4 X 6
= 24
5 X 6 = 30
2x7 =
14
3X7
= 21
4x7
= 28
5 X 7 = 35
2x8 =
16
3x8
= 24
4x8
= 32
5 X 8 = 40
2x9 =
18
3x9
= 27
4x9
= 36
5 X 9 = 45
6x0 =
0
7x0
= 0
8x0
= 0
9x0=0
6x1 =
6
7 X 1
= 7
8 X 1
= 8
9x1=9
6x2 =
12
7x2
= 14
8x2
= 16
9 X 2 = 18
6x3 =
18
7x3
= 21
8x3
= 24
9 X 3 = 27
6x4 =
24
7x4
= 28
8x4
= 32
9 X 4 = 36
6x5 =
30
7x5
= 35
8x5
= 40
9 X 5 = 45
6 X G =
36
7x6
= 42
8x6
= 48
9 X 6 = 54
6x7 =
42
7x7
= 49
8x7
= 56
9 X 7 = 63
6x8 =
48
7x8
= 56
8x8
= 04
9 X 8 = 72
6x9 =
54
7x9
= 63
8x9
= 72
9 X 9 = 81
PROBIiEM I.
77. To multiply any number by any other number
less than 10.
EXAINIPLES.
1. Multiply 357 by 7.
MULTIPLICATIOX. 57
OPERATION. VERIFICATION.
357 357
49
350
2100
7 357
357
357 vide 68 and 75.
357
357
2490 357
2499
7 times 7 units of the first order are 49 units of tlie
first order; that is, 9 units of i\iQ first order and 4 of the
second.
7 times 5 units of the second order are 35' units of the
second order; that is, 5 units of the second order, and 3
of the third.
7 times 3 units of the third order are 21 units of the
tliird order; that is, 1 unit of the third order and 2 of
the fourth.
The sum of these partial products is the product
required.
This operation may be contracted thus :
7 times 7 are 49. Write down the 9 units operation
of the first order and mentally add the four oerr
units of the second order to the 35 units of 7
the same order, making 39. Write down
the 9 units of the second order, and men-
tally add the 3 units of the third order to
the 21 units of the same order, making 24.
Hence,
58
MULTIPLICATION.
RULE,
1. Write the midtiplier under the midtipliccmd, so that
units of the same order may stand under each other.
2. Ifultijyly the right-hand figure of the midtiplicand
hy the multiplier^ and set doivn the figure of tlie product
representing units of the first order under the column of
units of that order, and add the figure representing units
of the second order to the product of the second figure of
the multiplicand hy the multii^lier. Set down the figure
representing units of the second order under the units of
the multiplicand of that order, and add the figure repre-
senting units of the third order to the next product, and so
on till all the figures of the midtiplicand have been mul-
tij)lied. The result is the product required.
(2.) (3.) (4.) (5.)
Multiply 1736 4530 7106 2400
By 3 4 5 6
Ans. 5208
18120
35530 14400
Multiply 303479
By 2
(7.)
9854321
: 7
(8.)
123456789
9
Ans. 606958
68980247
1111111101
9. Multiply
10. Multiply
11. Multiply
12. Multiply
13. Multiply
14. Multiply
456031 by 3.
32467 by 8.
10054 by 5.
999999 by 9.
5432 by 7.
142857 by 7.
Ans. 1368093.
Ans. 259736.
Ans. 50270.
A71S. 8999991.
Ans. 38024.
Ans. 999999.
MULTIPLICATION.
59
15. Multiply 101010 by 9. A71S. 909090.
16. Multiply 3421 by 4. Ans. 13684.
17. Multiply 123456789 by 2, 3, 4, 5, 6, 7, 8, 9.
18. Multiply 987654321 by 2, 3, 4, 5, 6, 7, 8, 9. ^
PROB1.EM11.
78. To multiply any number by a unit of any order,
that is, by 10, 100, 1000, etc.
RULE.
An7iex as many ciphers to the right of the multiplicand
as there are ciphers in the multiplier. — (Vide 38, II.)
EXAMPLES.
1.
Multiply 357 by 1.— (Vide 76
>')
Ans. 357,
2.
Multiply 357 by 10.
Am. 3570,
3.
Multiply 357 by 100.
Ans. 35700,
4.
Multiply 358 by 1000.
Ans. 358000,
5.
Multiply 3476 by 100.
Ans. 347600,
6.
Multiply 35760 by 10.
Ans. 357600,
7.
Multiply 473 by lOOOOOO.
A:
ns. 473000000,
8.
Multiply 473000 by 1000.
Ans. 473000000,
9.
Multiply 473000000 by 1.
A:
ns. 473000000,
PROBIiEM III.
79. To multiply any number by a figure represent-
ing units of any order,
(1.) Consider the figure as representing units of the
first order, and zvrite the numbers as in 77.
(2.) After multiplying annex the ciphers, and the result
will he the j^roduct required.
60
MULTIPLICATION.
Multiply 357
By 40
Ans. 14280
(5.)
Multiply 4530
By
E X A JI P L E S .
(2.) (3.)
357 1736
200 70
30
71400 121520
(6.)
4530
900
(4.)
1736
400
694400
(7.)
3456
7000
Ans. 135900
24192000
A71S. 6321000.
A71S. 355300.
4077000
8. Multiply 9030 by 700.
9. Multiply 7106 by 50.
10. Multiply 303479 by 20000.
Ans. 6069580000.
11. Multiply 987654321 by 900000.
Ans. 888888888900000.
PROBIiEM IV.
80. To multiply one number by another.
EXAMPLES.
(1.) (2.)
Multiply ... 357 Multiply . .
By . . ; . . ^ By ... .
Vide 77, Ex. 3,
4530
934
Vide 77, Ex. 1, 2499
Vide 79, Ex. 1, 1428
Vide 79, Ex. 2, 714
1812
Vide 79, Ex. 5, 1359
Vide 79,, Ex. 6, 4077
Ans.
88179
Ans.
4231020
Remark. — The cipher of 79, Ex. 1, need not appear in (he opera-
tion, and so of the ciphers in the other examples referred to. Hence,
I
MULTIPLICATIOX.
61
1. Write iJie niimhers so that the riglit-ltand significant
figures of the multiplicand and multiplier may stand
under each other.
2. Multiply the multiplicand hy each figure of the mul-
tiplier, placing the first figure of each product directly
tinder the figure used in multiplying.
3. Add the several products together and annex to their
sum all the ciphers on the right of both factors. The
result is the product required.
Remark 1. — The second point of the rule does not apply to the
ciphers on the right of either factor.
Multiply
By .
Ans.
(3.)
(4.)
1476
470
Multiply .
By . .
. . 34300
. . 4310
10332
5904
343
1029
1372
fip.q790
Ans. . . 147833000
Remark 2. — When ciphers occur between the significant figures
of the multiplier, their product into the multiplicand need not ap-
pear in the operation.
(5.)
(6.)
Multiply
By. .
. 459
. 307
3213
1377
Multiply
By . ,
2134
5004
8536
10670
Ans 140913 Ans.
7. Multiply 46834 by 4060.
8. Multiply 47042 by 47042.
. . . 10678536
Ans. 190146040.
Ans. 2212949764.
62
MULTIPLICATION.
9. Multiply 123 by 125.
10. Multiply 328 by 67.
11. Multiply 75432 by 47.
12. Multiply 678954 by 24.
13. Multiply 789563 by 570.
14. Multiply 1579126 by 1710.
15. Multiply 67853 by 8765.
16. Multiply 3678543 by 4567.
17. Multiply 492 by 625.
18. Multiply 1312 by 335.
19. Multiply 603456 by 94.
20. Multiply 1357908 by 144.
21. Multiply 2368689 by 190.
22. Multiply 8432 by 6350.
23. Multiply 27496 by 1658.
24. Multiply 82488 by 555.
81. When one or both factors
Ans. 15375.
Ans. 21976.
Ans. 3545304.
Ans. 16294896.
Ans. 450050910.
^Tis. 2700305460.
Ans. 594731545.
Ans. 16799905881.
A71S. 307500.
Ans. 439520.
Ans. 56724864.
Ans. 195538752.
Ans. 450050910.
Ans. 53543200.
A71S. 45588368.
Ans. 45780840.
are decimals.
EXAMPLES.
(1-)
Multiply 357
By ..... . 24.7
2499
1428
714
Ans 8817.9
In this example, since the mul-
tiplier is decreased ten times,
(vide 38, II,) the product ought
to be decreased ten times, which
is done by placing one figure
to the right of the decimal
point.
(2.)
Multiply 85.7
By 24.7
2499
1428
714
Ans 881.79
In this example, since the mul-
tijDlicand is also decreased ten
times, (vide 80, Ex. 1,) the pro-
duct ought to be decreased ten
times more than in Ex. 1, which
is done by putting the decimal
point one place further to the left
MULTIPLICATION.
63
Hence :
(1.) Proceed 'precisely as in 80.
(2.) Place the decimal ijoint in the lyroduct so as to cause
as many figures to stand on its right as there are figures
on the right of the point in both the factors, supplying
any deficiency by prefixing ciphers.
(3.) (4.) (5.)
Multiply 45.9 4.59 .459
By 3.07 3.07 .307
3213 3213
3213
1377 1377
1377
Ans. 140.913 14.0913
.140913
(6.) (7.)
(8.)
Multiply .0008 .00008
.000008
By .0007 .007
.07
Ans. .00000056 .00000056
.00000056
(9.)
(10.)
Multiply .0716
.1234
By 1.326
1234
Ans. .0949416 152.2756
11. Multiply $3.57 by 7.
Ans. $24.99.
12. Multiply ^3.57 by 40.
Ans. $142.80.
13. Multiply $3.57 by 200.
Ans. $714.00.
14. Multiply ?3.57 by .7
Ans. $2,499.
15. Multiply 171.61 by 365. Ans. $26137.65.
16. Multiply $0.93 by 63.
Ans. $58.59.
17. Multiply $13.75 by 43.
Ans. $591.25.
18. Multiply $4.68 by 169.
Ans. $790.92.
64
MULTIPLICATION.
19. Mult
20. Mult
21. Mult
22. Mult:
23. Mult
24. Mult
25. Mult
26. Mult
27. Mult
28. Mult
29. Mult
30. Mult
31. Mult
32. Mult
ply
ply
ply
ply
ply
ply
ply
ply
ply
ply
ply
ply
ply
,^0.057 by 84C).
1132.55 by 369"!
^0.299 by 69.
§69.748 by 144.
§3.75 by 47.
$6.79 by 163.
§1.375 by 19.
§4.57 by 18.
§15.89 by 9.
§0.75 by 125.
§58.90 by 45.
§0.058 by .07
§0.904 by .025
§1.287 by .9
Ans, §48.222.
Ans. §48910.95.
Ans. §20.631.
Ans. §10043.712.
Ans. §176.25.
Ans. §1106.77.
Ans. §26.125.
Ans. §82.26
Ans. §143.01.
A71S. §93.75.
Ans. §2650.50.
Ans. §0.00406.
Ans. §0.022600.
Ans. §1.1583.
82. In Multiplication the j-^'^'oduct is always of the
same name as the multiplicand^ and the multiplier in the
operation must be considered as simply a number without
name. Thus: §75X47=-§3525.— (Vide 166.)
PKACTICAL APPLICATION.
1. What will 47 oxen cost, at §75 each? '
Ans. $3525.
2. If a man walk 23 miles a day, how far will he
walk in 17 days? Ans. 391 miles.
3. If a vessel sail 451 miles a day, how far will it
sail in 9 days ? Ans. 4059 miles.
4. What will 495 yards of cloth cost, at 11 dollars a
yard?— (Vide 75.) Am. §5445.
5. What will 569 hogsheads of molasses cost at $37
each? Ans. §21053.
MULTir LIGATION. '65
6. What will 451 bales of cotton cost at |53 per bale?
Alls. §23903.
7. There arrived in market 18 wagons during one
■week, each wagon containing 6 bales of cotton, worth
|56 per bale. How much was the cotton worth ?
Ans. §6048.
8. Bought 8 bales of cotton, each bale containing
530 pounds, worth §0.13 per pound. How much did I
pay for it? . A^is. §551.20.
9. I purchased at one time 3 bales of cotton, each
bale weighing 554 pounds, at §0.11 per pound; at
another time, 5 bales, each weighing 535 pounds, at
§0.12 per pound. What was the whole worth ?
Ans. §503.82.
10. I sold the cotton in the preceding example at
§0.115 per pound. Did I make or lose, and how much?
Ans. lost §5.065.
, 11. I bought 27 hogsheads of molasses, each contain-
ing 63 gallons, at §0.53 per gallon. How^ much did 1
pay for it? Ans. §901.53.
12. I sold a sack of hops, weighing 396 pounds, at
§0.113 per pound. How much did I get for it?
Ans. §44.748.
13. What cost 5342 barrels of flour at §8.50 per
barrel ? A71S. §45407.
14. One chest of tea contains 69 pounds, and costs
§0.299 per pound; another chest contains 74 pounds,
and costs §0.274 per pound. How much is lost by sell-
ing the whole at §0.28 per pound? Ans. §0.867
15. What cost 169 boxes of oranges at §6.71 per
box? Ans. §1133.99.
66 MULTIPLICATIOX.
16. What cost 357 barrels of potatoes at §2.47 per
barrel? Ans. ^SS1.19.
17. I purchase 243 casks of butter, each containing
57 pounds, at §0.34 per pound, and sell the same at
S0.40 per pound. How much do I gain ?
Ans, §831.06.
18. My vineyard produces 5342 bottles of wine, at a
cost of §0.37 per bottle, and I retail the wine at §1.25
per bottle. How much do I clear? Ans. §4700.96.
19. Two passenger trains of cars meet at a station,
and each train runs, on an average, 37 miles an hour.
Suppose them to start at the same time, how far apart
will they be at the end of 17 hours ?
Ans. 1258 miles.
20. A passenger train and a freight train of cars
start from a given station and run in the same direction.
The passenger train moves at the rate of 37 miles an
hour, and the freight train 19 miles an hour. How far
apart will they be at the end of 13 hours?
A71S. 234 miles.
How far apart had they run in different directions?
Ans. 728 miles.
21. A drover bought 357 oxen, at the rate of §47 a
head. In going to market, 12 oxen fell through a bridge
and were killed. The cost of driving the remainder to
market was §5 a head. The oxen were then sold at an
average of §49 each. What was the loss ?
A71S. §1599.
22. How much would have been gained by selling the
oxen in the preceding example at §55 a head?
Ans.Un.
MULTIPLICATION.
67
MERCHANTS' BILLS.
83. A Bill is a written statement of articles bought
or sold, their prices, entire cost, etc.
Buffalo, April 26, 1865.
Messrs. J. H. Reed & Co., Chicago, 111.,
Bought of D. Ransom & Co.,
Terms cash. 121 Main Street.
10
5
Gross Trask's Magnetic Ointmeut, @ $21
" Ransom's Hive Syrup Tolu, @ $24
. $210
. 120
. 160
. 160
. 95
00
00
5
5
" Mrs. Winslow's Soothing Syrup, @ $32
" Brown's Bronchial Troches, @ $32
00
00
5
" Judson's Mt. Herb Pills, @ $19
00
$745
00
Received payment. ,
D. Ransom & Co.,
By John M. Sabin, Cl'k.
10 Cases per Lake Shore Railroad.
(2.)
New York, Jan. 1, 1865.
Mr. James Bryant, Mobile, Ala.,
Bought of A. Smith & Co.
24
Pounds of Tea, @ $0.63
$15jl2
55
Barrels of Potatoes, @ $5
275 00
98
Pounds of Raising @ $0.45
41
807
85
85
Barrels of Shad, @ $9.50
50
^1139
47
6'p^i-'i
^
68 MULTIPLICATION.
(8.)
New York, Jan. 1, 1865.
Mr. Lewis Rollings,
• To Otis Howe & Co., Dr.
1864.
Sept. 13. To 30 doz. Wool Hose @ $1.50
" 13. " 6 prs. Gloves @ $0.75
" 13. " 4 doz. Napkins @ $1.20
Oct. 12. " 4 " Shirt Bosoms @ $2.40
" 12. " 22 yds. Drilling : .@ $0.10
" 12. " 6 " Broadciotli @ $4.00
$90.10
Received payment.
S. Billings,
For Otis Howe & Co.
- • (4.)
^ St. Louis, July 15, 1865
Mr. S. H. DeCamp.
To A. T. Campbell, Dr.
1865.
Jan. 5. To 12 doz. Scythes @ $10.00
" 9. " 6J " Hoes @ $7.00
Feb. 1. " 10 " Rakes @ $1.80
March 4. " 3 Plows @ $11.00
" 4. " 7 doz. Pitchforks @ $9.50
" 1. " 9 « Padlocks @ $24.00
« 12. " 1 Coffee-mill @ $5.00
July 15. Settled by due bill $504.00
A. T. Campbell.
DIVISION. 69
DIVISION.
84. Division is the operation of finding how many
times one number is contained in another.
85. The number to be divided is called the dividend.
80. The member by which to divide is called the di-
visor.
8T. The number of times the divisor is contained in
the dividend is called the quotient.
88. IVlien there is a number left after dividing, it is
called the remainder.
SIGN.
89. The sign -^- is known as the sigri of division, and
when placed between two numbers, it signifies that the
former is to be divided by the latter. Thus, the ex-
pression,
40-f-5=8,
is read, forty divided by five equal eight, or five in forty
eight times.
OO. Operations in division are carried on under two
forms. Thus :
(1.) (2.) (3.)
5)13 5)13(2 13 1 5
-:. o 10 10 T
2-3 — —
3 3
70
DIVISION.
The first is known as the method of Short Division.
The second is known as the method of Long Di-
vision.
In either of these forms, 13 is the dividend, 5 is the
divisor, 2 is the quotient, and 3 the remainder.
In the operations of Long Division, the divisor is some-
times written as in (3.)
Ol. The remainder added to the product of iJie divisor
and quotient produces the dividend. Thus :
5x2+3=13.
(1.) Any number divided by itself produces 1 ; and
(2.) Any number divided by 1 produces the number
itself. Thus, 7^7=1, and 7-^1=7.
TABLE OF
division.
2-4-2=1
3 -f-3 = 1
4 -r-4= 1
5-4-5 = 1
4-^2 = 2
6 -r-3 = 2
8-^4 = 2
10 -^ 5 = 2
6h-2 = 3
9^3 = 3
12 -T-4 = 3
15 -=- 5 = 3
8-^2 = 4
12 -H 3 = 4
16 --4 = 4
20 H- 5 = 4
10 -i- 2 = 5
15 -V- 3 = 5
20 ^ 4 = 5
25 -4- 5 = 5
12 -V- 2 = 6
18 -V- 3 == 6
24 -H 4 = 6
30 -4- 5 = 6
14 ~ 2 = 7
21 -4- 3 = 7
28 ~ 4 = 7
35 -4- 5 = 7
16 -^ 2 = 8
24 -^ 3 = 8
32 -T- 4 = 8
40 -4- 5 = 8
18 -r- 2 = 9
27 -r- 3 = 9
36 ~ 4 = 9
45 -4- 5 = 9
6 -=-6 = 1
7 -T-7 = 1
8-4-8 = 1
9-4-9 = 1
12 -- 6 = 2
14 -^ 7 = 2
16 ~ 8 = 2
18 -4- 9 = 2
18 -^ 6 = 3
21 -4- 7 = 3
24 H- 8 = 3
27 -T- 9 = 3
24 ^ 6 = 4
28 -^ 7 = 4
32 -4- 8 = 4
36 -4- 9 = 4
30 ^ 6 = 5
35 -i- 7 = 5
40 -4- 8 = 5
45 -4- 9 = 5
36 -T- 6 = G
42 -v- 7 = 6
48 -T- 8 = 6
54 -4- 9 = 6
42 ~ 6 = 7
49 -V- 7 = 7
56 -4- 8 = 7
63 -4- 9 = 7
48 -f- 6 = 8
56 -^ 7 = 8
64 -^ 8 = 8
72 -4- 9 = 8
54 -r- 6 = 9
63 -^ 7 = 9
72 -:- 8 = 9
81 -^ 9 = 9
DIVISION. 71
SHORT DIVISION.
02. To divide a number by a number less than 10.
EXAMPLES.
(1.)
(2.)
(3.)
7)2499
5)35530
3)3698
357 7106 1232-2
7 is contained in 24 units of the tldrd order, 3 units
of the same order and 3 over. Write down the 3 units
of the third order, and prefix the 3 over to the 9 units
of the second order, making 39.
7 is contained in 39 units of the second order 5 units
of the same order and 4 over. Write down the 5 units
of the second order, and prefix the 4 over to the 9 units
of the first order, making 49.
7 is contained in 49 units of the first order , 7 units of
the same order. Then 357 is the quotient. — (Vide 77,
Ex. 1.)
In the second example, we may then say at once, 5 in
35 give 7; 5 in 5 give 1 ; 5 in 3 give 0; 5 in 30 give 6.
—(Vide 77, Ex. 4.)
In the third example, 3 in 3 give 1 ; 3 in 6 give 2 ; 3
in 9 give 3 ; 3 in 8 give 2, and 2 remainder. Hence-,
B u L E .
1. Write the divisor on the left of the dividend, as in
90,(1.)
2. Find hoiv many times the divisor is contained in
the left-hand figure, or, if that is smaller than the divisor^
72
DIVISION.
in two of the left-hand figures of the dividend, mid ivrite
the quotient diredhj under the figure of the lowest order
used.
3. Prefix the remainder to the figure of the next lower
order, and divide as before, continuing the tvorJc till all
the figures have been used. The result is the quotient re-
quired.
Remark 1. — The final remainder, if tliere is any, is to be written
as in 90, (1.)
(4.) (5.) (6.)
2)606958 7)68980247 9)1111111101
303479 Vide 77, Ex. 7.
Vide 77, Ex.
7.
Divide 1368093 by 3.
Ex.9.
.8.
Divide 259736 by 8.
Ex. 10.
9.
Divide 8999991 by 9.
Ex. 12.
10.
Divide 5208 by 3.
Ex.2.
11.
Divide 38024 by 7.
Ex. 13.
12.
Divide 999999 by 7.
Ex. 14.
13.
Divide 13684 by 4.
Ex. 16.
14.
Divide 340974 by 9. •
Ans. 37886.
15.
Divide 10101 by 3.
Ans. 3367.
16.
Divide 270192 by 6.
'Ans. 45032.
17.
Divide 3530380 by 5.
Ans. 706076.
18.
Divide 4236456 by 6.
Ans. 706076.
19.
Divide 4942532 by 6.
Ans.
20.
Divide 818181 by 9.
Ans. 90909.
21.
Divide 103701 by 3.
Ans. 34567.
22.
Divide 2751840 by 1, 2, 3, 4,
etc.
23.
Divide $4725 by 4.
Jlem. §1.
24.
Divide 13257 by 8.
Pern. |1.
DIVISION.
25.
Divide $4321 by 2.
Rem. $1.
26.
Divide 16721 by 3.
Rem. $1.
27.
Divide $3454 by 5.
Rem. $4.
28.
Divide $2348 by 7.
Rem. $3.
29.
Divide 4532 by 9.
Rem. 5.
30.
Divide 17623 by 3.
Rem. 1.
31.
Divide 30407 by 5.
Rem. 2.
32.
Divide 40321 by 2.
Rem. 1.
83.
Divide 76541 by 9.
'Rem. 5.
34.
Divide $451 by 5.
Rem. $1.
73
* Remark 2. — The divisor may be written under the remainder
with a line between them, the whole being considered as forming
a part of the quotient. Thus, (vide 134,)
(35.) (36.) (37.)
2)7 3)16 8)245
3i 5i 81f
The answer of (35) is read three and one half. —
(Vide 8.)
The answer of (36) is read five and one third.
The answer of (37) is read eighty-one and two thirds.
88.
Divide 4725 by 4.
Ans. 1181 J.
39.
Divide 8890 by 4.
Ans. 972f .
40.
Divide 5341 by 5.
Ans. 1068J.
41.
Divide 8459 by 5.
Ans. 691f.
42.
Divide 3005 by 6.
Ans. 5001.
43.
Divide $451 by 2.
Ans. $225J.
44.
Divide $650 by 3.
Ans. $216f .
45.
Divide $121 by 4.
Ans. $301.
46.
Divide $154 by 4.
Ans. $38f .
47.
Divide $3459 by 7.
7
Ans. $4944.
74 DIVISION.
48. Divide P258 by 8. ^^s. *407f .
49. Divide $1111 by 9. Ans. |123|.
50. Divide $2222 by 5. Ans. §444f .
51. Divide $3333 by 6. Ans. $555|.
52. Divide $4444 by 7. Ans. $634f .
53. Divide $5005 by 8. Ans. $625|.
54. Divide $3003 by 9. Ans. $333f .
55.- Divide $221 by 2. Ans. $110|.
Remark 8;— r-Ciphers may be annexed to tlie dividend, and the
division continued till there is no remainder, or it may terminate
at any convenient point. The figures of the quotient, after annexing
ciphers, are decimals. Thus,
(56.) (57.) (58.)
4)375.00 3)$5678.000 9)$4573.000
93.75 $1892.6661 $508,111^
The answer of (57) is 1892 dollars 66 cents 6 mills
and two-thirds of a mill.
59. Divide 4725 by 4. ^ns. 1181.25.
60. Divide 3890 by 4. Ans. 972.5.
61. Divide 5341 by 5. ^ws. 1068.2.
62. Divide 3459 by 5. Ans. 691.8.
63. Divide 3005 by 6. Ans. 500.83|.
64. Divide $451 by 2. Ans. $225.50.
65. Divide $4357 by 6. ^m. $726.166|.
66. Divide $1 by 2, 3, 4, 5, 6, 7, 8, 9.
Ans. $0.50, $0,331, $0.25, $0.20, $0.166|, $0.142f,
$0,125, $0.111f
03.' When the divisor, dividend, or both contain
decimals,
1. If (he numher of decimal figures in the divisor
DIVISION. 75
exceeds that in the dividend^ make it equal in both hy
annexing ciphers as decimals to the dividend.
2. Divide as in 92, and place the decimal point in the
quotient, so as to cause as many figures to stand on its
right as all the decimal figures used in the dividend exceed
those in the divisor, supplying any deficiency of quotient
figures hy prefixing ciphers. — (Vide 81.)
EXAMPLES.
Divide 249.9 by 7.
OPERATION.
7)249.9
35.7
Here the dividend has one decimal figure, the divisor
none.
2. Divide 2499 by .7.
OPERATION.
.7)2499.0 (Vide 93, 1.)
3570
Here the dividend has one decimal figure, and the
divisor one. The quotient is an integral number.
3. Divide 24.99 by .07.
OPERATION.
.07)24.99
357
Here the dividend has two decimal figures, and
76 DIVISION.
the divisor two, and the quotient is then an integral
number.
4. Divide 249.9 by .0007.
OPERATION.
.0007)249.9000 (Vide 93, 1.)
357000
Here the dividend has four decimal figures, and"^ .3
divisor four. „
5. Divide .2499 by 7.
OPERATION.
7).2499
.0357 (Vide 93, 2, last clause.)
Here the dividend has four decimal figures, the divisor
none. One cipher had to be prefixed to the three quo-
tient figures.
6. Divide 52.08 by .03.
OPERATION.
.03)52.08
1736 Ans.
.03
52.08 Proof
'. (Vide 91.)
7.
8.
9.
10.
11.
Divide 375 by A
Divide 430 by .008
Divide 4725 by .4
Divide 32570 by .08
Divide 43.21 by .002
Ans. 937.5
Ans. 53750,
Ans. 11812.5
Ans. 407125.
Ans. 21605.
DIVISION. 77
12. Divide 7.621 by .0003. Am. 25403.331.
13. Divide .00214 by .002. Am. 1.07,
14. Divide .03017 by .007. Am. 4.31.
15. Divide 93.276 by .007. Am. 13325.142f .
16. Divide 48.35 by .005. Am. 9670.
17. Divide 14535 by 9, .9, .09, .009, .0009, .00009,
and .000009. Am. 1615, 16150, 161500, etc.
B4:. To divide an integral number or decimal by a
UMt of any order, that is, by 10, 100, 1000, etc.,
T Remove the decimal point of the dividend as many
figures to the left as are indicated hy the ciphers in the
divisor.— {VidiQ 38, II.)
EXAMPLES.
1. Divide 3570 by 10, (vide 78, Ex. 2.)
Am. 357.
2. Divide 35700 by 100. Am. 357.
3. Divide 357 by 10. Am. 35.7.
4. Divide 357 by 100. Am. 3.57.
5. Divide 35.7 by 10. Am. 3.57.
6. Divide 31.4 by 100. Am. .314.
7. Divide §451.30 by 10. Am. $45.13.
8. Divide §4321 by 100. Am. §43.21.
9. Divide §23456 by 1000. Am. §23.456.
10. Divide §1.00 among 10 men, 100 men, 1000
men. Am. §0.10, §0.01, §0.001.
11. Divide §45.20 equally among 10 men; 100 men;
1000 men. -Aws. §4.52; §0.452; §0.0452.
12. Divide §3120 equally among 10000 men.
A71S. §0.312.
78
DIVISION.
LONa DIVISION.
05. To divide a number by any number greater
than 10.
EXAMPLES.
1. Divide 1716 by 11.
(1.) We find how many times
11 is contained in 17, and write
the quotient figure 1 to the right
of the dividend.
(2.) Multiply the divisor 11
by the quotient figure 1, and
place the product 11 under 17.
(3.) Subtract 11 from 17, and
OPERATION.
11)1716(156
11
61
55
66
Am.
place the remainder 6 under 11.
(4.) Bring down the next q
figure 1 of the dividend to the
right of the 6, and proceed with the result 61 exactly as
with 17. That is,
(1.) 11 in 61 are 5, which is placed in the quotient.
(2.) Then 5 times 11 are 55, which is placed under
61. -
(3.) 55 from 61 are 6.
(4.) Bring down the next figure 6 of the dividend.
In the resulting number 66 the divisor 11 is contained
Hence,
6 times, with no remainder.
RULE,
(1.) Find how many times the whole divisor is con-
tained in the same niuiiher of figures on (he left of the
DIVISION. 79
dividend considered as representing so many units ; or,
if this number is smaller than the divisor, find how
many times the divisor is contained in a number of figures
of the dividend^ greater by one than the number of figures
in itself, and place the quotient on the right of the divi-
dend.
(2.) Multiply the whole divisor by this quotient figure,
and place the p)roduct under the figures of the dividend
just compared ivith the divisor.
(3.) Subtract the product from the figures above it.
(4.) To the right of the difference bring doion the next
figure of the dividend.
(5.) If the new dividend is noiv smaller titan the divi-
sor, place a cipher in the quotient, and bring down the next
figure of the dividend; but if the dividend is equal to or
larger than the divisor, proceed in obtaining the second
figure of the quotient as ivith the first, and so on till all
the figures are brought down.
Remark 1. — Notice that no product can be greater than the
figures of the dividend above it.
Remark 2. — Notice that no difference can be equal to or greater
than the divisor.
2. Divide 1716 by 12. Ans. 143.
3. Divide 1716 by 13. Ans. 132.
4. Divide 3360 by 14. Ajis. 240.
5. Divide 3360 by 15. Ans. 224.
6. Divide 3360 by 16. Ans. 210.
7. Divide 7429 by 17. Ans. 4S7.
8. Divide 7429 by 19. Ans. 391.
9. Divide 7429 by 23. Ans. 323.
10. Divide 26691 by 21. Ans. 1271.
80 DIVISION.
11. Divide 7371 bj 91. Arts. SI.
(12.) (13.)
91)7371(81 184)56488(307
728 552
91 1288 (Vide Rule, 5.)
91 1288
0 0
14. Divide 56488 by 92. Ajis. 614.
15. Divide 84487 by 97. Ans. 871.
16. Divide 24108 by 98. Ans, 246.
17. Divide 24108 by 246. Ans. 98.
18. Divide 5768 by 56. A^is. 103.
19. Divide 48071 by 53. . Ans. 907.
20. Divide 2448 by 16, 17, and 18.
Ans. 153, 144, and 136.
21. Divide 6072 by 22, 23, and 24.
Ans. 276, 264, and 253.
22. Divide 5544 by 36, 44, and 56.
Ans. 154, 126, and 99.
23. Divide 12259 by 13, 23, and 41.
Ans. 943, 533, and 299.
(24.) (25.)
509)461663(907 999)708291(709
4581 6993
3563 8991
3563 8991
26. Divide 73647 by 147 and 167.
Ans. 501 and 441.
DIVISION. 81
27. Divide 36942 by 131 and 141.
Ans. 282 and 262.
28. Divide 526587 by 191 and 919.
Ans. 2757 and 573.
29. Divide 3203055 by 711 and 901.
Ans, 4505 and 3555.
30. Divide 193581 by 137 and 157.
Ans. 1413 and 1233.
31. Divide 404278 by 431 and 134
Ans. 938 and 3017.
32. Divide 670592745 by 54321.
OPERATION.
670592745154321
54321 12345
127382
108642 (Yide 90, (3.)
187407
162963
244444
217284
271605
271605
33. Divide 45780840 by 82488. Ans. 555-
34. Divide 1234549380 by 12345. Ans. 100004.
35. Divide 497477808387 by 987.
Ans. 504030201.
36. Divide 49419533647761876 by 9876.
Ans. 5004003002001.
82 DIVISION.
Remark 3. — If there are cipliers on the right of the divisor,
they may be neglected in the operation, together with a like num-
ber of figures on the right of the dividend.
37. Divide 34705 by
700.
38. Divide 34705 by
760.
OPERATIONS
3.
(37.)
(38.)
7'00)347'05
76^0)3470'5(45
49—405
304
430
380
505
The remainder of (37) is 405, and that of (38) is 505.
39. Divide 3521 by 200. Rem, 121.
40. Divide 3521 by 30. Bern. 11.
41. Divide 4561 by 40. Rem. 1.
42. Divide 3457 by 50. Rem. 7.
43. Divide 7654 by 70. Rem 24.
44. Divide 3420 by 90. Arts. 38.
45. Divide 1716000 by 12000. Ans. 143.
46. Divide 1716000 by 130000. . Rem. 26000.
47. Divide 3360000 by 1400000. Rem. 560000.
48. Divide 73647000 by 1470000. Rem. 147000.
49. Divide 123456789 by 2000000.
Rem. 1456789.
Remark 4. — In decimals observe the Rule in 93 and 92, Rem. 3.
50. Divide 143 by 25.
51. Divide 143 by 125.
DIVISION. 1
(50.)
25)143.00(5.72
125
OPERATIONS.
(51.)
125)143.000(1.144
125
180
175
50
50
180
125
550
500
0
500
500
83
52. Divide 206.166492 by 4.123.
53. Divide .102048 by 3189.
OPERATIONS.
(52) (53)
206.166492| 4.123 .102048|3189
206.15 50.004 9567 .000032
16492 (Yide 90, (3.) 6378
16492 6378
54.
Divide 4567 by 25, 125, and 625.
-
Ans. 182. 68, etc.
55.
Divide 1460 by 16, 32, and 64.
Ans. 91. 25, etc.
56.
Divide 7623 by 40, 80, and 1600.
Ans. 190. 575, etc.
57.
Divide 143 by 15.
58.
Divide 100 by 24.
84 I DIVISION.
' OPERATIONS.
(57.) (58.)
15)143.00(9.53 X 24)100.00(4.16||
135 \ 96
80 (Vide 92, Rem. 2.) 40
75 24
50 , 160
45 ^ 144
5 16
59. Divide 140.913 by 3.07 and 45.9.
60. Divide 281.826 hj 30.7 and 4.59.
Ans. 9.18 and 61.4.
61. Divi^ .0949416 by 1.326. Ans. .0716.
62. Divide 152.^756 by .1234. Ans. 1234.
63. Divide $14|.80 equally among 40 men.
Ans. 13.57 each.
64. Divide §714.00 equally among 200 men.
Ans. 13.57 each.
65. Divide §2.499 equally among 7 men.
Ans. §0.357 each.
66. Divide $26137.65 equally among 365 men.
Ans. $71.61 each.
67. Divide $58.59 equally among 63 men.
Ans. $0.93 each.
68. Divide $10043.712 equally among 144 men.
Ans. $69,748 each.
69. Divide $176.25 equally among 47 men.
Ans. $3.75 each.
DIVISION. 85
70. Divide $48910.95 equally among 369 men.
Ans. $132.55 each.
71. Divide |20.631 equally among 69 men.
Ans. fO.299 each.
96. In Division the quotient is always of the same
name as the dividend; and the divisor in the operation
must be considered as simply a number without name.
Thus, $3525^47-?75.— (Vide 167.)
PRACTICAL APPLICATIONS.
1. If 47 oxen cost $3525, what will one ox cost?
Ans. $75.
2. If a man can walk 391 miles in 17 days, how far
can he walk in one day ? Ans. 23 miles.
3. A vessel has sailed 4059 miles in 9 days. What
was the average rate per day? Ans. 451 miles.
4. I sell cloth to the amount of $5445, at the rate of
$11 per yard. How many yards are sold ?
Ans. 495 yards.
5. Eleven men divide 5445 boards equally among
them. How many does each have ?
Ans. 495 boards.
6. If 5445 marbles are equally divided among 495
boys, how many does each have ? Ans. 11 marbles.
7. If 495 yards of cloth are sold for $5445, what is
the average price per yard? Ans. $11.
8. If 569 hogsheads of molasses cost $21053, what is
the price per hogshead ? Ans. $37.
9. I buy molasses to the amount of $21053, at the
rate of $37 per hogshead. How many hogsheads are
bought ?
86 DIVISION.
10. If $21053 are equally divided among 37 men,
how many dollars does each man receive?
11. There arrived in market a number of wagons,
each loaded with 12 bales of cotton, worth |56 per bale.
The cotton was sold for $12096. How many wagons
came to market?
12. A train of 18 cars brought down the Mobile and
Ohio Railroad cotton to the amount of $12096, at the
rate of $56 per bale. How many bales did each car
carry, supposing the cotton equally distributed?
13. Eighteen cars took into Memphis $12096 worth
of cotton, each car being loaded with 12 bales. What
was the cotton valued at per bale?
14. The exact distance round the earth at the equa-
tor, is 24897.883 miles, which is divided into 360
degrees. What is the length of each degree ?
Ans. 69.161 miles.
15. How long would it take a man to travel round
the earth at the equator, traveling at the rate of 60
miles per day ? A718. 414.9647 J days.
16. Since the earth turns on its axis once in 24 hours,
how far does any point on the equator move per hour ?
Per minute? Per second?
Last Ans. .2881699 mHeS.
17. When the moon is 240,000 miles from the earth,
how long would it take a car to make the distance, at
the rate of 45 miles per hour ? Ans. 5333 Jf hours.
18. When the sun is 95,000,000 miles from the earth,
how long would a car be occupied in making the dis-
tance, running at the rate of 57 miles per hour ?
Ans. 16666661 f hours.
PROPERTIES OP INTEGRAL NUMBERS. 87
t
PROPERTIES OF INTEGRAL NUMBERS,
DEFINITIONS.
97. Any number is exactly divisible by another when
the q^tient is an integral number. Thus, 51 is exactly
divisible by 17, because the quotient 3 is an integral
number. — (Vide 7.) But 51 is not exactly divisible by
25, because the quotient, 2j^^ contains a fractional
number. — (Yide 8.)
98. An EVEN NUMBER is one that is exactly divisible
by 2. The even numbers, then, are 2, 4, 6, 8, 10, etc.
99. An ODD NUMBER is one that is not exactly divisible
by 2. The odd numbers are, then, 1, 3, 5, 7, 9, 11, etc.
100. A PRIME NUMBER is One that is exactly divisible
by no number except itself and 1.
Remark 1. — No even number^ except 2, can he a prime number.
Remark 2. — The prime numbers less than 100, are
1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
101. A COMPOSITE NUMBER is the product of two or
more prime numbers. — (Vide 74.) Thus,
24=2X2X3X2.
Remark. — Any composite number is exactly divisible by either of its
prime factors, or by the product of any two or more of them. Thus,
30 is divisible by 2, 3, or 5, or by 2X3=::6, 2X5=10, 3X5=15.
88 PROPERTIES OP INTEGRAL NUMBERS.
10!S. Any two numbers are said to be prime with
respect to each other when no factor of the one is a factor
of the other. Thus, 26=2X13, and 63-=3x3x7, are
prime with respect to each other.
103. A COMMON DIVISOR of two or more numbers is
a number which exactly divides each of them. Thus, 2 is
a common divisor of 12 and 20.
Remark. — Numbers which are prime with respect to each pther, have
no common divisor other than 1.
104. The GREATEST COMMON DIVISOR of two or more
numbers is the greatest number which exactly divides
each of them. Thus, 4 is the greatest common divisor of
12 and 20.
105. A MULTIPLE of a number is the product of that
number by some other number. Thus, the multiples of
2 are 4, 6, 8, 10, etc. The multiples of 5 are 10, 15,
20, etc. The multiples of 36 are 72, 108, 144, etc.
106. A COMMON MULTIPLE of two or more numbers
is a number that is exactly divisible by each of them.
Thus, 84 is a common multiple of 2, 3, 6, and 7, because
it is exactly divisible by each of them.
107. The LEAST COMMON MULTIPLE, of two or more
numbers is the least number that is exactly divisible by
them. Thus, 84 is the least common multiple of 7, 12,
and 14, because it is the least number exactly divisible
by each of them. ^
Remark. — Every number is the least common multiple of all its prime
factors, or of all its divisors. Thus, 42 is the least common multiple
of 2, 3, and 7, its prime factors, or of 2, 3, 6, 7, 14, and 21, all its
divisors, except 1 and 42, which may be included.
plloperties of integral numbers. 89
Properties of the Integral Numbers, from 2 to 13,
inclusive.
108. Every number, and no other, is exactly divisible
by 2, wJiose rigid-hand figure is a 0, or an even number.
Thus,
120, 374, 336, 95678, are exactly divisible by 2.
341, 753, 255, 24683, are not exactly divisible by 2.
101]>.. Every number, and no other, is exactly divisible
by 3, the sum of whose digits is divisible by 3. Thus,
1035, 1305, 1350, 5031, are exactly divisible by 3,
because 1+0+3+5=9 is exactly divisible by 3.
liO. Every number, arid no other, is exactly divisible
by 4, tvhose two right-hand figures, considered as repre-
senting a number less than 100, are exactly divisible
by 4. Thus,
120, 324, 428336, are exactly divisible by 4, because
20, 24, 36, are exactly divisible by 4. But
121, 463, 782354, are not exactly divisible by 4, because
21, 63, 54, are not exactly divisible by 4.
111. Every number, and no other, is exactly divisible
by 5, whose right-hand figure is a 0, or a S. Thus,
3240, 4535, etc., are exactly divisible by 5.
112. Every numher, and no other, is exactly divisible
by 6, which is even, and exactly divisible by 3. Thus,
534, 2136, 6348, etc., are exactly divisible by 6.
113. Every number, and no other, is exactly divisi-
ble by 7, when the difference, of the sum of the products
produced by multiplying the figures standing in the odd
PERIODS, by 1, 3, 2, respectively, counting from right to
90 PROPERTIES OF INTEGRAL NUMBERS.
leftj and the sum of the products of the even periods,
found in the same way, is exactly divisible by 7. Thus,
6, 883, 905,
is exactly divisible by 7, because the difference between
the sum of 5+18+6=29=5Xl+9x2+6Xl=products
in odd periods; and of 3+24+16=43=3x14-8X3+
8x2=products in even periods, which is 14, is exactly
divisible by 7.
Remark 1. — In practice, the multiplication and addition should
be done mentally, and as soon as the sum amounts to 7 or more
omit the 7, and use the surplus. Thus,
5 and 4 are 2 and 6 are 1.
3 and 3 are 6 and 2 are 1.
Remark 2. — If the figure 7 or a 0 is found in the number it need
not be multiplied. Thus, 34, 707, 205.
5 and 4 are 2 and 4 are 6 and 2 are 1;
and the number is not exactly divisible by 7, but will have a re-
mainder of 1.
114. Every number, and no other, is exactly divisi-
ble by 8, whose three right-hand figures, considered as
representing a number less than 1000, are exactly di-
visible by 8. Thus,
387528; 135752, etc., are exactly divisible by 8, because
528; 752, are exactly divisible by 8.
115. Every number, and no other, is exactly divisi-
ble by 9, the sum of whose digits is divisible by 9.
Thus,
14454, 44451, 54414, etc., are exactly divisible by 9,
because 4+4+4+1+5=18, is exactly divisible by 9.
IIG. Every number, and no other, is exactly divisi-
ble by 11, when the difference of the sujn of the figures
PROPERTIES OF INTEGRAL NUMBERS. 91
standing in the odd places^ and the sum of the figures
standing in the even places, counting either way, is
exactly divisible by 11. Thus, 64532146723 is exactly
divisible by 11, because the difference between 27=6+5
+2+4+7+3, sum in odd places, and 16=4+3+1+6
+2, sum in even places, which is 11, is exactly divisible
by 11.
117. Every number, and no other, is exactly divisible
by 12, which is exactly divisible by 3 and 4.
118. The proposition in reference to 7 is true also of
13, if the numbers 1, 10, 9, are substituted respectively
for 1, .3, 2.
Remark 1. — Any number composed of two full periods^ with the same
figures in each, and the same arrangement, is exactly divisible by 7 or 13.
144144, 305305, 101101, 352352, 111111, are eacli exactly divisi-
ble by 7 or 13.
Remark 2. — The number 1001, or any multiple of it, is exactly di-
visible by 7, 11, or 13.
FACTORING.
119. To resolve a number into its prime factors,
1. Divide the number by any prime number that will
exactly divide it.
2. Divide the quotient by any prime number that will
exactly divide it.
3. Continue thus till a quotient of 1 is obtained.
The several divisors are the prime factors sought,
EXAMPLES.
1. What are the prime factors of 13860 .
92
PROPERTIES OF INTEGRAL NUMBERS.
Vide 108.
Vide 109.
Vide 111.
Vide 113.
Vide 116.
OPERATION.
2 13860
2 6930
, 3 3465
5 1155
7
11
231
33
The prime factors of
13860 are 2, 2, 3, 3,
5, 7, 11.
2. What are the prime factors of 4, 6, S, 12, 14, and
16? A)is. 2 and 2, etc.
3. What are the prime factors of 18, 20, 22, 26, 28,
and 30 ? A71S. 2, 3 and 3, etc.
4. What are the prime factors of 32, 34, 36, 38, 40,
and 42? Ans. 2,2, 2, 2, and 2.
5. What are the prime factors of 44, 45, 46, 48, 49,
and 50 ? Ans. 2, 2, and 11, etc.
6. What are the prime factors of all the composite
numbers between 50 and 100 ?— (Vide 100, Rem. 2.)
A71S. 51=3X17, 52=2X2X13, etc.
7. What are the prime factors of 121, 144, 169, 196,
225, 256, 289, 324, 361, 400, 441, and 484.
Ans. 121=11X11, etc.
8. What are the prime factors of 143, 187, 209, 253,
319, 341, 451, 561, 737, 913, and 957.
Ans. 143=11X13, etc.
PROPERTIES OF INTEGRAL NUMBERS. 93
9. What are the prime factors of 371, 413, 469, 497,
623, 1001, 3003, 1309, 1463, and 1771?
Last Ans. 7, 11 and 23.
10. What are the prime factors of 2940, 4620, 5460,
7140, 7690, 930, 1330, 1610, 6020, and 4350?
Last Ans. 2, 3,5, 5 and 29.
11. Wliat are the prime factors of 744 ?
Ans. 2, 2, 2, 3 and 31.
12. What are the prime factors of 1680 ?
Ans. 2, 2, 2, 2, 3, 5 and 7.
13. What are the prime factors of 636 ?
Ans. 2, 2, 3 and 53.
14. What are the prime factors of 1080?
A71S. 2, 2, 2, 3, 3, 3 and 5.
15. What are the prime factors of 5000?
Ans. 2, 2, 2, 5, 5, 5, and 5.
16. What are the prime factors of 221, 299, and 387?
Ans. 13 and 17, etc.
17. What are the prime factors of 2431? A^is.
GREATEST COMMON DIVISOR.
ISO. To find the greatest common divisor of two or
more numbers,
Resolve the numbers into their prime factors, and then
find the 2?roduct of such as are common to all the num-
bers. The result will be the greatest common divisor
sought.
EXAMPLES.
1. What is the greatest common divisor of 168, 420,
and 6006?
94
PROPERTIES OF INTEGRAL NUMBERS.
OPERATION
168
2
420
84
2
210
42
3
105
21
5
35
7
7
7
1
1
2
6006
3
3003
7
1001
11
143
13
13
The factors common to all the numbers are 2, 3, and
7. Hence, 2X3X7 = 42, is their greatest common
divisor.
2. What is the greatest common divisor of 2, 4, and
6 ? Ans. 2.
3. What is the greatest common divisor of 4, 6, and
8? ' Ans. 2.
4. What is the greatest common divisor of 6, 8, and
10 ? Ans. 2.
5. What is the greatest common divisor of 2, 4, and
8? ^7is. 2.
6. What is the greatest common divisor of 3, 6, and
9? Ans. 3.
7. What is the greatest common divisor of 4, 8, and
12? Ans. 4.
8. What is the greatest common divisor of 8, 12,
and 20? A7is.4. .
9. What is the greatest common divisor of 12, 18,
and 24? Am. 6.
L
PROPERTIES OF INTEGRAL TSTUMBERS. 95
10. What is the greatest common divisor of 24, 36,
and 48 ? Ans. 12.
11. AVhat is the greatest common divisor of 14, 21,
and 35 ? Ans. 7.
12. What is the greatest common divisor of 26, 39,
and 52? Aiis. IS.
13. What is the greatest common divisor of 16, 24,
and 48? ^ Ans.S.
14. What is the greatest common divisor of 252, 180,
and 288 ? Ans. 36.
• 15. What is the greatest common divisor of 144,
196, and 256 ? Ans. 4.
16. What is the greatest common divisor of 744,
1680, 636, and 1080 ?— (Vide 119, Examples 11, 12,
13, and 14.) Ans. 2X2X3=12.
17. What is the greatest common divisor of 375,
1100, 120, and 1440? Ans. 5.
18. What is the greatest common divisor of 780,
1560, 720, and 1008? Ans. 12.
19. What is the greatest common divisor of 144,
196, 256, and 324? Ans. 4.
20. What is the greatest ^common divisor of 143,
187, 209, and 253? Ans. 11.
21. W^hat is the greatest common divisor of 216,
408, 740, and 810 ? Ans. 2.
22. What is the greatest common divisor of 187,
209, 52, and 161 ?— (Vide 103, Rem.) Ans. 1 .
121. The greatest common divisor of two numbers is
the greatest common divisor of the less of the two num-
bers, and the difference found by taking the largest
multiple of the less number, which is smaller than the
96 PROPERTIES OF INTEGRAL NUMBERS.
greater, from the greater number. Thus, the greatest
common divisor of
36 and 136,
is also the greatest common divisor of
36 and 136— 108-=28,
108 being 3X36, the largest multiple of 36 less than
136. (Vide 105.) This must be so. For any divisor
of 36 also divides 36x3, and since it divides 36X3, if
it divides 36x3+28, it must divide 28. There can,
then, be no common divisor of 36, and 36x3+28, -which
is not a common divisor of 36 and 28. Therefore the
greatest common divisor of 36 and 28 is the greatest
common divisor of 36 and 136. Hence,
\'22. To find the greatest common divisor of two
numbers,
1. Write the numbers on a line far enough from each
other to draw one or two perpendicular lines between
them.
2. Find the greatest multiple of the smaller, that is less
than the larger, and subtract it from the larger number.
,i.--.3. Find the greatest multiple of the remainder, less
than the smaller number, and subtract it from the smaller
number.
4. Find the greatest multiple of this remainder, less
than the previous remainder, and subtract as before, and
continue the work till there is no remainder.
The last integral remainder will be the greatest common
divisor sought.
EXAMPLES.
1. What is the greatest common divisor of 36 and
136 ?
PROPERTIES OF INTEGRAL NUMBERS.
97
OPERATIONS.
(1.)
(2.)
36
28
8
8
136
108
28
24
36
28
8
8
136
108
28
24
0 4 Am. 0 4 Ans.
Remark. — The multiples are found precisely as in division, and
the quotient figures may be retained as in (1), or neglected as in (2).
2. What is the greatest common divisor of 285 and
465 ? Ans. 15.
3. What is the greatest common divisor of 532 and
1274 ? Ans. 14.
4. What is the greatest common divisor of 693 and
1815 ? Ans. 33.
OPERATIONS.
285
465
532$
21274
693
1815
180
285
420$
U064
429
1386
105
180
112]
L 210
264
429
75
105
981
L 112
165
264
30
75
14^
r 98
99
165
30
60
98
66
99
0
15
0
33
66
66
0
5. What is the greatest common divisor of 1054 and
1426? Ans. 62.
9
98 PROPERTIES OF INTEGRAL NUMBERS.
6. What is the greatest common divisor of 725 and
3175? Ans,2^.
7. What is the greatest common divisor of 4585 and
6685 ? Am. 35.
8. What is the greatest common divisor of 4605 and
5505? Ans.U.
9. What is the greatest common divisor of 636 and
1080? of 744 and 1680? of 972 and 1260? of 3471
and 1869 ? Ans. 12, 24, 36, and 267.
10. What is the greatest common divisor of 1137 and
9475? of 3447 and 9575? of 2359 and 8425? of 1903
and 4325 ? Ans. 379, 383, 337, and 173.
11. What is the greatest common divisor of 117869
and 137773? ^ns. 311.
Remark, — If there are inore than two numbers, ,find the greatest
common divisor of any two of them, and then of this divisor and a third
number, and so on till all the numbers have been used. The last integral
remainder will be the greatest common divisor of all the members.
12. What is the greatest common divisor of 285, 465,
and 35?— (Vide Ex. 2.) Ans. 5.
13. What is the greatest common divisor of 532,
1274, and 21?— (Vide Ex. 3.) Ans. 7.
14. What is the greatest common divisor of 1815,
693, 66, and 88?— (Vide Ex. 4.) Ans. 11.
15. What is the greatest common divisor of 620,
1116, and 1488 ? Ans. 124.
16. What is the greatest common divisor of 1054,
1426, and 2263? Ans. 31.
17. What is the greatest common divisor of 233, 587,
and 653?— (Vide 103, Rem.) Ans. 1.
18. What is the greatest common divisor of 739,
503, and 439? * Ans. 1,
I
E^
PROPERTIES' OF INTEGRAL NUMBERS. 99
19. What is the greatest common divisor of 97343,
139639, and 206193? Ans. 311.
LEAST COMMON MULTIPLE.
123. To find the least common multiple of two or
more numbers,
1. Write the numbers in a line, with convenient in-
tervals.
2. Divide by any prime number that ivill exactly divide
any two or more of them, and write the quotients and
numbers not exactly divisible in a line below the given
numbers.
3. Divide this line by any prime number as before, and
continue to divide till no prime number, except 1, will
exactly divide any two numbers in the line.
4. Multiply the divisors and numbers left in the last
line together.
The product will be the least common multiple
sought.
Remark, — If any of the given numbers will exactly divide any other
of them, it may be neglected in the operation, and not affect the result.
EXAMPLES.
1. What is the least common multiple of 4, 8, 9,
and 21.
OPERATION.
8, 9, 21
8, 3, 7
Hence, 3X8x3X7=504, is the least common mul-
tiple required. '
100
I'KOPEIITIES OF INTEGRAL NUMBERS.
The figure 4 is neglected in the operationj because it
will exactly divide 8.
2. What is the least common multiple of 8, 64, and
72? ^^s. 576.
3. What is the least common multiple of 26, 39, and
52 ? A71S. 156.
4. What is the least common multiple of 14, 56, and
196?
(2.)
2
64,
72
2
32,
36
2
16,
18
8,
9
OPE
13
RATIONS.
(3.)
39, 52
3, 4
A71S. 392.
(4.)
2
2
56,
196
28,
98
14,
49
2,
7
(2.) •
2X2X2X8X9=576.
(3.)
13X3X4=156.
(4.)
2X2X2X7X7=392.
5. What is the least common multiple of 2, 3, 4, 5,
6,7,8,9,12?
OPERATION.
2
5, 7, 8, 9, 12
2
5, 7, 4, 9, 6
8
5, 7, 2, 9, 3
X
5, 7, 2, 3, 1
6X7X2X3=2520 Ans
PROPERTIES OF INTEGRAL NUMBERS. 101
6. What is the least common multiple of 7, 8, and
14? of 5, 25, and 50? of 2, 4, and 6?
Ans. 56, 50, and 12.
7. What is the least common multiple of 4, 6, and 8 ?
of 6, 8, and 10 ? of 2, 4, and 8 ? Ans. 24, 120, and 8.
8. What is the least common multiple of 3, 6, and 9 ?
of 4, 8, and 12 ? of 8, 12, and 20 ?
Ans. 18, 24, and 120.
9. What is the least common multiple of 12, 18, and
24 ? of 24, 36, and 48 ? of 8, 32, and 16 ?
Ans. 72, 144, and 32.
10. What is 'the least common multiple of 8, 18, and
36 ? of 21, 42, and 14 ? of 7, 14, and 70 ?
Ans. 72, 42, and 70.
11. What is the least common multiple of 9, 18, and
27 ? of 12, 16, and 20 ? of 5, 10, and 15 ?
Ans. 54, 240, and 30.
12. What is the least common multiple of 2, 6, and
8 ? of 7, 14, and 21 ? of 3, 4, and 5 ?
Ans. 24, 42, and 60 ?
13. What is the least common multiple of 2, 3, and
4 ? of 2, 5, and 7 ? of 3, 7, and 11.
_ Ans. 12, 70, and 231.
14. What is the least common multiple of 14, 28,
and 98? of 8, 14, and 35 ? of 24, 25 and 32?
Ans. 196, 280, and 2400.
15. What is the least common multiple of 63, 12, 84?
of 54, 63, 81 ? of 21, 35, 84 ? Ans. 252, 1134, 420.
16. "\^niat is the least common multiple of Q6, 143,
55 ? of 144, 196, 128 ? of 18, 45, 63 ?
Ans. 4290, 56448, 630.
102 PROPERTIES OF INTEGRAL NUMBERS.
17. What is the least common multiple of 20, 35, 80 ?
of 16, 24, 56 ? of 26, 39, 65 ? Ans. 560, 336, 390.
18. What is the least common multiple of 34, 51, 85 ?
of 57, 95, 133? of 69,115,161?
Ans. 510, 1995, 2415.
19. What is the least common multiple of 8, 7, 10,
14 ? of 2, 6, 7, 29 ? of 14, 21, 35, 49 ?
Ans. 280, 1218, 1470.
20. What is the least common multiple of 272, 238,
204, 170 ? Ans. 28560.
21. What is the least common multiple of 11, 12, 13,
14, 15, 16, 17, 18, 19, 20 ? Aiis. 232792560.
134. To find the least common multiple of two
numbers,
(1.) Find the greatest common divisor of the numhei^s.
(2.) Divide one of the members by this divisor, and
multiply the quotient by the other.
The product will be the least common multiple of the
numbers.
EXAMPLES.
1. What is the least common multiple of 1903 and
4325 ?
OPERATION.
1903
1557
346
346
0
4325
3806
519
346
173
Then, 1903-v-173x4325-:47575 Ans,
Or, 4325-^173Xl903--47575 Aris.
PROPERTIES OF INTEGRAL NUMBERS. 103
2. What is the least common multiple of 3471 and
1869 ?— (Vide 122, Ex. 9.) Ans. 24297.
3. What is the least common multiple of 1137 and
9475 ?— (Vide 122, Ex. 10.) Ans. 28425.
4. What is the least common multiple of 3447 and
9575? J.7i.§. 86175.
5. What is the least common multiple of 2359 and
8425 ? Ans. 58975.
6. What is the least common multiple of 117869 and
137773? * ^ris. 52215967.
125. Practical Applications.
1. The diJBference between two numbers is 7, and the
less number is 25. What is the greater? Ans. 32.
2. The difference between two numbers is 19, and the
less number is 43. What is the greater? Ans. 62.
3. The difference between two numbers is 184, and
the less is 325. What is the greater? Ans. 509.
4. The difference between two numbers is 7, and the
greater is 32. What is the less? Ans. 25.
5. The difference between two numbers is 19, and the
greater is 62. What is the less? Ans. 43.
6. The difference between two numbers is 184, and
the greater is 509. What is the less? Ans. 325.
7. The difference between two numbers is 7, and the
less is 25. What is their sura? Ans. 57.
8. The difference between two numbers is 19, and
the less is 43. What is their sum ? Ans. 105.
9. The difference between two numbers is 184, and
the greater is 509. What is their sum ? Aiis. 834.
104 PROPERTIES or INTEGRAL NUMBERS.
10. The sum of two numbers is 105, and tlieir differ-
ence 19. What are the numbers ? Ans. 62 and 43.
(1.) Add the difference to the sum, and divide by 2.
(2.) Subtract the difference from the sum, and divide
by 2.
The result will be the numbers. Thus,
(1.) (2.)
105 105
19 . 19
2)124 2) 86
62=Greater. 43:^Lcss.
11. The sum of tAvo numbers is 5, and their difference
1.4. What are the numbers ? Ans. 3.2 and 1.8.
12. The sum of two numbers is 783, and their difference
141. What are the numbers ? A71S. 462 and 321.
13. The sum of two numbers is 46.4, and the greater
is 29.3. What is the less? Ans. 17.1.
14. There are ^47.32 in two boxes. One of them
contains §24.85. How much money in the other?
Ans. §22.47.
15. I have §323.67 in two purses. One of them
contains §125.63. How much in the other?
Ans. §198.04.
16. I have two purses. One of them contains
§198.04, which is more money than is in the other by
§72.41. How much does it contain ? Ans. 125.63.
17. My money is in two purses, both of which con-
tain §323.67, and there are §72.41 more in the one tlian
in the other. How many dollars in each purse ?
PROPERTIES • OF INTEGRAL NUMBERS. 105
18. Two men together own 3521.25 acres of land,
but one of them owns 45.75 acres more than the other.
How many acres does each man own ?
Ans. 1783.5 and 1737.75.
19. If to a certain number I add 45, the sum will be
223. What is the number? Ans. 178.
20. If from a certain number I take 34, the remain-
der will be 213. What is the number ? Ans. 247.
21. If to a certain number I add 14, and then sub-
tract 75 the result will be 268. What is the number?
Ans. 329.
22. If from a certain number I subtract 123, and
then add 329, the result will be 930. What is the num-
ber ? Ans. 724.
23. The divisor of a number is 45, and the quotient
is 73. What is the dividend? Ans. 3285.
24. The divisor of a number is 73, and the quotient
is 45. What is the dividend ?
25. The dividend is 3285, and the quotiei\t is. 73.
What is the divisor ?
26. The dividend is 3285, and the quotient is 45.
What is the divisor?
27. The multiplier is 45, and the \product 3285.
What is the multiplicand?
28. The multiplicand is 45, and the product 3285.
What is the multiplier ? • .^
29. If a certain number is divided by 321^ the quo-
tient will be 23. What is the number ? , . ^
30. If a cej^tain number is multiplied by 321, the
product will be 7383. What is the number?
31. If a certain number is multiplied by 4, and the
106 PllOPERTIES OF INTEGRAL NUMBERS.
product is then divided by 7, the quotient will be 16.
What is the number ? A^is. 28.
32. If a certain number is divided by 7, and the quo-
tient is then multiplied by 4, and the product increased
by 46, the sum diminished by 37, the result will be 25.
What is the number?
33. If to a number you add 65, and from the sum
subtract 38, divide the diiference by 2, multiply the
quotient by 3, the result will be 141. What is the
number? Ans. 67.
34. The dividend is 251, the divisor 13. What is the
remainder ?
35. The dividend is 251, the quotient 19, and the
remainder 4. What is the divisor ?
36. The divisor is 13, the quotient 19, the remainder
4. What is the dividend ?
37. ^he divisor is 13, the dividen4 251, the remain-
der 4. What is the quotient ?
3^ Wliat is the least number of marbles that can be
divided eqiially among 2, 3, 4, or 6 boys? Ans. 12.
39. What is the least number of dollars that can be
divided equally among 8, 14, or 21 men? Ans. 168.
40. A can dig 7 rods of ditch per day; B caia dig 13
rods, and C 14 rods in the same time. Wjbat isAe least
number of rods that will make a number^of JBl day's
^ork for each of the three men? '* f-^^* ■^^•2*
41. A/gentleman has 145 gallons, of /ftatawbst', 203
gallons ^ Madeira, and 319 gallons of Sci^ppjernong,
and he desires, to fill a number of casks of equal size
without mixing or wp^ting the wine. How many gallons
must each cask hold (—(Vide 121.) Ans. 29 or 1.
PROPERTIES OE INTEGRAL NUMBERS. 107
42. A farmer has 482 bushels of corn, 622 bushels of
wheat, and 758 bushels of barley. He wishes to fill a
number of sacks of equal capacity, not mixing the grain,
or leaving any out. IIo^y many bushels must each sack
hold? Ans. 1 or 2 bushels.
43. A merchant has 64 silver dollars, 72 half dollars,
and 144 quarters. He wishes to place an equal number
of each in several drawers, not mixing them, or leaving
any out. What is the least number of drawers that will
answer the purpose ? Ans. 35.
44. A certain number, on being divided by 11, 12,
13, 14, 15, 16, 17, 18, 19, and 20, respectively, has a
remainder on each division just one less than the divisor.
What is the number ?— (Vide 123, Ex. 21.)
Ans. 232792559.
45. A, B, C, and D start together, and travel the
same way, round an island 500 miles in circumference.
A goes 8 miles an hour, B 12, C 16, and D 20. What
is the least number of hours that will bring them to-
o;ether aoiain ? Ans. 125.
How many times round the island will each have
traveled? Ans. A 2, B 3, C 4, and D 5 times.
46. Suppose three railroad trains to start at the same
time from Quito, and to run round the earth on the
equator at the rate of 3G^ 60, and 75 miles an hour re-
spectively. What is the least number of days in which
all will arrive at Quito at the same time ? — (Vide 96,
Ex. 14.) Ans. 69.1608 days.
How many times round the earth will each train have
gone? Ans. First 2, second 4, and third 5 times.
108 FRACTIONS.
FRACTIONS
NATURE OF FRACTIONS.
126. If an apple is divided into two equal parts, each
part is said to be a half of the whole apple.
If an orange is divided into tJwee equal parts, each
part is said to be a third of the whole orange.
If a line is divided into four equal iJarts, each part is
said to be di, fourth of the whole line.
127. If any quantity whatever is divided into a given
nuraher of equal parts, each pai^t takes a name lohich
indicates the nurdher of parts into tuhich the quantity is
divided. Thus,
A half indicates a division into two equal parts.
A third indicates a division into three equal p>arts.
A fourth indicates a division into four equal p)arts.
A fifth, sixth, seventh, eighth, ninth, etc., indicate,
when applied to any quantity, that it has been divided
into five, six, seven, eight, nine, etc., equal parts.
12S. If an apple is divided into any number of equal
parts, each part is a whole part or unit, and the word
one may therefore be applied to it. (Vide 4.) Thus,
one half, one third, one fourth, one fifth, one tenth, one
twentieth, etc.
More than one part may be indicated, just as more
than one of any other quantity is indicated. Thus, two
thirds, two fourths, three fourths, two halves, four
fourths, seven tenths, nineteeii twentieths, etc., any of
which expressions is called a fraction. Hence,
FRACTIONS. 109
(1.) A FRACTION represents one or more than one of
the equal parts of a unit. — (Vide 8.)
(2.) The UNIT OF A FRACTION is the whole quantity
from which the fraction is derived.
(3.) A FRACTIONAL UNIT is ONE of the equal parts of
the whole quantity that is divided. — (Yide 38, I.)
NOTATION OF FRACTIONS.
120. The unit of a fraction may always be repre-
sented by the figure 1.
130. A fractional unit is represented by drawing a
horizontal line under the figure 1, and placing under-
neath it the figure denoting the number of parts into
which the unit of the fraction is divided. Thus,
One half is represented by J.
One third is " " -J.
One fourth is " " J.
One fifth is " " i-
One sixth is " " %.
One seventh is " " 4-
One twentieth is " " . j*^.
One eighty-fifth is represented by -i^.
Remark. — The reciprocal of a number is represented by placing
the number under the figure 1 in the manner of a fractional unit.
Thus, the reciprocal of 2 is ^; of 3 is \] of 100 is j-^^, etc.
131. Any given number of fractional units is repre-
sented by writing the given number above the line in
place of the figure 1. Thus,
Two thirds is represented by |.
Two fourths is " " f .
Three fo\irths is " " f .
110
FRACTIONS
23
S5'
Two fifths is represented by
Three fifths is " "
Four fifths is " "
Seven twentieths is " "
Twenty-three eighty-fifths is represented by
XS2. Properly a fraction represents a less number of
fractional units than is contained in the unit of the
fraction. The unit of the fraction may, however, be
represented in the form of a fraction, by writing that
figure above the line which denotes the entire number
of fractional units contained in it. Thus, two halves
is represented by §, three thirds by |, etc.
133. A greater number of fractional units than is
contained in the unit of the fraction, may also be repre-
sented in the form of a fraction. Thus,
Three halves is represented by §.
Four halves is " "...... 4
Five halves is
Four thirds is
Five thirds is
Six thirds is
Seven thirds is
Seventeen sixths is
134. An integral number is jo
thus, (vide 92, Rem. 2,)
One and one half is represented by
One and one third is represented by
Two and one half is " "
Three and one fifth is " "
Five and three fifths is " "
Eight and six thirteenths is represented by
ned to a fraction
21.
5|.
8A-
FRACTIONS. Ill
135. When one quantity is to be divided by another,
the dividend and divisor may be written in the form of
a fraction. Thus,
Seven divided by five is represented by .... J
One half divided by three is represented by ... |
One divided by one half is represented by • • • i
One third divided by one fifth is represented by . . f
3"
2
Two divided by five and one third is represented by gy
Two and one half divided by three and one third is
represented by rf
136. Since any fractional unit is a whole part of the
unit of the fraction, it may itself he divided into any
number of equal parts, and each part into any number of
other equal parts, and so on to any extent whatever.
Thus, a half dollar may be divided into two equal parts,
and each of these into five other equal parts, etc.
This division is indicated by figures, thus :
One half of one half is indicated by ... J of J.
One fifth of one half is " "... i of J.
One fifth of one seventh is " " ... i of 4.
One fifth of one third of one fourth is indicated
i>y J of J- of 1.
NUMERATOR AND DENOMINATOR.
137. In every fraction, the figure above the line
indicates the number of fractional units taken. It is
thence called the numerator of the fraction.
112 FK ACTIONS.
138. In every fraction the figure below tlie line in-
dicates the number of equal parts into which the unit of
the fraction is divided. It therefore determines the
name to be applied to the fractional unit, and is thence
called the denominator of the fraction.
Remark 1. — The numerator and denominator taken together are
called the terms of a fraction.
Remark 2. — The terms of a fraction are said to be inverted when
the numerator takes the place of the denominator, and the reverse.
Thus, the fraction f inverted becomes |.
Remark 3. — Every fraction is said to be in its lowest terms when
they are prime with respect to each other. — (Vide 102.)
Remark 4. — When a quantity is written in the form of a fraction^
the part above the principal line is still called the numerator^ and
1
that below, the denominator. Thus, in the expression | , one half is
the numerator, and one third the denominator.
CLASSIFICATION OF FRACTIONS.
139. A Simple Fraction is one whose terms are each
single integral numbers. Thus, J, |, f, f, y, |J, etc.,.
are simple fractions.
140. A Proper Fraction is a simple fraction whose
numerator is less than the denominator. Thus, J, |, f ,
?> A? IJj ^*^-? ^^® proper fractions. — (Vide 132.)
141. An Improper Fraction is a simple fraction whose
numerator is equal to, or greater than, the denominator.
Thus, I, I, I, f, f, V? if J ^tc., are improper fractions. —
(Vide 133.)
142. A Compound Fraction is an expression consist-
ing of two or more simple fractions, united by the word
OF. thus, I of i, I of J, \ of I of i of I of f, are
compound fractions. — (Vide 136.)
FRACTIONS. ' 113
143. A Mixed Fraction is an integral number united
to a proper fraction. Thus, 1^, Ij, 2^, 31, 8^^, 9^1,
etc., arc mixed fractions. — (Vide 134.)
144. A Complex Fraction is a quantity written in
the form of a fraction, in which one of its terms is a
fraction, or both. Thus, tt^ v? vjvt,^,^^ are complex
fractions. — (Vide 135.)
VALUE OF A FRACTION.
145. The value of a fraction is the quotient of the
numerator divided by the denominator.
Remark 1. — The value of a. proper fraction is less than 1.
Remark 2. — The value of an improper fraction is equal to or
greater than 1. Thus, f=l; |=1 ; f=l^; V=2j etc.
Remark 3. — The value of a fraction is such a part of the nu-
merator as is denoted by the reciprocal of the denominator. Thus,
1=1 of 5; -|=i of 4; -|=i- of 4; |=i of 5, etc.— (Vide 127 and
130, Rem.) "
PROPOSITIONS.
I. The value of a fraction is not changed by multi-
plying or dividing both terms by the same number.
II. The value of a fraction is multiplied when the
numerator is multiplied or denominator divided.
III. The value of a fraction is divided when the nu-
merator is divided or denominator multiplied.
REDUCTION OF FRACTIONS.
140. The reduction of a fraction consists in changing
its form, or the value of its terms, without altering the
value of the fraction.
10
114
FRACTIONS
147. To reduce a simple fraction to its lowest terms,
Divide the numerator and denominator hy their greatest
common divisor , or cancel the factors common to the nu-
merator and denominator.
EXAMPLES.
1. Reduce f to its lowest terms. Ans. J.
2. Reduce §, f , ^%, |, /^-, Jg, /^, and >f , each to its
lowest terms. Ans -f , f, |, |, |, f, J, and f .
3. Reduce f J, f§, if J, |||, if^ ||, and |f, each to
its lowest terms. Arts. J, f , |, g, ^, f , and y%.
OPERATION.
(Vide 115.) 9|i-||=ig==-Mn5.
4. Reduce H, -j3^<V, ^^(j. JS. IS. ??, and ^%%, each to
its lowest terms. Ans. |, f, |, |, f, |, and f.
^ "Rprlnpp 18 105 84 9_9 78 117 o^lfl 19^ paph
o. xveuute -3 q, 175? T40' les? Tsn? Tirs? '^^^^ 325? ^^^^
to its lowest terms. Ans. f.
6"RprInf»P IT 3 8 6 9 116 155 J2 5 9 104 nQoli+nif^
. xieauce -34, ^t^, ^^j T45j ths? 296? 1 it? ^^^^^ ^^ ^^^
lowest terms. Ans. J, |, |, etc.
7 PprlnPA 39 51 69 87 169 183 213 popVl tn
*• xieauce g^, g-g, yyg, y45, 25^? sTJSJ 355> ^^^^i ^^
its lowest terms. Ans. |.
8. Reduce f ||g to its lowest terms.
OPERATION.
(Viae iiy, j^x. iv.; 4^2 0—2x2x3X5X7X11 ' '
Remark. — In practice a line may be drawn across the common
factors, and the factors in each term not crossed must be multi-
plied together. Thus,
9. Reduce -f f g to its lowest terms.
FRACTIONS. 115
OPERATION.
,4 6_^X3>aXl3_3 9 A^
10. Reduce J-JiJ to its lowest terms. Ans. {J.
11. Reduce ?J|J to its lowest terms. Aiis ^|f.
12. Reduce jii J to its lowest terms. A')is, -J|.
13. Reduce Iff § to its lowest terms. Ans. [f.
14. Reduce Jitlfl to its lowest terms.
OPERATION.
(Vide 122, Ex. 11.) 3ii|J.J.|B..|^
15. Reduce Iff ? to its lowest terms.
16. Reduce -JJ|| to its lowest terms.
17. Reduce J|§f to its lowest terms.
18. Reduce |f-|J to its lowest terms.
19. Reduce |||| to its lowest terms.
20. Reduce } J|| and j%\% to their lowest terms.
Ans. hi and ||.
148. To reduce a simple fraction to another fraction
having a given denominator,
(1.) If necessary, reduce the fraction to its lowest terms.
(2.) Divide the proposed denominator by the denomi-
nator of the reduced fraction.
(3.) Midtiply both terms of the reduced fraction by the
quotient.
EXAMPLES.
1. Reduce \l\% to a fraction whose denominator
shall be 69.
OPERATIONS.
(1.) (2.) (3.)
— 23X3 ^^ ^*
379 J^n
44 3 -^^
* ■
Ans.
t\-
Ans.
s^-
Ans.
M-
Ans.
A-
Ans.
2\-
116
FRACTIONS.
2. Reduce y°o to fractions whose denominators shall
be 9, 15, 18, 21, and 27. Ans. f , J g, jf, if, and Jf.
3. Reduce !§ to fractions whose denominators shall
be 15, 25, 30, 35, 40, 45, etc.
477.9 -9- J S 18 2 1 pf«
4. Reduce f, ;|, g, and /q, to fractions whose denomi-
nators shall each be 60. Ans. |J, |g, |g, and |f.
5. Reduce |, Z^, 2§5 ^'^i^d i\? t<^ fractions whose de-
nominators shall each be 30. A71S. fg, §J, -Jf, and -Ig.
6. Reduce |J, J|, y'/g, and J|, to fractions whose
denominators shall all be 105. Ans. /q\, -^^-g, etc.
7. Reduce f i, 4§, Jf I' ^^^^ iii' *^ fractions whose
denominators shall be 280. Ans. Jl^, etc.
8. Reduce -J|, ||, ||, and [if, to fractions whose
denominators shall be 60. Ans. |J, etc.
9. Reduce f |, iff, -|f }, and ||f, to fractions whose
denominators shall be 504. Ans. |§J, etc.
10. Reduce Jf |, o^, f |, and /g^, to fractions whose
denominators shall be 126. Ans. j^^^, etc.
a fraction whose denominator
Ans. If.
a fraction whose denominator
Ans. \2.
to fractions whose denominators shall
Ans. f, y, y, etc.
14. Reduce 13, 14, 15, 16, 17, 18, 19,^ and 20 to
fractions whose denominators shall all be 17.
Ans. ^iV> ¥-7% etc.
15. Reduce 11, 12, 21, 22, 23, 24, and 25 to fr^-
tions whose denominators shall be 19.
Ans. Yg% t/> etc.
11. Reduce Jff J
to
shall be 52.
12. Reduce 3=f
to ;
shall be 4.
13. Reduce 5 to
fraci
be 1, 2, 3, 4, 5, 6, 7,
etc.
FRACTIONS. 117
16. Reduce 19 to fractions.Tvhose denominators shall
be 11, 12, 13, 14, etc., to 20. Ans. \\9, ^^^^ etc.
17. How many half dollars in 23 dollars ?
Ans. 46 half dollars.
18. How many quarters in 23 dollars ?
Ans. 92 quarters.
149. To reduce a mixed number to an improper
fraction which shall be in its lowest terms,
(1.) Reduce the fractional part to its lowest terms, if it
is not so given.
(2.) Multiply the whole number by the denominator of
the reduced fraction, and add to the product the nu-
merator.
(3.) Write the sum over the reduced denomiiiator.
EXAMPLES.
1. Reduce 23j|fg, 15ff§g, and 7|4§, to improper
fractions in their lowest terms. — (Vide 147, Ex. 9 and
10; also, 148, Ex. 1.)
OPERATIONS.
(1.) (2.) (3.)
23i§!S-23H 154fiS-15H 7f4g=r|f
SgV Ans. Y_6 J^^jg. 4_2_4 ^^^s.
2. Reduce 13f, 15|, 17|, and I^^q, to improper
fractions in their lowest terms.
Ans. y, V? V^ and V-
S. Reduce 21f , 23/o, 27i§, and 31|f, to improper
fractions in their lowest terms.
J/*,<? 6 5 7 1 13 8 ori,l 1 27
JLns. 3,3, 5 , aiiu ^ .
118 FRACTIONS.
4. Reduce 17j J, 16||, 12f |, and 19; J |, to improper
fractions in their lowest terms.
Ans. %^, y^, V, and %K
5. Reduce 14||, 15|i, IGyVs? and ITyVs, to improper
fractions in their lowest terms. Aiis. \^, ''/, etc.
6. Reduce lSj%, ITy^, 19tV, and 2d.^\, to improper
fractions in their lowest terms. A7is. \y, etc.
7. Reduce 124jf|, 256/r\^ 211|ff-g, and 112j-||i,
to improper fractions in their lowest terms.
J^a 1120 1795 3600 and^^^^
150. To reduce an improper fraction to an integral
or mixed number,
Divide the numerator hy the denominator, and place
the excess of fractional units to the right of the quotient.
Remark. — Let it be understood that, unless special direction is
given to the contrary, all answers are to be given in their lowest
terms.
^-dB-l^A^FL E S .
1. Reduce '4'* to an improper fraction. A71S. 3^.
2. Reduce |, f, |, y , y, y, ^/, to integral or mixed
numbers. Ans. 4, 4, 4.^, 2^, 2i, 3, 3].
3. Reduce \4, V? 'P? 'i¥' ^V^ ^^^d \8^, to mixed
numbers. Ans. 13i, 15|, 17|, 18|, 21|, 23f .
4. Reduce ^^f, YsS W? ¥? Wj to mixed numbers.
J.WS. 23^1, etc.
5. Reduce 1//, ^^{, 4_y, \¥? and 1/^2^ to integral
numbers. Ans. 13, 14, etc.
6. Reduce W> W? ¥eS \V> ¥2S to mixed num-
bers. Ans. ISJ^, I7J5, 20/g, 20 Jj, 20i.
7. Reduce W? \¥j W? \V? ^^^ W? to mixed
numbers. j-ws. 19, \, etc.
FRACTIONS. 119
8. Reduce -%^^, '"'^S^^, ^V/^ and 5f|f^ to mixed
numbers. Ans. 3574, 1228^^, etc.
ADDITION OF FKACTIONS.
151. To add two or more proper fractions together,
(1.) Reduce the fractions to their lowest terms ^ if they
are not so given. — (Vide 147.)
(2.) Find the least common multiple of the reduced
denominators, — (Vide 123.)
(3.) Reduce each fraction to one which shall have a
denominator denoted hy the least common multiple. — (Vide
148.)
(4.) Add the numerators of the resulting fractions, and
if the sum placed over the denominator is an improper
fraction, reduce it to an integral or mixed number. —
(Vide 150.)
Remark. — If the given fractions all have the same denominator,
their numerators should, of course, be added at once by (4.)
EXAMPLES.
1. Add together f , |, |, and f^.
OPERxVTION.
f+i+i+A =Given fractions.
i+l+f+l (Vide 147, Ex. 1 and 2.)
(Vide 148, Ex. 4.) iHiHiS+ig= W^Sfi Ans,
2. Add together f , f^, J §, and ^-^.—(yide 148, Ex.
5.) ^ Ans. 2t3.
3. Add together | J, Jf . iVa. and J]-. Ans. 2-f^%.
4. Add together if, fj-, -,\%, and ^.. Ans. 2f.
5. Add together §, f i, |-f, and |i. Ans. IJ.
120 FRACTIONS.
6. Add together §, y\, -^^^ and ^V- ^'^s. Ijoo-
7. Add together A 8, 1.0 5;_8_4_^ and /gV -^^s. 2|.
8. Add together 4, f , f , and 4- -^^s. If.
9. Add together -J, f, -|, and |. ^ws. 1-J.
10. Add together I, J, i, and J. Ans. l^J.
11. Add together J, |, f , and |. ^ns. 2jf .
12. Add together |, -/g, |, and j4_^. JlWS. 2/^.
13. Add together 4, Z^, ^\, and 3%-. ^tis. Jf g.
14. Add together f |, i?-f , ifi, and -|if.
^/^§^ 3229^
15. Add together h^, /,%, ff, and ^%\.
Ans. 1{^.
16. Add together |, -j^, 4, f , and ^^g. J.?zs. 2/^.
17. Add together y/^g and ^Z^^.— (Vide 124, Ex.
1.) Ans. 44J45.
18. Add together -^^^j and yj§^. ^tis. of oIt-.
19 Add together y 1^377 and 9 yV?- ^ns. 5/4^0 5.
Kemark. — When mixed numbers are required to be added, add
the sum of the fractional parts to the sum of the integral numbers.
20. Add together 3j and 4j; also, 7j and 6| ; also,
14| atid 135.
OPERATIONS.
(1.) (2.) (3.)
3i=
=3^
n=n
14i=14|f
H=
=4A
^=^
13f=13|§
7A
Ans.
14i Ans.
28|f Ans.
21.
Add together 13f and 14|.
Ans. 28J.
22.
Add together
171 and 18 fV
Ans. 36iJ.
23.
Add together
21f and 23,8^-.
Ans. 45 J.
24.
Add together
17||andl6|?.
Aws. 34^.
FRACTIONS. 121
25. Add together 4j, 3j, 4i, and 6|. Ans. 18io §.
26. Add together 7^, ISj^^-, 16|, and 5j.
Ans. 42f 3..
27. Add together 4i, 2i, 28|f, and 29j-J.
28. Add together 9i, 8^^, 7o^5, and 65.
^7^s. 64|.
Ans. 89f f.
29. Add together 1 J and 2i ; also, Sj and 4J ; also,
2| and Ij; also, Ij and 2j.
^"*' ^15? '"30? "^Bf ^40*
30. Add together 4Jq and 3^^; also, 2 J^ and 5-J;
also, 5J3 and 7^^. Ans. 7/- 7/^; 12/^.
31. Add together 2^ and 1 J^ 5 ^^so, 3Jj and 4J- ;
also, 51 and 3J. ^ns. 3|f ; 7if ; 8/0-
32. Add together 3f and 4f ; also. If and 5| ; also,
4|and6|. Am.7U; 6fi; llji.
SUBTRACTION OF FRACTIONS.
XS2. To subtract one proper fraction from another,
(1.) Reduce the fractions to their lowest terms, if they
are not so given.
(2.) Find the least common multiple of the reduced de-
nominators, and reduce each fraction to the denominator
denoted by it.
(3.) Take the numerator of the subtrahend from that
of the minuend, and place the difference over the denomi-
nator.— (Vide 150, Rem.)
EXAMPLES.
. From 1
take
2\-
2.
From
1%
take
1%
11
3.
From
Jl take
hh
^
>•"'
-
12-2 FRACTIONS.
OPERATIONS.
(1.) (2.) (8.)
8 — /f 1% — 1\ if — ii Given fractions.
1. .3-_i 13.
3 8 4 1^
(Vide 147.)
i-f-^ A718. l-i=^iAns. ||-i|-3^^^s. (Vide 148.1
4. What is the value of |-A? of |-|? of jl.—^^'i
of A-^r' ^^^^- 8%; T6; §§; f-
5. What is the value of i— A? of J— J^V of ^—^J
6. What is the value of y%— §'? of Ji — J? of Ji— |?
7. What is the value of /f— 1\? of |— ^? of 4— _i-?
of 6 189 J ^,9 1. 7. 37. 1
8. From {J.j take «|. ^?«8. J,.
9. From f || take f|9, j^^s^ _i^^
10. From Hf take f|J. ^^s. 1.
11. From ^^%% take 5%. J.^s. ^.
12. From 34^^ take y^V?- ^^s- ^eVf 5-
13. From ^g^^^ take ^f^^. A7is. ^gV^^.
14.' From j^%^ take 34^^,-. J.ws. ^ J/g^.
15. From ^/g^ take ^4^-5. ^ws. olfl^.
Remark 1. — A proper fraction is taken from 1 by writing the dif-
ference between the numerator and denominator over the denominator.
Thus, 1—3^1 since f — | = 1 (Vide 132.)
And 1—^^=3^ since \l—^^=3^
16. From 1 take §, |, J, J, a, |, f , |, |, and f
yd 790 1 1 4 1 2 p + p
yiAts. 25 -3, 5, 2, 3, eio.
17. From 1 take J§, /„ J|, ,*t, /„ Jg, and |J.
^-
18. From 3 take §, |,
Ans
•5^0
1 5
? 2 2"?
iJ,
etc,
1,
4,i,
5
A,
and
sV
4??s.
Ol,
01
04
25,
etc,
FRACTIONS.
19. Findthe value of 9— J§; of 24— ^^. ; of37— '«;
of 86-/^. Ans. 8/^; 23||; 36 Ji; 85/^.
20. Find the value of 4— f ; of 121— J§ ; of 187—^^ ;
of 2145-IJ-. Ans. 3|; 120,^^; 186,3^; 2144f.
21. From 8 take 24.. 22. From 4j take 3J-.
23. From 5i take 3
OPERATIONS.
(21.) (22.) (23.)
8 4j-4i 5i=5§
51 Ans. 1| Ans. 1| Ans.
The first two examples need no explanation. In the
last, having reduced the fractional parts to the same
denominator, the numerator of the upper fraction is
added to the difference hetiveen the terms of the lower
fraction for the numerator of the ansiver.
This is the same as adding a fractional unit, in terms
of the reduced fraction, to the upper fraction, and then
subtracting the lower fraction. Thus, |+§ — i=i.
The true difference is preserved by adding 1 to the
lower integral number, before taking i.t from that above
it. Hence,
Remark 2. — Always proceed in this way when the fractional
part of a mixed subtrahend is greater than that of the minuend.
24. From 28i take l^. ■ Ayis. 14j
25. From 36ji take 17f . Ans. 18f
26. From 45| take 23/^. Ans. 21|.
27. From 34i take 17J J. Ans. 16f .
28. From 123| take 45/^. Ans. 1^^^,
4
FRACTIONS.
29.
From 106| take 103g.
Ans. 3 J.
30.
From 165/j- take 63^^.
Ans. 102//^-.
31.
From 71Ji take 3f J.
Ans.QlUi.
32.
From 19^- take lOff.
Alls. 8|||.
33.
From 55j% take 37i-gi.
Ans. 17lil.
34.
From 25jV5 take 18|4|.
Ans. 6i|.
35.
From 122 J take 16 j|.
Ans. 105f f .
36.
From 1345f take 237|.
Ans. 1107f i.
37.
From 1000 take 555 j\.
A71S. 4:4:4: ^^j.
Remark 3. — Improper fractions may be subtracted precisely like
proper fractions, but it is generally much better to reduce them to
mixed numbers before subtracting. Thus,
.8 1 — 8 8—1 44—1 42 (Vide Ex. 24.)
MULTIPLICATION OF FRACTIONS.
153. To multiply a simple fraction by an integral
number,
(1.) Multiply the numerator by the integral number,
and plaee the product over the denominator; or, (vide
145, II,) _ . .
(2.) Divide the denominator by the integral number, if
it is exactly divisible, and place the quotient under the
numerator.
Remark 1; — All answers should be integral numbers, mixed
numbers, or proper fractions. — (Vide 150, Rem.)
EXAMPLES
1. Multiply I by 10. Ans. 6.
3. Multiply f by 6.
4. Multiply T^y by 2.
Ans. 1\.
Ans. fo-.
6. Multiply Jy by 5. Ans. 2\.
6. Multiply ^jy by 4. Ans. 1^.
7. Multiply \l by 15. Ans. ^.
8. Multiply V by 14. Ans. 24.
FRACTIONS. 125
9. Multiply I by 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Ans. If, 2|, 3i, etc.
10. Multiply 1^ by 11, 12, 13, 14, 15, 16, 17, 18,
and 19. Ans. 7f^g, 8i, etc.
11. Multiply JJ by 20, 25, 30, 35, 40, 45, 50, and
100. Ans. 13f , 17, etc.
Remark 2. — Any simple fraction multiplied by its denominator
produces the numerator for a product. Thus,
VX5-17; f|fX163=2o9.
154. To multiply a mixed number by an integral
number,
(1.) Multiply the fractional part as in 153.
(2.) Multiply the integral part as in 80.
(3.) Add the products together.
EXAMPLES.
1. Multiply 4| by 10. 2. Multiply 7| by 6.
3. Multiply 9/5 by 5
•
OPERATIONS.
(1.) (2.)
(3.)
4f (vide Ex. 1, 153.) 7i (vide Ex. 3.)
10 6
9/g (vide Ex. 5.)
5
46 Ans. 44 i Ans.
47| A71S.
4. Multiply 27/3 by 2.
Am. 54} J.
i 5. Multiply 29/0 by 4.
Ans. 117|.
6. Multiply 121 § by 15.
Ans. 1831.
7. Multiply 15f by 5.
Ans. 78.
8. Multiply 13| by 18.
Ans. 250.
9. Multiply 14/^ by 34.
Ans. 486.
126 FRACTIONS.
10. Multiply 1311 by 11, 12, 13, 14, 15, 16, 17, 18,
and 19. Last Ans. 260^^^.
Remark, — A complex fraction may be reduced to a simple frac-
tion bj/ multiplying both terms of the fraction by the least common mul-
tiple of the denominators of the fractional part s^.
11. Reduce |, -f, and ~^ to simple fractions.
OPERATIONS.
(1.) (2.) (3.)
3X4 T^^^^- 6ix6-3^^^'^* 3^-3^X10-31-11 ^''^-
• 1 1 1 1 1 -2_ 33 ' <
12. Reduce rf, rf, r|, and rf, to simple fractions.
^3 ^S ^1 ^¥
^^s- hh A. h and |.
13. Reduce J, r? "?5 and -_% to simple fractions.
J_?2S. -/o5 14, I, and ^%.
14. Reduce |, ?^, | |, ^, and ^^, to simple
f^^^ti^^l- ^ ^ ^^.. 4?, il, Y, etc.
155. To multiply an integral number by a simple
fraction, '
(1.) Divide the integral number hy the denominator
of the fraction, and multiply the quotient hy the nu-
merator ; or,
(2.) Multiply the integral number by the numerator of
the fraction, and divide the p)roduct by the denominator.
EXAMPLES.
1. Multiply 20 by f .
2. Multiply 60 by ^\.
3. Multiply 76 by J |.
Ans. 15,
Ans. 16,
Ans. 26,
FRACTIONS. 127
4. Multiply 169 by J|.
5. Multiply 5 by y\.
6. Multiply 320 by /g.
7. Multiply 480 by j|.
8. Multiply 75 by f .
9. Multiply 87 by ^f .
10. Multiply 147 by /g.
11. Multiply 1728 by J,
5 7 ^5 9_ 1 ;
^ns.
156.
J.?25
^2f
J-TiS.
180.
Ans.
440.
Ans.
64f.
Ans
■.39.
Ans.
lOi.
, and
U'
Last Ans. 1692.
156. To multiply two or more simple fractions to-
gether,
Place the product of the numerators over that of the
denominators.
Remark 1. — The rule applies also to tlie reduction of compound
fractions to simple ones,
EXAMPLES.
1. Multiply I by | ; that is, find the value of g of |.
OPERATION.
fX?-/5^^^^-
2.
Multiply i by |.
Ans. |,
3.
Multiply 1 by f .
, Ans. IJ,
4.
Multiply f by ji.
^ns. if,
5.
Multiply ^ by II
Ans. If?,
6.
Multiply f by |.
^^s. If.
7.
Multiply ,^3 by y.
Ans. .v..
8.
•
Multiply 1 by f
J.WS. if.
9.
Multiply II by If.
^r^s. f if
10.
Multiply 1 by | of f.
Ans. 3^0.
11.
Multiply ,3, of 1 by f .
^?js. -igl,
12.
Multiply i of 1 by ^' of |>.
Ans. 4||.
128 FRACTIONS.
13. Multiply V of i ^J A- ^^^' ioi-
14. Multiply f of I of /j by V._(yi(ie 147, Ex. 9.)
Remark 2. — All the factors common to the numerators and de-
nominators should be canceled before multiplying.
15. Multiply f of V of V ^J t\-
OPERATION.
f X V X V X i\. Given fractions.
^ ^^ ^X^ ^ ^X2X2 ^ ^X^~^'
16. Multiply f I of fl of J by /g. ^tzs. f.
. 17. Slultiply f § of 1/ of V by J^. J.m. 8.
■ _18. Find tlje value of fX VX^j^. A71S. 511 1.
19. Findthe^valueof JfXfJXfXlXV-
' A71S. 58f §.
V » 20. Find the value of ;|gXVXifgX|X-|Xf.
. \ /. ' ^^«- 24j3y.
# '21, Find the value of ^X V-X|fgX4Xi>^f
* % >-^ ■' . A71S. s^di.
♦ , ^2. -^Firid the value of %^ X 3^- >^10. Ans^20. '
GENERAL RULE. V '^ 5f
157. To multiply fractional numbers, ^. *^ • '
(1.) In compound fractions consider the word of as a
sign of multiplication.
(2.) Reduce complex fractions, integral and mixed
numbers, to simple fractions.
(3.) Cancel all the factors common to the numerators
and denominators.
(4.) Multiply the remaining factors of the numerators
FRACTIOXS. 129
iog ether, and also those of the deiiominators, forming a
simple fraction of the products. — (Vide 153, Rem. 1.)
Remark. — The best mode of canceling common factors will be
learned by practice. It is hardly ever necessary to resolve each
term into lis prime factors.
EXAMPLES.
1. Multiply 2^X61x31X1^3X2x1, forming a simple
inber.
OPERATION.
fraction or integral nuinber,
^ ^ ^ 73 1 // 1
Draw a line across 4 and 32, writing 8 in place of 32.
2. Multiply 2 J by 2h # Ans. 61.
3. Multiply 61 by 61. ^ ^ ^ -Ans. Z^-^^.
4. Multiply 31 by H. r4Nl Ans. l^.
5. Multiply 2i by ^. ^ ^ ^ Ai^. 1^\.
6. Jiultiply 5^ by:5i/' Ans. 30i.
"7.' fli^tiply XQV^y 16^. Ans. 272i.
8. Multiply 4i; byv4i.^i% Ans. 20i.
9. Multiply 7i by 1'^ig > Ans. h^.
10. Multiply ^ by^3j t ^ .Aws. 8f .
11. Multiply i\ hj!%. Ans. 1.
,12. Multiply 8^ by 3 J^. Ans. 2B.
13. Multiply 91 by 2^\. Ans. 18J.
JL4. Multiply 3} by 3^^. ' Ans. 9 J.
15. Multiply 21 by 4^6. Ans. 8f.
16. Find the value of I4x||-X^.
OPERATION.
fifXTViXlg. (Vide 154, Rem.)
mX4 5X10 5X3 _ ^ 3 .
113X5^3X47 ^5X2~-^^- ^''''
X) niACTIONS.
J
17. Find the value of ?X^x4 Am. |.
18. Find the value of ^X^fX^f • Am. 1/A.
19. Find the value of ^X#,X^X207.
Am. 16|.
20. Find the value of 4 X 8 ^^ X f X V X 24 J.
J.72S. 614^.
21. Find the value of |^x||xgx^.
Am. If.
22. Find the value of f X^^X^xp'. Am. ^%^^%.
23. Find the value of V/XyX— ^xSx^ of ^
" 465 3^ 2^ '
X5i. ^^s. 132.
24. Find the value of 1 § x 3^0,0 x § X | X ^s"" X 15.
Ans. 120.
25. Find the value of T\X|X|Xy. Am. 1^.
26.. Find the value of IxO/^Xf X4|X8|.
Am. 240|.
27. Find the value of f gX }iX V0VXI8.
Ans. 9.
28. Find the value of 5iX5i XS^. Ans. 166|.
29. Find the value of 3^ x^lxS^ X3|.
A71S. 150 Jg
DIVISION OF FRACTIONS.
158. To divide a simple fraction by an integral
number,
(1.) Divide the numerator by the integral number, if it
is exactly divisible, and place the quotient over the de-
nominator; otherwise,
FRACTIONS. 131
(2.) Multiply the denominator hy the integral nitmher,
and place the product under the numerator. — (Vide
145, III.)
EXERCISES.
5. Divide | by 5. Am. -^.
6. Divide y by 4. Ans. 1^^.
7. Divide %^ by 13. Ans. If.
8. Divide f| by 19. Ans. ^.
2. Divide ^% by 18. Ans. ^^
3. Divide y by 36. Ans. ^\.
4. Divide ^ by 51. Ans. ji-^.
9. Divide JiJ§ by 11, 12, 13, 14, 15, 16, 17, and 18.
A.71S. 2 2 0' 2 4 05 ^^^'
10. Divide ^^g^ ]by 17; s^o^e by 18; ^^^^ by 19, and
36ihv1Q ' An9 §' 1^- g and J 9
150. To divide a mixed number by an integral
number.
Reduce the mixed number to an improper fraction, and
then divide as in 158.
EXERCISES.
1. Divide 2J by 5.— (Vide Ex. 5, 158.)
2. Divide 7 J by 8. Ans.'\l.
3. Divide 9| by 12. Ans. f .
4. Divide Ij by 2. Jltis. f.
5. Divide 5f by 4. J.?is. l/^-
6. Divide 2-i by 5. A71S. /g.
7. Divide If by 36. Ans. ^^.
8. Divide 41 by 6. ' Ans. ^^.
9. Divide ^ by 11. J.7is. f .
10. Divide 5j by 13. A71S. §.
Remark 1. — If the dividend is larger than the divisor, its inte-
gral part may be first divided for the integral part of the answer;
then divide the remainder united ^ the fractional part of the divi-
dend for the fractional part of the answer.
132 FRACTIONS.
11. Divide 27-1 by 5; 71^ by 8; and 153f by 12.
OPERATIONS.
(1.) (2.)
(3.)
5)271 8)711
12)153f
5/5(yideEx.l) 8li(yideEx.i
2.) 12|(VideEx.3.)
12. Divide 54i| by 2.
Ans. 27/^.
13. Divide 117i by 4.
Ans. Ex. 5, 154.
14. Divide 1831 by 15.
Ans. Ex. 6, 154.
15. Divide 150/g by 11.
. Ans.lS}^.
16. Divide 1641 by 12.
Ans. im.
17. Divide 471 by 5.
A71S. 9/5.
18. Divide 441 by 6.
Ans. 7|.
19. Divide 100 by 3.
Ans. 331.
20. Divide 1274J by 11,
12,1^
5, 14, 15, 16, 17, 18,
and 19.
Ans.
115||, 106^4, etc.
Remark 2. — If the divisor is a composite number, it is usually
best to divide first by one of the component parts, and the resulting
quotient by another part, and so on till all the component parts are
used.
21. Divide 4567f by 25, 32 and 51.
OPEKATIONS.
(1-)
(2.)
(3.)-
5)4567|
4)4567|
3)4567f
5) 913>-J
Ans.
8)1141}i
142 i»A
Ans.
17)1522/^
182tV0
89i-Jf Ans.
22. Divide 34027f by 36, 42, 65, 56, 72, and 49.
Ans. 945,^6, 810, Vg, ^23,^6, etc.
FRACTIONS. 133
23. Divide 72431 i by 15, 21, 28, and 35.
Ans, 4828f 3, 3449 Z^, etc.
160. To divide an integral number by a simple
fraction,
Multiply the whole niimher hy the denominator of the
fraction, and divide the product by the numerator ; or,
Invert the divisor, and then multiply t?s m 155, (2.) —
(\^ide 138, Rem. 2.)
EXERCISES.
1.
Divide 15 by f.
Ans. 20.
2.
Divide 16 by j\.
Am
f. Ex. 2, 155.
3.
Divide 26 by ^f.
Ans. Ex. 3.
4.
Divide 156 by a|.
Ans. Ex. 4.
5.
Divide 3 by 5.
^ns. 2i.
6.
Divide 49 by J.
Ans. 28.
7.
Divide 57 by '/.
J.71S. 24.
8.
Divide 32 by V-
J.??s. 114.
9.
Divide 45 by J, |,
f.
h
h\h
1 6
16?
V, and ^%.
Ans. 90, 67J, 60, etc.
10. Divide 4290 by f f , VoS ^^^l f |.
^ns. 2015, etc., (vide 123, Ex. 16.)
11. Divide 28560 by ^p, ^f ^ YrS and V/-
^ns. 525> 840, 1540, 2184.
161. To divide one simple fraction by another,
Invert the divisor, and then place the product of the
numerators over that of the denominators.
EXAMPLES.
1. Divide f by |.
2 Divide | by |.
Ans. \
Ans.
134
14
FRACTIONS.
3.
Divide | J by |.
Ans. |.
4.
Divide §1 by J J.
Ans. If.
5.
Divide f^_o.^^ II .
JLtzs. ^^.
6.
Divide 3y, by i|.
A71S. |.
7.
Divide Jgf by |f.
^Tis. /r-
8.
Divide | by f .
Ans. Jf .
9.
Divide -/ by |.
^/^s. 10.
10.
Divide J| by V-
J.n.s. 1^\.
11.
Divide If? by ]J.
Ans. ;|.
12.
Divide |f | by If.
^7ZS. if.
13.
Divide J by f.
Ans.
Ex. 14, 154.
14.,
Divide ^% by j%.
Ans.
Ex. 13, 154.
GENERAL RULE.
162. To divide fractional numbers,
(1.) In compound fractions consider the tvord of as a
sign of multiplication.
(2.) Reduce complex fractions^ integral and mixed
numbers, to simple fractions.
(3.) Invert each of the reduced or given simple frac-
tions considered as divisors,
(4.) Cancel all the factors common to the numerators
and denominators of the simple fractions.
(5.) Multiply the remaining factors of the numerators
together, and also those of the denominators, forming a
simple fraction of the products. — (Vide 153, Rem. 1.)
EXAMPLES.
1. Divide 4 of § of 5i by if of 48.
OPERATION.
4X|XVXifX^xV5=/6 ^ns.
FRACTIONS. 135
2. Divide f of | by J of 2^. Ans. Jf.
3. Divide 39-jig by 6\. Ans. Gj.
4. Divide 272} by 16\, Am. 16j.
5. Divide 3061 by 17 J. Ans. 17^.
6. Divide f of J/ by ^\ Ans. 1.
• 7. Divide 1419^X11X^5 by ^. ^?is. 2f.
8. Divide i|x^ by ^,Xl8f . Ans. f .
9. Divide 4iX^" by %^X~. Ans. I.
10. Find the value of ^X-f divided by ^XlOj.
Ans. ii^.
11. Find the value. of |X^ divided by ?X^X^|
X^-lf . ^^^ ^ • ^.;. i-. '
12. Find the value of ilX^Xyi divided by ^X
167 ^ 6f
13. Find the value of 2i- divided by 5i.
Ans. Ex. 14, 154.
14. Find the value of ?±-^. Ans. Ex. 14, 154.
15. Find the value of ^4- -^^s- lA-
16. Reduce | to a fraction whose denominator shall
be 4.— (Vide 148.) Ans. ^.
17. Reduce J| to a fraction whose denominator shall
be 2J.— O^ide 154, Ex. 12.) Ans. ^.
3
18. Reduce J J to a fraction whose denominator shall
be 5i. Ans. rr.
X^. ^ns. 5if.
H
136 FRACTIONS.
REDUCTION OF COMMON FRACTIONS TO DECIMAL
FRACTIONS.
163. To reduce a simple fraction to a decimal
fraction,
Divide the numerator hy the denominator. — (Vide 92,
Rem. 3, and 145.)
EXAMPLES.
1. Reduce J, |, |, and ]|, to decimal fractions.
OPERATIONS.
(1.) (2.) (3.) (4.)
2)1.0 . 4)3.00 8)5.000 16)13.0000
.5 Ans. .75 Ans. .625 Ans. .8125 Ans.
2. Reduce ^-q, ^l^, iJif? ^^^ IS *^ decimal frac-
tions.— (Vide 94 and 95, Rem. 3.)
Ans. .1, .03, 1.728, and .925.
3. Reduce -||, H, ff, ^^-3%, ^i^, i^%, and if to deci-
mal fractions.
Ans. .4, .85, .5, .025, .128, .096, and .36.
4. Reduce ^y^, J/^, ,-%, /„ hi, |f, and ,-Ji^ to
decimal fractions.
Ans. .06640625, .08, .4, .25, .48, .0765625, .006875.
5. Reduce ^fS n^, f§, i§, 0^, ||, and W to
decimal fractions.
Ans. 30.25, 35.2, 2.0625, 1.2, 4.09375,1.171875, 5.08.
6Rp(lnPP 3 26 7 210 80 24 nrifl 5 1 f^ /Ippi'mnl
fractions. Ans. 1.5, 2.5 .28, 8.75, .128, 1.6, and .75.
7. Reduce 1^, 3j, 81, 6/^, 30f, 35f, and 7S to
mixed decimals.
^?zs 1.5, 3.25, 8.2, 6.7, 30.75, 35.6, and 7.625.
FRACTIONS. 137
8. What is the value of ^3^, $4|, $7f , $Sl, and ^6\ |?
Ans. $3.50, §4.625, |7.75, §8.20, and §6.81i.
Remark. — If the denominator of a fraction in its lowest terms
contains a prime factor other than 2 or 5, the value of the fraction
can not be exactly expressed by a decimal. The exact value may,
however, be preserved by placing the excess of fractional units to
the right of the quotient, stopping the division at pleasure.
9. Reduce ^, |, f.j, f, |, and y\ ^^ mixed decimals.
Ans. .331, .16§, .416f , .66|, .83J, and .583i.
10. Reduce |, J^, fj, |, f J, |f, and || to mixed
decimals.
Ans. .4281, .06|, .846/3, .33^, 1.16f , .66|, and 1.66f .
11. What is the value of $7^, |4|, |6|, $8J, and
§12 J^?
A71S. §7.33J, |4.83i, |6.66|, ?8.773, and |12.06|.
164. To reduce a decimal fraction to a common
fraction in its lowest terms,
(1.) For the numerator of the fraction, write the
figures composing the given number.
(2.) For the denominator, write 1, with as many
cij)he7's annexed as there are decimal places in the given
number.
(3.) Reduce the resulting fraction to its loivcst terms.
EXAMPLES.
1. Reduce .5, .75, .625, and .8125 to simple fractions.
OPEKATIONS.
(1.) (2.) (3.) (4.)
,»,=jAns. j\%=\Am. {ii-,=lAm. -^^yii,^\lArw.
12
138 FRACTIONS.
2. Reduce .06640625 and 30.25 to simple fractions.
OPERATIONS.
(1.) (2.) or (2.)
3. Reduce .25, .85, .55, .125, .135, and .325 to simple
fractions. Aiis. i, JJ, J i, -J, i^\, and i-§.
4. Reduce .025, .0085, .9375, .0008, and .16 to
simple fractions. Ans. J^, ^ij^, i|, ^ J^o' and 5*^.
5. Reduce .34375, .1328125, and .203125 to simple
fractions. Ans. J J, y^g, and ||.
6. Reduce $3.50, $4,625, $7.75, §8.20, and $3.40 to
dollars. Ans. $3j, $4|, $7f , $8|, and $3f.
Remark. — "When there is an irreducible fraction at the end of
the decimal,
(1.) Consider the given number as integral^ and reduce it to an im-
proper fraction.
(2.) Annex to the denominator as many ciphers as there are decimal
p)laces in the given number.
(3.) Reduce the resulting fraction to its lowest terms.
7. Reduce .428|, .066|, and .83j to simple fractions.
OPERATIONS.
(1.) (2.) (3.)
.428f .066| .83i
3000 S-d/Mo 200 1 J*iQ 2 50 5 A'i'}^
8. Reduce .47J, 4.7|, 47.33i, .047^, and .43J to
simple or mixed fractions.
^ns. ^5^5, 4j-i, 47^, tJJ^, and J^.
9. Reduce .473j, .33^, .45|, .125^, and .25| to simple
fractions. Ans. -jVo> irs^ ft? /^^j ^^^ «o-
FRACTIONS. 189
10. Reduce §7.33|, |4.83|, |7.25J, |0.754f, and
|4.50f to dollars.
Ans. $7J, Uh ^n, Psih and |4f|.
11. Add together $240,172, §120.75f , and §255.136f .
Ans. $616,066^.
12. Add together $5.87|, $3,187^, and |2|.
^ ^718. $11,687^.
13. Add together |, .066f , and .8].
OPERATION.
1= .4281
.066|
M= .833A
1.328f=:lf g Ans.
14. Add together f , .18|, and 7-|.
Ans. 8.7652|=8f |i.
15. From $25.41 take $17|. Ans. $7.66.
16. From $28,026 take $19.15|. Ans. $8.872f .
17. From $12.25 take $8i. J^ns. $3.75.
18. From $5| take $4.25. Ans. $1,08 «.
19.* From $6i take $5.25. Aiis. $1.00.
20. From $7f take $1|. " ^?^s. $6.58'.
165. Problems Involving Preceding Principles.
1. If a horse consume ^ a bushel of oats in one day,
I of a bushel in another, | of a bushel in another, and
f of a bushel in another, how many bushels are con-
sumed in the four days? — (Vide 151, Ex. 1.)
Ans. 2|g bushels-
140 FKACTIONS.
2. If I buy 4 of ^ yard of ribbon at one store, y\ at
another, o^- ^^ another, and -f^ at another, hoAV many
yards have been purchased? — (Vide 151, Ex. 13.)
Ans. if J yards.
3. A man bought two pieces of cloth, one containing
13f yards, and the other 14| yards. How many yards
in both pieces ?— (Vide 151, Ex. 21.)
Ans. 28^ yards.
4. In one pile of wood I have 4 J cords; in another
3J cords; in another 4i; and in another 6^ cords.
How many cords of wood in the four piles? — (Ex. 25)
Ans. 18j2§ cords.
5. If I make purchases to the amount of 'fl7|-| at
one time, and §16|| at another, how many dollars have
I expended in all?— (Ex. 24.) Ans. $34i.
6. If I travel 47 f miles in one day, 33 ig in another,
and 19 in another, how many miles have I traveled in
all? Ans. 100 miles.
7. Add together §4.25i, §3.37 J, §6.753, and §7.52 j^,.
Ans. §21.91. '
8. Add together §240.17^, §120.75|, and §255.13f .
A9is. §616.06J2.
9. Add together §16.254, §40.20|, §13.60 J, and
§24.035. Ans. §94.094 J.
10. What is the difference between § of a dollar and
2'j- of a dollar?— (Vide 152, Ex. 1.) A7is. §0.16|.
11. What is the difference between §3.00 and -| of a
dollar ? Ans. §2.25.
12. From a cask of wine containing 8 gallons, 2J
gallons were drawn. What quantity remained in the
cask?— (Vide 152, Ex. 21.) "^Ans. 5j gallons.
FRACTIONS. 141
13. If I purchase flour at 3 J dollars per barrel, and
sell it for 5| dollars, ^'hat doJ^ gain? Ayis. |1.83j.
14. A farmer sold 55 y^j- ajp^ from a farm of 100 acres.
IIow many acre^did lie s^ own? ^ A71S. 4:4: f^ acres.
15. If I buy a'p'iece of land for |1 03.33 J, and sell
the same for $106.66 f, Avhat do I gain ? Ans. $3.33 J.
16. A merchant bought a piece of cloth containing
123| yards, and from it sold 45j'^(j yards. How many
yards remained? Ans. 78 /^ yards.
17. A railroad train has 13 1 hours in which to run
550 miles. Having run lOj hours, the conductor finds
that only 41 6| miles have been made. What distance
is yet to be run, and in what time?
Ans. 133 J miles in 2| hours.
18. From 1000 yards of cloth I sold at one time
479j'^g yards, and at another 275 1. How many yards
have I still on hand? A71S. 245 /g.
19. A merchant bought at one time 234| yards of
cloth, at another time 753y'^Q yards, and sold of the two
lots 843^^ yards. How much cloth has he yet on hand?
Ans. 145 i yards.
20. I bought 30 cords of wood for $105 J-, and sold
17 cords of the wood for $65,25. If I sell the remain-
ing 13 cords for $45^, how much do I gain in the trans-
actions? Ans. $5,585.
21. The sum of two numbers is 4b^ ; one of them is
23/2-. What is the other ?— (Vide 151, Ex. 23.)
Ans. 21|.
22. The difference between two numbers is J, and
the less is 4|. What is the greater? Ans. 4|.
23. The difference between two numbers is 13?, and
142 FRACTIONS.
the greater is 28 f|. What is the less? — (Vide 151,
Ex. 20, (3.) ^ .^ A71S. 14f .
24. The difference betA^kn two nuiflbers is 184i, and
the greater is 509^. What il the sumVf^the two num-
bers?—(Vide 125"rte*x. 9.) . " • ' Ans. 8343^.
25. The sum of two numbers is 34^3, and one of
them is 16||. What is the other? Ans. ITJ.
26. The sum of four numbers is 89ff. Three of the
numbers are 65, 7^^, and SjL. What is the fourth?
Ans. 9^.
27. A farmer, who had wheat worth |4325.75, sold at
one time 120 bushels for $135 J ; at another time 45J
"imshels for §70 1 ; and at another time 87| bushels for
/, »; ii60. What is the value of the 4000 bushels he still
. finds he has on hand? Ans. $4019.58 J.
'^^ *^ 4 lOO, To find the cost of a number of things, when
jy^ 1!||e cost of one i^ given,
**^^ iKidtipm the cost of one hy the .numher of tilings. The
t* ^ plbduct ^vi'il be the cost of the w^hole. — (Vide 82.)
^ •*. ^*28f^If*lt yard of cloth cost | of a dollar, what will
*'*-4^;^ards cost? (Vide 153, Ex. 1.) 15 yards? 20?
40? J.ns. $6.00, etc.
29. If a pound of butter cost | of a dollar, what
will 6 pounds cost? (Vide 153, Ex. 3.) 8 pounds?
12? 20? Alls. $2.25, etc.
30. If a pound of raisins cost ^^ of a dollar, what
will 5 pounds cost? (Vide 153, Ex. 5.) 10 pounds?
12? 15? 20? Ans. $2.33 J, etc.
31. At J§ of a dollar a pound, what will 15 pounds
of beef cost? (Vide Ex. 7.) 20 pounds? 30? 45? 60?
Ans. $3.25, etc.
FRACTIONS. 143
32. At J I of a dollar a bushel, what will 20 bushels
of apples cost? (Vide Ex. 11.) 25 bushels? 30? 35?
40? Ans. §13.60, etc.
33. If a ton of hay cost |27i^3, what will be the
cost of 2 tons? (Vide 154, Ex. 4.) 3 tons? 4? 5?
12? Ans. 154.76; I, etc.
34. If one coat cost 13 J | dollars, what will be the
cost of 11 coats? 12? 13? 14? 15?
Ans. $150,561, etc.
35. If a ton of hay cost $20, what will | of a ton
cost? (Vide 155, Ex.'l.) | of a ton? ^^'i -/^?
' '\ ^ ^ ^^ Ans. $15.00, etc.
36. If a bale of cott<)^ cost ,^6p, what will /g of a
bale cost ? (Vide Ex. 2.) " ^\'? .^% ? 'i § ? |J ?
Ans. $16.00, etc.
37. When flour is woYtJi $5 a barrel, what is the
value of j^5 of a barrel? o%? -/^ ? J§?
Ans. $2,331, etc.
38. If a flock of sheep is wbrth $1728, what are |
of it worth"? ^'^ l'^ A? -9 ? 13? 4 7 7'
Ans. $1382.40, etc.
39. If an acre of land is worth $128, what are -| of
it worth? ^'^ -%.*? -K'^ -5- *? Si? is?
Ans. $112.00, etc.
40. If a yard of cloth cost § of a dollar, what are |
of a yard worth?— (Vide 156, Ex. 1.) Ans. $0.174 .
41. If a yard of cloth cost f of a dollar, what are f
of a yard worth? Ans. $0,174.
42. If a pound of tea cost \ of a dollar, what cost
f of a pound? (Vide 156, Ex.~2.) | of a pound? 4?
i ? ^ ? A71S. 37i cents, etc.
144 FRACTIONS.
43. If silk is worth || of a dollar a yard, what are ^\
of a yard worth? Ji? ^\ ? ff ? Ans. ?0.112|0f.
44. What will -^\ of a pound of tea cost at J of
a dollar a pound? f? |? |? J of ^ of a dollar a
pound? • Last Ans. $0.07 o\.
45. At 2 J cents each, what will be the cost of 2h
apples? (Vide 157, Ex. 2.) Sj? 4j? 4i? 12i ? 12i?
Ans. 6\ cents, etc.
46. At 16^ dollars an acre, what will 16^ acres of
land cost? 25|? SO-J? 401? 75|? JLtis. |272.25.
167. To find the cost of one thing when the number
of things and the cost of the whole are given,
Divide the cost of the tvhole hy the number of things.
The quotient will be the cost of one. — (Vide 96.)
47. If 2 yards of cloth cost f of a dollar, what will
one yard cost?— (Vide 158.) Ayis. $0.16|.
48. If 18 yards of ribbon cost ^^ of a dollar, what
will one yard cost? Ans. 2\ cents.
49. If 5 sheep cost 27j dollars, what will be the
price of one sheep? — (Vide 159, Ex. 11, (1.)
Ans. |5.46f .
50. If 8 yards of broadcloth are worth 71 J dollars,
what is one yard worth ? Ans. -$8.9 If
51. If 12 acres of land cost 153| dollars, what will
one acre cost? Ans. §12.80.
52. If 6 pounds of butter cost 2 J dollars, what will
one pound cost? Ans. 37^ cents.
53. If 9 bushels of wheat cost 10^- dollars, what will
one bushel cost? Ans. $1.16|.
54. If 4 acres of land cost fll7|. Avliat will one acre
of the same land cost? Ans. §29.45.
FRACTIOXS. l-i5
55. I have four small farms containing, respectively,
11, 12, 13, and 14 acres, and I value each farm at
|1274j. What is the value per acre of each farm ?
What is the average price per acre ?
Ans. $115.8G^*j-, $10G.208i-, $98.038yfij, $91,035^, $101.96.
56. If I of a ton of hay cost §15, what is the price
per ton ?— (Vide 160, Ex. 1.) Ans. §20.
57. If y'^W of a man's salary per month amounts to
§84, what is his salary per ^ear? Ans. §1728.
58. A gentleman divided j\ of a lot of marbles
among 4 boys, giving each boy 21 marbles. How many
marbles would each boy have received if the whole lot
had been divided ? Ans. 36 marbles.
59. If J of a bale of cotton are worth §48, what is
one bale worth? Aois. §54.85|.
167^. To find the number of things when the cost .•
of one is given,
Divide the cost of all hy the cost of one thing. The
quotient will be the number of things? — (Vide 96.)
60. At 12f dollars an acre, how many acres of land
can be bought for 153 1 dollars? Ans. 12 acres.
61. At I of a dollar a pound, how many pounds of
butter can be bought for 2 J- dollars. Ans. 6 pounds.
62. If cherries are worth 7| cents a quart, how many
quarts can be bought for 1^^^ dollars?
Ans. 16 quarts.
63. How many sheep can be bought for 27-| dollars,
if the average price is §5.46f . Ans. 5 sheep.
64. At 2 J- dollars a yard, how many yards of cloth
can be bought for ^^-^ dollars ? Ans. 3 J yards.
13
146 FKACTIONS.
65. If a man boards at 3| dollars a week, how long,
can he board for 188. i dollars? Ans. 52 Avecks.
66. At 2 dollars a yard, how many yards can be
bought for 5i dollars? ^ws. 2| yards.
67. If I pay 656J dollars for 75 tons of coal, what
is the price per ton ?
68. If a ton of coal is worth 8| dollars, what will bo
the cost of 80 tons ?
69. If I pay §656.25 for 75 tons of coal, what will
be the cost of 80 tons ?
70. If 75 tons of coal cost §656.25, how many tons
can be bought for §700 ?
71. Ii 80 tons of coal cost §700, how many tons can
be bought for §656.25 ?
72. I^ 75| yards of cloth cost §643.87|, how much
^nust I pay for 36 yards?
73. When §306 are paid for 36 yards of cloth, how
many yards can be purchased for §643 J ?
74. For 36 yards of cloth 306 dollars were paid.
How many yards of the same cloth can be purchased
for §643.875 ?
75. For 75|.jards of cloth' I pay §643.87^. How
many yards can I get for §306 ?
76. How many coats, each containing 1| yards of
cloth, can be made of 18| yards?
77. What will ten barrels of apples cost at 1 J dollars
per barrel ?
78. If 10 boxes of oranges cost 18j dollars, what is
the price per box ?
79. If 10 pounds of copper cost §18.75, what num-
ber of pounds can be had for §171.25.
FiiACTlONS. 147
80. If ^p3|P^ifcds of copper can be liad for 171 J
dollars, howma^y pounds will 18| dollars buy?
81. A merchant sells | of his ship. What part of it
does he still own^ Ans. J.
82. A merchant owning | of a ship sells J of his
interest. What part of the ship does he still own ? —
(Vide 156, Ex. 2.) Ans. |.
83. Having | of an apple, I give away half of it.
What part of the apple is now gone? Ans. |.
84. Having | of a ship, I sell half my interest for
9420 dollars. What is the whole ship worth at the same
rate?
85. If a §hip is worth |25120, what are | of her
worth ? What are | worth ? ' g- ? Ans. -|=|21980.
86. If I of a ship are worth $15700, what is the
value of I ? ^ '
87. If J of a bale of cotton are worth 48 dollars,
what is the value of f of a bale? Ans. $41^.
88. What fraction is that to which if i be added, the
sum will be 1 ? 2 ? 3 ? 4 ? 5 ? Ans. f , |, etc.
89. What fraction is that to which if J^be added,
the sum will be 1 ? 20 ? 45 T 100 ? Ans. ~'^, %%\ etc.
90. What number is tllaj which, if it be taken from
57, will leave a remainder of | ? | ? ^ J ? J| ?
A71S. 56|, 56|, etc.
91. What number is that which, on being added to
357/5, will make the sum 455?? ^72^5? 9570^*5?
Ans. 983 J, etc.
92. What fraction is that to which if |- of f be
added, the sum will be 1 ? 15 ? 8| ? 4,^o ^
hsist Ans. 4i§.
\
148 FIIACTIOXS.
93. The sum of two numbers is 47-J-, an{>tlie differ-
ence 7l. What are the numbers ? — (Vide 125, Ex. 10.)
A71S. 21^^ and 19iJ.
94. If a certain number be divided by 2 J, and the
quotient be multiplied by 8i, the product diminished by
5 4, the difference increased by 7^, the sum will be 62 f.
What is the number? Ans. 18-|.
95. I have a fortieth interest in an oil well, and am
willing to sell half of it for $1340. What is the value
of my interest, and of the whole well, at the same rate ?
Ans. 12680 and $107200.
3 T
96.
What
is
i
of 360 ?
1 of 720?
ii
of 378? -
of 1643?
Am
f. 270, 600,
351
, and 636.
97.
What
is
H
of
38? 1
3 of 331?
Ans. ^\8^,
ifo
29f
f 32|?
, and 31 1.
> 98. 270 is I of what number ? 600 is § of what num-
ber? 351 is H of what number?
99. 636 is Jf of what number? j\%is i| of what
number? 29| is J J of what number? 31j is || of
what number?
100. J^of 68 is If of what number? Ans. 58.
101. 11 of 341 is J J of how many times 5 ?
Ans. 781 times 5.
102. i| of 49| is If of 'how many thirds of 18?
Alts. '9 thirds of 18.
103. On a trip from New Orleans to New York, I
expend i of my money, and still have »^270. What did
the trip cost me ? Ans. §90.
104. During a storm a captain threw overboard ^ of
his cargo of cotton, and still lias 600 bales on board.
How many bales were thrown overboard? Ans. 120.
FRACTIONS. 149
105. If, after reserving ^^ of my wine, I sell 351
gallons, how many gallons do I reserve ? Ans. 27.
106. Multiply the fractions J and | by the least
common multiple of their denominators.
Ans. 3 and 4.
107. Multiply the fractions | and J by the least
common multiple of their denominators.
Ans. 8 and 9.
108. Multiply the fractions y\ and ^^ Ja^he least
common multiple of their denominators. ^^B
Ans. 12 and 5.
109. Multiply 35 1 and 15 1 by the least common
multiple of their denominators. Ans. 214 and 91.
110. Multiply ^, -i, and | by the least common mul-
tiple of their denominators. Ans. 6, 4, and 9.
111. Multiply ^, I, y^Q, and §, by the least common
multiple of their denominators.
Ans. 15, 20, 9, and 12.
112. Multiply j J-, JJ, /g, and | by the least common
multiple of their denominators.
Ans. 44, 33, 32, and 24.
113. Which fraction is the greater, /^ or -f^ ?
Ans. ^
Remark. — Of two fractions, that wliich gives the gi'eater product
on multiplying both by the least common multiple of the denomi-
nators is the greater.
114. Which fraction is the greater, J J or J J ? -J-f or
i J ? and by how much ? A71S. ] ^ by ,1 3 ; J J by -,- j^.
115. Which fraction is greatest, J, /j, or f^
Alls. /_.
116. If A and B together can do /^ of a piece of
150 FRACTIONS.
work in one day, and A alone can do J of it in one day,
what part can B do in one day? Ans. J.
117. If A and B together can do a piece of work in
2| days, and A alone can do it in 4 days, in what time
can B alone do the work ? Ans. 5 days.
118. A and B can do ^^ of a piece of work in one
day; A and C can do j% pf'the same work in one day;
B and C can do J J in one day. What part of the work
could al^M^ether^do in one day ? What part could A
alone d^^Bn'e day ? What part could B alone do in
one day 'f^ What part could C alone do in one day ?
A7is.A\\, ij; Ai; Bi; Ci.
119. A can do a piece of work in 4 days, B in 5
days, and C in 6 days. What part of the work can A
and B together do in one day ? What part B and C
together ? What part A and C together ? What part
can all together do in one day ? In how many days can
all together do the work? Last Ans. Iff days.
120. A cistern has three pipes. The first will fill it
in 2 hours, the second in 3 hours, the third in 4 hours.
In what time will the cistern be filled when the three
pipes are running together ?
Ans. In ]| of an hour.
121. A cistern has 3 pipes, two at the top and one
at the bottom. One of the top pipes would fill it in 5
hours, the other in 6; but the pipe at the bottom
empties it in 8| hours. In what time will the cistern
be filled when the pipes are running together ?
Ans. In 4 hours.
122. A man and his wife could drink a cask of beer
in 10 iays. In the absence of the man it lasted his
COMPOUND NUMBERS. 151
-wife 30 days. How long Avould the man be occupied in
drinking it? A7is. 15 days.
123. A, B, and C could do a piece of work in .f'^
days; A, B, and D in | days; A, C, and D in J§ days;
B, C, and D in j § days. In what time could they all
do the work, and in what time could each man do it
alone ?
Ans. All in Jf days; A in 1; B in 2; C in 3; and D in 4 days.
COMPOUND NUMBERS.
DEFINITIONS.
168. An ABSTRACT NUMBER is a number whose unit
has no name other than that given it as a mere number.
Thus, 5, 29, 3i, are abstract numbers.
169. A CONCRETE or DENOMINATE NUMBER is a num-
ber ivkose unit has a name other than that given it as a
mere number. Thus, 5 dollars, 29 feet, 3 J apples, are
concrete numbers.
ITO. A SIMPLE NUMBER IS a unit or collection of units
of the same kind. A simple number is either abstract
or co7icrete. Thus, 5, 5 dollars; 3^, 3^ apples, are
simple numbers.
171. A COMPOUND NUMBER is a number consisting of
two or more concrete numbers of different unit values, but
reducible to a simfle number. Thus, 3 feet 4 inches is
a compound number, and equal to 40 inches, which is a
simple number; 8 dollars 5 cents = 805 cents.
1
f
152 COMPOUND NUMBEilS.
TABLES OF COMPOUND NUMBERS.
M 0 N E Y.
172. United States Money is the national currency
of the United States. The relative value of its differ-
ent units or denominations has already been given. —
(Vide 39-43.)
173. English Money is the national currency of
Great Britain. The units or denominations are named
Guinea, Pound, Crown, Shilling, Penny, and Farthing.
TABLE.
4 farthings (far.) make 1 penny, abbreviated d. (denarius.)
12 pence " 1 shilUng, " s. (solidus.)
20 shillings " 1 pound, " £. (hbra.)
21 shillings " 1 guinea, " G.
5 shillings " 1 crown, " Cr.
Remark 1. — The Pound, also called Sovereign, is the Primary
Unit of English Money, and is Avortli $4.84. X Crown is worth
§1.21; a Shilling, $0,242; a Penny, $0.0201; a Farthing, G^Jj mills.
Remark 2. — The Sovereign is 22 carats fine, the other 2 parts
being copper; it weighs 5 dwts. 3-J-|J- gx*s.
174. French Money is the currency of the Empire
of France. The Franc is the unit, and is worth 18 1
cents.
EXERCISES.
1. In 2 dollars how many cents? 3? 4? 5? 15? IJ?
2. In 200 cents how many dollars? 300? 400? 500:
1500? 150?
3. In U eagles how many mills? 2? 2^? 3}?
4. In 15000 mills how many eagles? 20000? 25000?
32500?
COMPOUND NUMBERS. 153
5. In 2 pounds how many shillings ? 3? 4? 5? 7?
13? 15? 25?
6. In 40 shillings how many pence? 60? 80? 100?
140? 260? 300? 500?
7. In 480 pence how many farthings? 720? 960?
1200? 1680? 3120? 3600? 6000?
8. In 1920 farthings how many pence? 2880? 3840?
4800? 6720? 12480? 14400? 24000?
9. In 480 pence how many 'shillings? 720? 960?
12.00? 1880? 3120? 3600? 6000?
10. In 40 shillings how many crowns? 60? 80? 100?
140? 260? 300'? 500?
WEiaHT.
175. Troy Weight is used in weighing gold, silver,
and precious stones. The units are named Grain,
Pennyweight, Ounce, and Pound.
TABLE.
24 grains (gr.) make 1 pennyweight, abbreviated dwt.
20 pennyweights " 1 ounce, " oz.
12 ounces " 1 pound, " lb.
Remark 1. — The Pound is the Primary Unit of Troy Weight, and
is determined by the weiglxt of 22.794422 cubic inches of distilled
water.
JIemark 2. — The carat by which the diamond is weighed and
valued is equal to 4 grains,
EXERCISES.
1. In 2 lbs. how many grs.? 3?
4? 5? 6? 13? 15? 27?
3. In 23 oz. how many grs.?
25? 26? 27? 28? 29?
2. In 11520 grs. how many lbs.?
17280? 23040? 28800? 74880?
4. In 11040 grs. how many oz.?
12000? 12480? 12960?
ail
COMPOUND NUMBERS.
5.|[n 1 lb. how many|oz.? how
Vmany dwt.? how many gr. ?
7. In 3 J lb. how many gr. ?
6. In 5760 gr. how many dwt.?
how many oz.? how many lb.?
8. In 20160 gr. how many lb.?
/170) Avoirdupois Weight is used in weighing gro-
ceries, and cheap commodities of every description.
fThc units are named Dram, Ounce, Pound, Quarter,
Hundredweight, and Ton.
abb]
^eviatc
)d oz.
a
lb.
u
qr.
ht,
n
cwt.
u
T.
TABLE.
16 drams" .) n^^ke 1 ounce,
16 ounces.'. -^ 1 pound,
25 pounds ^ 1 quarter,
4 quarters ^ 1 hundredweight,
20 hundredweight " 1 ton.
Remark 1. — The Pound is the Primary Unit of Avoirdupois
Weight, and is determined by the weight of 27.701554 cubic inches
of distilled water.
Remark 2. — The Troy Pound is the same as \ji lbs. Avoirdu-
pois, and the Troy Ounce is the same as ^|| oz. Avoirdupois.
EXERCISES.
1. In 1 T. how many dr.?
3. In 1 cwt. how many oz.?
5. In 17^. how many dr.?
7. In 37 lb. how many dr.?
9. In 47 dr. how many/oz.?
11. In 13 cwt. how many qr.?
13. In 113 T. how many dr.?
33? 45? 127? 254?
15.' In 21 lb. how many oz.? 23?
17? 13? 15? 19?
2. In 512000 dr. how many T.?
4. In 1600 oz. how many cwt.?
6. In 272 dr. how many oz.?
8. In 9472 dr. how many lb.?
10. In 18800 oz. how many qr.?
12. In 52 qr. how many cwt.?
14. In 57856000 dr. how many T.?
16896000? 23040000? G5024000?
16. In 336 oz. how many lb.?
368? 272? 208? 240? 304?
^7^ At 1 cent per ounce, what v/ill a ton of raisins
cost? Jns. P20.00.
18. At 50 cents per pound, what will 7 tons of butter
cost? Ans. $7000.00.
/V^
COMPOUND NUxMBERS.
155
19. At 5^0 cents per pound, how many tons of lard
can be bought for $35000 ? ' ^ Ans. 85.
177. Apothecaries Weight is used in mixing medi-
cines. The units arc named Grain, Scruple, Dram,
Ounce, and Pound.
TABLE.
20 grains (gr.) make 1 scruple, marked . . 9
3 scruples " 1 dram, " • . 5
8 drams " 1 ounce, " • • ^
12 ounces " 1 pound, " . . lb
Remark. — The Pound is the Primary Unit, and is the same as
the Pound Troy.
EXERCISES.
I. In 2 ft) how many §? 5? 7?
9? 11? 13? 15?
3. In 2 ft) how many 5? 6?
8? 10? 12? 14?
5. In 15 ft) how many 9? 17'
19? 21? 23? 25? 29?
7. In 16 ft) how many gr.? 18
20? 22? 24? 2G? 28?
9. In ^ lb how many 5? -J? |
II. In f of an § how many 9
t? F I? t\?
13. In y'j of a 5 ^^^^ many
2. In 24 § how many ft)? 60?
84? 108? 132? 156? 180?
4. In 192 3 how many ft)? 576?
768? 960? 1152?
6. In 4320 g how many ft)?
4890? 5472? 6048? 6G24? 7200?
8. In 23040 gr. how many ft)?
34560? 74880? 104440?
10. In 48 5 how many ft)? 32?
24? IG? 13f?
12. In 18 9 how many § ? 16?
20? 21? 10?
14. In 35 gr. how many 5? IG?
54? 45? 21?
LINEAR MEASURE.
178. Long Measure is used in measuring the length
of all quantities, except cloth. The units are named
Inch, Foot, Yard, Rod, Furlong, Mile, and League.
156 COMPOUND NUMBERS.
TABLE.
12 inches (in.) make 1 foot, abbreviated ft.
" 1 yard, " yd.
" 1 rod, " r.
" 1 furlong, " fur.
" 1 mile, " m.
" 1 league, " 1.
IvEMARK 1. — The Imperial Yard is the standard of English linear
measure, and is determined by the length of the pendulum vibra-
ting once a second at London, temperature 62} degrees Fah. This
length is 39.1393 inches.
Remark 2. — Gunter's chain, used in surveying land, is 4 rods
long, and consists of 100 links.
EXERCISES.
3
feet
H
yards
:0
rods
8
furlongs
3
miles
1. In 1 m. how many in.?
3. In 7 fur. how many in.?
5. In 2 r. how many yd.? 3?
4? 5? 6? 75?
7, In 3 r. how many in.? 12?
13? 15? 17? 19?
9. In 1 fur. how many in.?
11. In 1 1. how many ft.?
2. In 63360 in. how many m.?
4. In 55440 in. how many fur.?
6. In 11 yd. how many r.'i
16^? 22? 27J? 33? 412}?
8. In 594 in. hov/ many r.^
2376? 2574? 2970? 3366? 3762?
10. In 7920 in. how many fur.?
12. In 15840 ft. how many 1.?
13. How many inches through the earth from pole to
pole?— (Vide 67, Ex. 54.) Ans. 500478929.28.
14. How many inches through the earth at the equa-
tor? Ans. 502143776.64.
15. In 1 link of Gunter's chain how many inches?
Ans. 7 If inches.
16. In 1 mile how many chains? Ans. 80 chains.
179. Cloth Measure is used in measuring goods
bought or sold by the yard. The units are named Inch,
Nail, Quarter, Yard, Ell Flemish, Ell English, and Ell
French.
COMPOU^^'D NUMBERS.
157
2:^ inches
4
ike
nails
quarters
quarters
quarters
quarters
TAELE.
1 nail,
abbrev.
na.
1 quarter,
qr.
1 yard,
yd.
1 ell Flemish,
E. Fl.
1 ell English,
E. E. ^
1 • ell French,
E. F.
of this measure is
that of Long
Remark 1. — The stau
Measure.
Remark 2. — In mercantile practice only the yard and quarter
are in general use.
EXERCISES.
1. In 1 yd. how many in.?
3. In 3 qr. how many in.?
5. In 5 E. Fl. how many in.?
7. In 7 E. E. how many in.?
9. In 8 E. Fl. how many in.?
11. In 24 E. Fl. how many yd.?
13. In 70 E. E. how many yd.?
15. In 120 yd. how many E. FL?
17. In 1 E. F. how many in.?
2. In 30 in. how many yd.?
4. In 27 in. how many qr.?
6. In 135 in. how many E. FL?
8. In 315 in. how many E. E.?
10. In 216 in. how many E. FL?
12. In 18 yd. how many E. FL?
14. In 87^ yd. how many E. E. ?
16. In 160 E. FL how many yd.?
18. In 54 in. hoAV many E. F.?
19. In 18360 inches how many quarters? yards?
ells Flemish ? ells English? ells French?
Ans. 2040 ; 510 ; 680 ; 408 ; 340.
20. What will -| of a yard of cloth cost at 7 cts. per
nail ? Atis. 98 cts.
21. What will ^^ of a yard of calico cost at 15 cts.
per nail ? Aiis. 12 cts.
22. In 2.5 feet how many inches? Aiis. 30 in.
23. In 3.75 furlongs how many rods? An§. 150 r.
24. What is the number of miles from the Equator to
the North Pole?— (Vide 67, Ex. 61.)
Ans. 6213.824 m.
158
COMPOUND IsUMBERS.
SUPERFICIAL OR SQUARE MEASURE.
ISO. Square Measure is used in measuring sur-
faces; as land, plastering, etc. The units are named
Square Inch, Square Foot, Square Yard, Square Rod,
Rood, Acre, and Square Mile.
9 square feet
30J square yards
40 square rods
4 roods
G40 acres
1 Inch.
sq. yd.
sq. r.
R.
A.
M.
TABLE.
144 square inches (sq. in.) make 1 square foot, abb. sq. ft.
" 1 square yard,
" 1 square rod,
" 1 rood,
" 1. acre,
" 1 square mile,
Remark 1. — The standard is the same
as that of Long Measure.
Remark 2. — 16 square rods make 1
i_i square chain, and 10 square chains make
^ 1 acre.
g Remark 8, — The figure in the margin
is exactly 1 square inch; that is, it is
1 linear inch on each side. 144 such
squares are equivalent to a square foot,
however the arrangement may be. They
are equal to a square foot when arranged so as to make another
square.
EXERCISES.
1 SQUARE INCH.
1 L\Gir.
1. In 1 A. how many sq. in.?
3. In 1 A. how many sq. r.?
2? 3? 4? 5?
5. In 1 sq. r. how many sq. ft.?
2? 5? 8?
7. In 1 R. liow many sq. yd.?
3? 13? 17?
2. In 6272640 sq. in. how many
A.?
4. In IGO sq. r. how many
A.? 320? 480? 640? 800?
6. In 272J sq. ft. how many
sq. r.? 544^? 1361^^? 2178?
8. In 1210 sq. yd. how many
R.? 3030'' ir)7:50? 20570?
COMPOUND NUMBER!
159
0. la 1 sq. r. how many sq.
in.?
11. In 1 A. lioAv many sq.
ft.?
13. In 1 sq. yd. how many sq.
in.? 121? 242?
10. In 39204 sq. in. how many
sq. r.?
12. In 130680 sq. ft. how many
A.?
14. In 1296 sq. in, how many
sq. yd.? 156816? 313632?
15. In 252| A. how many sq. I 16. In 2524 sq. ch. how many
ch.? A.?
17. In -^^ m. how many fur.? 18. In ^j fur. how many r.?
Ans. 5^j. I Ans. Z^j.
SOLID MEASURE.
181. Cubic Measure is used in measuring solids, as
timber, earth, and such other things as have length,
breadth, and thickness. The units are named Cubic
Inch, Cubic Foot, Cubic Yard, Ton, Cord Foot, and
Cord. -
TABLE.
1728 cubic inches (cu. in.) make 1 cubic foot, abb. cu. ft.
ton, "
ton of shipping, "
cord foot, "
cord, "
cu. yd.
T.
T.
T.ofS.
CO. ft.
CO.
27 cubic feet " 1 cubic yard,
40 feet of round timber " 1 ton,
50 feet of hewn timber " 1
42 cubic feet " 1
16 cubic feet " 1
8 cord feet " 1
Remark 1. — The standard of this meas-
ure is that of Long Measure.
Remark 2. — A cube is a solid bounded
by 6 equal squares.
Remark 3. — The figure in the margin
represents an exact cubic inch. Its
squares are called faces, and the bounda-
ries of the faces are called edr/es. Each
edge represents 1 linear inch. Each edge
of a cubic foot contains 12 linear inches, so that there are
12x12x12: that is, 1728 cubic inches in a cubic foot.
/
/'
1 Cubic Inch.
/
Y 1 Linear Inch.
/
160
COMPOU]S'D NUMBERS.
EXERCISES,
1. In 1 CO. liow many cu. ft.?
2? 3? 4? 10?
1-V?
3. In 1 CO. liow many cu. in.?
6? 6? 7? ^? -I? f?
5. In 3.125 CO. liow many cu. ft. ?
7. In 1 cu. yd. how many cu.
2. In 128 cu. ft. how many co.?
256? 384? 512? 1280? 96? 192?
4. In 221184 cu. in. how many
CO.? 27648? 41472?
6. In 400 cu. ft. how many co.?
8. In 46656<»cu. in. how many
cu. yd.? 3888? 324? 36?
182. Wine Measure is used for measuring alh
liquors, except ale, beer, and milk. The units are
named Gill, Pint, Quart, Gallon, Tierce, Barrel, Hogs-
head, Pipe, and Tun.
TABLE.
4 gills (gi.) make 1 pint, abbreviated pt.
2 pints
4 quarts
31J- gallons
42 gallons
63
2
gallons
hogsheads
pipe
1 quart, "
1 gallon, "
1 barrel, "
1 tierce, ^"
1 hogshead, "
1 pipe, "
1 tun. ^
qt.
gal.
bbl.
ti.
hhd.
py
Remark. — The Wine- Gallon contains 231 cubic inched
1. In 1 tun how many hhd.?
gal.? qt.? pt.? gi.?
3, In 13 gal. how many gi.?
15? 17? 19? 21? 23?
6. In 5 tuns how many gi.? 7?
9? 11? 13? 17?
7. In 3 bbl. how many gal.?
4? 7? 10? 13? 16?
9. In 1 hhd. how many bbl.?
11. In 126 ti. how many hlid.?
13. In 2.5 qt. how many gal.?
15. In 23.625 gal. how many hhd ?
EXERCISES.
2. In 8064 ^\A\(
pi
(ii€ , -1 i
hhd.? pi.? tuns?
4. In 416 gi. how many gal.?
480? 544? 608? 072? ^?
6. In 40320 gi. how m^y= tuns/
56448? 72-576? 88704?
8. In 94J.,gal. how many bbl.?
126? 220i'j^l5? 409 J?
10. In 1«"0 bbl. how many hhd.?
12. In 168 bbl. how many ti.?
14. In .625 gal. how many qt.?
16. In .375 hhd. how many gnl.?
3^
COMrOUXD XUxMBERS. 161
183. Ale or Beer Measure is used for measuring
ale, beer and milk. The units are named Pint, Quart,
Gallon, Barrel, and Hogshead.
table.
2 pints (pt.) make 1 quart, abbreviated qt.
4 quarts " 1 gallon, " ' gal.
36 gallons " 1 barrel, " bbl.
\\ barrels . " 1 hogshea'd, " hhd.
Remark. — The Beer Gallon contains 282 cubic inches.
EXERCISES.
1. In I hhd. how many gal.?
qt..? pt.? /
3. In 693 Beer Gal. how many-
Wine Gal.? 2079?
6. In Jg of a gal. how many pt. ?
7. In -^-^ of a hhd. how mamy
pt.? ' '"^
9. In \ of a bbl. how many
qt.?
2. In 432 pt. how many qt.?
gal.? bbl.?
4. In 846 Wine Gal. how many
Beer Gal.? 2538?
6. In I a pt. how many gal.?
8. In 1 pt. what part of a
hhd.?
10. In 36 qt. what part of a
bbl.?
184. Dry Measure is used in m^easuring such arti-
cles as grain, fruit, etc. The units are named Pint,
Quart, Peck, Bushel, and Quarter.
TABLE.
2 pints (pt.) make 1 quart, abbreviated qt.
8 quarts " 1 peek, " pk.
4 pecks " 1 bushel, " bu.
8 bushels " 1 quarter, " qr.
Remark. — The Winchester Bushel is a cylinder, 18^- inches
internal diameter, and 8 inches deep. It contains 2150.4 cubic
inches.
14
162 COMPOUJS^D XUMBEilS.
EXERCISES.
1. What cost 25 quarters of wheat at 90 cents per
bushel? A71S. $180.
2. At 90 cents per bushel how many quarters of
wheat can be bought for $360 ? Ans. 50 quarters.
3. What cost 17 bushels of apples at 27 cts. a peck?
4. At 27 cents a peck, how many bushels of apples
can be bought for $18.36 ?
5. What must be paid for 7 bushels of chestnuts at
3 cents a pint? Ans. $13.44.
6. What cost I of a pint of blackberries at $3.20 per
bushel? Ans. 3 cents.
7. W^hat cost 25-J bushels of potatoes at 20 cents a
peck? ^ns. $20.70.
8. How many bushels of potatoes can I buy for
$41.40, at 2-J cents a quart? Ans. 51|- bushels.
TIME.
185. The units of Time are named Second, Minute,
Hour, Day, Week, Month, Year, Century.
TABLE.
GO seconds (sec.) make 1 minute, abbreviated m.
60 minutes " 1 hour, " h.
24 hours " 1 civil day, " d.
% days " 1 week, " w.
12 months " 1 year, " y.
Remark 1. — The standard unit of time is the period occupied by
the eurth in making one revolution on its axis, wliich period is
called a Sidereal Day and consists of 23 h. 5G m, 4 sec.
Remark 2.— The Tropical Year consistR of 305 d. 5 h. 4S m. 47.57
S3C.
COM IHJ V .\ D N U M 13 ERS .
163
REMAini: 3.— The Civil, Legal, or Julian Year consists of 365
days, except Leap Year, "tvhich consists of 300 days.
Remark 4. — Every year which is exactly divisible by 4 is a
Leap Year, excepting those centennial years not exactly divisible
by 400. Thus, 1868 will be a Leap Year; 1900 will not be a Leap
Year; but the year 2000 will be a Leap Year.
TABLE OF THE MGNTIH
Mouth.
Abb.
Order.
Xo. D.
Mouth.
Abb.
Order.
Xo. D.
January.
Jan.
1st.
31.
July.
7th.
31.
February.
Feb.
2d.
28.
August.
Aug.
8th.
31.
March.
Mar.
3d.
31.
September,
Sept.
9th.
30.
April.
Apr.
4th.
30.
October.
Oct.
10th.
31.
May.
5th.
31.
November.
Nov.
11th.
30.
June.
6th.
30.
December.
Dec.
r2th.
31.
Remark 5. — In Leap Year, February has 29 days.
Remark 6. — In finding the interval between two dates, it is
customary to consider the months as having 30 days each.
TABLE
Showing the time in days from any day in one month to the corref^pond-
ing day in another month.
January...
February...
March
April
Mav
June
•'^ny
AugXTSt
Sepr'ember.
October
November .
December..
365
1 334
■306!
!275|
i245
l214!
184
153
122
92
61
31
31
365
337
306
59
28
365
334 365
276|304l335
245I273!304
21o;2!3274
184 212 243
89
61
30
365
90;120il51181
59
31
120 150
1334 365
304 335
122
91
61
30
365
212243
1811212
153;184
122 153
t\^
153
123
92
62
181J212
151 i 182
120;i51
90:i21
242!273!303
2121243:273
18l!212;242
151:182212
92
61
31
365
334
123
92
62
31
365
2731304 334
242i273 303
214|245j275
183l214|244
!153 1841214
153{1
122
92
61
30
[365
13341365
3041335
273|304
243:274:3041 3351 365
1231153
92J122
61 91
3l| 61
30
164 COMPOUND NUMBERS.
EXERCISES.
1. How many days from January 10th to June
10th? Ans, 151.
Find January in the left-hand column, and follow the line to
the right till you come to June,
2. How many days from February 6th to May
6th? Ans. 89.
3. How many days from January 1st to July
4th? ■ Ans. 184.
Here add 3 to the tabular number, which is 181.
4. How many days from December 25th to July
10th? Ans. 197.
Here subtract 15 from the tabular number, 212,
5. How many days from January 17, 1868, to July
17, 1868? Ans. 182.
Here add 1 to the tabular number for Leap Year, as the dates
include the month of February,
CIRCULAR MEASURE.
186. Circular Measure is used in estimating Lati-
tude and Longitude, and in measuring the relative dis-
tances of the Planets and other heavenly bodies. The
units are named Second, Minute, Degree, Sign, and
Circumference of Circle.
TABLE.
60 seconds (") make 1 minute, marked '
60 minutes " 1 degree, " °
30 degrees " 1 bign, abbreviated S.
12 signs make 1 circumference of circlcj abb. circ.
COMPOUND NU3IBE11S.
166
Remark 1. — The length of a degree measured on the equatoi- is
69.161 m. The length of a degree measured on a meridian is
69.042 m.
Remark 2. — A degree contains 60 geo-
graphic miles.
Remark 3. — Since every circumference of
a circle contains 360 degrees, the length of
the degree varies as the diameter of the cir-
cle varies. The circumference of a circle
is always about 3.1416 times its diameter.
Remark 4. — The Celestial Equator is di-
vided into 12 Signs; their names are Aries,
Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Saggitarius,
Capricornus, Aquarius, and Pisces.
EXERCISES.
1. In 1 circle how many degrees? minutes? seconds?
2. What is the length of 1' along the equator?
AiiB. 1.15268 m.
3. What is the length of V along a meridian ?
Am. 1.1507 m.
4. What is the length of V along the equator?
An%. .01921.
5. What is the length of V^ along a meridian ?
Am, .01918 nearly.
6. If the semi-circumference of a circle is 3.1415926-
535897932, etc., inches in length, as it really is when
the diameter is 2 inches, what is the length of 1' ?
Am. .000290888298665721 in.
7. What is the circumference of a circle whose diam-
eter is 10 inches? Am. 31.4159 in.
8. What is the circumference of a circle whose diam-
eter is 10,000,000 miles?
Am. 31415926.535 m.
166
COMPOUND ^'UMBERS.
KEDUCTION OF COMPOUND NUMBERS.
187. Reduction in Aritlimetic consists in making
some change in the method of i^epresenting a quantity.
Hence, reduction makes no change upon the value of a
quantity.
188. It has been seen that many simple concrete
numbers may be reduced to other concrete numbers of
a lower unit value By multijMcation, Thus,
2 £=40 s., because 20x2=40. (Vide 174, Ex. 5.)
189. *lt has also been seen that simple concrete
numbers may be reduced to other simple concrete num-
bers of a higher unit value by division. Thus,
72 in.= 6 ft., because 72-f-12=6. (Vide 178.)
190- To reduce a compound
number to a simple concrete
number,
Arrange the different units com-
posing the compound number in a
horizontal line^ and over each place
the number connecting it with the
next higher unit.
Multiply the units of the highest
value hy that number which stands
over the next lower units, and to the
])roduct add the same loiver units;
then multiply the sum by the num-
ber standing over the next lower
units, etc., continuing the work till
the lowest units given have been
added.
EXAMPLES.
1. Reduce 17 £ 68. Od. 8 far. to
fartliinss.
191. To reduce a simple con-
crete number to a compound
number.
Divide the given number by that
number which connects it ivith the
unit of the next higher value, plac-
ing the remainder, if there be any,
to the right.
Continue the work till the unit
of the highest value is reached, or,
till the next divisor would be greater
than the dividend.
Ulc last quotient and the several
remainders, written in the order of
their tmit values, will be the com'
2^ound number.
E X A ai P L E S .
2. Reduce 1G599 farthings to
pounds, etc.
CO M PO L X D >: U:.I BERS .
167
b.
7.
9.
11.
grai
13.
15.
17.
19.
21.
23.
25.
OPERATION.
20 12 4
17 5 9 3
20
345
12
4149
4
16599 far. Ans.
Reduce 20 <£ 15 s. 9d. 3 far.
Reduce 240£ Os. 7d. 2 far.
Reduce 17s. Ifar. to far.
Reduce lib. loz.ldwt, Igr.
Reduce 17 lb. 5 dwt. to
ns.
Reduce Soz, 3 dwt. 3gr.
Reduce 7 T. 15 oz. to oz.
Reduce 3T. 7cwt. 3qr.
Reduce 3qr. 11 oz. 13 dr.
Reduce 1 lb 3 § 55.
Reduce lib 2 3 to grains.
Reduce 5r. 4 yd. 2 ft. 7 iu.
OPERATION.
5K 3 12
5 4 2 7
54
3U
3^
96^
12
OPERATION.
1G599
12
20
4149—3
345—9
4.
6.
8.
10.
12.
etc.
14.
16.
18.
20.
22.
24.
26.
17—5
17-£ 5s. 9d. 3far. Ai
Reduce 19959 far. to £, etc.
Reduce 230430 far. to £.
Reduce 817 far. to s., etc.
Reduce 6265 gr. to pounds.
Reduce 98040 gr. to pounds,
Reduce 1515gr. to oz., etc.
Reduce 224015 oz. to T., etc.
Reduce 27;igr. to T., etc.
Reduce 19389 dr. to qr., etc.
Reduce 7500 gr. to To, etc.
Reduce 5800 gr. to lb, etc.
Reduce 1165 in. to r., etc.
OPERATION.
1165
97—1
32—1
64 =lialf yds.
5— 4 J
1165 iry.AriS. ^/Js. 5r. 4)/^yd.lft.lin.=5r. 4j'd.2ft.7m.
27. Reduce 401. 6 fur. 2 in. to I 28. Reduce 7650722 in. to
inches. | leagues, etc.
29. Reduce 22 fur. IGr. 3yd. | 30. Reduce 1479-4 ft. to fur-
1ft. to ft-et. longs, etc.
168
31. Reduce 4 m. 7 fur. 20 r 16
ft, to inches.
33. Reducel yd. Iqr, 2na. 7}in.
35. Reduce 4 tuns 5hhd. 3qt.
to quarts.
37. Reduce Itun Igal. 3qt. to
gills.
39. Reduce 47bbl. 18 gal. of
ale to pints.
41. Reduce 15 bu. 2pk. 7qt. to
quarts.
43. Reduce 9bu. 5qt. Ipt. to
pints.
45. Reduce 14 A. IR. 17 r. to
rods.
47. Reduce 17 A. 3R. 12 r. to
square feet.
49. Reduce 3 da. 55 m. to min-
utes.
51. Reduce 9S. 13° 25^ to sec-
onds.
63. Reduce 25° 14^ V^ to sec-
onds.
55. Reduce 5 fur. 3 r. 10 ft. 6 in.
to inches.
57. Reduce 21$ 3 m. to mills.
COMPOUND NUMBERS.
32. Reduce 313032 in. to miles,
etc.
34. Reduce 50 in. to yards, etc.
36. Reduce 5295 qt. to tuns,
etc.
38. Reduce 8120 gi. to tuns,
etc.
40. Reduce 13680pt. to barrels,
etc.
42. Reduce 503 qt. to bushels,
etc.
44. Reduce 587 pt. to bushels,
etc.
46. Reduce 2297 r. to acres,
etc.
48. Reduce 776457 sq. ft. to
acres, etc.
50. Reduce 4375m. to days,
etc.
52. Reduce 1020300^^ to signs,
etc.
54. Reduce 90847^'' to degrees,
etc.
56. Reduce 40320 in. to fur-
longs.
68. Reduce 21003 m. to dollars.
59. What cost 1 lb. 1 oz. 1 dwt. 1 gr. of gold, at 3^
cts. per grain? Ans. 208.83 J.
60. If gold is worth 3 J cts. per. grain, hoy>^ many
pounds can be bought for $626.50 ?
Ans. 3 lb. 3 oz. 3 dwt. 3 gr.
61. What cost 17 lb. 5 dwt. of silver, at 31^ cts.
per dwt. ?
62. What weight of silver can be bought for |1260.-
93|-, at the rate of 31 J cts. per dwt?
COMPOUXD NUMBERS. 1G9
63. What will 3 T. 7 cwt. 3 qr. of rice cost at 3d.
English money per- pound ?
64. How much rice at 3d. per pound can be bought
for £84 13s. 9d.?
65. At the rate of |0.060J per pound, how many tons
of hay can be had for $4D9.887i ?
66. What cost 4 m. 7 fur. 20 r. 16 ft. of railroad, at
an expense of |3.78f| per ft.?
67. How many miles of railroad can be built for
^98810.60f §, at an expense of 3f | dollars per ft. ?
68. What will 4 tuns 5 hhd. 3 qt. of molasses cost,
at 15 cts. per qt. ?
69. What number of tuns of claret can be had for
$794.25, at the rate of 7J cts. per pt,?
70. What will 15 bu. 2 pk. 7 qt. of chestnuts amount
to in dollars, at 3 farthings English money per pt. ?
71. How many bushels of chestnuts can be had for
§15.215-1 at the rate of IJd. per qt.?
72. If a rod of land produce 3i pounds of cotton,
how many bales can be taken from 14 A. 1 R. 17 r. of
land, at 500 pounds to the bale ?
73. If a rod of land produces 3| pounds of cotton,
what number of acres will produce 14yYo bales ?
74. A gentleman having 17 A. 3 R. 10 r. of land, laid
it off in lots, each containing 25 rods, and sold them at
|450 per lot. How much did he get for his land?
75. If I sell a quantity of land at the rate of $450
for 25 sq. r., and obtain $51300, how many acres do I
sell?
192. To reduce a denominate 103. To rediioe a compound
fraction to a compound number, number to a denominate fraction,
Multiply the fraction by thenum- Reduce the unit of the proposed
15
170
COMPOUND NUMBERS.
her zvhich connects it with the next
loicer imitj and the fractional part
of the product by the number which
connects it with the next lower unit,
and so on till the lowest unit of the
table is reached.
The several integral parts of the
product will form the compound
number, retaining the fractional
part, if there be any, of the last
quotient.
EXAMPLES.
compound number.
OPERATION.
11)56(5 (Vide 180, Ex. 17.)
65
1
40
11)40(3 (Vide 180, Ex. 18.)
7
m
ll)115i(10
no
5J
12
11)66(6
66
Ans. 5 fur. 3r. lOft. Gin.
fraction to the lowest units men-
tioned in the compound number for
the denominator of the required
fraction.
Reduce the compound number to
the same units for the numerator of
ths fraction.
Reduce the fraction to its lowest
terms.
Remark. — If there is a fraction
connected loith the lowest units, mul-
tiply both parts by the denominator
of the fraction before reducing.
examples.
2. Reduce 5 fur. 3r. 10 ft. 6 in.
to a denominate fraction.
operation.
40
16>^ 12
1
8
5 3
40
10 6
8
40
203
16J
320
16^
3359*
12
5280
12
40320
63360
(Vide 178, Ex. 1, and 190, Ex.55.)
COMPOUND NrMBERS.
171
3. Reduce |ra. to a compound
number.
^ 5. Reduce y'gCwt. to a com-
pound number.
7. Reduce f T. to a compound
number.
9. Reduce ^£ to a compound
number.
11. Reduce |s. to a compound
number.
13. Reduce ^d. to a compound
number.
15. Reduce iituns to a com-
pound number.
17. Reduce flilid. to a com-
pound number.
19. Reduce f bbl. wine to a
compound number.
21. Reduce If A. to a com-
pound number.
23. Reduce 17|fA. to a com-
pound number.
4. Reduce 3 fur. 22 r. 3 ft. Sin.
to fraction of mile.
6. Reduce Iqr. 181b. 12 oz. to
fraction of hundredweight.
8. Reduce 8cwt. 2qr. 71b. 2 oz.
4idr., etc.
10. Reduce 8s. 6 d. 3ffar., etc.
6 3f
OPERATION.
OPEEATION.
3
20
1
20
8
12
7)60(8
66
20
12
102
4
4
240
411f
12
4
2880
7)48(6
42
960
7
6
6720
(191 Rem.)
4
|f|^=3£ Ans.
7)74(3^
'
8s. 6d. 33 far. Ans.
12. Reduce 4d. 2 far. to frac-
tion of a shilling.
14. Reduce If- far. to fraction
of a penny.
16. Reduce Ipi. Ihhd. 42 gal.
to fraction of a tun.
18. Reduce 23gal. 2qt. Ipt.,
etc.
20. Reduce 18 gal. 3qt. Ipt.
f gi., etc.
22. Reduce 1 A. IR. 28fr. to
fraction of acre.
24. Reduce 17 A. 3R. 12 r. to
fraction of acre.
172
COMPOUND XL'MEERS.
25. Reduce ^-^ yr. to a com-
pound number.
27. Reduce |- w. to a compound
number.
29. Reduce | h. to a compound
number.
31. Reduce ^bu. to a compound
number.
33. Reduce || bu. to a com-
pound number.
35. Reduce 3| lb. Troy to a
compound number.
2(i. Reduce TOO
fraction of year.
da. 12 h. to
28. Reduce 2 da. 19 h. 12 m. to
fraction of week.
30. Reduce 22 m. 30 sec. to
fraction of hour.
32. Reduce 3 pk. 2qt. IJpt. to
fraction of bushel.
34. Reduce 1 pk. 6 qt. § pt. to
fraction of bushel.
36. Reduce 3 lb. 10 oz. to frac-
tion of pound,
38. Reduce 9 dwt. 9 gr. to frac-
pound number. tion of pennyweight.
39. If I ride 5 fur. 3 r. 10 ft. 6 in. in a railroad car,
what ought to be my exact fare at the rate of 11 cts. a
mile ? Ans. 7 cts.
40. What cost the iron on a track measuring 3 fur.
22 r. 3 ft. 8 in., at the rate of §4500 per mile ?
Ans. §2000.
41. Sold 1 pi. 1 hhd. 42 gal. of molasses at the rate
of 175.60 per tun. What did I get ? Ans. $69.30.
42. A grocer bought 8 cwt. 2 qr. 7 lb. 2 oz. 44 dr. of
coffee at §9.50 per cwt., and sold it at a retail price of 16
cts. per lb. How much did he make? A7is. §55.71 f.
43. Bought 1 pk. 6 qt. f pt. of chestnuts at the rate
of §2.88 per bushel, and retailed them at 9 cts. per
quart. Do I make or lose ?
44. If a boy could count 6000 marbles in an hour,
how many could he count in 22 m. 30 sec. ?
Ans. 2250.
45. If I buy iron at §45 per ton, and sell it at 2J cts.
per pound, what do I gain by selling 13 cwt. 2 qr. 15
lb.? Ans.^S.41h
COMPOUND NUMBERS.
173
194- To reduce a compound
number to a decimal fraction,
Divide the lowest units by that
number which connects them with
the next higher^ and annex the quo-
tient as a decimal to the given num-
ber of those higher units.
Continue to divide till the units
required are reached.
The last quotient will be the re-
quired decimal.
EXAMPLES.
1. Reduce 23 gal. 2 qt. 1 pt. to
a decimal in hogshead.
OPERATION.
2.5. (Vide 182, Ex. 13.
63123.625 (Vide 182, Ex. 15.)
.375 hhd. Ans.
(Vide 192, Ex. 17.)
3, Reduce 3 fur. 22 r. 3 ft. 8 in.
to decimal fraction of mile.
5. Reduce 1 qr. 18 lb. 12 oz. to
fraction of hundredweight.
7. Reduce 8 s. 6d. 3^ far. to
fraction of pound sterling,
9. Reduce 15 £ 10s. 9d. to
fraction of pound sterling.
11. Reduce 18 h. 9 m. to frac-
tion of day.
13. Reduce 5 cwt. 2 qr, 15 lb. to
fraction of ton.
195. To reduce a denominate
decimal fraction to a compound
number.
Multiply the given decimal frac-
tion by the number which connects
it with the next lower units, and the
decimal part of the product by the
number connecting it tvith the next
lower units, and so on till the lowest
unit is reached. The integral parts
of the products will form the num-
ber required.
EXAMPLES.
2. Reduce .375 hhd. to a com-
pound number.
OPERATION.
.375 hhd.
63
23.625 gal.
4
2,500 qt.
2
1.000 pt.
Ans. 23 gal. 2qt. Ipt.
4. Reduce .44| m to a com-
pound number.
6. Reduce .4375 cwt. to a com-
pound number.
8. Reduce .42857^ £ to a com-
pound number.
10. Reduce 15.5375 <£ to a com-
pound number.
12, Reduce .75625 da. to a com-
pound number.
14. Reduce .2825 T. to a com-
pound uumbci'.
174: COMPOUND N^UMBERS.
15. Reduce 3 ft. 9 in. to frac-
tion of yard.
17. Reduce 3 pk. 2 qt. lipt. to
fraction of bushel.
19. Reduce 22 m. 30 sec. to
fraction of hour.
21. Reduce 2 da. 19 h. 12 m. to
fraction of week.
23. Reduce 3° 30^ 36^^ to frac-
tion of degree.
16. Reduce 1,25 yd. to a com-
pound number.
18. Reduce .83^ bu. to a com-
pound number.
20. Reduce .375 h. to a com-
pound number.
22. Reduce .4 w. to a compound
number.
24. Reduce 3.51° to a compound
number.
25. What will 23 gal. 2 qt. 1 pt. of wine cost at $60
per hhd. ? A71S. $22.50.
26. What will the above wine bring at $1 per gal. ?
Alls. §23.625.
27. What will 3 pk. 2 qt. IJ pt. of corn cost at 90
cts. per bushel ? Ans. 75 cts.
28. If I buy 13 A. 2 R. 35 r. of land at $17.28 per
acre, and sell it in lots of 1 rood each at 12 dollars a
lot, how much do I make? Ans. $421.44.
29. What will .2825 tons of rice bring at 7 cts. per
lb.? at $1.50 per qr.? at $5 per cwt. ?
Ans.$S9.bD; $33.90; $28.25.
30. At $25 per acre, how much land can be bought
for $648.75 ? Aits. 25.95 A.-=25 A. 3 R. 32 r.
31. Bought 18 cwt. 1 qr. 18 !b. of tea at $65 per cwt.,
and sell the same at 75 cts. per lb. How much do I
I make? Ans. $184.30.
32. Bought 56 hhd. 16 gal. 3 qt. of molasses at
$46 per hhd. What did it amount to?
Alls. $2588.23.
33. I sell the above molasses at 87J cts. per gal.
What do I gain ? Ans. $513.43.
34. A planter sold 270 bales of cotton at $60 per
COMPOUND NUMBERS. 175
bale, and invested the proceeds in land at $28 per acre. ,
How much land did he purchase ?
Ans. 578 A. 2 R. llf r.
35. A merchant imported 325 yards of silk, at an
expense of £1 4s. 6d. per yard, and desires to clear
$1200 in retailing it. What must be the price per yard ?
(Vide 173, Rem. 1.) A7is. $9,621.
36. I import 275 yards of French broadcloth, at an
expense of 24 francs per yard, and clear, in retailing it,
$500. What do I charge per yard? (Vide 174.)
Ans, $6.23.
ADDITION OF COMPOUND NUMBERS.
190. To add compound numbers,
(1.) Wrile the units of the same name under each other ^
and place over each column that number which connects
its unit with the next higher unit.
(2.) Add the coluynn of units of the least value, and
divide the sum hy that number which stands over it,
placing the remainder under the column.
(3.) Add the column of units of the next higher value,
including the quotient of . the preceding division, and
divide by the number which stands over it, placing the re-
mainder under the column.
(4.) Add all the colmnns of units in the same ivay,
writing down, hoivever, the entire sum of the column
ivhich can have no number over it.
' EXAMPLES.
1. Add together 20£ 15s. 9d. 3 far. ; 240c£ 7d. 2 far.;
and 17s. 1 far.
176 COMPOUND NUMBERS.
2. Add together 40 1. 6 fur. 2 in.; 2 m. 6 fur. 16 r.
3 yd. 1ft.; 4 m. 7 fur. 20 r. 16 ft.; 11. 2 m. 7 fur. 13 r.
1 ft. 11 in.
OPERATIONS.
(1-) (2.)
20
12
4
20
15
9
3
240
0
7
2
17
0
1
40 5% 3 12
40 0
6
0
0
0
2
2
6
16
3
1
4
7
20
0
16
1 2
7
13
0
1
11
Ans. 261 £ 13 s. 5d. 2 far.
441. 2 m. 3 fur. 10 r. 3 ^yd. 1ft. 1 in.
Ans. 441. 2 m. 3 fur. 10 r. 3 yd. 2 ft. 7 in., since ^ yd.=l ft. 6 in.
3. Add together 1 hhd. 25 gal. 3 qt. 1 pt. ; 8 hhd. 2 qt. ;
3 hhd. 27 gal. 1 pt. ; and 21 hhd. and 1 pt.
Ans. 8 T. 1 hhd. 53 gal. 2 qt. 1 pt.
4. Add together 13 bu. 2 pk. 7 qt. 1 pt. ; 150 bu. 1 pk.
5 qt.; 200 bu. 3 pk. 5 qt. 1 pt. Ans. 365 bu. 2 qt.
5. Add together J J of a tun, § of a hhd., and | of a
bbl.
OPERATION.
2 2 f)3 4 2
\i tun =
1 hhd.-
f bbl.=:
1 1 42
23
18
0
2
3
0 (Vide 192, Ex. 15.)
1 ( " 192, " 17.)
li ( " 192, " 19.)
ItunO 0 21 gal. 2 qt. J pt.
6. Add together | £ g s. and f d.
Ans. 8s. lid. 3|far.
7. Add together 1| A. and 17f § A.
Ans.^lO A. 1 R. 4 r.
8. Add together /jj yr. § w. and | h.
Ans. 112 da. 7 h. 34 m. 30 sec.
COMPOUND NUMBERS. 177
9. Add together § bu. and || bu.
Ans. 1 bu. 1 pk. 1 qt.
10. Add together j i m. and f fur.
Ans. 7 fur. 26 r. 5 ft. 4,% in.
11. Add together 15.5375£ and .4285|£. (Vide 195,
Ex. 8 and 10.) Ans. 15£ 19s. 3d. 3|far.
12. Add together .4 bu. and .7 pk.
Ans. 2 i)k. 2 1 qt.
13. Add together 5.88125 A. and4 A. 2 R. 35 r.
Ans. 10 A. 2 R. 16 r.
14. Add together J lb. Troj and .583J oz.
Ans. 6 oz. 11 dwt. 16 gr.
15. Add together .875£ and .75s. Ans. 18s. 3d.
SUBTRACTION OF COMPOUND NUMBERS.
197. To subtract compound numbers,
(1.) Write the nitmhers, as in 196.
(2.) Subtract the units of the loioest value in the sub-
irahend from the corresponding units of the 7ninuend,
and iJ^lace the difference under the same column; but if
the number in the subtrahend is larger than that in the
minuend, add the number standing over the column to the
number in the minuend, and subtract from the sum the
number in the subtrahend.
(3.) If tfte jurmher standii^g above any column has
been employed in the subtraction, add 1 to the units of
next higher value in the subtrahend ; after which proceed
exactly as with the preceding column, and so on till all
the columns have been subtracted.
178 COMPOUND NUMBERS.
EXAMPLES.
1. From 21 r. 3 ft. 5 in. take 17 r. 16 ft. 9 in.
(1.)
12
OPERATIONS.
A7IS.
(2.)
12
21 3
17 16
5
9 . .
21
18
3
0
5
3
3r. 24ft.
Sin.
3r.
3ft.
2.
Remark. — The first result is easily reduced to tlie second by ob-
serving that 6 in.= ^ ft.
2. From 3 fur. 29 r. 2 yd. 1 ft. take 1 fur. 39 r. 3 yd.
2 ft. A71S. 1 fur. 29 r. 4 yd. 6 in.
3. From 63 T. 1 hhd. 15 gal. take 19 T. 3 lihd. 17
gal. A71S. 43 T. 1 lilid. 61 gal.
4. From 8 bu. 3 pk. 1 qt. take 3 bu. 2 pk. 7 qt.
Ans. 5 bu. 2 qt.
5. From 25° 4' 27^^ take 17° 20^ 40^^
A71S. 7° 43' 47'^
6. JV^hat is the difference between 40 m. and 39 m.
7 fur. 39 r. 16 ft. 7 in. ? Ans. 1 inch.
7. From 77° 0' 15'^ take 71° 3' 30^'.
Ans. 5° 56' 45''.
8. From 85° 30' take 77° 0' 15". Ans. ^ 29' 45".
9. From 86° 49' 3" take 79° 55' 38".
Ans. 6° 53' 25".
10. Washington is 77° 0' 15" West Longitude, and
Boston 71° 3' 30" West Longitude. What is the differ-
ence in the Longitude of these places ?
Ans. 5° 56' 45".
COMPOUND NUMBERS. 179
11. Louisville is 85° 30' W. L., and Mobile 88° 1' 29"
W. L. What is the difference in the Longitude of these
places ? Ans. 2° 31' 29''.
12. The City of Mexico is in North Latitude 19° 25'
45", and Cincinnati is in N. L. 39° 5' 54". What is the
difference ? Ans. 19° 40' 9".
13. The difference of time between Greenwich and
Milledgeville, Ga., is 5 h. 33 m. 19 sec. When it is
noon at Greenwich, what is the time at Milledgeville ?
Ans. 26 m. 41 sec. past 6 A. M.
Remark. — The time of a place being given, the time of all places
east of it is later, and of all places zcest^ earlier.
198. To find the time between two dates,
Write the year, the order of the month, (Vide 185,
Rem. 4,) and the day of the month of each date, respect-
ively, under each oilier.
Then proceed as in 197.
■ '
EXAMPLES.
1. Find the
:, 1867.
time
from
January
27,
1865,
to
July
OPERATION.
12 30
1867
1865
7 4
1 27
Ans. 2yr. 5 m. 7 da.
2. Find the time from July 4, 1865, to August 1,
1866. Ans. 1 yr. 27 da.
3. Find the time from August 1, 1869, to September
9, 1871. Ans. 2 yr. 1 m. 8 da.
180 COMPOUND NUMBERS.
4. Find the time from November 15, 1866, to Decem-
ber 8, 1879. Ans. 13 yr. 0 m. 23 da.
5. Find the time from January 27, 1866, to Septem-
ber 9, 1871. Ans. 5 yr. 7 m. 12 da.
6. The Independence of the United States was de-
clared July 4, 1776. What interval has passed on Jan-
uary 1, 1867? A71S. 90 yr. 5 m. 27 da.
Remark.— The true interval found by the table, (185, Rem. 6,)
is 90 yr. 181 da.)
7. A merchant bought at one time 8120 gills of wine,
at another J J of a tun, at another | of a hhd., and at
another time .375 of a hhd. What did the whole cost
at 11.00 per gallon ? Ans. $532.00.
8. Bought at one time 17 yd. 3 qr. 2 na. of broad-
cloth; at another time lo^% yd.; at another 87.8125
yd.; at another 27 yd. Ij qr. ; and at another time
29.375 yd. What did the whole cost at 5 J dollars per
yard? Ans. ^$965.93 1 .
9. From a lot of land containing 10 A. 3 R. ip r., I
sell at one time 1 A. 2 R. 13 r.; at another time 2 A. 2 R.
5r. I gave §600.53 J- for the land; sold the first lot
at $70 per acre, and the second lot at $75 per acre.
For how much per acre could I sell what remains, and
lose nothing ? Ans. $44.78.
10. Sold corn in three lots, viz:. 13 bu. 2 pk. 7 qt.
1 pt., at 60 cts. per bu.; 150 bu. 1 pk. 5 qt., at 50 cts.
per bu. ; 200 bu. 3 pk. 5 qt. 1 pt., at 55 cts. per bu.
What should I have gained by selling all the corn at 56
cts. per bu. Ans. $10.48|J.
11. A load of hay weighs 43 cwt. 2 qr. 18 lb., includ-
ing the wagon which weighs 9 cwt. 3 qr. 23 lb. What
is the hay worth at $7.50 per ton?
COMPOUND NUMBERS. 181
12. While in London, I paid for a vest 1£ 13s. 4d.; a
coat, 7£ 12s. 9d. ; pants, 2£ 3s. 9d. ; boots and hat,
9£ 8s. How many dollars did the whole come to ?
Am. $101.115f .
13. If from a cask of molasses containing 118 gal.,
20 gal. 1 pt. leak out, for how much must the remainder
be sold per gallon to lose nothing, the whole cost having
been $75? Ajis. |0.766.
MULTIPLICATION OF COMPOUND NUMBERS.
199. To multiply a compound number.
Multiply each of the simple numhei^s composing the
compound number hy the multiplier^ reducing lower to
higher units, as in Addition of Compound Numbers.
EXAMPLES.
1. Multiply 3 m. 2 fur. 4 r. 2 ft. 5 in. by 13.
OPERATION.
8 40 16)^ 12
5
13
42m. 3fur. 13r. 14A-ft. 5in.
Ans. 42 m. 3 fur. 13 r. 14 ft. 11 in., since \ ft.=6 in.
2. Multiply 12£ 4s. 6d. 2far. by 13.
Ans. 158£ 19s. 2far.
3. Multiply 1 lb. 9 oz. 13 dwt. by 15.
Ans. 27 lb. 15 dwt.
4. Multiply 19 cwt. 3 qr. 23 lb. by 18.
Ans. 359 cwt. 2 qr. 14 lb.
182 COMPOUND NUMBERS.
5. Multiply 18 gal. 3 qt. 1 pt. by 27.
Ans. 509 gal. 2 qt. 1 pt.
6. Multiply 365 da. 5 h. 48 m. 47.57 sec. by 7.
Alls. 2556 da. 16 h. 41 m. 32.99 sec.
7. Multiply 4 A. 3 R. 20 r. 4 yd. 7 ft. 47 in. by 84.
OPERATION.
4 40 30)^ 9 144
4
3
20
4
7
47
7
34
0
21
2f
6
il
12
409A. 2R. 13r. lOfyd. 3ft. 60 in.
Ans. 409A. 2R. 13r. lly^. 1ft. 24in., since 36in.=^ft. and 2| ft.=Jyd.
8. Multiply 4 cwt. 1 qr. 7 lb. 6 oz. 6.5 dr. by 44.
Ans. 9 T. 10 cwt. 1 qr. 9 oz. 14 dr.
9. Multiply 4 cwt. 18| lb. by 476.
Ans. 99 T. 12 cwt. 6 lb.
10. Multiply 21 m. 65 r. 13 ft. by 5.
Ans. 106 m. 8 r. 15 ft. 6 in.
11. Multiply 1 sq. r. 57 sq. ft. 55 sq. in. by 7.
Ans. 8 sq. r. 129 sq. ft. 61 sq. in.
12. Multiply 13£ 5s. 4.75d. by 24.
Ans. 31 8£ 9s. 6d.
13. Multiply 81£ 14s 9d. by 80.
Ans. 6539£.
14. Multiply 13s. 6d. Ifar. by 519.
Ans. 350£ 17s. 3d. 3far.
15. Multiply 17 cwt. 3 qr. 10 lb. by 60.
Ans. 1071 cwt,
I
COMPOUND NUMBERS. l«o
16. If a bale of cotton Is worth 12£ 14s. 6d. 2far.,
what will be the cost of 13 bales at the same rate?
Ans. 1800.79.
17. I own 16 lots of land, each containing 5 A. 3 R.
20 r. What is the land worth at $40 per acre ?
Ans. §3760.
18. If a man travel 24 m. 4 fur. 4 r. per day, what
will it cost to travel 5 days at the rate of 12 J cts. per
mile ? Ans. |15.32.
19. What will 9 casks of sugar cost at $12 per cwt.,
each cask weighing 8 cwt. 2 qr. 12 lb. ? Ans. $930.96.
20. If one silver cup weighs 8 oz. 4 dwt. 10 gr., what
will 6 cups, each of the same weight, be worth at the
rate of $1.25 per ounce ? Ans. $61.66.
21. What will 9 pieces of broadcloth cost at 5 dollars
a yard, each piece containing 29 yd. 2 qr. 3 na. ?
Ans. $1335.93|.
22. A steamship in crossing the Atlantic makes an
average distance of 250 m. 3 fur. per day. How far
will she sail in 9 days? Ans. 2253 m* 3 fur.
23. I send to market 5 casks of wine, each containing
123 gal. 2 qt. 1 pt. What is the wine worth at 1 dollar
per gallon ? Ans. $618.12 J.
24. A ship sails on the line of the Equator 5 days,
at the rate of 2° 15' 20^' per day. How many miles
does the ship make? (Vide 186, Rem. 1.)
Ans. 779.98 miles.
25. Suppose the ship in the preceding problem had
sailed the same number of days on a meridian, what
would have been the distance made?
Ans. 778.64 m.
184 COMPOUND NUMBERS.
DIVISION OF COMPOUND NUMBERS.
200. To divide a compound number,
(1.) Divide that term of the compound number which
has the highest U7iit value by the divisor^ and write the
quotient as the highest term of the answer.
(2.) Reduce the remainder, if there he any, to the next
loiver unit, adding that term of the dividend which has
the same unit value.
(3.) Divide the sum as before, and 2:)roceed in the same
manner till the lowest units are reached.
EXAMPLES.
1. Divide 165£ 9s. 2 far. by 13.
OPERATIONS.
20 12 4 20 12 4
13)165 9 0 2 13)165 9 0 2 (12 £ 14s. Gd. 2 far. .4ns.
150
A)is. 12 £ 14s. 6d. 2 far.
9
Analysis: 165£-f-13=12£ 20 Remark. — The processes by
and 9£ Rem. 9£x20-j-9s. ~~ Short and Long Division are
=]89s. (More exact, 20s. -.qo essentially the same. The
X9-{-9s.= 189s.) 189s.-r- operation by Long Division
13 = 148. and 7s. Rem. 7s. 7 is preferable till it has become
Xl2=84d. _2^ quite familiar, since no figures
84d.-i-13— 6d. and 6d. 34 requiring attention are sup-
Rem. 6d.X4-f2far.= 26 78 pressed.
far. Finally, 26 far.-^13
=2 far. 5
26
2. Divide 42 m. 3 fur. 13 r. 14 ft. 11 in. by 13.
Ans. 199, Ex. 1.
COMPOUND NUMBERS. 185
3. Divide 27 lb. 15 clwt. by 15. Ans. 199, Ex. 3.
4. Divide 359 cwt. 2 qr. 14 lb. by 18.
Ans. 199, Ex. 4.
5. Divide 509 gal. 2 qt. 1 pt. by 27.
Ans. 199, Ex. 5.
6. Divide 2556 da. 16 li. 41 m. 32.99 sec. by 7.
(Vide 185, Rem. 2.) Ans. 199, Ex. 6.
7. Divide 84° 18^ by 15. Aiis. 5° 37' 12^-
8. Divide 83° 19' 45'' by 15. Aiis. 5° 33' 19".
9. Divide 88° 1' 29" by*'l5. Ans. 5° 52' 5.9".
10. Divide 86° 49' 3" by 15. Ans. 5° 47' 16.2".
11. If 4 of a ship be worth 235£ 16s. lid., what is
the whole ship worth in United States money?
A71S. $7990.46.
12. If I of a ship be worth 943£ 7s. 8d.-, what is the
whole ship worth in francs ? Ans. 43485.48 francs.
13. If 8 bbl. of flour cost 2£ 12s., \vhat will 29 bbl.
cost? Ans. $45,617.
14. I bought a tract of land containing 486 A. 2 R.
30 r. for §1000; subsequently I divide the land into 12
farms, and sell 11 of these farms for |6 an acre, reserv-
ing the twelfth as a homestead. How much do I make
over and above the homestead? A7is. |1676.78-|.
15. From 7 acres of land I harvest 299 bu. 1 pk. 7 qt.
of wheat. The tillage of the land cost me $2 an acre,
and I sell the wheat at |1.25 per bushel. What do I
clear from each acre? A71S. $51.47||.
16. If 15T. 7 cwt. 2qr. 181b. of cotton cost $3384.48,
what will 1 lb. cost ? What 1 qr. ? What 1 cwt. ? What
1 T. ? What would 100 bales be worth, each bale
weighing 511 lb.? Ans. 1 lb. is worth 11 cts.
16
186 COMPOUND NUMBERS.
LONGITUDE IN TIME.
201. To cliange °, \ ", of Longitude, into h. m. sec.
of Time,
(1.) Every point of the Earth's surface, except the
poles, moves through 360° in 24 hours. Hence,
15° of Lon.=l hour of Time.
(2.) Now 15°=900^ and lhour==60 minutes. Hence,
15' of Lon.=l m. of Time.
(3.) But 15^:^900^ and 1 m.=60 sec. Hence,
15'' of Lon.=l sec. of Time. Therefore,
Divide the °, ', ", hy 15, and consider tlie terms' of the
quotient as h. m. sec.
EXAMPLES.
1. The city of Lexington is 84° 18' W. L. When it
is noon at Lexthgton, Avhat time is it at Greenwich?
(Vide 200, Ex. 7, and 197, Ex. 13, Rem.)
Ans. 37 m. 12 sec. past 5 P. M.
Remark. — The Meridian from wliicli Longitude is reckoned •
passes through Greenwich near London.
2. Milledgeville is 83° 19' 45" W. L. When it is
noon at Greenwich, what is the time at Milledgeville ?
(Vide 197, Ex. 13.) Ans, 26 m. 41 sec. past 6 A. M.
3. When it is noon at Lexington, what time is it at
Milledgeville? Ans. 3 m. 53 sec. P. M.
4. When it is noon at Milledgeville, what is the time
at Lexington? Ans. 11 o'clock 56 m. 7 sec. A. M.
5. The city of Mobile is 88° 1' 29" W. L. When it
is noon at Mobile, what is the time at Greenwich?
Ans. 5 o'clock 52 m. 5.9 sec. P. M.
C03IP0UND NUMBERS. 187
6. Louisville is 85° 30' W. L. What is the difference
in time between Lexington and Louisville ?
7. Washington is 77° 0' 15'' W. L. When it is noon
at Washington, what is the time at Louisville ?
8. Boston is 71° 3' 30'' W. L. When it is noon at
Washington, what is the time at Boston ?
Ans. 2S m. 47 sec. P. M.
9. The Cape of Good Hope is 18° 29' E. L. When
it is noon at Washington, what is the time at Good
Hope ? Ans. 6 o'clock 21 m. 57 sec. P. M.
10. San Francisco is 122° 26' 48" W. L. When it is
8 o'clock 9 m. 47.2 sec. P. M. at London, what is the
time' at San Francisco ? Ans. Noon.
11. New York is 74° 0' 3" W. L. When it is noon
at New York, what is the time at all the places men-
tioned in the preceding problems ?
12. On the morning of November 15, 1859, a re-
markable meteor was seen at New York, Albany, Wash-
ington, and Fredericksburg. At Washington the time
was 9 o'clock and 30 m. What w^as the time at the other
places, Albany being 73° 44' 39" W. L., and Fredericks-
burg 77° 38' W^ L. ?
ANALYSIS BY ALIQUOT PARTS.
202. An aliquot part of any number is an exact
half, third, fourth, etc., of the number. Thus,
12J- cts. is an aliquot part of ^1, because 12J is J- of 100.
2*^ is J of 4.
6 is J of 24.
5 is 1 of 20.
121 is 1 of 25.
2 R.
u
a
u
lA,
u
6 gr.
u
((
u
1 dwt.,
ii
5 s.
ii
u
u
1£,
'i.
21 lbs.
u
u
«
1 qr.,
a
188
COMPOUND NUMBERS.
EXAMPLES.
- 1. What cost 39 A. 2 R. 15 r. of land, at §87.375
per acre.
OPERATION.
§87.375
39
Therefore,
Price of 1 A. is . . .
" 39 A. is . .
" 2 R. is . . .
" 10 r. is . . .
" 5 r. is . . .
Price of 39 A. 2 R. 15 r. is |3459.503f ^ Ans.
2. What cost 39 A. 2 R. 15 r. of land, at |139.80
per acre? ylii-s. 15535.206 J-.
3. What cost 176 A. 3 R. 25 r. of land, at §75.375
3407.625
43.687.1
. J of 1 A.
5.460 }i
. I of 2 R.
2.73011
. i of 10 r.
per acre
Ans. §13334.308^9.
4. What cost 20 A. 2 R. 24 r. of land, at §30 per
acre? Ans. §619.50.
5. What cost 10 yd. 3 qr. 2 na. of silk, at §1.80 per
yard ?
OPERATION.
Price of 1 yd. is
.80
10
Therefore,
a
10 yd. is
. 18.00
<£
2 qr. is .
. .90 .
. i of 1 yd.
u
1 qr. is .
. .45 .
. i of 2 qr.
a
2 na. is .
. .225 .
. J of 1 qr.
Price of 10 yd. 3 qr. 2 na. is §19.575 Ans.
6. What cost 15 yd. 2 qr. 3 na. of cloth at 25 cents
per yard? Ans. ^S.02^^^.
COMPOUND NUMBERS.
189
7. What cost 25 yd. 1 qr. 3 na. of broadcloth, at $5.50
per yard? A/i5. $139.90 1.
8. What cost 67 bu. 3 pk. 7 qt. of cranberries, at §2
per bushel?
OPERATION.
Price of 1 bu. . . §2.00 Therefore,
67
" 67 bu. .
. 134.00
^' 2pk. .
1.00
1 bu.
" Ipk. .
.50
i of 2 pk
" 4 qt. .
.25
i of 1 pk
" 2 qt. .
.125
I of 4 qt.
" 1 qt. .
.0625 .
\ of 2 qt.
Price of 67 bu. 3 pk. 7 qt. $135.9375 Ans.
9. What cost 125 bu. 3 pk. 1 qt. of wheat, at 87^ cts.
per bushel? Ans. $110,058.
10. What cost 25 bu. 1 pk. 3 qt. of clover seed, at
$5.00 per bushel? Ans. $126.72.
11. What cost 503 bu. 4 qt. of corn, at 43j- cts. per
bushel?. ^7?s. $220,117.
12. What cost 76 bu. 1 qt. of peas, at $1.66| per
bushel? J.ns. $126.72.
13. What cost 10 bu. 3 pk. of apples, at 50 cts. per
bushel? Ans. $5.37^.
14. What cost 25 bu. 1 pk. of potatoes, at 35 cts. per
bushel? A71S. $8.83f.
15. What cost Ibu. 1 pk. 1 qt. of chestnuts, at $1.00
per bushel? Ans. $1.28 J.
16. What cost 17 cwt. 3 qr. 23 lb. of hay, at $13
per ton?
190
COMPOUND NUMBERS.
OPERATION.
Price of
IT....
10 cwt. . .
. $13.00
. therefore,
iC
6.50
. J of 1 T.
u
5 cwt.
3.25
. i of 10 cwt
a
2 cwt.
1.30
J of 10 cwt.
a
2 qr. .
.325
1 of 2 cwt.
u
1 qr. . .
.1625 .
i of 2 qr.
a
5 1b. .
.0325
I of 1 qr.
a
15 1b. . .
.0975 .
3 times 5 lb.
a
lib. . .
.0065 .
I of 5 lb.
Ci
2 1b. .
.013
2 times 1 lb
Price of 17 cwt. 3 qr. 231b. $11,687 Ans.
17. What cost 3 T. 10 cwt. 3 qr. of iron, at $30,375
per ton? Ans. $107.45.
18. What cost 16 boxes of sugar, each box contain-
ing 4 cwt. 3 qr. 18 lb., at $6.65 per hundredweight?
Ans. $524,552.
19. What cost 9 casks of sugar, each cask weighing 8
cwt. 2 qr. 12 lb., at $12 per cwt. ? A7is. 199, Ex. 19.
20. What cost 16 cwt. 3 qr. 21 lb. 6 oz. of rice, at
$7.00 per hundredweight? Ans. $118.75.
21. What cost 5 cwt. 2 qr. of hay, at $27 per ton?
Ans. $7.42f
22. What cost 2 T. 3 cwt. 3 qr. of hay, at $30 per
ton? ^ws. $65,625.
23. What cost 7 T. 15 cwt. 1 qr. of hay, at $40 per
ton? Ans. $S10.bO.
24. What cost the iron on a track measuring 3 fur.
22 r. 3ft. 8 in., at the rate of $4500 per mile?
COMPOUND NUMBERS.
191
OPERATION.
Price of 1 m.
§4500.00
Therefore,
2 fur. .
. 1125.00
. J of a mile
1 fur. .
562.50
. J of 2 fur.
20 r.
281.25
. J of 1 fur.
2r.
28.125
. /^ of 20 r.
3 ft. .
2.556 f9-
. Jf of 2 r.
6 in. .
0.426/,
1 of 3 ft.
2 in. .
0.142 J, .
I of 6 in.
Price ofS fur. 22 r. 3 ft. 8 in. 12000,000 (192-3; Ex. 40.)
25. If I ride 5 fur. 3 r. 10 ft. 6 in. in a railroad car,
what ought to be my exact fare, at the rate of 11 cts.
per mile? Ans. 192-3; Ex. 39.
26. What will 23 gal. 2 qt. 1 pt. of wine cost, at §60
per hogshead? Ans. 194-5; Ex. 25.
27. What cost 11 hhd. 17 gal. 2 qt. of wine, at |49.77
per hogshead? Ans. §561.29 J.
28. If one silver cup weighs 8 oz. 4 dwt. 10 gr., what
will 6 cups, each of the same weight, be worth, at the
rate of §1.25 per ounce? Ans. 199; Ex. 20.
29. What will 537 'bushels of wheat cost, at §1.374
per bushel?
OPERATION.
Price at §1.00 per bu.
" 25 cts. "
§537.00
134.25
1 of §1.00.
121 cts. "
67.125 . h of 25 cts.
Price at §1.37.
§738.375 A71S.
192
COMPOUND NUMBERS.
80. What cost 327 bushels of potatoes, at 62^ cts.
per bushel? 50 cts.= i of §1; 121 = i of 50 cts.
Ans. 1204.375.
31. What cost 453 bushels of corn, at 87 i cts. per
bushel? 87i'cts.= 50 cts.-{-25 cts.+ 12i cts.
Ans. 1396.375.
32. What cost 1999 gal. of wine, at |1.62i per
gallon? JLws. 13248.375.
33. What cost 5794 yd. of cloth, at |3.16| per yard?
16| cts.= i- of $1.00. Ans. $18347. 66f.
34. What cost 3579 yards of cloth, at $1.12i- per
yard? $1.18|? |2.26? $3.37J? 18| cts.=3 times 6| cts.
=j\ of $1.00. First Ans. $4026.375.
35. What cost 2468 gal. of wine, at $1.43} per
gallon? $2.50? $3.62 J? $4.56i? 43|- cts.=:25 cts.+l2^
ctB.+6^ cts. Last Ans. $11260.25.
36. What cost 3 T. 10 cwt. 3 qr. of iron, at 6£ 4s.
6d. per ton?
OPERATION.
Price of 1 T. . . 6£ 4s. 6d. Therefore,
3
a
3T- .
. 18 13
6
u
10 cwt. .
. 3 2
3 .
. J of IT.
(.6
2qr. .
3
l/o
J-^ of 10 cwt
a
Iqr. .
1
m
. ^i of 2 qr.
Priceof3T. lOcwt. 3qr.22£ Os. 5 J^d-, or $106.58.
37. What cost 40 yd. 3 qr. 1 na. of broadcloth, at 1 £
per yard? Ans. 40 £. 16s. 3d.=-$197.53i.
38. What cost 25 bu. 8 pk. 5 qt. of wheat, at 5 s. 6 d.
per bushel ? Ans. 7£ 2 s. 5 J ^ d---$34.48l3^..
OPERATION.
3 for 1 jr. is
. . §325.00
2
" 2 yr. is
. . 650.00
" 3 m. is
. . 81.25
" 10 da. is
. . 9.02J-
COMPOLXi) XUMBEllS. 193
39 If I pay $325 for the use of a certain sum of
money 1 year, what ought I to pay for the use of the
same money 2 yr. 3 m. 10 da.?
Therefore,
. 1 of 1 yr.
. i of 3 m.
Use for 2 yr. 3 m. 10 da. is $740.27-J Ans. (185 ; Rem. 6.)
40. At the rate of $125 per year, what ought I to
pay for a sum of money 1 yr. 3. m. 15 da. ?
A71S, $161.45f.
41. At the rate of SI 75 per year, what ought I to
pay for a sum of money 1 yr. 4 m. 21 da. ?
A71S. $243.54^.
42. At the rate of $400 per year, what ought I to
pay for a sum of money 1 yr. 5 m. 7 da. ?
A71S. $574.44^,
43. At the rate of $525 per year, what ought I to
pay for a sum of money 2 yr. 3 m. 12 da. ?
Ans. $1198.75.
44. At the rate of $1000 per year, what ought I to
pay for a sum of money 5 yr. 5 m. 5 da.?
A71S. $5430.55|.
45. At the rate of $2000 per year, what ought I to
pay for a sum of money 10 yr. 10 m. 10 da. ?
Ans. $21722.22|.
46. At the rate of $750 per year, what ought I to
pay for a sum of money 3 yr. 7 m. 19 da.?
17
194
COMPOUND NUMBERS.
OrERATION.
J for 1 yr. is .
. .?750.00
3
Therefore,
" 3 yr. is .
. 2250.00
" 6 m. is .
. 375.00 .
• J of 1 .yr-
" 1 m. is .
62.50 .
1 of 6 m.
" 15 da. is .
31.25 .
. I of 1 m.
" 3 da. is .
6.25 .
. 1 of 15 da
" 1 da. is .
2.08J .
. J of 3 da.
Use for 3 yr. 7 m. 19 da. |2727.08J Ans.
47. At the rate of §1000 per year, what ought I to
pay for a sum of money 5 yr. 9 m. 27 da.?
Ans. S5825.00.
48. At the rate of |2500 per year, what ought I to
pay for a sum of money 4 yr. 11 m. 25 da. ?
Am. §12465.27|.
49. At the rate of §825 per year, what ought I to
pay for a sum of money 6 yr. 2 m. 18 da.?
Ans. §5128.75.
50. The rent of a farm is §1150 per year. What
ought to be paid during an occupancy of 7 yr. 7 m.?
Ans. §8720.831 .
51. What ought I to pay for the use of §5000 during
4 yr. 3 m. 20 da., at the rate of §400 per year?
J.7is..§1722.22|.
52. What ought I to pay for §1850 during 3 m. 3 da.,
at the rate of §148 per year? Ans. §38.23 1.
53. What ought I to pay for §9215 during 3 m. 3 da.,
at the rate of §737.20 per year? Ans. §190.44 J.
54. What ought I to pay for the use of §8000 from
COMPOUND NUMBERS. 195
January 27, 1866, to July 4, 1868, at the rate of $640
per year?— (198; Ex. 1.) Ans. $1559.111.
55. What ought I to pay for the use of $7250 from
July 4, 1865, to August 1, 1866, at the rate of $580
per year? Ans. $623.50.
56. What ought I to pay for the use of $5280 from
August 1, 1869, to September 9, 1871, at the rate of
$422.40 per year? Ans. 889.38|.
57. If I pay 12 cts. per year for the use of $1, "what
ought I to pay for its use 3 yr. 5 m. 15 da. ?
Ans. $0,415.
58. If I pay $99.75 per year for the use of $950,
what ought I to pay for the use of the same sum 2 yr.
4m. 20 da.? Ans. $238.29^.
59. What ought I to pay for the use of $421.40 for
3 yr. 5 m. 15 da., at the rate of $25,284 per year?
OPERATION.
Use for 1 yr. is . . $25,284
3
a
3 yr. is .
75.852
a
4 m. is
8.428
. 1 of 1 yr.
a
1 m. is
2.107
. 1 of 4 m.
u
15 da. is
. , 1.0535
. 1 of 1 m.
Use for 3 yr. 5 m. 15 da. $87.4405
203. Review in Addition.
1. Find the value of 123+45+2004. Ans. 2172.
2. Find the value of 21+105+710. Ans. 836.
3. Find the value of 7+90+1041. Aiis. 1138.
196 COMPOUND NUMBERS.
4. Find the value of 50+75+432ia ^m 43335.
5. Find the value of ?1 2 3+^45+12 004.
Ans. $2172.
6. Find the value of 21 £+105 £+710 £.
Ans. 836 £.
7. Find the value of 7 m.+90 m.+1041 m.
Ans. 1138 ra.
8. Find the value of 50 pk.+75 qt.+43210 pt.
Ans. 44160 pt.
9. Find the value of .123+.45+.2004. Ans. .7734.
10. Find the value of .21+.105+.71. J.ns. 1.025.
11. Find the value of .7+.9+.1041. Ans. 1.7041.
12. Find the value of .5+.75+.4321. Ans. 1.6821.
13. Find the value of $0.123+S0.45+S0.2004.
Ans. $0.7734.
14. Find the value of .21 £+.105 £+.71 £.
Ans. 1.025 £.
15. Find the value of .7 1.+.9 m.+ .1041 fur.
Ans. 24.1041 fur.
16. Find the value of .5 pk.+.75 qt.+.4321 pt.
A71S. 9.9321 pt.
17. Find the value of 1.23+4.5+20.04.
Ans. 25.77.
18. Find the value of 2.1+1.05+7.1. Ans. 10.25.
19. Find the value of 7+90+10.41. Ans. 107.41.
20. Find the value of 50+7.5+.4321.
Ans. 57.9321.
21. Find the value of |1.23+$4.50+|20.04.
A71S. $25.77.
22. Find the value of 2.1 £+1.05 £+7.1 £.
Ans. 10.25 £.
COMPOUND NUMBERS. 197
23. Find the value of 7 l.-f9 m.+10.41 fur.
Ans. 250.41 fur.
24. Find the value of 50 pk.+7.5 qt.+.4321 pt.
A71S. 815.4321 pt.
25. Find the value of |+i+|. ^ns. IJ.
26. Find the value of J + | + |. Ans. If §.
27. Find the value of ^^-{-f^-\-^^. Ans. l^gf .
28. Find the value of ii-hyVs + Z/s- -^ns. f i§. '
29. Find the value of ||+li+li. Ans. |1.12i.
30. Find the value of i £+| s.-[-| d.
Ans. 128J d.
31. Find the value of y^^ 1.+/^- m.+ ^g fur.
Ans. lOif fur.
32. Find the value of ^| pk.+ fVs ^t-+i¥3 P^-
Ans. 4|| pt.
33. Find the value of 123|+45|+2004|.
Ans. 2173|.
34. Find the value of 21i+105H-710|.
Ans. 837|g.
35. Find the value of 7i\+90/o+1041/5.
A71S. 1139^i§.
36. Find the value of 50^J+75/5V+43210//3.
Ans. 43335|i§.
37. Find the value of 1.231+4.5-1+20.041.
Ans. 25.791.
38. Find the value of 2.1i+1.05|+7.10|.
Ans. 10.31/5.
39. Find the value of .7f,-+9.0/3+10.41/5.
^ Ans. 20.18H§.
40. Find the value of 5.0iJ+7.5jV5+4321.0//3.
Ans. 4333.56.\.
198 COMPOUND NUMEERi.
41. Find the value of $1.23|+§4.51i+§20.04|.
A71S. $25,791.
42. Find the value of 2.15X+1.05|£+T.108£.
A71S. 10.3141 £.
43. Find the value of .7i\ r.+9.0/^ ft.H- J in.
Ans. 1 r. 4 ft. 7 in.
44. Find the value of 5.0^| A.+7.5jV5 R.+l r.
Ans. 6 A. 3 R. 21.48 r.
45. Add together l£ 2s. 6d.; 9£ 8s. 9d.; 12£ 13s.
2d.; and 4£ 7s. 3d. Ans. 27£ lis. 8d.
46. Add together 15 r. 16 ft. 6 in.; 33 r. 14 ft. 7 in.;
and 19 r. 8 ft. 9 in. Ans. 69 r. 6 ft. 10 in.
47. Add together 5 A. 2 R. 7 r. ; 8 A. 3 R. 32 r. ; and
9 A. 1 R. 27 r. Ans. 23 A. 3 R. 26 r.
48. Add together 3 hhd. 62 gal. 3 qt.; 2 hhd. 16 gal.
1 qt.; 3 hhd. 57 gal. 2 qt.; 3 hhd. 45 gal. 3 qt.; 2 hhd.
59 gal. 3 qt. ; and 3 hhd. 39 gal. 2 qt.
Ans. 20 hhd. 29 gal. 2 qt.
204. Review in Subtraction.
1. From 1001 take 763. Ans. 238.
2. From 3999 take 455. Ans. 3544.
3. From 31015 take 9999. Ans. 21016.
4. From 11111 take 7778. Ans. 3333.
5. From $245 take $78. Ans. $167.
6. From 24 pears take 19 pears. Ans. 5 pears.
7. From 91 apples take 74 apples. Ans. 17 apples.
8. From 375 da. take 109 da. Ans. 266 days.
9. From .763 take .1001. Ans. .6629.
10. From .455 take .3999. Ans. .0551.
11. From .45 take .0073. Ans. .4427.
COMPOUND NUMBERS. 199
12. From .39 take .039. Ans. .351.
13. From §0.37 take |0.227. Ans. |0.143.
14. From .5 pears take .25 pears. Ans. .25 pears.
15. From .91 apples take .74 apples.
Ans. .17 apples.
16. From .67 lb. take .4937 lb. Ans. .1763 lb.
17. From 10.01 take 7.63. Ans. 2.38.
18. From 39.99 take .455. Ans. 39.535.
19. From 3.8 take 1.005. Ans. 2.795.
20. From 37 take 19.04. Ans. 17.96.
21. From |3.25 take $2.12. Ans. |1.13.
22. From 3.5 da, take 2.85 da. Ans. .65 days.
23. From 43.7 m. take 27.43 m. Ans. 16.27 m. '
24. From 9.5 sec. take 4.67 sec. Ans. 4.83 sec.
25. Find the value of 3— ^ ; of 4— |. Ans. 2^ ; 3}.
26. Find the value of |— | ; of |— ^. Ans. | ; |.
27. Find the value of 3^—2 J ; of 4i— 3l.
Ans. 1|; IgL.
28. Find the value of 8— 3^ ; of 12—41.
Ans. 4i ; 7|.
29. Find the value of 3.8— 2j ; of 37— 19-^^
Ans. 1.3 ; 17.96.
30. Find the value of 4.3^—2.331; of 84— 3.14f.
Ans. 2.01f ; 5.
31. Findthe value of 42— .OOOJ; 37—4.000'".
Ans. 41.9991; 32.999 J.
32. Find the value of 860.4581— 25.1 J.
Ans. 835.3248f .
33. From | h. take i m. Ans. 29 m. 45 sec.
34. From i da. take J h. Ans. 5. h. 45 m.
35. From | m. take * in. Ans. 191 r. 16 ft. 5J in.
200
COMPOUND NUMBERS.
36. From
J A. take J
sq. m.
».75i.
A71S. 19r. 272 ft. 35f in.
Ans. $0,495/5.
Ans. 7 ft. 8|g in.
Alts. 2 gal. -i qt.
A718. |1.00/g.
Ans. $12.48.
37. From §1.25| take
38. From 8.3 J ft. take 7.0| in.
39. From 2J gal. take J qt.
40. From 7i francs take 371 cts.
41. From 3|£ take 3} dollars.
42. From 4i bu. take 3j pk. J.ws. 3 bu. 2 pk. 4 qt.
43. From 9 m. take 8 m. 7 fur. 39 r. 16 ft. 5 in.
Ans. 1 inch.
44. From 8 m. 7 fur. 16 ft. 5 in. take 5 m. 6 yd. 11 in.
A71S. 3 m. 6 fur. 39 r. 14 ft.
45. From 1£ 21s. 13d. 5 far. take 5s. 3d. 7 far.
Ans. l£16s. 9d. 2 far.
46. From 9 bu. 3 pk. 5 qt. take 4 bu. 7 pk. 15 qt.
Ans. 3 bu. 2 pk. 6 qt.
I305. Review in Multiplication and Division.
1. Multiply 45 by 45.
3. Multiply 101 by 1001.
5. Multiply .45 by .45.
7. Multiply .101 by .1001.
9. Multiply 4.5 by 45.
11. Multiply .0009 by .0009.
13. Multiply $45 by 75.
15. Multiply 365 da. by 17.
17. Multiply 375.61b. by .125.
19. Multiply 03.5 bu. by .78.
21. Multiply 74 by ^f-y.
23. Multiply vjVj by 370.
25. Multiply H by ff.
27. Multiply 4 by zll.
29. Multiply 6^| by 24.
31. Multiply 4^ by 3f
Am. 145.
2. Divide 2025 by 45.
4. Divide 101101 by 1001.
6. Divide .2025 by .45.
8. Divide .0101101 by .1001.
10. Divide 202.5 by 4.5.
12. Divide .00000081 by .0009.
14. Divide $3375 by 75.
10. Divide 6205 da. by 17.
18. Divide 4G.95 by .125.
20. Divide 49.53 bu. by .78.
22. Divide 6 by ^i^^.
24. Divide i||o by -870.
20. Divide j% by j^^.
28. Divide 14 by 3i|. ,
30. Divide 7| by 3.
32. Divide 14^ by S^V
Ans. 415.
{J03il'0UXD ^' UMBERS. 201
33. Multiply .OOOi by 4.8.
85. Multiply S.Of by 2.00i.
37. Multiply 375f r. by |.
39. Multiply 63^ A. by |f
41. Multiply 3.75ir. by 51.
43. Multiply 8.61 bu. by 129.
31. Divide .0016 by .0001.
36. Divide 6.031 1| by 2.00^.
38. Divide 46 |i r. by i.
40. Divide 49.53 A. by ff.
42. Divide 191f i by 51.
44. Divide 1112f bu. by 8|.
45. Multiply 26 £ 14 s. 8d. 3 far. by 11.
Ans. 294: £ 2 s. Od. Ifar.
46. Multiply 1 m. 1 fur. 1 r. 1 ft. 1 in. by 27.
Ans. 30 m. 3 fur. 28 r. 12 ft. 9 in.
47. Multiply 8 T. 1 cwt. 3 qr. 7 lb. 5 oz. 6 dr. by 3^
Ans. 28 T. 6 cwt. 1 qr. 13 lb. 2 oz. 13 dr.
48. Multiply 365 da. 5 li. 48 m. 47.57 sec. by 6.
Ans. 2191 da. 10 h. 52 m. 45!'42 sec.
49. Add together 10 apples, 4 pears, and 6 peaches
(Vide 171.)
50. Add together 3 ft. and 4 in. Ans. 40 inches.
51. From §100 take 4 bu. 2 pk. 1 qt The problem
is absurd. Why?
52. I buy 4 bu. 2 pk. 1 qt, of cherries at 50 cts. per
qt., and pay for them out of a §100 bill. How much
money do I receive in change? Aiis. §27.50.
53. Multiply 25 cts. by 25 cts.— (Vide 82.)
54. Divide 25 rails by 25 rails.— (Vide 96.)
55. Reduce each of the following expressions to a
compound number: 3.23125 £; S^Vo^; 3.23|£; 64.-
625s.; 64gs; 64.6is.; 775 id.; 775.5 d.; 3102 far.
Ans. 3£4s. 7d. 2 far.
56. Reduce 1 yr. 2 mo. 6 da. to months and decimals
of a month. 1 yr. 3 mo. 9 da.; 2 yr. 4 mo. 12 da.; and
3 yr. 6 mo. 15 da.
J.ns. 14.2 mo. ; 15.3 mo.; 28.4mo.; 42.5 mo.
202 PERCENTAGE.
PERCENTAGE.
206. Percentage embraces those operations in
which numbers are compared with 100 as a unit.
SOT. Per centum, in Latin, signifies by the hundred,
but the contraction per cent, is used, for the most part,
as a synonym of the word hundredth or hundredths.
Thus,
1 per cent, of 25 is the same as 1 hundredth of 25.
2 per cent, of 50 is the same as 2 hundredths of 50.
25 per cent, of a number is the same as 25 hundredths
of it.
The sign ^ is the same as the contraction per cent.
The RATE PER CENT, is the number of hundredths.
Thus,
■ 5 ^ of 400 is read 5 per cent, of 400, and has the
same meaning as 5 hundredths of 400, that is, yj^ of
400, or .05 of 400.
7 % of a number is 7 hundredths of it.
1% of a number is f hundredths of it, that is ^f^
of it.
^iOS. To represent decimally any given rate per
cent.,
Write the given rate as so many hundredths.
EXAMPLES.
1. Represent the following rates per cent, decimally,
viz: 1%; 2%; 12%; 10%; 25%; 47%; 100%;
125%; 200%; 1000%; 1275%.
Ans..O\] .02; .12; .10; .25; .47; 1.00; 1.25; 2.00; 10.00; 12.75.
TERCENTAGE. 203
2. Represent decimally h%;i%; ~l%; Jo%> i o % ;
A71S. .005; .0025; .001; .0005; .001; .0002; .0075; .004.
3. Represent decimally 1^^; 2\%; 3j%; lj^%;
^i\fo; 11%; H%; H%.
Ans. .015; .0225; .032; .0105; .061; .075; .051; .041.
4. Represent decimally ^^-^ ; ^%%; |%; 125|%;
4Jg%. ^7is. .00025; .00075; .00375 ; 1.25|; .040625.
209. To find what rate per cent, a given decimal
represents,
Multiply the decimal hy 100 and reduce the result^ if
necessary^ as in 164.
EXAMPLES. .
1. What rate per cent, does .01 represent? .04? .05?
.07? .1? .2? Ans,l%',4.%', 5%; 1 %; 10%; 20%.
2. What rate per cent, does .00428f represent? .000|?
.008 1 ? (Vide 164, Ex. 7.) Ans, | %; -^^%', %%.
3. What rate per cent, does .001 represent? .0075?
.041^? 1.00? Ans, ^\%; f %; 4i%; 100%.
4. What rate ^er cent, does .007 represent? .0007?
1.07? Ans. /^%; tJo%; 107%.
!S10. To find the per cent, (or percentage) of a num-
ber at a given rate,
Multiply the given number hy the rate per cent, ivrit-
ten as a decimal. The product will be the per cent.
required.
EXAMPLES.
1. Find 3% of 25; -J% of |500 ; and 1.^^% of
7000 lbs. of coff'ee.
204 PERCENTAGE.
OPERATIONS.
25 $500 7000 lb.
.03 .005 .0105
.75=1 Ans. 12.500 Ans. 73.5000=-73i lb. Ans.
2. Find 1 % of $120 ; 2 % of $410 ; 7 % of 140 bu. ;
9% of 555 da. (211, Ex. 2.)
3. Find 6% of 333 fur.; 5% of 400 m.; 1% of
$1000; J % of $3333.
4. Find 1% of $7000; n% of $9000; 2^% of $700;
J^% of $10000.
5. Find 2% of$125;7^;ll%;25%;75%;100%;
150%.
6. Find 11% of Gets.; 7cts.; $1.20; $1.75; $3.50;
$7.62i; $9.45.
7. Find 25% of $25; 33i % of $500; 16f % of
$8000.
8. Find 101 J % of $505; 202 J % of $404; 1000%
of $23.10.
9. Received at Mobile from New Orleans 200 lihds.
of sugar ; but in discharging the cargo 2 % of the sugar
was lost. How much remains to be sold?
Ans. 196 hhds.
10. The steamer Indiana started from Vicksburg with
500 bales of cotton on board. On the trip to New Or-
leans 14% of the cotton was transferred to another boat.
How much remained on board? Ans. 430 bales.
11. Bought at New Orleans 75 hhds. of molasses, but
on its arrival at Cincinnati 4% is missing. How much
remains to be sold? Ans. 72. hhds.
12. Shipped at Havana for New York 1250 boxes of
PERCENTAGE. 205
oranges; on the passage 14 fo of the oranges Avere
thrown overboard. How many boxes arrived in New
York? Ans. 1075 boxes.
13. On a trip from New Orleans to New York I ex-
pend 25% of my money. I started with. §360. How
much did I have on arriving at New York? (Vide 167,
Ex. 103.) Ans. §270.
14. During a storm a captain threw overboard 16| ^
x)f a cargo of cotton. He left New Orleans with 720
bales; with how many did he arrive at Liverpool^ (167,
Ex. 104.) Ans. 600 bales.
15. My wine made during the year 1865 was 378
gallons. Reserving 7^%, I sell the remainder at §4.50
per gallon. What did the crop bring me? (167, Ex.
105.) Ans. §1579.50.
16. My wine made during the year 1866 w^as 450
gallons. Reserving 10 ^ for private use, I sell the re-
mainder at §3.75 per gallon. Which year was most
profitable, 1865 or 1866, and by how much?
Ans. 1865, by §60.75.
17. I buy sugar for §1700, and sell so as to clear 5 ^
on the cost. How much do I get? Ans. §1785.
18. A merchant failing, pays his creditors 30^. He
owes A §2500; B §4000; and C §4500. What will each
receive? Ans. A §750; B §1200; and C §1350.
19. I pay 10% of my salary for board; i% for
washing ; l2 % for clothes, and 8 % for other expenses.
How much do I clear from a salary of §2000 ?
Ans. §1395.
Sll. To find the rate per cent, of a number at a
given percentage,
' c
}
206 PEllCEXTAGE.
Divide the given percentage hy the number of which
the rate per cent, is required. The quotient will repre-
sent the rate per cent, decimally, and may be changed
as in 209.
EXAMPLES.
1. What rate % of 25 is .75? of $500 is $2.50? of
7000 lb. of coffee is 73. \ lb. of coffee?
OPERATIONS.
(1.) " (2.) (3.)
25).75 500)2.50 7000)73.5
.03=% Ans. .005= J % Aiis. .0105=1 ^.^^^ -^ns.
2. What rate % of $120 is $1.20? of $410 is $8.20?
of 140 bu. is 9| bu. ? of 555 da. is 49 ^ § da. ?
3. What rate % of 333 fur. is 19|~§fur.? of 400 m.
is 20 m.? of $1000 is $5.? of $3333 is $11.11?
4. What rate % of $7000 is $17.50? of $9000 is
$135? of $700 is $15f ? of $10000 is $5?
5. What rate % of $125 is $2.50? $8.75? $13.75?
$31.25? $93.75? $125? $187.50?
6. What rate % of 6 cts. is j% m. ? of 7 cts. is 1 j^ m.?
of $1.20 is If cts.? of $1.75 is 2| cts.? of $3.50 Is 5J-
cts.? of $7.62i is 11/5 cts.? of $9.45 is 14/^ cts.?
7. What rate % of $25 is $6 J? of $500 is $166|?
of $8000 is $1333| ?
8. What rate % of $505 is $511.06? of $404 is
$818.10? of $23.10 is $231?
9. From a ship having on board 200 bales of cotton,
4 bales fell into the sea, and wer^ lost. What rate %
were these four bnles of the whole number on board?
rEllCEXTAGE. 207
10. A boy commenced play with 200 marbles, and
ended with 196. What was his rate ^ of loss?
11. A ship sailed from New Orleans for Liverpool
with a cargo of 500 bales of cotton. The ship reached
her port with only 430 bales, the remainder "having been
thrown overboard. With what rate % of her cargo did
she reach Liverpool, and what rate % had been thrown
into the sea?
12. Out of 1250 boxes of oranges shipped at Havana,
1075 arrived in New York. What rate % of the cargo
had been lost ?
13. I started from Charleston with $360, and arrived
in Quebec with §270. What rate % of my money had
been expended in the trip?
14. Out of my wine made in 1865, which was 378
gallons, I sold a quantity amounting, at the rate of ?4j
per gallon, to $1579|, having reserved the remainder
for private usfe. What rate ^ on the whole wine was
reserved ?
15.. My wine made in 1866 had increased at the rate
of 19 2^- ^/o on that made in 1865, but tHe market being
dull I sold a portion of it for §3.75 per gallon, clearing
§60.75 less than on the previous year. What rate ^ of
the wine of 1866 remained unsold?
!S1!^. To find a number on which, at a given rate ^,
a given percentage may be obtained.
Divide the given 'percentage hy the given rate per cent,
expressed decimally. .The quotient Avill be the required
number.
208 ^ PERCENTAGE.
EXAMPLES.
1. What number is that of which 3% is .75? 1% of
how many dollars is $2.50 ? l^^ % of how many pounds
is 73i pounds?
OPERATIONS.
(1.) (2.) (3.)
.03).75 .005)12.500 .0105)73.5000 lb.
25 Ans. 1500 Ans. 7000 lb. Ans.
2. 8^ of what number will produce 125?
Am. 1562.50.
3. 8^ of how many dollars will produce |175?
A71S. 12187.50.
4. 5 ^ of how many dollars will produce |400 ?
Ans. 18000.
5. 4^ of how many dollars will produce §250?
Ans. 16250.
6. On a pleasure excursion I spend $90, which I find
is 25 fo of the money with which I started. How much
money have I still on hand? Ans. (211, Ex. 13.)
7. If I pay 8 ^ of a sum of money for its use during
a year, and thereby pay |525, what amount of money
do I have the use of? Ans. $6562.50.
8. If I pay 6^ of Sb sum of money for its use during
a year, and thereby pay $750, what amount of money
do I have the use of? Ans. $12500.
9. I borrow $350 for 1 year, and agree to pay 7%
of the sum for its use. How much do I pay ?
Ans. $24.50.
10. If I pay $24.50 for the use of $350 for 1 year,
what rate % on the money do I pay? * Ans. 7%,
PERCENTAGE. 209
11. What amount of money can I get the use of for
1 year by paying $24.50, that being 7 ^ on the money
borrowed? Ans. $350.
S13. A number being given which is a given rate
per cent, more than another number, to find that other
number,
Divide the given number by \ -\- the rate per cent,
written as a decimal.
EXAMPLES.
1. 560 is 12 fo more than a certain number. What
is that number?
Remark. — Thp question is precisely the operation.
same as this; 560 is iif of what number?
and (560-112) X 100=560-1.12. 1.12)560.00
500 Ans.
2. 1000 is 33| fo more than a certain number. What
is that number? 1000-^1.33|=| of 1000.
Ans. 750.
3. $150 is 20% more than what sum? Ans. $125.
4. $140 is 16? fo more than what sum?
Ans. $120.
214. A number being given which is a given rate p'er
cent, less than another number, to find that other number.
Divide the given number by X — the rate per cent, ivr it-
ten as a decimal.
EXAMPLES.
1. 270 is 25 % less than what operation.
number? 1-^9
Remark. — The question is the same as
this: 270 is ^^-^ of what number? and (270 •'^5)270.00
-f-75) X 100=270— 75. (Vide 167. Ex. 98.) 36O im
18
210 PERCENTAGE.
2. $1.40 is 30% less than what sum?
Ans. $2.00.
3. $4.50 is 25% less than what sum?
A71S. $6.00
4. $8.75 is 33J% less than what sum? (| of 8.75.)
Ans. $13.12J.
APPLICATIONS OF PERCENTAGE.
215. Insurance is a contract made between parties,
by which the one binds itself, for a consideration, to re-
imburse the other for losses of property occasioned by
fires, or other casualties.
(1.) The party taking the risk is called the Under-
writer.
(2.) The party protected is called the Insured.
(3.) The Policy is the written contract of insurance.
(4.) The Premium is the sum paid for insurance.
(5.) The premium is usually a percentage on the value
of the property insured, and is paid at the time the pol-
icy is drawn.
' 310. To find the Premium, when the amount insured
and the rate per cent, of insurance are given.
Multiply the amount insured hy the rate %, wiHtten
decimally. — (Vide 210.)
EXAMPLES.
1. What premium must be paid annually for insuring
a house worth $4500, at -J % ? Ans. $11.25.
2. What is the premium on a cargo of cotton valued
at $2500, at 1 % ? Ans. $3,125.
PEllCE^'TAGE. 211
3. What is the premium on a cargo of goods carried
from New York to Mobile, the goods being valued at
112500, and insured at Ij % ? Ans. $187.50.
4. I have a house worth §4500, and insure it for | of
its value at 1|% per annum. What is the expense of
insurance, including $2.00 for the policy?
Ans. §50.00.
5. I lose by fire four houses, valued at §25000. On
this property I had paid annually, during five years, a
premium of lj% on the entire value. The policy be-
ing good, what have I saved by insuring the property?
Ans. §28437.50.
217. To find for what sum a policy must be taken
out, at a given rate per cent., to cover both property and
premium,
Divide the sum for wJiich the property is to be insured
hy 1 — the rate ^, written as a decimal. — (Yide 214.)
EXAMPLES.
1. Shipped a cargo of flour from New York to Mata-
moras, valued at §23940. For what sum must it be in-
sured to cover the value of the flour and the premium,
the rate of insurance being 5 ^ ? Ans. §25200.
2. For what sum must I insure §45000 worth of cot-
ton, shipped from Yicksburg to New Orleans, so as to
cover the cotton and premium, the rate of insurance
being 2 J % ? Ans. §46153 |f
3. The premium for insuring a house is ^50, including
§2.00 for the policy. The rate of insurance is If %.
What is the value of the house ?— (Yide 212.)
Ans. §3000.
212 PERCENTAGE.
318. Commission is a sum allowed to a Commission
Merchant, Agent, or Factor, by a Principal, for his
services in buying or selling goods. The Agent, if re-
siding in a different part of the country, or in a foreign
country, is called a Consignee; the goods shipped, a
Consignment; the Principal, a Consignor.
The commission is usually a given percentage of the
money involved in the purchase or sale.
210. To find the commission on a given sum, at a
given rate per cent..
Multiply the given sum hy the given rate ^, written as
a decimal. — (Vide 210.)
EXAMPLES.
1. I receive an order in Mobile, from Liverpool, for
the purchase of 300 bales of cotton. The cotton I buy
weighs on an average 500 pounds to the bale, and costs
in Mobile 11 J cts. per lb. What is my commission, at
the rate of 1^% on the cost of the cotton?
Ans. §253.12^.
2. A commission merchant sells goods to the amount
of §4375, on which he receives a rate of 2 ^. 'To what
does his commission amount? Ans. |87.50.
3. What is the commission on a purchase of 50 bales
of cotton, at 500 pounds to the bale, and costing 10| cts.
per pound, the commission being IJ % ?
Ans. §45.39Jg.
330. To find the commission when the amount in-
cludes the sum to be invested, and also the commission,
Divide the given amount by l-\- the rate ^, ivritten as
a dscimal, and subtract the result from the given amount.
<^ ■^^
EXAMPLES.
•L3
1. I receive in Mobile |7500, with which to purchase
cotton. My commission is to be 2 ^ on the purchase,
which is to be deducted from the money. What is my
commission? Ans. $147.05] |.
2. Suppose I had received $22400, with the under-
standing that my commission should be 2j ^ on the
amount purchased. AVhat amount should I have to ex-
pend for cotton? Ans. $21853.658.
3. I send a commission merchant $1000, with which
to buy cotton, after deducting his commission of 5 J^ on
the money invested. What amount is invested in cotton,
and how much is retained as commission?
Ans. $952.38 and $47.62.
221. Stock is money belonging to a collection of in-
dividuals, called a Corporation, authorized by law to do
business together.
(1.) The owners of the stock are called Stockholders.
(2.) A Share is one of the equal parts into which the
stock is divided. Such a part is usually $100.
(3.) A Certificate is a written evidence of ownership
of stock.
(4.) The par value of stock is the number of dollars
mentioned in each share.
(5.) The market value of stock is the number of dol-
lars a share w^ll bring when sold for cash.
(6.) Stock is above par, or below par, according as the
market value is above or below the par value.
(7.) If above par, stock is at a premium; if below
par, it is at a discount.
sft
PERCENTAGE.
22S. To find the value of stock at a given rate per
cent, premium,
Multiply the par value hy l-\- the rate ^ ivritten as a
decimal.
EXAMPLES.
1. What is the value of 17 shares of stock, at 5 ^ ,
?
premium
OPERATION.
11700
1.05
85.00 ^"^^ ■/
1700 ^-^ '^^
?^
$1785.00 Ans. (Vide 210, Ex. 17.)
2. What is the value of 14 shares of railroad stock,
at a premium of 7 % ? Ans. $1498.
3. What is the value of $32000 in State bonds, at a
premium of J ^ ? ' Ans. $32040.
223. To find the value of stock at a given rate per
cent, discount.
Multiply the par value hy 1 — the rate % written as a
decimal.
EXAMPLES.
1. What is the value of 17 shares of stock at 5 %
discount ?
OPERATION.
$1700
.95
85.00
1530.0
$1615.00
PERCENTAGE. 215
2. What is the value of 14 shares of railroad stock, at
a discount of 7 % ? Ans. §1302.
3. What is the value of 12 J shares of stock, at a dis-
count of 14 % ?— (210, Ex. 12.) Ans. |1075.
4. What is the value of |13000 in State bonds, at a
discount of 8% ? Ans. $11960.
5. I invest $10000 in railroad stock at the par value.
In a year the stock depreciates 3 fo, and, fearing a
further decline, I sell all my certificates. What is my
loss? Ans. $300.
6. If I buy 15 shares of stock at a premium oi Sfo,
and sell at a discount of 3%, what do I lose?
Ans. $90.
7. If I buy 18 shares of railroad stock at 5 ^ below
par, and sell at 7 % above par, what do I gain ?
!S34. Brokerage is the percentage charged by money
dealers, called Brokers, for negotiating Bills of Exchange.
Brokers all deal in stocks and other monetary matters.
225. To find the brokerage on a given sum.
Multiply the sum hy the rate fo written as a decimal.
EXAMPLES.
1. Wishing to rais,e an amount of money, I sell to a
broker 100 shares of railroad stock at a discount of i %.
What is the amount of brokerage? Ans. $25.00.
2. What must I pay a New Orleans broker for cashing
bills on New York to the amount of $5000, brokerage
at the rate of > % ? Ans. $25.00.
22%. Profit and Loss are terms used by merchants
and other business men in reference to the purchase and
sale of goods.
2r6 PERCENTAGE.
(1.) The cost is the price paid for an article.
(2.) The selling price is the amount received for an
article.
(3.) The profit is the amount received less the cost.
(4.) The loss is the cost less the amount received.
2217. To find the profit or loss when the cost price and
the rate per cent, of profit or loss are given,
Multiply the cost price by the rate fo written as a
decimal. The result will be the profit or loss.
EXAMPLES.
1. A merchant bought goods for ^500, and sold them
at a profit of 12^. What does he clear? Ans. $60.
2. If I buy goods for |750, and sell them at a profit
of 33A %, what do I clear ? Ans. ,^250.
3. What is the profit on oil, valued at $175, retailed
at 25% above the cost? Aiis. $43.75.
4. Buy sugar for $700, $800, and $1000; clear 25%
on the first lot, 33^ % on the second, and lose 50 % on
the third. How did I come out of the trade?
Ans. Lost $58 J.
228. To find the rate per cent, of profit or loss when
the cost and selling prices are givQn,
Divide the difference hetiveen the cost and selling prices
hy the cost price. Change the quotient by 209.
EXAMPLES.
1. If I buy goods for $500, and sell the same for
$560, what is the rate % of profit?
2. If I buy a quantity of flour for $750, and sell it
for $1000, what is the rate % of profit?
PERCENTAGE. £17
f
3. If I buy flour at |4 per barrel, and sell it at §5.50
per barrel, what is the rat^ fo of profit?
OPERAftoNS.
(1.) (^.) (3.)
§560 11000 $5.50
500 750 4.00
500) 60.00(.12 750) 250.00(.33J 4.)1.50(.37i
60.00 250.00 1.50
Arts. 12 fc- Ans. 33|%. Ans.S1l%>
4. If I purchase tea at 60 cts. per pound, and sell it
at 90 cts., what is the rate ^ of profit? Ajis. 50^.
5. If I buy 40 yards of broadcloth at §2.50 per yard,
and sell the whole for $120, what is the rate ^ of profit?
Ans. 20%.
6. I have in my storehouse 300 barrels of damaged
flour, which cost me §1450. I am willing to sell the lot
at §4 per barrel. What w^ould be the rate fo of loss ?
Ans, nj^fo.
7. Cost price §1.20, selling price §1.50. Rate % of
profit? Ans, 229, Ex. 5.
8. Cost price §1.25, selling price §1.75. Bate % of
profit ?
9. Cost price §1.40, selling, price §2.00. Rate % of
profit ?
10. Cost price §4.50, selling price §6.00 Rate % of
profit ?
11. Cost price §6.00, selling price §4.50. Rate.^ of
loss?
12. Cost price §2.00, selling price §1.40. Rate % of
loss?
19
PERCENTAGE.
229. To find the selling price, when the cost price is
known, so that a given per cent, may be made or lost,
(1.) If profit is to be made, multiply tlie cost^rice hy
l-\- the rate ^, ivritten as u decimal. (Vide 222.) .
(2.) If loss is to be susfained, multiply the cost price
hy 1 — the rate fo, ivritten as a decimal. (Vide 223.)
EXAMPLES.
1. I buy goods for ^500, and propose to clear 12 fo-
What must be the selling price? |500xl.l2.
Ans. $560.
2. If I buy flour for §750, and in the sale of it clear
33| ^, what is my selling price?
$750X1.33J=§750X-|. Ans. |1000.
3. I buy cloth at §4 per yard, and wish to make a
profit of 37| % on the cost. What must be my selling
price per yard? §4Xl.37i-=PX V-
Ans. $5.50
4. I have in store 300 barrels of flour, which cost
$1450. It being damaged, I am willing to lose VI J^fc
What must I charge per barrel? Ans. $4.
5. Cost price $1.20, rate % of profit 25. What is
the selling price ?
6. Cost price $1.25, rate % of profit 40. What is
the selling price ?
7. Cost price $1.40, rate % of profit, 42f . What is
the selling price?
8. Cost price $4.50, rate % of profit 33 J. What is
the selling price?
9. Cost price $6.00, rate % of loss 25. What is the
selling price?
PERCENTAGE. 219
10. Cost price 12.00, rate % of loss 30. What is the
selling price?
11. To make 12 » % profit, for how much must I sell
cloth that cost 16 cts. per yard? 24 cts.? 32 cts.? 64 cts.?
72 cts.? 88 cts.? Am. % of 16 cts.=18 cts., etc.
12. To make 16f % profit, what must a merchant
mark calico that cost 36 cts. per yard? 42 cts.? 54 cts.?
72 cts.? 11.26? $1.50? $1.80? .
13. To make 33^^, what must I mark books which
cost 24 cts.? 27 cts.? 30 cts.? 42 cts.? §1.02? $1.05?
$1.08? $1.24? $1.44?
230. To find the cost price, when the selling price
and the rate per cent, of profit or loss are given.
(1.) If a profit has been made, divide the Belling 'price
hy\-\- the rate % written as a deciftiaL (Vide 213.)
(2.) If a loss has been sustained, divide the selling
pnce hy 1 — the rate % written as a decimal, (Vide 214.)
EXAMPLES.
1. Selling price $560, rate % of profit 12. What is
the cost price?
2. Selling price $1000, rate % of profit 33^. What
is the cost?
3. Selling price $5.50, rate ^ of profit 37|. What
is the cost?
4. Selling price $1.50, rate % of profit 25. What
is the cost?
5. Selling price $1.75, rate % of profit 40. What
is the cost?
6. Selling price $2.00, rate % of profit 42f . What
is the cost?
220 PERCENTAGE.
7. Selling price $4:.50, rate % of loss 25. What is
the cost?
231. Duties or Customs are sums of money assessed
by government upon imported goods.
(1.) Specific duties are assessed upon goods at a cer-
tain rate per hogshead, gallon, bale, etc., with no refer-
ence to their value.
(2.) Ad valorem duties are a certain percentage of the
cost of goods.
(3.) An Invoice is a written account of the goods of a
cargo containing a statement of the cost of each article
in the currency of the country whence imported.
EXAMPLES.
1. The invoice of 'a cargo of goods which arrived in
Mobile from Liverpool, contained the following among
other items :
325 yd. Broadcloth cost 26 s. sterling per yd.
623 yd. Muslin " 4 s. " "
600 yd. Lace " Is. lOd. " "
975 yd. Carpeting " 6 s. " "
1280 yd. " " 4s. 8d. " "
The duty on the broadcloth was 15 ^ ; on the muslin
12| fo ; on lace 12 J ^ ; carpeting 15 %. What was
the amount of duty in United States money?
Ans. 1844.58.
232. Miscellaneous Examples.
1. A and B invest $550 in a speculation of which A fur-
nishes $330, and B the balance. They gain $70. What
is the rate % of profit on the money invested ? What is
the share of each ? Ans. 12 f\ %-, A $42; B $28.
PERCENTAGE. 221
2. A man failing owes A §175; B $500; C $600; D
1210; E $42.50; F $20; and G $10. His property is
sold for $934.50. What is the rate % of loss? What
is the share of each creditor?
Ans. Loss 40 per cent. A loses $70; B $200; C $240; D $84.00;
E $17; F $8; G $4.
3. A bankrupt owes A $500; B $1200; and C S4300.
The net cash proceeds of his estate amount to only
$1500. What rate ^ does he pay on his debts? What
does each creditor receive?
Am. 2h%, A $125; B $300; C $1075.
4. If the money and effects of a bankrupt amount to
$3361.74, and he is indebted to A $1782.24, to B
$1540.76, and to C $2371.17, how much will each re-
ceive? Ans. A $1052.20; B $909.64; C $1399.90.
5. I send a commission merchant $1000, with which
to buy cotton. If I allow him 5 % commission on the
money sent, how much will he have to expend in cot-
ton? ^Tis. $950.
6. How much railroad stock can be obtained for
$3860, when the stock is at a discount of ^l%t
Ans. $4000.
7. What rate % of $700000 is $700? Ans. J^%. .
8. What rate % of $450000 is $2250? Ans. \%.
9. A certain town, whose property is valued at $750-
000, proposes to raise a tax of $1875. What will be
the rate % ? Ans. J % , or 2^ mills on the dollar.
10. AVhat will be the tax of a man whose property
is valued at $12000, at the rate of i % ? Ans. $30.
11. A city agrees to loan a railroad company $1000-
000, which amount is to be raised on a property valued
222 PERCENTAGE.
at $175000000. What is the rate % of taxation, and
what does A pay, whose property is taxed for |35000 ?
Ans. 4%. A pays $200.
12. Sold tea at 90 cts. per pound, and gained 20 %.
What % should I have gained had I sold it for $1.00
per pound? Am. 33i %,
13. I sell tea at $1.28 per pound, and thereby lose
20 % . What would be gained or lost ^ by selling the
same tea at $1.68 per pound? Am. Profit of 5%.
14. iBy selling coffee at 67 J cts. per pound I make a
profit of 12J%, but I desire to make 30%. What must
be my selling price per pound? An%. 78 cts.
15. I bought a quantity of broadcloth for $2.59 per
yard, but on measuring it I find it falls short 12|% in
length. What must be my selling price per yard in
order to clear 12 1% on the real cpst? Am. $3.33.
16. I bought a quantity of calico at 40 J cts. per yd.;
but 10% of the calico proved to be damaged, and 10%
of the balance was lost by bad debts, and yet I cleared
10 % on the cost. What was the selling price per
yard? An%. 55 cts.
INTEREST.
1333. Interest is a percentage paid for the use of
money.
(1.) The Principal is the money for which interest is
paid.
(2.) The Amount is the sum of the principal and
interest.
(3.) The Rate per cent, per annum is the number of
cents paid fur the use of 1 tlollar for a year.
PERCENTAGE. 223
(4.) The Time is the period for which interest is paid.
Thus,
July 4, 1865, A borrowed of B $7250, agreeing to
pay at the rate of 8^ per annum. August 1, 1866,
he paid $623.50.
The principal is $7250; the interest is $623.50; the
amount is $7873.50; the rate per cent, per annum is
8; the time, 1 yr. 27 da. (Vide 198, Ex. 2; and 202,
Ex. 55.)
Resiark 1. — The rate per cent, in tlie various states is estab-
lished by law, and is thence called the legal rate; a higher than
legal rate is usury. The legal rate in Maine, New Hampshire,
Vermont, Massachusetts, Rhode Island, Connecticut, New Jersey,
Pennsylvania, Delaware, Maryland, Virginia, North Carolina,
Tennessee, Kentucky, •••Ohio, ^Indiana, •••Illinois, *Iowa, "^Nebraska,
^Missouri, *Kansas, ^Arkansas, -^Mississippi, Florida, District of
Columbia, and debts in favor of the United States, is 6 per cent.;
^Michigan, New York, Minnesota, Georgia, and South Carolina, 7
per cent.; Alabama and Texas, 8 per cent.; California, 10 per cent.;
Louisiana, 5 per cent. By special contract, parties, in the states
marked * can take interest as high as 10 per cent.
Remark 2. — The month, in computing interest, is regarded as
having 30 days. Custom has made it lawful. (Vide 185, Rem. 6.)
!334. To find the interest of any principal for 1 year,
at any given rate per cent, per annum.
Multiply the principal hy the rate fo per annum^ writ-
ten as a decimal.
EXAMPLES.
1. What is the interest of $6500 for 1 year, at 1 %
per annum? 2%? 3%? 4%? etc., to 12%?
Ans. $65; $130; $195; $260, etc. . . $780.
2. At 12 % per annum, what is the interest of $1041|
forlyear? $1458i? $3333i? $4375? $6250? $8333i?
$20833^? $6875? ^ws. $125; $175, etc. . $825.
224 PERCENTAGE.
3. At Sfo per annum, what is the interest of §5000
for 1 year? $1850? |9215? $8000? $7250? $5280?
Ans. 202, Ex. 51. . . 56.
4. What is the amount of $500, at 2^ per annum,
for 1 year? $250 at 5%? $375 at 6%? $475 at 6%?
$450 at 7 % ? $1250 at 8 % ? (Vide 233, (2.)
Ans. $^10; $262.50. . . $1350.
5. What is the amount of $1.00, at 5^ per annum,
fori year? 6%? 7%? 8%? 10%?
Ans. $1.05; $1.06; $1.07; $1.08; $1.10.
6. What is the amount of $1.05, at 5% per annum,
for 1 year? $1.06 at 6% ? $1.07 at 7% ? $1.08 at 8% ?
$1.10 at 10%?
Ans. $1.10i; $1.12/^; $1.1449; $1.1664; $1.21.
' 235. To find the interest of $1.00, at 12 % per
annum, for any given time.
EXAMPLES.
1. Find the interest of $1.00, at 12% per annum, for
3 yr. 5 mo. 15 da.
ANALYSIS.
(1.) 12 per cent, per annum, means 12 cents for the use of $1
one year. (Vide 233, (3.)
(2.) 12 cents per year gives a rate of 1 cent per month.
(3.) 10 mills (1 cent) for 30 days (1 month) gives a rate of 1
mill for 3 days.
The interest of $1.00 for 3 yr. 5 mo. is therefore 3Xl2-f-5 $0.41
The interest of $1.00 for 15 da. is . . 15-5-3 . . 0.005
The interest of $1.00 for 3 yr. 5 mo. 15 da. is (202, Ex. 57.) $0,415
Hence,
(1.) Call the months in the given time so many cents.
(2.) Call one third of the days so many mills.
PERCENTAGE.
225
Tlie sum of tliese results will be the interest required.
Find the interest of |1.00, at 12 ^ per annum, for
the following times. The work should be done mentally.
(Vide, also, 205, Ex. 56.)
2. 1 yr. 2 mo. 6 da.
Ans. §0.142.
3. 1 yr. 3 mo. 9 da.
Ans. $0,153.
4. 2 yr. 4 mo. 12 da.
.4ns. $0,284.
5. 3 yr. 6 mo. 15 da.
Ans. $0,425.
6. 4 yr. 1 mo. 18 da.
Ans. $0,496.
7. 5 yr. 3 mo. 21 da.
Ans. $0,637.
8. 6 yr. 7 mo. 24 da.
Ans. $0,798.
9. 1 yr. 3 mo. 29 da.
Ans. $0.159f.
10. Oyr. 4 mo. 12 da.
Ans. $0,044.
11. Oyr. 8 mo. 3 da.
Ans. $0,081.
12. 0 yr. 0 mo. 24 da.
Ans. $0,008.
13. 20 yr. 1 mo. 3 da.
Ans. $2,411.
14. 1 yr. 1 mo. 27 da.
A71S. $0,139.
15. 8yr. 4 mo. 0 da.
Ans. $1.00.
16. 2 yr. 1 mo. 1 da.
Ans. $0.25O|.
17. 3 yr. 2 mo. 2 da.
Ans. $0.380f.
18. 4 yr. 3 mo. 4 da.
Ans. $0.511|.
19. 5 yr. 7 mo. 11 da.
A71S. $0.673f.
20. 3 yr. 4 mo. 25 da.
A71S. $0.4081-.
21. 5 yr. 3 mo. 9 da.
.4ns. $0,633.
22. 0 yr. 1 mo. 1 da.
A71S. $0.0101
23. 0 yr. 8 mo. 2 da.
Ans. $0.080f.
24. 12 yr. 0 mo. 0 da.
Ans. $1.44.
25. 25 yr. 5 mo. 0 da.
Ans. $3,052.
S30. To find the interest on any principal, for any
time, at any rate per cent, per annum,
Multiply the principal by the interest^of §1 at 12% per
annum for the given time. (Vide 235.)
(1.) The interest at 1% is one twelfth that at 12%.
(2.) The interests at 2%, 3%, 4%, and 6% are, re-
spectively, one sixth, one fourth, one third, and one half
that at 12%.
226 ' PERCENTAGE.
(3.) The interests at S% and 9 % are, respectively, two
thirds and three fourths that at 12 %.
(4.) The interest at any rate whatever may he found hy
multiplying that at Ifo hy the numher representing the
rate % per annum.
EXAMPLES.
1. What is the interest of $421.40 for 3 yr. 5 mo.
15 da., at the rate of 6^ per annum?
OPERATION.
1421.40
(Vide 235, Ex. 1.) .415
210700
42140
168560
(Vide 235, (2.) 2)174.88100
(Vide 202, Ex. 59.) |87.4405 Ans.
2. What is the interest of $19.35 for lyr. 2 mo. 6 da.,
at the rate of 6^ per annum? Ans. $1,374.
3. What is the interest of $17.21 for 1 yr. 3 mo. 9 da.,
at the rate of 6^ per annum? Ans. $1.316565.
4. What is the interest of $140.10 for 2 yr. 4 mo.
12 da., at the rate of 6 ^ per annum ?
Ans. $19.8942.
5. What is the interest of $75.15 for 3 yr. 6 mo.
15 da., at the rate of 6^ per annum?
Ans. $15.96||.
6. What is the interest of $1000 for 4 yr. 1 mo.
18 da., at the rate of 6% per annum?
Ans. $248.
PERCENTAGE. 227
7. What is the interest of |175 for 5 jr. 3 mo. 21 da.,
at the rate of 6% per annum? Ans. |55.73|.
8. What is the interest of $2141 for 6 yr. 7 mo. 24
da., at the rate of 6% per annum? Aiis. §854.259.
9. What is the interest of $1041| for 1 yr. 3 mo. 15
da., at the rate of 6^ per annum? Ans. |80.72iJ.
10. What is the interest of |1458| for 1 yr. 4 mo. 21
da., at the rate of 6^ per annum? Ans. $121.77y'2.
11. What is the interest of ?3333| for 1 yr. 5 mo.
7 da., at the rate of 6 ^ per annum ?
Ans. $287.22f .
12. What is the interest of $4375 for 2 yr. 3 mo. 12
da., at the rate of 6% per annum? Ans. $599. 37^.
13. What is the interest of $421.40, from January 1,
1865, to June 16, 1868, at 6% per annum? 3^? 4%?
Last Ans. $58.293|..
14. What is the interest of $19.35, from April 4,
1866, to June 10, 1867, at 2 % ? 3 % ? 4 % ?
Last Ans. $0,916.
15. What is the interest of $25.14, from February 6,
1866, to April 3, 1867, at 6% ? 3% ? 4^ ?
First Ans. $1,747.
16. What is the interest of $200, from March 14,
1865, to July 14, 1873, at 6 % ? 3 % ? 4 % ?
First Ans. $100.00. •
17. What is the interest of $525, from May 17, 1868,
to June 18, 1870, at 6%? at 3%? at 4%?
First Ans. $65.71i.
18. What is the interest of $700, from August 14,
1865, to October 16, 1868, at 8%? at 2%? at 4%?
• Ans. $177,644
228 PEIICENTAGE.
19. What is the interest of §925.16, from October 15,
1865, to January 19, 1870, at 8%? at 2%? at 4%?
Ans. $315,377.
20. What is the interest of $375, from December 8,
1869, to Julj 19, 1875, at '8%? at 2%? at 4^?
Ans. $168,416.
21. What is the interest of $1400, from November 4,
1868, to March 29, 1872, at.8%? at 2%? at 4%?
Ans. $381,111.
22. What is the interest of $2000, from September
10, 1869, to February 5, 1873, at 8%? at 2%? at 4%?
Ans. $544,444.
23. What is the interest of $4062.50, from July 5,
1866, to October 15, 1868, at 8^ per annum?
Ans. 202, Ex. 39.
24. What is the interest of $1562.50, from June 6,
1870, to September 21, 1871, at 8%? 2%? 4%?
Ans. 202, Ex. 40.
25. What is the interest of $2187.50, from August
10, 1868, to January 1, 1870, at 8 % ? 2 % ? 4 % ?
Ans. 202, Ex. 41.
26. What is the interest of $8000, from January 1,
1866, to June 8, 1867, at 5 % ? 10 % ?
Ans. 202, Ex. 42.
27. What is the interest of $10500, from February
1, 1867, to May 13, 1869, at 5 % ? 10 % ?
Ans. 202, Ex. 43.
28. What is the interest of $15000, from March 4,
1868, to October 23, 1871, at 5 % ? 10 % ?
Ans. 202, Ex. 46.
PERCENTAGE. 229
29. What is the interest of ^20000, from April 9,
1867, to February 6, 1873, at 5%? at 10%?
Ans. 202, Ex. 47.
30. What is the interest of |50000, from May 16,
1865, to May 11, 1870, at 5%? at 10%?
Am. $12465.27?.
31. What is the interest of $425.30, from March 4,
1866, to May 19, 1868, at 7%? at 3j%?
Ans. $65,744.
32. What is the interest of $510.83, from March 21,
1867, to December 30, 1867, at 7%? at 3^-%?
Ans. $27,713.
33. What is the interest of $170, from June 19, 1865,
to July 1, 1866, at 7%? at 3^%? Ans. $12,296.
34. What is the interest of $966, from January 1,
1867, to March 20, 1869, at 7%? at 3j%?
Ans. $150,078.
35. What is the interest of $213.27, from August 15,
1872, to March 13, 1875, at 7%? at 3J%?
Ans. $38,483.
36. What is the interest of $426.50, from Sept. 4,
1868, to May 4, 1874, at 9 %? 4i- %? Ans. $217,515.
37. What is the amount of $164.06, from July 4,
1864, to February 15, 1870, at 3%? 1^%? 6%? 9%?
4J %? (Vide 233, (2.) Ans. $191.69 ; $177.87, etc.
38. What is the amount of $120.10, for 8 yr. 4 mo.,
at 12% per annum? 100 yr., at 1%? 50 yr., at 2%?
Ans. $240.20.
39. What is the amount of $120.10, for 12 yr. 6 mo.,
at 8 % per annum? 25 yr., at 4 % ? 33 yr. 4 mo., at
3 % ? Ans. $240.20.
230 PERCENTAGE.
40. What is the amount of $120.10, for 16 yr. 8 mo.,
at 6% per annum? for 20 yr., at 5%? 12 yr., at 8|%?
Ans. $240.20.
41. What is the amount of $450, from January 1,
1865, to March 16, 1865, at 8^ per annum?
Ans. $457.50.
42. What is the amount of $382.50, from March 16,
1865, to January 1, 1866, at 8^ per annum?
An§. $406,725.
43. What is the amount of $306,725, from January
1, 1866, to April 4, 1866, at 8% per annum?
Ans. $313,064.
44. What is the amount of $113,064, from April 4,
1866, to January 1, 1867, at 8^ per annum?
- Ans. $119,772.
45. What is the amount of $700, from January 1,
1865, to July 28, 1865, at 6% per annum?
Ans. $724.15.
46. What is the amount of $624.1 5, from July 28,1865,
to April 4, 1866, at 6^ per annum? Ans. $649.74.
47. What is the amount of $149.74, from April 4,
1866, to January 1, 1867, at 6^ per annum?
Ans. $156.40.
48. What is the amount of $3000, from Jan. 1, 1865,
to April 1, 1-865, at 10% per annum? Ans. $3075.
49. What is the amount of $2075, from April 1,
1865, to January 1, 1866, at 10% per annum?
Ans. $2230.625.
50. What is the amount of $1230.625, from January
1, 1866, to January 1, 1867, at 10% per annum?
Ans. $1353.68f.
PERCENTAGE. 231
51. What is tlie amount of $620.53, for 4 mo. 6 da.,
at 5% per annum? Arts. |631.39.
52. What is the amount of |1123.60, for 8 mo. 15 da.,
at 6% per annum? Ans. |1171.353.
53. What is the amount of $1531.301, for 3 mo.
24 da., at 7% per annum? Ans. $1565.24.
54. What is the amount of $4709.25, for 5 mo. 27
da., at 10% per annum? Ans. $4940.788.
2S*7. To find interest for days, counting 365 to the year,
Multiply one year's interest (vide 234) by the number
of days in the time, and divide the product by 365.
EXAMPLES.
1. What is the interest of $6500, from April 15 to
December 15, 1866, at 6 %? (Vide 185, Rem. 6 ; Table.)
$390X244--365. J^ns. $260.71.
2. At 6fo per annum, what is the interest of $375
from January 1 to March 15? $22.50 x 73 -4-365 = i of
$22.50. Ans. $4.50.
3. What is the interest of $1000, for 365 days, at the
rate of 6% for 360 days? $60Xftf=$60Xf|.
Ans". $60,831.
4. What is the interest of $1000, for 360 days, at the
rate of 6 ^ for 365 days ? $60Xf |f=$60X-ff •
Ans. $59.18.
5. What is the interest of $500000, for 365 days, at
the rate of 6% for 360 days? Ans. $30416f.
6. What is the interest of $500000, for 360 days, at
the rate of 6% for 365 days? Ans. $29589^^5.
7. What is the interest of $500000, for 90 days, at
the rate of 6% per annum of 360 days? Ans. $7500.
232 PERCENTAGE."
8. What is the interest of $500000, for 90 days, at
the rate of 6% per annum of 365 days?
Ans. $7397.26.
9. What is the interest of a §1000 bond, for 75 days,
at the rate of 7.30 fo per annum of 365 days ? 20 cts.
per day. Ans. $15.00.
10. What is the interest of a $100 bond, for 67 days,
at the rate of 7.30 fo per annum of 365 days ? 2 cts.
per day. Ans. $1.34.
11. What is the interest of a $10000 bond, for 93
days, at the rate of 7.30 % per annum of 365 days?
$2 per day. AnS. $186.
Remark. — In New York, interest for years and months is com-
puted by the rule under sec. 236, but for the odd days by the rule
under this section, 237.
12. What is the interest on a note of $1000, in New
York, having run from March 4, 1865, to March 25,
1866? • ^ns. $74.03.
13. What wbuld be the interest of the above note
computed in Kentucky? Ans. $63.50.
PROBLEMS IN INTEREST.
238. To find the principal, when the time, rate per
cent., and interest are given.
Divide the given interest hy the interest o/i $1.00 at
the given rate and time.
EXAMPLES.
1. The interest of a certain sum, at 6 % per annum, for
Syr. 5 mo. 15 da. is $87.4405. What is the principal?
(Vide 236, Ex. 1.) $87.4405-.2076. Ans. $421.40.
PERCENTAGE. 233
2. The interest of a certain sum at 6 ^ per annum,
for 1 yr. 2 mo. 6 da. is ^1.374. What is the principal?
Ans. 236, Ex. 2.
239. To find the principal, when the time, rate per
cent., and amount are given,
Divide the given amount bij the amount of §1.00 at
the given rate and time.
EXAMPLES.
1. The amount of a certain sum at 6% per annum,
for 3yr. 5 mo. 15 da. is §508.8405. What is the prin-
cipal? $508.8405--1.2075. Ans. $421.40.
2. The amount of a certain sum, at 8^ per annum,
from January 1, 1865, to March 16, 1865, is $457.50.
What is the principal ? Ans. 236, Ex. 41.
3. The amount of a certain sum, at 10% per annum,
from January 1, 1866, to January 1, 1867, is $1353. 68|.
What is the principal? Ans. 236, Ex. 50.
240. To find the time, when the principal, interest,
and rate per cent, are given.
Divide the given interest by the interest on the principal
at the given rate for 1 year.
EXAMPLES.
1. The interest of $421.40, at 6 % per annum, is
$87.4405. What is the time? 87.4405 --25.284 =3.458^
yr.=3yr. 5mo. loda. (Vide 195.)
2. The interest of $75.15, at 6 % per annum, is
$15.96|-§. What is the time?
Ans. 236, Ex. 5.
20
234 PERCENTAGE.
3. The interest of §525 to June 18, 1870, at 6 %
per annum, is $65.71 1. What is the date from which
interest is computed? Ans. 236, Ex. 17.
4. The interest on §100 is also §100. What has the
time been at the rate of 1 % per annum? 2%? 3%?
4%? 5%? 6%?
5. The interest on §120.10 is also §120.10. What
has the time been at the rate of 6 ^ per annum?
8%? 12%? Ans. 236, Ex. 38, 39, 40.
6. In what time will a^iy sum double itself at a rate of
5% per annum? 7%? 9%? 4i%?
Ans. 20 yr., etc.
7. In what time will any sum triple itself at the rate
of 6% per annum? Aiis. 33 yr. 4 mo.
S41. To find the rate per cent., when the principal,
interest, and time are given,
Divide the given interest hy the intei^est on the principal
at Ifo p^T annum for the given time.
EXAMPLES.
1. The interest of §421.4t) for 3yr. 5 mo. 15 da. is
§87.4405. What is the rate % per annum? 87.4405--
14.5734^ Ans. 6 % .
2. The interest of §700 from August 14, 1865, to
October 16, 1868, is §177.64|. What is the rate %
per annum? Ans. 236, Ex. 18.
3. The interest of §750 for 3 yr. 4 mo. is §162.50.
What is the rate % per annum? Ans. 6j%.
4. The interest of §950 for 2 yr. 4 mo. 20 da. is
§238.29 J. What is the rate % per annum? (Vide 202,
Ex. 58.) ' Ans. 10J%.
PERCEXTAGE. 235
PRESENT Yv^OKTH.
242. The Present Worth of a sum of money due
at some future time, is a principal ^vliicli, being put at
interest at the time \)f payment, Avill amount to the sum
at the time it is due.
The Discount is the diiference between the present
worth and the sum of money due.
243. To find the present worth of a sum of money
for a given time and rate per cent, per annum,
Divide the given sum by the amount of fl.OO at the
given rate and time. (Vide 239.) The quotient is the
present worth.
EXAMPLES.
1. What is the present worth of |508.8405, due in Syr.
5 mo. 15 da., at the rate of G ^ per annum? §508.8405-^-
1.2075. Alls. S421.40.
2. What is the discount of §457.50, due March 16,
1865, but paid January 1, 1865, at 8% per annum?
Ans. ^7.50.
3. What is the discount on §1353.68 1, due January
1, 1867, but paid January 1, 1866, at 10 J/^ per annum?
A71S. $123,061.
4. What is the discount of $1800, due 1 yr. 3 mo.
hence, at the rate of 6 ^ per annum ?
Ans. $125,582.
5. What is the discount of $475, due 1 yr. hence, at
the rate of 7% per annum? Ans. $31,075.
6. What is the discount of $2500, due 3 mo. hence,
when money is worth 4J%? A71S. $27.81.
236 PERCENTAGE.
7. Four notes are due as follows : §900 in 6 mo. ;
$2700 in 1 yr.; |3900 in 1 yr. 6mo.; and |4200 in 2yr.
If the notes are all paid at the present time, what will
be the entire discount, at 6$^ ? Ans. ?951.06.
8. What is the discount of §400, due three months
hence, but paid now, at the rate of 12^ per annum?
S%2 r^/ot 6%? 5^,?
>lns. §11.65; §7.84; §6.88; §5.91; §4.94.
BANK DISCOUNT.
244. Bank Discount is a deduction made by a bank
upon a sum of money borrowed of it, at the time the
money is taken.
(1.) By custom, this discount is the interest on the
face of the note for the time it is drawn, increased by
three days, called Days of Grace,
(2.) The Proceeds of a bank note is what remains
after the discount has been deducted.
245. To find the bank discount on a note or draft,
Find the interest on the face of the note for three days
more than the time for ivhich it is draivn.
EXAMPLES.
1. What is the bank discount on a note of §400, dis-
counted for 90 days, at 8^; ? 6 % ? 2 of $400X.031; | of
$.loox.03i. Ans. §8.26f ; §6.20.
2. Find the proceeds of a note of §150, discounted
forGOdays, at 6%? 7%? 8%? 10%? 1 of $i50x.02i;
-i^jj of $150X.021, etc.
^715. §148.42i; §148.16.1; §147.90; §147.37i.
TEJICENTAGE. 237
3. Find the proceeds of a note of §177.75, discounted
for 90 days, at 6^ per annum. Ans. §175.00.
4. Find the proceeds of a note of §505.305, discounted
for 60 days, at 6^ per annum. A7is. §500.
5. Find the proceeds of a note of §375.00, discounted
for 30 days, at 6^ per annum. Ans. §372. 93|
246. To make a note of which, when discounted, the
proceeds shall be a given sum.
Divide the given sum by the proceeds on §1.00 at the
given time and rate per cent, per annum.
EXAMPLES.
1. I wish to obtain §175 for 90 days. For what sum
must my note be drawn, at 6 ^^ per annum ?
1.00-§0.0155=§0.9845, then §175-^.9845--§177.755.
2. I buy produce worth §500, but it will be 60 days
before I shall have money with which to pay for it.
For what sum must I draw, at 6 ^ per annum ?
Ans. §505.305.
3. I buy cotton to the amount of §372.93 J, and bor-
row money at the bank with which to pay for it. For
what sum do I draw, payable in 30 days, at 6^ per
annum? Ans. §375.
PROMISSORY NOTES.
247. A Promissory Note is a promise in writing to
pay a certain sum of money to a person, named in the
note, or order, or to the bearer.
(1.) The Drawer of a note is the person signing it.
(2.) The Payee of a note is the person to whom the
money is to be paid.
238 PERCENTAGE.
(3.) The Indorser of a note is a person who guaran-
tees the payment of it. He does this by writing his
name on the back of the paper on which the note is
written.
(4.) An Indorsement is an acknowledgment on the
back of the note that, at a given date, a part of the
money was paid.
(5.) The Face of a note is the sum promised to be
paid.
S48. To find the amount due at the maturity of a
note upon which one or more indorsements have been
made,
I. When the time of the note is one year or
less,
(1.) Find the amount of the face of the note from its
date to the time of maturity/.
(2.) Find the amount of each payment from the time it
was made till the time the note matures.
(3.) Subtract the sum of the amounts of all the pay-
ments from the amount of the face of the note. (Vide
233, (2.)
EXAMPLES.
$500. Richmond, Jan. 1, 1865.
(1.) Ninety days after date, I promise to pay to the
order of Frank H. Ransom Five Hundred Dollars,
with interest, value received.
John M. Sabin.
Indorsements : January 20, $100; February 10, $50;
February 25, $100; March 1, $150.
What was due at maturity ?
PERCENTAGE.
239
OPERATION.
A^mottnt of JJ500 for 93 days is
§507.75
" §100 " 74 "
§101.231
" §50 '' 53 "
50.44^
§100 " 38 "
100.631
" §150 " 34 "
150.85
Sum of the amounts of payments,
.§403.15§
Sum due at maturity, April 4, 1865, §104.59^
II. When the time of the note is more than 1 year,
(1.) Find the amount of the face of the note to the date
of the first payment, and deduct the payment.
(2.) Find the amount of the remainder to the date
of the next payment, and, after deducting the payment,
find the amount of the remainder to the date of the
next indorsement, and so on till the last payment is
reached.
(3.) Find the amount of the remainder, on deducting
the last payment, from the date of that payment till the
time the note matures.
Remark. — Unless a payment or payments are eqnal to or exceed
the interest due at the date of the last, they must be added to the
succeeding payment, and the sum considered as a single payment.
In business, therefore, no payment should be made on a note unless
it exceeds th« interest then due.
§450. Mobile, Jan. 1, 1865.
(2.) Two years after date, I promise to pay to the order
of James Boone Four Hundred and Fifty Dollars, with
interest, value received. David Miller.
Indorsements: March 16, 1865, §75; January 1,
1866, SlOO; April 4, 1866, §200.
What was due at maturity?
240 PERCENTAGE.
OPERATION.
Amount of $450 to March 16, is (236, Ex. 41.) |457.50
75.00
Payment deducted is 382.50
24.225
Amount of $382.50 to Jan. 1, is (236, Ex. 42,) 406.725
100.000
Pa^^ment deducted is 306.725
6.339
Amount of $306,725 to April 4, is (236, Ex. 43,) 313.064
200.000
Payment deducted is 113.064
6.708
Amount of $113,064 to Jan. 1, '62, (236, Ex. 44) $119,772
$700. Louisville, Jan. 1, 1865.
(3.) Two years after date, for value received, I
promise to pay A. B., or order, Seven Hundred Dollars,
with interest. M. Greene.
Indorsements: July 28, 1865, $100; April 4, 1866,
$500.
What was due at maturity? (Vide 236, Ex. 45, 46,
47.) Ans. $156.40.
$3000. San Francisco, Jan. 1, 1865.
(4.) Two years after date, for value received, I promise
to pay James Monroe, or order, Three Thousand Dol-
lars, .with interest.
Indorsements: April 1, 1865, $1000; January 1,
1866, $1000.
What was due at maturity? Ans. $1353.68|.
PEKCEXTAGE. 241
S.SOO. Mobile, June 10, 1865.
(5.) June 2, 1866, for value received, I promise to
pay S. S. Bryant, or order, Three Hundred Dollars, with
interest. P. Hamilton.
Indorsements : January 20, 1866, §116 ; March 2,
1866, §49.50; April 26, 1866, §85.
What was due at maturity by both rules ?
Ans. I. $67.89; II. §68.17.
§1000. Galveston, Jan. 1, 1869.
(6.) July 1, 1870, for value received, I promise to pay
C. Q. M., or order. One Thousand Dollars, with interest.
Wm. Daniel.
Indorsements: July 1, 1869, §30; Jan. 1, 1870, §470
What was due at maturity by both rules?
Ans. 11. §603.20 ; I. §598.80.
§400. Buffalo, Jan. 1, 1870.
(7.) One year after date, for value received, I promise
to pay N. Stacy, or order, Four Hundred Dollars, with
interest. M. M. DeYoung.
Indorsements: March 16, 1870, §200; July 1, §100.
What was due at maturity ?
Ans. II. §113.88 ; I. §112.25.
COMPOUND INTEREST.
249. Compound Interest is interest on the principal,
and then, after the interest becomes due, on the amount.
(Vide 233, (2.)
examples.
1. What is the amount of §1.00 at compound interest,
for3yr., atS^perannum? 6%? 7%? 8%? 10%?
21
242
PEllCENTAGE.
OPERATIONS.
$1.05X1.05X1.05=11.157625 Ans.
$1.06Xl.06Xl.06-=:|1.191016 Ans.
$1.07Xl.07Xl.07=$1.225043 Ans.
$1.08Xl.08Xl.08==$1.259712 Ans.
$1.10X1.10X1.10=$1.331 (234, Ex. 5, G.)
by suljtracting
Remark. — The compound interest is obtained
$1 from the amounts.
TABLE,
Showing the amount of §1.00 at compound interest, for any number
of years from 1 to 25, at 5, 6, 7, 8, and 10 per cent.
Yeaks
5 Per Cent.
G Per Cent.
7 Per Cent.
8 Per Cent.
10 Per Cent.
"ijMooo"
1
1.050000
1.060000
1.0700U0
1.080000
2
1.102500
1.123600
1.144900
1.166400
1.210000
3
1.157625
1.191016
1.225043
1.259712
1.331000
4
1.215506
1.262477
1.310796
1.360488
1.464100
5
1.276282
1.338226
1.402551
1.469328
1.610510
6
1.340096
1.418519
1.500730
1.586874
1.771561
7
1.407100
1.503630
1.605781
1.713824
1.948717
8
1.477455
1.593848
1.718186
1.850930
2.143589
9
1.551328
1.689479
1.838459
1.999004
2.357948
10
1.628895
1.790848
1.967151
2.158924
2.593742
11
1.710339
1.898299
2.104851
2.331638
2.853117
12
1.795856
2.012196
2.252191
2.518170
3.138428
13
1.885649
2.132928
2.409845
2.719623
3.452271
14
1.979932
2.260904
2.578534
2.937193
3.797498
15
2.078928
2.396558
2.759031
3.172169
4.177248
16
2.182875
2.540352
2.952163
3.425942
4.594973
17
2.292018
2.692773
3.158815
3.700018
5.054470
18
2.406619
2.854339
3.379932
3.996019
5.559917
19
2.526950
3.025600
3.616527
4.315701
6.115909
20
2.653298
3.207135
3.869084
4.660957
6.727500
21 ■
2.785963
3.399564
4.140562
5.033834
7.400250
22
2.925261
3.603537
4.430402
5.436540
8.140275
23
8.071524
3.819750
4.740530
5.871404
8.954302
24
8.225100
4.048935
5.072367
6.341181
9.849733
25
3.386354
4.291871
5.427433
6.848475
10.834700
PERCENTAGE. 243
S50. To find the amount at compound interest of
any principal, at any rate ^ per annum, and for any
given time,
(1.) For the given integral number of years, multiply
the principal hy the amount of |1.00, for the same time
and rate %.
(2.) 0)1 this amount find the amount for the months
and days, as in §236.
Remark. — The compound interest will be the diflference between
the amount and the given principal.
EXAMPLES.
1. Find the compound interest of §400 for 9 yr., at
bfo per annum. §400Xl.551328=f620.5312.
Ans. $220.53.
2. Find the compound interest of $400 for 9 yr. 4 mo.
6 da., at 5% per annum. (Vide 236, Ex. 51.)
Ans. $231.39.
3. Find the compound interest for $1000 for 2 yr.
8 mo. 15 da., at 6 % per annum. (Vide 236, Ex. 52.)
Ans. $171,353.
4. Find the compound interest of $1250 for 3yr. 3 mo.
24 da., at 7% per annum. (Vide 236, Ex. 53.)
Ans. $315.24.
5. Find the compound interest of $700 for 20 yr.
5 mo. 27 da., at 10 % per annum. (Vide 236, Ex. 54.)
Ans. $4240.788.
244 n.vTio.
RATIO.
251. Ratio is the quotient obtained by dividing one
number by another of the same kind. Thus,
The ratio of 5 to 15 is '^f=^, commonly expressed
by 5 : 15.
(1.) The two numbers forming a ratio are together
called terms.
(2.) The first term is called the antecedent.
(3.) The second term is called the consequent.
(4.) The antecedent and consequent form, a couplet.
(5.) The value of a ratio is the quotient of the conse-
quent divided hy the antecedent.
(G.) A ratio is in its simjjlest or loivcst terms -when the
terms are integral and prhyie tvith resjject to each other.
(Vide 102.)
252. To reduce a ratio to its lowest terms,
(1.) If fractions are involved, multijjlt/ the terms hy the
least common multiple of the denominators of the frac-
tions. (Yide 107.)
(2.) Divide the resulting terms hy their greatest common
divisor^ (vide 104,) or cancel such factors as are common
to hoth terms,
EXAMPLES.
1. Reduce 15:20; 14:21; 16:24 to their lowest
terms, and find their values. Values !{ ; li ; IJ.
2. Reduce 9:63; 26:169; 34:187 to their lowest
terms, and find their values. Values 1\ 6^; 5^.
RATIO.
245
3. Reduce 5| : 4f ; 2i : If ; and Jf : 3i to their lowest
terms, and find their values.
OPERATIONS.
(2.)
21 : If
5|:4f
119 : 102 68 : 48
7 : 6 Value f
4'. Reduce | : I ; j
and find their values.
(3.)
21 : 16 Value J f ,
M:3i
52 : 351
4
27 Value 6^,
1 .
4 '
2 . 7
6 • 1 Oi
to their lowest terms,
Values i; f; If.
5. Reduce 4i:5i; 6j:7i; and 4^ : 2|f to their
lowest terms, and find their values.
VahcesV^; Ig^; and |.
6.
Reduce 3i
.92 . 4.1
:f; and J:
: 7^ to their lowest
term
s, etc.
Values
3 ? T6 ? ^"^ ^g-
7.
Reduce 5 :
2.5; .3:21; and 1/^
^ : 3.4 to their low-
est terms, etc.
OPERATIONS.
(1.)
(2.)
(3.)
5;
:2.5
.3:21
l/,:3.4
50:
:25
3 : 210
17:34
2 ;
: 1 Value J.
1:70
Value 70. '
1:2 Vahie2.
8.
Reduce .5 :
.2; Mi:
.25 ; and .4
: .7 to their lowest
terms, etc.
Values, Ex. 4.
9.
What is the ratio of 50 cts. : 20 cts.? 33 1 cts. : 25
cts.?
40 : 70 cts.
?
Ans. f; I; If.
10. What is the ratio of 14 hu. : 35 bu.? 2 qt. : 8 qt.?
30 sec. : 50 sec? Ajis. 2^-; 4; f.
11. What is the ratio of 2 qt. : 3 pk.? 30 sec. :7m.?
lpt.:l gal.?— (Vide 251.)
246 RATIO.
OPERATIONS.
(1.) (2.) (3.)
2 qt. : 3 pk, 30 sec. :7 m. 1 pt. : 1 gal.
2 qt. : 24 qt. 30 sec. : 420 sec. 1 pt. : 4 qt.
1 qt. : 12 qt. Value 12. 1 sec. : 14 sec. Value 14. 1 pt. : 8 pt. Value 8.
12. What is the ratio of 1 mile to 5 fur. 3 r. 10 ft
6 in.?— (193, Ex. 2.) Ans. j\.
13. What is the ratio of 1 mile to 3 fur. 22 r. 3 ft.
8in.?— (193, Ex. 4.) Ans.^^.
PROPORTION.
353. A Proportion is an equality/ of ratios.
The equality is indicated by four dots or a double
colon written between the couplets. Thus,
5: 15:: 6: 18
is a proportion, and is read 5 is to 15 as 6 is to 18, the
meaning of which is that 15-^-5 is the same as 18-^-6;
that is, M=:ig^
(1.) The first and last terms of a proportion are called
extremes.
(2.) The second and third terms are called means.
(3.) The first and second terms form the first couplet.
(4.) The third ' and fourth terms form the second
couplet. Thus, in the proportion above,
5 and 18 are the extremes; 15 and 6 the means; 5
and 15 the first couplet; 6 and 18 the second couplet;
5 and 6 the two antecedents, and 15 and 18 the tAvo
consequents.
!354. Proposition. — If four numbers are in propor-
tion, the product of the extremes' is equal to the product
of the means. Thus,
RATIO. 247
From any proportion as 3 : 9 : : 7 : 21,
we have by 251 and 253, |=V) ^^^ ^J multiply-
ing both of these fractions by the least common multiple
of their denominators we have 9x7=3x21, and it is
evident that any proportion may be treated in a similar
way.
Remark. — In any proportion, if the second term is less than the
first, the fourth will be less than the third, and if the second term is
greater than the first, the fourth will be greater than the third.
S55. Proposition. — If in any j^'^oportion the terms
of either couplet he multiplied or divided hy the same
number, the proportion luill not he destroyed. Thus, from
the proportion,
7 : 21 : : 8 : 24 we have
1 : 3 : : 8 : 24 or 7 : 21 : : 1 : 3
1^256. Proposition. — If, in any proportion, the two
antecedents or the two consequents he midtiplied or divided
hy the same numher, the proportion will not he destroyed.
Thus : From the proportion
7 : 8 : : 21 : 24 we have
1 : 8 : : 3 : 24 or 7 : 1 : : 21 : 3
257. Problem. — The two extremes of a proportion
and one mean being given, to find the other mean,
(1.) Reduce the given terms as'loiv as possible hy 255
and 256.
(2.) Divide the product of the resulting extremes hy the
mean. The quotient will be the other mean.
EXAMPLES.
1. Given the terms 7 : 21 : : : 24, to find the other
mean.
248
RATIO.
OPERATION.
Divide first couplet by 7. 7 : 21
Divide consequents by 3. 1:3
1: 1
2. Given the terms 7 :
mean.
:24
: 24 (Vide 255.)
: 8 Ans. (Vide 256.)
8 : 24, to find the other
Ans. 21.
3. Given, 10 : 14 : :
: 35, to find the other mean.
Ans. 25.
4. Given, 10 : : : 25 : 35, to find the other mean.
Ans.
5. Given, 85 : 102 : : : 306, to find the other mean.
A71S. 255.
6. Given, J : | : : : f , to find the other mean.
OPERATION.
Multiply first couplet by 6. A : |
Divide consequents by 2. 3:4
3:2
-I (Vide 1671 Ex. 106.)
f (Vide 255.)
I (Vide 256.) Ans. j^^.
7. Given, J : : : ^^^ : §, to find the other mean.
8. Given, -/g : /,-
9. Given, f : 24 :
Ans. f .
: 100, to find the other mean.
Ans. 120.
: 20, to find the other mean.
Ans. j\.
10. Given, | : ^5_ . . . 20, to find the other mean.
258. The two means of a proportion and one extreme
being given to find the other extreme,
(1.) Reduce the terms as low as possible by 255 and
256.
(2.) Divide the product of the resulting means by the
extreme. The quotient will be the other extreme.
RATIO. 249
EXAMPLES.
1. Given of : 4f : : 21, to find the fourth term.
OPERATION.
Multiply couplet by 21. 5| : 4f : : 21 (Vide 252, Ex. 3.)
Divide couplet by 17. 119 : 102 : : 21 (Vide 255.)
Divide antecedents by 7. 7 : 6 : : 21 (Vide 256.)
1 : 6 : : 3 : 18 Ans.
2. Given, 1 : 2 : : 3, to find the fourth term. A7is. 6.
3. Given, 3:9:: 12, to find the fourth term.
Ans. 36.
4. Given, 4 : 16 r : 15, to find the fourth term.
Ans, 60.
5. Given, ^% : /o : : 120, to find the fourth term.
Ans. 100.
6. Given, 1 : | : : J, to find the fourth term.
Ans. I.
7. Given, 9 : 18 : : ^, to find the fourth term.
A71S. 1.
8. Given, 35| : 15^ : : 4, to find the fourth term.
Ans. IjV^.
9. Given, 4^ : 22^ : : 9, to find the fourth term.
Ans. 45.
10. Given, ^^ : 28? : : 3, to find the fourth term.
Ans. 27.
11. Given, 2.5 : 45 : : 63, to find the fourth term.
A71S. 1134.
12. Given, 4.5 : 22.5 : : 9, to find the fourth term.
A71S. 45.
13. Given, J-i : ^J- : : ^%, to find the fourth term.
Ans. §.
250 RATIO.
14. Given, 5 : f : : |, to find the fourth term.
Ans. /j.
15. Given 7| : 6 : : 5|, to find the fourth term.
Ans. 4j|-f.
16. Given, 6 : 7| : : 5f , to find the fourth term.
Ans. 6§§.
17. Given, 1 : 2| : : 2|, to find the fourth term.
Ans. 6i.
18. Given, 75| : 36 : : 643|, to find the fourth term.
Ans. 306.
19. Given, 1.50 : 1.00 : : .30, to find the fourth term.
Ans. .20.
20. Given, 3X32 : 5X16 :: 120, to find the fourth
term. Ans. 100.
21. Given, 8X21 : 56X6 :: 10, to find the fourth
term. Ans. 20.
22. Given, 2\ : If : ; |f , to find the fourth term.
Ans. ||.
23. Given, • :^%:'. 120 : 100, to find the first term.
A71S. /g.
RULE OF THREE.
259. Every problem in proportion involves at least
three quantities, so related to each other that a fourth
may be found from them.
260. The statement of a problem consists in properly
arranging the three quantities mentioned in it, so as to
form the first, second, 'dud third terms of a proportion.
261. The Rule of Three consists of directions by
which a problem in proportion may be stated. They
are as follows :
HATIO. 251
(1.) Of the three quantities mentioned, make that the
THIRD TERM wMcJl JiaS the SAME NAME aS the ANSWER
required.
(2.) Of the two remaining quantities, make the
GREATER the SECOND TERM if the ANSWER ought to be
GREATER than the third term; make the less the sec-
ond term if the answer ought to be less tha7i the third
TERM. — (Vide 254, Rem.)
(3.) Place the remaining quantity for the first term.
The fourth term, found by 258, will be the answer,
examples.
1. If 5f lb. of sugar cost 21 cts„ what will 4!^ lb. cost?
(1.) 21 cts. has the same name which the answer
should have.
(2.) 4| lb. will evidently cost less than 21 cts., the
jorice of 5f lb.
(5.) 5| should then be the first term, 4? the second,
and 21 cts. the third.
Thus 5§ : 4^ : : 21 (vide 258, Ex. 1.) Ans. 18 cts.
2. If j\ of a quantity of sugar cost S120, what will
/j of the same quantity cost?— (See 167 J, Ex. 113,
Rem.; and 258, Ex. 5.) Ans. |100.
3. If 3 yd. of cloth cost $12, what will 9 yd. cost?
A71S. §36.
4. If 4 lb. of rice cost 15 cts., what will 16 lb. cost?
Ans. 60 cts.
5. If 1 lb. of tea cost J of a dollar, what will J lb.
cost? Ans. 16| cts.
6. If a staff 9 feet high cast a shadow \ of A foot
-OJ RATIO.
LONG, what will be the length of the shadow of a post
which is l^feet high f Ans. 258, Ex. 7.
7. If a tree 35| ft. high casts a shadow 4 ft. long,
what will be the length of the shadow of a tree 15 1 ft.
Mgh 1 Am. 1 ^V? ^.=1 ft. .8 ^V^ in.
8. If a staff 10 ft. high casts a shadow 12 ft. in lengthy
what will be the hight of a tree whose shadow measures
70 ft. ? Am. 58 J ft.=58 ft. 4 in.
9. If 34 yd. of cloth cost ^3, what will 28| yd. cost?
Am. |27.
10. If 2.5 A. of land cost |63, what will 45 A. cost?
Am. §1134.
11. If 45 A. of land cost' |1134, what will 2.5 A.
cost ? Am.
12. If 45 A. of land cost §1134, what quantity can
be bought for §63 ? 1134 : 63 : : 45 Am.
13. If 4.5 yd. of cloth cost §9, what will 22.5 yd.
cost ?
14. If 25.5 yd. of cloth cost §45, what will 4.5 yd.
cost?
15. If J J of a pound of butter cost -^^ of a dollar,
how much will ij of a pound cost?
16. If W of a lb. of butter cost 40 cts., what will
jj lb. cost?
17. If \\ of a lb. of butter cost 53 1 cts., what quan-
tity can be bought for 40 cts.? An8. 8| oz.
18. If 8 J oz. of butter can be bought for 40 cts., what
quantity can be purchased for 53 J cts.
An%. 11J4 oz.
19. If 6 yd. of cloth cost §4.55||, what will 71 yd.
cost ?
RATIO. 253
20. If 75 yd. of cloth cost ^5.60, Avhat will G yd.
cost?
21. If 1 yd. of carpeting cost |2-J, what will 2 J yd.
cost?
22. If 2 J yd. carpeting cost ^6.25, what will .one yd.
cost?
23. If 75f yd. of cloth cost |643|, what will 3G yd.
cost?— (Vide 167i, Ex. 72; and 258, Ex. 18.)
24. If 1 coat require If yd. of cloth, how many coats
can be made of 18 J yd.? Ans. 10 coats.
25. If 10 lb. of copper cost ^18.75, what number of
pounds can be had for ^171.25?— (Vide 167J, Ex. 79
and 80.)
26. If f of a ship are worth $15700,^ what is the
value of i of the ship? • Ans. 167}, Ex. 85.
27. If 1 lb. of butter cost 3 pence, how many tons
can be bought for 84 £ 13 s. 9d.?
Ans. 191, Ex. 63.
28. If I buy cloth at §1.50 and sell it for §1.20,
what shoiild I lose on §1.00?— (Vide 258, Ex. 19.)
Ans. §0.20.
29. If I buy cloth at §1.25 and sell it for §1.75,
what should I make on §1.00?— (Vide 228, Ex. 8.)
Ans. 40 cts.
30. If I pay at the rate of 11 cts. per mile, how far
can I ride for 7 cts.? Ans. 193, Ex. 39.
31. If I sell a quantity of land at the rate of §450
for 25 sq. r., and obtain §51300, how many acres do I
sell? Ans. 191, Ex. 74:.
32. If 2.5 lb. of tobacco cost 75 cts., how much will
185 pounds cost? Ans. §55.50.
254 RATIO.
33. If 2 oz. of silver cost ^2M, wliat will f oz. cost?
Ans. $0.84.
34. If 7 lb. of sugar cost 75 cts., what will 12 pounds
cost? Ans. $1,284.
35. If 141 tons of coal cost 85 £, what will 94 tons
cost? Ans. $274.26.
36. If the interest of $19.35 is $2.7477, what will be
the interest of $17.21 for the same time and rate per
cent.? Ar^s. $2.44382.
37. If the interest of $17.21 is $2.63313, what will
be the interest of $140.10 for the same time and rate
per cent.? Ans. $21.4353.
38. If 4 men can do a piece of work in 100 days, in
how many days would 5 men do the same work?
Ans. 80 days.
39. If 20 men can do a piece of work in J of a day,
how^ long would it take 2 men to do the same w^ork?
Ans. 5 days.
40. If 60 men can do a piece of work in 8 days, how
many men would perform the same w^ork in 20 ^ays?
Ans. 24 men.
41. If 2 men cai> dig a ditch in 40 days, in how many
days would 10 men dig the same ditch?
A71S. 8 days.
42. Having read 120 pages of a book, I find that I
have still to read | of the book. How many pages does
the book contain?
§ : I : : 120 Ans. 300 pages.
43. A person owning | of a coal mine sells | of his
share for $400. What is the mine worth at this rate?
Ans. $1000.
iiATio. 255
44. If 10 covfs eat 8 tons of hay in a given time,
how many cows would eat 56 tons in the same time?
Ans. 70 cows.
45. If 70 cows eat a certain quantity of hay in 6
weeks, how many coavs would eat the same hay in 21
weeks? Ans. 20 cows.
46. If 20 men have been at work 18 days to con-
struct a given length of railroad, how many days will
76 men require to construct the same length of road ?
Ans. 4i| days.
47. If, by making 10 hours a day, a certain number
of men complete a work in 18 days, how many days
would be required to complete a similar work, at 12
hours a day? Ans. 15 days.
48. If 18 days are required to construct 500 feet of
railroad, how many days will be occupied on 1140 feet
of the same road? Aiis. 41^^ days.
49. If a passenger train of cars gain on a freight
train at the rate of 8 miles in 3 hours, how many hours
will it take to gain 60 miles? Ans. 22^ hours.
50. A passenger train of cars moves at the rate of
45 miles in 3 hours, and a freight train at the rate of
37 miles in the same time. If the freight train has 60
miles the start, in what time will it be overtaken ?
51. A hand car, running at the rate of only 1 mile
an hour, has 12 miles the start of a passenger train,
which runs at the rate of 12 miles an hour. In what
time will the hand car be overtaken?
Ans. 1 h. 5y\ m.
52. If the long hand of a clock move at the rate of
12 spaces an hour, and the short hand 1 sjmce an hour,
256 RATIO.
in what time will the long hand gain 12 spaces upon the
short hand?
53. At what time between 9 and 10 o'clock will the
hands of a clock be together?
Ans. 9 o'clock, 49 Jj- m.
S62. To state a j)roblem when it involves more than
three terms,
(1.) Of the quantities mentioned, maJce that the third
term which has the same name as the answer required.
(2.) Select tivo terms of the same name, and arrange
the couplet as though it were entirely disconnected with all
other conditions of the i^rohlem. (Vide 261, (2.)
(3.) Select two other terms of the same name, and ar-
range as before, and so on, till all the teyms are arranged.
Having canceled all the factors common to any ante-
cedent and consequent, (vide 255,) or common to either
one of the first terms and the third, (vide 256,) the co7i-
tinued product of the means divided hy that of the first
terms ivill he the ansiuer.
EXAMPLES.
1. If 10 cows eat 8 tons of hay in 6 weeks, how many
cows will eat 56 tons in 21 weeks?
It would take more cows to eat 56 tons than 8 tons,
and less cows to eat it in 21 weeks than 6 weeks.
STATEMENT.
(Vide 261, Ex. 44 and 45.) 8 : 66 .^ (vide 258, Ex. 21.)
21: 6"^"
SOLUTION.
Divide upper couplet by 8. 1-7..;^q (vide also 156, Ex. 22.)
Divide lower couplet by 3 and cancel the 7s. Ans. 20 cows.
RATIO. 257
2. If the wages of 6 men for 14 days be §84, what
will be the wages of 9 men for 16 days? Ans. ^144.
3. If 12 oz. of wool make 21 yd. of cloth which is
1^ yd. zvide, how many pounds will it take to make
150 yd. only 1 yd. wide? Ans. 30 lb.
4. If the interest of §19.35 for 1 yr. 2 mo. 6 c!a. is
§2.7477, what is the interest of §17.21 for 1 yr. 3 mo.
9 da., the rate per cent, being the same in each case ?
(Vide 205, Ex. 56; also 236, Ex. 3.) Ans. §2.63313.
5. If the interest of §140.10 for 2 yr. 4 mo. 12 da.
is §39.7884, what is the interest of §75.15 for 3 yr.
6 mo. 15 da., the rate per cent, being 12 in each case?
(Vide 205, Ex. 56 ; .236, Ex. 5.) Ans. §31.93|.
6. How much hay will 32 horses eat in 120 days, if
96 horses eat 3| tons in 7 J weeks? A^is. 2f tons.
7. If 6 laborers dig a ditch 34 yd. long in 10 days,
how many yards will 20 laborers dig in 15 days?
8. If 20 laborers dig a ditch 17-0 yd. long in 15 days,
how many days will 6 laborers require to dig a ditch
34 yd. long ? Ans.
9. If 14 men can reap 84 acres in 6 days, how many
men must be employed to reap 44 acres in 4 days?
10. If 20 men, by Avorking 10 hours a day, have been
employed 18 days in constructing 500 ft. of railroad,
how many days of 12 hours each must 76 men bo em^
ployed to construct 1140 feet of the same road.
STATEMENT.
(Vide 261, Ex. 46, 47, 48.) 76 : 20
(Vide 157, Ex. 27) 12
500
99
10:: 18
1140 A71S. 9 davs.
258 RATIO.
11. If 50 men, by working 5 hours a day, can «^'> 9A
cellars in 54 days, each cellar being 36 ft. lo-
wide, and 10 ft. deep, how many men can dig lb ctii.. s
in 27 days, each cellar being 48 ft. long, 28 ft. wide, and
9 ft. deep, by working 3 hours a day?
Ans. 200 men.
12. If 496 men, in 5^ days of 11 hours each, dig a
trench of 7 degrees of hardness, 465 ft. long, 3| ft. wide,
2| ft. deep, in how many days, of 9 hours long, will 24
men dig a trench, of 4 degrees of hardness, 337 i ft.
long, 5f ft. wide, and 3 J ft. deep?
Ans. 157, Ex. 23.
13. If 12 men can build a wall 30 ft. long, 6 ft. high,
and 3 ft. thick in 15 days, when the days are 12 hours
long, in what time will 60 men build a wall 300 ft. long,
8 ft. high, and 6 ft. thick, when they work only 8 hours
a day? ^?zs. 157, Ex. 24.
14. If 25 pears can be bought for 10 lemons, and 28
lemons for 18 pomegranates, and 1 pomegranate for 48
almonds, and 50 almonds for 70 chestnuts, and 108
chestnuts for 2 J cts., how many pears can I buy for
$1.35? Ans. 375 pears.
15. If the interest of J2187.50, from Aug. 10, 1868,
to Jan. 1, 1870, at 8 per cent, per annum, is |243.54^,
what is the interest of |10500, from Feb. 1, 1867, to
May 13, 1869, at the rate of 5 per cent, per annum?
STATEMENT.
218750:1050000
167:274 : : 243.54J
8:5
Ans. 236, Ex. 27.
KATIO. 250
2$IS. To divide a given number into parts "which
shall be proportional to given numbers,
(1.) If the given numbers are fractions, midfij^Ii/ them
all hy the least common multiple of the denominators.
(2.) 3Iake the sum of the results the first term of a
proportion, any one of the residts the second term, and
the given number the third term.
The fourth term, found by 258, will be one of the
required parts, and the others may be found in like
manner.
EXAMPLES.
1. Divide 49 into two parts which shall have the ratio
of I • ^
2
OPERATION.
iX6=3. (Vide 167J, Ex. 106.)
PROOF. fx6=4.
21+28=49. 7 : 3 : : 49 : 21 1st part.
21 : 28 : : J : I 7 : 4 : : 49 : 28 2d part.^
2. Divide 136 into two parts which shall be propor-
tional to the numbers | and f . (Vide 167J, Ex. 107.)
Ans. 64 and 72.
3. Divide 544 into two parts which shall be propor-
tional to the numbers /^ and ^^. (Vide 167J, Ex. 108.)
Ans. 384 and 160.
4. Divide 2135 into two parts which shall be pro-
portional to the numbers 35 § and 15^. (Vide 167 J,
Ex. 109.) Ans. 1498 and 637.
5. Divide 209 into three parts which shall be propor-
tional to the numbers J, |-, and |.
260
PvATIO.
OPEEATION.
^.X12=6. (Vide 167J, Ex. 110.)
J X 12-4.
fXl2-=9.
19: 6:: 209:66 1st part.
PROOF.
66+44+99=209.
66 : 44 :: J : J (vide 254.) 19 : 4 : : 209 : 44 2d part.
66 : 99 :: i : I 19 : 9 : : 209 : 99 3d part.
6. Divide 504 into four parts which shall be propor-
tional to the numbers J, f , j%, and f . (167J, Ex. 111.)
Ans. 135; 180; 81;lindl08.
7. Divide 30; 45; 75; 135; 180; and 750, each into
five parts which shall be proportional to the numbers 1,
2, 3, 4, and 5. Ans. 2; 4; 6; 8; 10, etc.
8. Divide 36 into three parts, so that J the first, | the
second, and i the third, shall all be equal to each other.
Remark, — The parts will evidently be as the numbers 2, 3, and 4.
Ans. 8; 12; and 16.
9. Divide 136 into two parts, so that -f of the first
and I of the second shall be equal.
Remabk. — The numbers will be as ii : i, or as 9 : 8.
A71S. 72 and 64.
10. A gold and a silver watch together cost §132, and
the gold watch costs 10 tijnes as much as the silver
watch. What did each cost? Ans. §120 and $12.
11. A's age is double B's, and B's is triple C's. The
sum of all their ages is 140. What is the age of each?
Remark. — Their ages are as the numbers, 1, 3, and G.
An8.A8i; B42; C 14.
12. A gentleman bought a certain number of oxen,
and double the number of cows; and also three tiiiics
RATIO. 261
as many sheep as cows. He gave ^50 each for oxen,
$25 each for coavs, and |3 each for sheep; the whole
costing §354. What number of each did he purchase ?
Ans. 3 oxen; 6 cows; 18 sheep.
13. A man paid |74 for a sheep, a cow, and an ox.
The cow was valued at 12 sheep, and the ox two cows.
What was the price of each?
Ans. sheep §2; cow §24; ox §48.
14. I wish to make a mixture of 360 pounds of tea,
using at the rate of 30 lb. worth 30 cts. a pound ; 11 lb.
worth 33 cts. a pound ; 23 lb. worth 67 cts. a pound ; and
26 lb. worth 86 cts. per pound. What quantity of each
kind must be used?
Ans. 120, 44, 92, and 104 lbs., respectively.
204. Partnership is an association of two or more
individuals for the transaction of business.
(1.) The partners constitute the company, firm, or
house.
(2.) The capital is the money invested by the company
in business.
(3.) The profit or loss to be shared is called a divi-
dend.
S05. To ascertain the dividend of the partners,
when the money of each has been invested the same
length of time.
Make the capital of the company the first term of a
projwrtion, the money of any partner the second term,
and the profit or loss the third term.
The fourth term, found by 258, will be the divi-
dend of the partner whose money forms the second
term.
262 RATIO.
EXAMPLES
1. A and B invest $550 in a speculation, of which
A furnishes $330 and B $220. They gain §70. What
is the dividend of each?
OPERATION.
550 : 330 : : $70 : A's share.
5: 3 ::$70:$42
Divide antecedents by 110.
550 : 220 : : $70 : B's share.
5: 2 ::$70:$28
(Vide 232, Ex. 1.) Ans. A $42; B $28.
2. A invests $300- in ti speculation, and B $400.
They gain $49. What is the share of each? (Vide
263, Ex. 1.) Ans. A $21 ; B $28.
3. A, B, and C form a partnership. A furnishes
$1200, B $1600, and C 2000. What is each partner's
share in a gain of $960?
Alls. A $240; B $320; C $400.
4. A, B, and C form a partnership; A furnishing
$800, B $1500, and C $3000. They gain $500. What
is the dividend of each?
Ans. A 75.471; B $141,509; C $283.02.
5. A and B form a partnership. A furnishes $1200,
and B $500. They gain $544. What is the share of
each? Ans. 263, Ex.- 3.
6. A, B, C, and D make up a purse to buy lottery
tickets. A furnishes $15, B $20, C $9, and D $12.
What is each one's share in a prize of $504?
Ans. 263, Ex. 6.
7. A, B, C, D, E, F, and G engage in an oil specula-
RATiO. . 263
tion. A furnislies §175 of the capital, B S500, C $600,
D ,^210, E ^42.50, F §20, and G §10. They expend
§623 in prospecting, and then give the matter up as a
total failure. What does each lose? ^ns. 232, Ex. 2.
8. Three gentlemen engage in a gold speculation. A
furnishes §500, B §1200, and C §4300. They clear
§1500. What is each man's share? A71S. 232, Ex. 3.
!S66. To ascertain the dividend of the partners, when
the money of each has been invested different lengths
of time,
(1.) Multiply each jjartner^s money hy the time it is
invested.
(2.) Make the swn of the ijroducts the first term of a
'proportion^ any one j^rodiict the second term, and the gain
or loss the third term.
The fourth term, found by 258, will be the share of
the partner whose product forms the second term.
EXAMPLES.
1. A and B are associated in trade. A has furnished
of the joi7it stock §330 for 5 months, and B §220 for 8
months. They gain §170.50. What is the share of
each?
OPERATION.
330X5=1650.
220X8=1760.
Divide the coup- 3410 : 1650 : : §170.50
let by 10, and the 2 : 165 : : §1
two antecedents by 3410 : 1760 : : §170.50
170.1 2: 176:: SI
A's share.
§82.50
B's share.
264 TvATIO.
2. A and B enter into partnership. At first A, with
a capital of §3G0, does business alone. At the expira-
tion of two months B comes in with a capital of §520,
and the partners do business together 5 months. The
profits of the concern, from the time A commenced
business, were $128. What should be the share of each?
Ans. A $63; B $65.
3. A, B, and C form a partnership. " A's part of the
capital is $4300, B's $2000, and C's $1500. At the end
of 2 months A withdrew with his stock. At the end
of another 2 months B withdrew with his stock. C con-
tinued the business alone for another 2 months, when
the entire profits were found to have been $1280. What
is the share of each?
A71S. A $430; B $400; C $450.
4. On the 1st of January, 1866, A commenced busi-
ness wdth a capital of $8000. On the 1st of July B
joins him with a capital of $16000. On the 1st of
July, 1867, it is found that $4000 have been cleared since
A began business. What is the share of each?
Ans. A $171^; B $2285|.
5. A, B, and C, at the end of a partnership, have
jointly $1000 in trade. A's stock has been in the busi-
ness 7 months, B's stock 8 months, and C's 12 months.
A's dividend is $21 ; B's $40; C's 24. What amount of
money did each invest?
A71S. A $300; B $500; C $200.
6. Two cousins, George and Frank Ransom, com-
menced business on the 1st of January, each partner
putting in $10000. On the 1st of June George in-
creased his stock by $2000, Frank withdravring the
RATIO. 265
same sura. On the 1st of September George withdrew
§4000, but Frank increased his by §3000. At the end
of the year they had made §6720. What was the divi-
dend of each? Ans, George §3360; Frank §3360.
EQUATION OF PAYMENTS.
267. To ascertain the mean time for the payment of
several sums due at different times,
(1.) Multipli/ each ijayment by its lime of credit.
(2.) Divide the sum of the products by the sum of the
payments. The quotient will be the mean time.
EXAMPLES.
1. I owe a merchant §30, due in 4 months; §40, due
in 5 months, and §50, due in 6 months. What is the
mean time for the payment of all the bills ?
OPERATION'.
30X4=120
40X5=200
50X6=300
120 ) 620(5J
Ans. 5^ months.
Remark. — The interests of the several sums, at any rate per cent,
per annum, for the given times, when added together, must be the
same as the interest of the sum of the payments for the mean time.
Hence the following rule:
(1.) Find the interest of each payment for its time of credit.
(2.) Divide the sum of the interests hy the interest of the sum of the
payments for 1 month. The quotient will be the mean time.
Remark. — If the rate of 12 per cent, per annum is selected, the
operation will be identical with that above.
23
266 RATIO.
2. A merchant owes §200, payable in 4 months ; $400,
in 5 months; |500, in 6 months; and §600, in 8 months.
What is the mean time of payment?
Ans. 6j\ months.
3. A merchant has given three notes to the same
creditor : §200, due in 2 months ; §200, in 4 months ;
and §200, in 6 months. What is the mean time of
payment ? Ans. 4 months.
4. I have several bills due at a store; §40 due in 20
days from January 1, 1866; §30 due in 40 days from
the same time; amd §50 due in 45 days. What is the
mean time of payment ?
A71S. 35/^ days; i. e. Feb. 4, 1866.
5. I buy a house and lot for §1600, with the under-
standing that I am to pay one fourth cloiV7i, one third of
the balance in 3 months, and the remainder in 6 months.
What would be the mean time for the payment of the
whole sum ? Ans. 3| months.
ALLIGATION MEDIAL.
268. To ascertain the mean price of a compound
consisting of ingredients of which the quantity and
value of each are given,
Divide the entire cost of the compound hy the sum of
the ingredients. The quotient will be the mean price.
EXAMPLES.
1. A merchant bought 160 gallons of wine at 40 cts.
per gallon ; 75 gallons at 60 cts. per gallon ; 225 gal-
lons at 48 cts. per gallon ; 40 gallons at 87 J cts. per
gallon, What wns the mean cost of the wine?
RATIO. 267
OPERATION.
160X.40 = 64.00
75X.60 = 45.00
225X.48 =-108.00
40X.87J= 35.00
500 ) 252.00(.505
252.00
Ans. 10.504 per gallon.
2. A farmer mixed wheat, viz : 5 bushels worth |1.10
per bushel; 10 bushels worth 60 cts. per bushel; 5
bushels worth 70 cts. per bushel. What is the mean
price of the mixture ? Ans. 75 cts. per bushel.
3. A wine merchant mixes wine, viz: 88 gallons of
Canary, worth 50 cts.' per gallon ; 88 gallons of Sherry,
worth 76 cts. per gallon ; and 48 gallons of Claret, worth
^1.75 per gallon. What is a gallon of the mixture
w^orth ? A71S. 87 cents.
4. A goldsmith mixes 7 ounces of gold 23 carats fine
with 3 ounces 16 carats fine, 3 oz. of 18, and 3 of 19
carats fine. What is the fineness of the mixture ?
Ans. 20 carats.
5. If I mix 30 lb. of tea at 30 cts. per lb.; 11 lb. at
33 cts. per lb. ; 23 lb. at 67 cts. per lb. ; and 26 lb. at
86 cts. per lb., what is one pound of the mixture worth?
Ans. 56 cents.
6. If I mix 11 lb. of tea at 30 cts.; 30 lb. at 33 cts.;
26 lb. at 67 cts. ; and 23 lb. at 86 cts., what is one pound
of the mixture worth? Ans. 56 cts.
7. If I mix 30 lb. of tea at 30 cts.; 41 lb. at 33 cts.;
23 lb. at 67 cts. ; and 49 lb. at 86 cts., what is one pound
of the mixture worth? Ans. 56 cts.
268
RATIO.
8. If I mix 41 lb. of tea at 30 cts. ; 30 lb. at 33 cts.;
26 lb. at 67 cts. ; and 49 lb. at 86 cts., what is one pound
of the mixture worth ? Aiis. 56 cts.
9. If I mix 41 lb. of tea at 30 cts.; 11 lb. at 33 cts. ;
49 lb. at 67 cts. ; and 26 lb. at 86 cts., Avhat is one pound
of the mixture worth ? Ans. 56 cts. •
10. If I mix 11 lb. of tea at 30 cts.; 41 lb. at 33 cts.;
49 lb. at 67 cts.; and 23 lb. at 86 cts., what is the mean
price? Ans. 56 cts.
11. During 14 hours on a certain day the mercury of
a thermometer stood as follows : 2 hours at 60 degrees ;
3 hours at 62 degrees ; 4 hours at 64 degrees ; 3 hours
at 67 degrees ; 1 hour at 72 degrees ; and 1 hour at 75
degrees. What was the mean temperature during the
time? Ans. 65 degrees.
ALLIGATION ALTERNATE.
260. To find the quantity that may be used of each
ingredient of a proposed compound, the price of each
being given, and the mean price of the compound.
EXAMPLES.
1. A farmer has wheat worth $1.10 per bu. ; wheat
worth 60 cts. per bu. ; and wheat worth 70 cts. per bu.
He desires to mix it so that a bushel may be worth 75
cents. What quantity of each may be used ?
75
110
60
70-
OPERATION.
(1) (2) (3) (4) (5)
15
5
3
1
4
35,
7
7
35
7
7
Am. 4 bu. at $1.10; 7 bu. at 60 cts.; and 7 bu. at 70 cts.
KATIO. 269
Explanation. — 110, which is greater than the mean rate, is
connected by a dotted line with 60, which is less than the mean
rate; also, 110 is connected with 70, forming another couplet,
one term of which is greater, and the other less than the mean
rate, 75.
In column (1), 15 is the difference between 60 and 75, and 35 is
the difference between 110 and 75 — each difference being opposite
the other term of the couplet.
In column (2), 5 is the difference between 75 and 70, and 35 is
the difference between 110 and 75 — each difference opposite the
other term of the couplet.
Column (3) is formed by dividing the terms of (1) by their
greatest common divisor. \ .
Column (4) is formed by dividing the terms of (2) b;y their
greatest common divisor.
Column (5) is formed by adding the corresponding terms of (3)
and (4).
Any number of answers may now be found.
OPERATION CONTINUED.
(6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
7
10
13
5
6
7
9
1
6
1
14
21
28
7
7
7
14
2
10
1
7
7
7
14
21
28
21
1
■ 5
4
Columns (6), (7), and (8) are found by multiplying the terms
of (3) by 2, 3, and 4, respectively, and adding the corresponding
terms of (4).
Columns (9), (10), and (11) are found by multiplying the terms
of (4) by 2, 3, and 4, respectively, and adding the corresponding
terms of (3).
Column (12) is found by multiplying the terms of (3) by 2, and
those of (4) by 3, and then adding the corresponding products.
Column (13) is that, of (6) divided by 7.
Column (14) is (13) multiplied by 5. (Vide 268, Ex. 2.)
Column (15) is (11) divided by 7. All the answers may be ver-
ified by 268, and as many others as one is curious to find.
2. I wish to mix tea at 30 cts. ; tea at 33 cts. ; tea at
67 cts. ; and tea at 86 cts. per pound, so as to make the
270
llATIO.
What quantity of
mixture worth 56 cts. per pound,
each can be used ?
OPERATIONS.
56
Multiply or divide the terms in columns (1) or (2)
by any numbers whatever (merely preserving their
ratio) for other answers. Thus,
(
;i.)
(2.)
(1) (2) (3) (1) (2) (3)
30 30
30
30 ,
11
11
33 1
11
11
56
33~4n
30
30
67 ••
23
23
67 -^
26
26
86 — 26
26
86 — '
23
23
(Vide
268,
Ex.
5.)
(Vide
268,
Ex. (
>.)
15
15
45
22
11
^
11
22
44
30
60
30
23
46
92
52
26
13
13
13
39
23
46
23
,30
56
130-
!33=1 I
|67— li
186 -^'26
(3.)
(1) (2) (3) (4)
30
23
56
30^
33
67-
(4.)
(1)(2)(3
30
30
(Vide 268, Ex. 7.)
(Vide 268, Ex. 8.
Change the numbers in columns (1), (2), and (3) at
pleasure, merely preserving their ratio, and add the cor-
responding terms, for other answers.
(5.)
(1) (2) (3) (4)
zw
30111
261
26
56
30.
33
67
86
(6.)
(1) (2) (3) (4) (5)
11
26
30
30
23126
(Vide 268, Ex. U.;
RATIO. 271
Having arranged tlie several prices and the mean price
as in the examples,
(1.) Connect in couplets each price that is less than the
mean price with one that is greater, hy lines which may
he readily distinguished from each other.
(2.) Select any couplet, and write the difference between
each of its terms and the mean price opposite the other
term, in a column to the right.
(3.) Treat each couplet in the same manner, forming
as many columns as there are couplets.
(4.) Add together the numbers standing opposite each
'price, and place the results in an additional column.
These sums will be the quantities that may be used
of the price opposite which they stand.
Remark 1. — To find other answers, divide or multiply the terms
of any column by any number whatever, and then add as in (4),
observing that it is only necessary that the two numbers in any
column preserve their ratio.
Remark 2. — The corresponding terms of any two or more col-
umns, containing answers, may be added together for other^
answers.
Remark 3. — If the prices are so connected that each price is
linked with only one other price, only one column to the right is
necessary. (See above, operations (1) and (2), where only one
column was necessary.
Remark 4. — When the mean price is the same as one of the
given prices, the latter need not be included in the operation.
3. A merchant would mix wines, worth 16 shillings,
18 shillings, and 22 shillings per gallon, so that the
mixture may be worth 20 shillings per gallon. What
quantity of each may be used ?
272
RATIO.
OPERATION.
20
The steps are precisely liko
those in Ex. 1.
(1) (2) (3) (4) (5)
16-
18-
22-
Ans. 1 gal. of IG; 1 of 18; and 3 of 22 shillings.
The following are among other answers easily obtained :
(6) (7) (8) (9) (10) (11) (13) (14) (15) (16)
2
1
1
2
1
1
4
2
2
1
3
2
3
4
1
1
1
3
5
*
9
1
1
1
2
3
4
2
4
9
10
5
7
9
4
5
6
8
14
10
28
(15) is half of (3) added to 9 times (4).
(16) is the sum of (11), (13), and (14). (See above,
Rem. 2.)
4. A merchant has wine at 40 cts. ; 75 cts. ; 60 cts. ;
and 48 cts., and he wishes to make a mixture of all
worth 50 cts. How many gallons can he use of each?
Remark 5. — Since the answers are limitless in number and va-
riety, it would be difficult to obtain those which might be given.
Let the pupil verify those which he may find by 268.
5. What quantity could be used of each to make the
mixture worth 70 cts. per gallon ? 45 cts. ? 48 ? 60 ?
(See Rem. 3.)
6. A merchant has three kinds of sugar, worth 6, 8,
and 15 cts. per pound. He wishes to make a mixture
worth 11 cts. a pound. What quantity of each may be
used ?
7. What quantity of each kind may he use to make
the mixture worth 12 cts.? 13 cts.? 14 cts.? 10 cts.?'
Sets.?
Remark 6. — If the quantity to be used of a given price is fixed,
multiply the terms of any answer, obtained as above, by the quo-
tient of the fixed quantity, divided by the number opposite the
RATIO. 273
price mentioned. Thus, if, in Ex. 1, it is required to use 21 bushels
at $1.10, multiply the terms of column (5) by 2_i__5i^ qj. column
(6) by V=3, or(7)by2TV, etc.
Remark 7, — If the entire quantity to be used is fixed, apply the
principle of 263 to the fixed quantity, using any answer as the pro-
portional numbers. (Vide Ex. 2, Operation (1); and 263, Ex. 14.)
8. A grocer has currants at 4 cts. ; 6 cts. ; 9 cts. ; and
11 cts. ; and he wishes to make a mixture of 240 lb.,
worth 8 cts. per pound. What quantity of each may
be used ?
Am, 72; 24; 48; 96, or 48; 48; 72; 72, etc.
POSITION.
270. Position is the operation of finding the true
answer to a problem through the aid of one or two as-
sumed answers.
(1.) Single Position assumes one ansiver.
(2.) Double Position assumes two answers.
271. Single position is applicable to those problems
in which, if any number is assumed as the answer, and
the steps, indicated by the problem, performed upon it,
the ratio of the result, and the result in the question., is
the same as that between the assumed ansiver and the true
answer. Hence, the
RULE.
(1.) Assume any convenient number to he the true answer,
(2.) Perform the operations indicated hy the pi^oblem
upon the assumed number.
(3.) Make the result the first term of a proportion, the
RESULT in the problem the secoimTterm, and the assumed
answer the third term.
The fourth term, found by 258, will be the true answer.
274 RATIO.
EXAMPLES.
1. The sum of | and -J- of a certain number is 87 J.
What is the number ?
OPERATION.
Suppose 24 to be the answer.
1 of 24 is 8
1 of 24 is 6
Then 14 : 87J : : 24
Multiply couplet by 2. (255.) 28 : 175 : : 24
Divide couplet by 7. (255.) 4 : 25 : : 24
Divide antecedents by 4.' (256.) 1 : 25 : : 6 : 150 Ans.
Proof.— I'of 150+i of 150=87^.
2. What number is that which being increased by A,
I, aiid i of itself, the result will be 75 ? Ans. 36.
Remark. — In order to avoid fractions, assume tlie least common
multiple of the denominators of the fractions, or any multiple
of it.
3. What number is that, the sum of | and J of which
is 376 ? Assume 45 or 90. A71S. 360.
4. What number is that which being increased by |,
I, and f of itself, the result will be 48i? Ans. 15.
5. The difference between the fifth and tenth of a
certain number is 17. What is the number?
Ans. 1^0.
6. The difference between | and | of a certain num-
ber is 15. What is the number? Ans. 180.
7. The rent of a farm is 20 per cent, greater this
year than last year. This year it is $1800. What was
the rent last year? Ans. $1500.
RATIO. 275
^ OPERATION.
Suppose $100 to be the rent last year.
20 per cent. (J) of §100 is 20
Then $120 : $1800 : : $100 : $1500
8. If I of a number be multiplied by 7, and | of the
number itself be added to the product, the result will be
219. What is the number? Ans. 45.
9. A gold watch is worth ten times as much as a
silver watch, and both together are worth $132. What
is each watch worth? Ans, 263, Ex. 10.
10. The rent of a farm is 10 J^ less this year than
last year. This year it is $1000. What was it last
year? Aiis. $1111 1.
11. The difference between | and | of a number is
38f. What is the number? Ans. 100.
12. The sum of | and § of a number is 920. What
is the number? Ans. 720.
13. One fifth of all the sheep I have die in 1866, and
three fourths of the remainder in 1867, when I have
only 20 sheep left. What number had I at first.
Ans. 100.
14. A certain gentleman, at the time of his marriage,
agreed to give his wife | of his estate, if at the time of
his death he left only a daughter, and if he left only a
son she should have I of his property ; but5 as it hap-
pened, he left a son and daughter, in consequence of
which the widow received in equity $2400 less than she
would have received if there had been only a daughter.
What would have been the wife's dowry if he had left
only a son? A71S. $2100.
276 KATIO.
S72. Double Position is applicable to those prob-
lems in which the difference between the true and first as-
sumed number is to the difference between the true and
second assumed number as the first error is to the second.
(1.) If the assumed numbers are both too large, or both
too small, the errors arising from them are said to be
ALIKE.
(2.) If one of the assumed numbers happens to be too
large, and the other too small, the errors are said to be
UNLIKE.
RULE,
(1.) Assume any convenient number, and perform the
ojjerations upon it indicated by the problem.
(2.) Take the differ eyice between the result and that
pointed out by the problem. This difference is the first
ERROR.
(3.) Assume a second number, and in like manner find
the SECOND ERROR.
(4.) If the errors, as shown by the results, are aljke,
multiply the first error by the second assumed num-
ber, and the second error i»^ the first assumed' nui^-
BER, and divide the difference of^ the products by the
difference of the errors. - The quotient is the^true an-
swer. But,
(5.) If the errors, as shown by the results, are unlike,
multijyly as before, and divide the sum of the products by
the SUM of the errors. The quotient will be the true
answer.
EXAMPLES.
1. A merchant expended ^1500 of his capital for the
support of his family a year. At the end of the year,
RATIO.
277
however, lie has added to the capital not expended a sum
equal to 3 times that part, thereby tripling his original
capital. With what sum did he begin the year?
OPEKATION.
Pirst assumed No. $2000 $3000
1500 1500
(2.)
Second assumed No.
$500X3=
$500
$1500
$1500
$4500=
=$1500X3
Result too small,
Capital X3=
$2000
$6000
$6000
$9000
Result too small.
Capital X3.
$6000-$2000=First
error. $4000
3000
$3000=Second error.
2000
12000000
6000000
60000
00
1000)6000000
$6000 A
ns.
See Rule (4.)
(1.)
First assumed No.
SECOND OPERATION.
$2400 $9000
1500 1500
(2.)
Second assumed No.
$900
$2700
$7500
$22500
Result too small,
Capital X3=
$3600
$7200
$30000
$27000
Result too large.
Capital X3.
$7200— $3600=First
error. $3600
9000
$3000 Second error.
2400
32400000
7200000
720000
0
6600)39600000
See Rule (5.)
$6000 Ans.
278 RATIO.
2. Divide tlic number 13 into two parts, so tliat 9
times one of the parts may be equal to 17 times the
other.
OPERATION.
(1-)
(2.)
Assume 10
9
and
3
17
Assume 7
9
and
6
17
90
51
51
63
102
63
.39
7
39
6
errors un
dike. 39
3
39
10
273
390
234
117
234
390
78)663 78)351
8^ one part. 4i other part.
3. Divide 100 into two parts, so that if \ of one of the
parts be subtracted from \ of the other, the remainder
may be 11. Ans. 24 and 76.
4. A and B had §85 between them, and f of A's
added to f of B's money make §60. How much had
each? Ans. A$50;B §35.
5. One half a certain number is the same as one third
another number. But if 5 is added to the first, and 10
to the second, then one fifth of the first is the same as
one eighth of the second. What are the numbers?
Ans. 20 and 30.
6. Triple a certain number is the same as double an-
other number. But if 10 is added to the first, and 5
subtracted from the second, then 5 times the sum will
be equal to 6 times the difi'erence. What are the num-
bers ? Ans. 20 and 30.
■^'^l'^s>\^^j>-4..^.
281
7. In a mixture of wine and Avater, J the whole +25^
gallons was wine; J the whole — 5 gallons was water.
What was the quantity of each in the mixture?
Ans. Wine 85; water 35 gallons.
8. A gentleman supported himself 3* years for 50 £ a
year, and at the end of each year added to that part of
his capital not thus expended a sum equal to | of this
part. At the end of 3 years his original capital was
doubled. With what sum did he begin business?
Ans. 74:0 £.
""Eemark. — 200 £ and 290 £ are convenient suppositions.
9. A boy had a number of marbles. He laid aside 2,
and then Avon in play as many as he had left. He then
laid aside 3, and Avon in play as many as were left. He
noAv lays aside 4, and, on winning as many as he had
left, finds he has 13 in all. How many did he begin
with? Ans. 5.
10. A gentleman has tAVO horses, and a saddle worth
$50. If the saddle be placed on the first horse it makes
his value double that of the second ; but if the saddle
be placed on the second horse, it makes his \'alue triple
that of the first. . What was the value of each?
Ans. First $30 ; second $40.
Remark. — If the first horse is assumed to be worth §70, then by
the frst condition of the problem the second horse must be worth
($504-§70)-r-2, or $60, and the second condition of the problem is
not filled by $100.
If the first horse is assumed to be worth $90, then the second
must be worth ($90-f$50)^2, or $70, and the second error will
be $150.
11. Divide 55 into tAvo parts, so that the less part di-
vided by the difference of the parts shall be 2.
Ans. 33 and 22.
278 ,^
RATIO.
12. A gentleman was asked the time of day, and re-
plied tliat I of the time past noon was equal to /g of
the time to midnight. What was the time?
An8. 12 minutes past 3 o'clock.
Remark.— In tlie first place f : -/j : : i : ^3 : : 11 : 4. (Vide 252.)
So that the reply was the same as if he had said that 11 times
the number of minutes past noon were equal to 4 times the number
of minutes to midnight.
13. In a mixture of corn and wheat, \ the whole -|-5
bushels was corn, and \ the whole +10 .bushels was
wheat. What was the quantity of each?
14. The sum of the first and second of three numbers
is 13; of the first and third 19; of the second and third
24. What are the numbers? An%. 4; 9; and 15.
15. A and B have the same income. A saves^ J of
his annually; but B, by spending |50 per annum more
than A, at the end of 6 years finds himself §150 in
debt. What is the income of each?
Am. $125 per year.
16. A commenced trade, and at the end of the third
year found his original stock tripled. Had his gains
been $1000 per year more than they actually were, he
would have doubled his stock each year. What was his
original stock ? An%. |1400.
17. Three persons. A, B, and C, were seen traveling
in the same direction. At first A and B were together,
and C 12 miles in advance of them. A goes 7, B 10,
and C 5 miles per hour. In what time will B be half-
way between A and C? How long before C will be mid-
way between A and B? How long since A was midway
between B and C?
Ann. Respectively, 1 h. 30 m., 3 h. 25 f m., and 12 h.
INVOLUTION. 281
INVOLUTION.
2^72. (1,) The first poiver of a number is the num-
be'r itself.
(2.) The second power or square of a number is the
number multiplied by itself. Thus, the square of 1 is
1X1=1; the square of 2 is 2x2^4; the square of 3
is 3X3=9; the square of 25 is 25x25=625.
(3.) The tliird power or cube of a number is the pro-
duct of a number taken as a factor three times. Thus,
the cube of 1 is 1X1X1=1; the cube of 2 is 2x2x2
=8; the cube of 3 is 3x3x3=27; the cube of 4 is
4X4X4=64.
(4.) Any given power of a number is indicated by a
small figure placed at the right and a little above the
number. Thus, 2* denotes that the fourth power of 2 is
to be taken, and that 2 is to be taken as a factor 4 times.
The small figure is called the exponent of the number
or index of the poiver.
The expression 2^=16, is read fourth poiver of 2
equals 16.
5573. Involution is the operation of finding any
givcn_pow^j' Uf i" "^^1^^^ er.
GENERAL EULE.
3Iultiply the number by itself till it is used as a factor
as many times as there are units in the exponent.
Remark. — Fractions should always be in their lowest terms be-
fore multiplying.
24
282 INVOLUTION.
EXAMPLES.
1. Find the values of 7^ 8^; 25^; and 125^
(1.)
1st power, 7
7
OPERATIONS
(2.)
8
8
64
8
512
(3.)
25
25
625
25
(4.)
125
125
Square, 49
7
15625
125
Cube, 343
15625
1953125
2. Find the values of 11^; 12^; 13^; lOP; 111 2;
and 1111^
3. Find the values of 14^; 15'; 16^; 17"^; 21*^; 24*;
and 27^
4. Find the values of 15^; 25^; 35^; 45^; 55'^; 65^;
75^; 852; 952^
Remark. — The value of these expressions may be found men-
tally. Thus,
152=10X20+25; 252=20X30+25; 352=30X40+25, etc.
5. Find the values of 15^; 25^; 35^; 45^; 55^; 65^;
753 ; 85^; 95^
6. Find the values of 2*; 3*; 4*; 5*; 6^; 7^ 8^
Ans. In Table, page 283.
7. Find the squares of f ; | ; £f^ ; and y'^g^^-.
^'^'5. J; i; -,^g; and -J^,
8. Find the squares of |; |J; ||; and ^|.
Ans. 4; /g; ^1; and ^4^.
9. Find the squares of 2^; 6J ; 31; 16}/, and 51.
A71S. 157, Ex. 2, etc.
10. Find the cube of 5^ and the 4th power of S^.
Ans. 157, Ex. 28 and 29.
INV'OLUTIOX.
283
11. Find the 4tli power of .025.
Ans. .000000390625.
12. Find the 5th power of .029.
Ans. .000000020511149.
13. Find the first nine powers of all the numbers
below 10.
TABLE.
02
o
o
n
I?
s
rji
a
H
2
1
2
3
4
5
6
7
8
9
1
4
9
16
25
36
49
64
81
1
8
27
G4
125
216
343
512
729
1
16
81
256
625
1296
2401
4096
6561
1
32
243
1024
3125
7776
16807
32768
59049
1
64
729
4096
15625
46656
117649
262144
531441
1
128
2187
16384
78125
279936
823543
2097152
4782969
1
256
6561
6553G
390625
1679616
5764801
16777216
43046721
1
612
19683
262144
1953125
10077696
40353607
134217728
387420489
274. Several useful facts may be gathered from the
table.
(1.) Any power of 1 is 1.
(2.) If the square of a number ends with 1, the num-
ber itself ends with 1 or 9.
(3.) If the square of a number ends w^ith 4, the num-
ber itself ends with 2 or 8.
(4.) If the square of a number ends with 6, the num-
ber itself ends with 4 or 6.
(5.) If the square of a number ends with 9, the num-
ber itself ends with 3 or 7.
(6.) If the cube of a number ends with either
1; 2; 3; 4; 5.; 6; 7; 8; 9, the number itself
ends with 1 ; 8; 7; 4; 5; 6; 3; 2; 9.
284 INVOLUTION.
(7.) The 5th power, 9th power, 13th power, etc., of a,
number ends Avith the same figure as the number itself.
(8.) It is evident that if any power of a number ends
with a 0, the number itself ends with a 0.
575. Any powder of a number, multiplied by any
other power of the same number, produces a power
whose index is the ^ sum of the exponents of the given
number. Thus 3"^X3*^3^ and 7^X7'=7^ which may
be verified by the table. Hence,
576. To find a given power of a number, when all
the low^er powers are not wanted,
3Iultiply any tivo or more powers iogetlier^ the sum of
whose indices equals the index of the required p>oicer.
EXAMPLES.
1. Find the 9th power of 3.
Ans. 3-^X3*-=243x81-=19683 Ans.
2. Find the 15th power of 2.
Ans. 2^X2'--256X 128-32768 Ans.
3. Find the 13th power of 3. Ans. 3^ X 3*^=1594323.
4. Find the 20th power of 2.
Ans. 2^X29X22=1048576-=512X512X4.
2T7. Any number squared produces a number con-
sisting of exactly double the figures in the number itself,
or of one less Hum double the figures. For, 1 is the
smallest number consisting of one figure, and 9 is the
largest; 10 is tlie smallest number consisting of two
figures, and 99 is the largest; 100 is the smallest of
tliree figures, and 999 the largest. Now we have
V^l ; 10---100 ; 100- -10000 ;
and 9---81 ; 99^=9801 ; 999^-998001 ; etc.
\
EVOLUTION. 285
ST8. Any number cubed produces a number consist-
ing of exactly triple the figures in the number itself, or of
one or two less than triple the figures. For,
P-= 1 ; 10^=1000 ; lOO^^-^lOOOOOO ;
d'=^n9 ; 99^--970299 ; 999'^=997002999.
EVOLUTION
S79. Any given root of a number is the number
whicli, on being taken as a factor as many times as is
indicated by the name of the root, will produce the
number itself. Thus :
(1.) The^Vs^ root of a number is the number itself.
(2.) Th'e second or square root of a number is one of
its two equal factors. Q^hus : the square root of 1 is 1,
because 1X1=1; the square root of 4 is 2, because
2X2=4; the square root of 9 is 3, because 3x3=9.
(3.) The cube or third root of a number is one of its
three equal factors. Thus : the cube root of 1 is 1, be-
cause 1X1X1=1; the cube I'oot of 8 is 2, because
2X2X2=8; the cube root of 27 is 3, because 3X3X
3=27; and the cube root of 64 is 4, because 4X4X
4=64.
(4.) The 4th; 5th; 6th, etc., root of a number is one
of its 4; 5; 6, etc., equal factors. Thus: the 4th root
of 1 is 1, because 1x1X1X1=1 ; the 4th root of 625
is 5, because 5x5X5X5=625; the 5th root of 7776 is
6, because 6x6x6x6x6=7776.
(5.) A perfect square is a number which has an exact
286 EVOLUTION.
SQUARE BOOT. Tlius, 16 ; 64; 81; and 144 are perfect
squares.
(6.) A perfect cube is a number wliicli has an exact
cube root. Thus, 8 ; 27 ; 64, etc., are perfect cubes.
(7.) Any given root of a number is indicated by a
fractional exponent, or by the aid of the character ]/,
called the radical sign. Thus :
The square root of 16 is indicated by 16^, or by
l/l6; the cube root of 8 is indicated by 8% or by V'8;
the 7th root of 2187 is indicated by 2187^ or by f 2187.
The figure at the left of the radical sign is called the
index of the root.
(8.) A surd is a number which requires a radical sign
or exponent to exactly express it. Thus, l/2; 1^4; 5^
are surds.
280. Evolution is the operation of finding any
given root of a number.
SQUARE ROOT.
281. The extraction of the square root of a number is
the operation of finding a number which multiplied by
itself will produce the given number.
282. The square root of any integral number which
is a perfect square and less than 400, may be found from
memory, or by inspecting Table I of the Appendix.
c
EXAMPLES.
1. What is the square root of 1? 4? 9? 16? 25?
36? (Vide 273, Ex. 13.)
EVOLUTION.
287
2. What is the square root of 49? 64? 81? 100?
121?
3. What is the square root of 196? 324? 361? 169?
4. What is the square root of 225? 289? 256? 400?
IIemark, — All the integral numbers less than 400, other than
those in the last four examples, have only approximate square roots;
that is, their square roots are surds. The same is true of all
numbers not perfect squares.
S83. The general method of extracting the square
root of a number may be readily understood from the
following operations. We will first square some num-
ber, as 37.
OPERATION FIRST.
(1.) (2.)
37 30+7
37 30+7
49
210
210
900
1369
30x7+7^
302+30X7
30^+2x30x7+7^
In (2) it is evident that 30x7 added to 30X7 is twice
30X7, or 2X30X7. To reverse these steps, that is, to
extract the square root of 1369, we proceed as follows:
OPERATION SECOND.
1369 1 37 30
9 ^
469 2x30+7
469
30^+2x30x7+7^ 30+7
302
2x30X7+7'
2x30x7+7'
67
Explanation, — By putting a point or dot over every alternate
figure of the given number, commencing with the right-hand
288 EVOLUTION.
figure, we not only can determine the number of figures of wliicli
the root will consist, (vide 277,) but we can also determine the
left-hand figure of the root; for the square root of the largest per-
fect square, less than 13, must be that figure. Now 9 is the largest
perfect square less than 13, and its square root is 3. This 3 is
written at the right and also at the left of the given number.
Having taken 9 from 13, the remainder, 469, is the same as 2X
30X7-j-7^. Now if we double the 3 just obtained, it in eifect pro-
duces 60, which is 2X30. If we now divide 469 by 60, and annex
the quotient to 6, and also to. the right of the root, as Operation (2)
points out, and then multiply 67 by 7, the exact remainder, 469,
will be produced whenever the given number is a perfect square.
Hence,
284. To extract the square root of any integral
number,
(1.) Divide the number into periods of ttvo figures each,
counting from right to left.
(2.) Fbid the greatest perfect square in tJie left-hand
period, and 'place its square root at the left and also at
the right of the given number. Subtract this greatest
perfect square from the left-hand p)eriod, and to the re-
mainder annex the next period to the right; the result is
the First Dividend.
(3.) Double the figure of the root just found and place
it to the left of the first dividend. See how many times
tills double figure, with, a cipher mentally annexed, is con-
tained in the first dividend. Place the quotient figure as
the second figure of the root, and also annex it to double
tJt^ first figure as the First Divisor.
(4.) Multiply the first divisor by the second figure of
the root, and subtract the ijroduct from the first dividend,
and to the remainder annex the next period to the right
for the Second Dividend.
EVOLUTION.
289
(5.) Double iJie part of the root now found and place
it to the left of the second dividend ; see how many ti7nes
this double root, luith a cipher mentally annexed, is con-
tained in the second dividend, and place the quotient at
the third figure of the root, and also to the right of double
the previous pa7i of the root, as the Second Divisor.
Proceed with the second divisor as with the first, and
continue the operation till all the periods have been used.
Remark 1. — Double the root already found is sometimes called
the trial divisor.
Remark 2. — It sometimes happens, as in Operation (2) below,
that in making the division for the next figure of the root the
quotient is too large. In such a case make the quotient figure so
small that the product will be less than the dividend.
Remark 3. — It sometimes happens, as in Ex. 15, Operations (1)
and (2) below, that a dividend is too small to contain the trial
divisor. In such a case place a cipher in the root and to the right
of the trial divisor, and then annex the 7iezt period to the dividend.
Remark 4. — In the case of decimals, the first two figures on the
right of the decimal point constitute one period, the next two
figures another, and so on.
EXAMPLES.
1. Find the square roots of 6084; 1521; and 98.01.
148
6084178 Ans.
49
1184
1184
OPERATIONS.
(2.)
152i|39 Ans.
9
69
621
621
18.9
(3.)
98.0i|9.9 Ans.
81
17.01
17.01
2. Find the square roots of 2025; 3969; 6241; and
6561. Ans. 45; 63; 79; and 81.
25
290 EVOLUTION.
3. Find the square roots of 1936; 2401; 8281; and
8836. Ans. 44; 49; 91; and 94.
4. Find the square roots of 10.24; 10.89; 11.56;
12.25. Ans. 3.2 ; 3.3 ; 3.4 ; 3.5.
5. Find the square roots of 6400 ; 2500 ; 841 ; and
729. Ans. 80 ; 50; 29; and 27.
6. What is the value of i/8281+t/6561+i/50.41.
Ans. 179.1.
7. What is the value of 1/10:89+^1444+1/18.49.
Ans. 45.6.
8. What is the value of t/5776— 1/40.96.
Ans. 69.6.
9. What is the value of l/5625— 1/3025.
Ans. 20.
10. What is the value of V 9025— l/l225.
Ans.-60.
11. What are the square roots of 21904 and 3564.09.
OPERATIONS.
1
21904 1
1
.48 Ans.
the square
5
109
118.7
(2.)
3564.09 59.7 ^ws.
25
24
119
96
1064
981
288
2304
2304
83.09
83.09
12. What are
roots of
332929 and 467856.
ins. 577 and 684.
13. Find the value of t/473344+i/48.8601.
Ans. 694.99.
EVOLUTION. 291
14. Find the value of 1-/44.0896+1/3080.25.
Ans. 62.14.
15. Find the square roots of 366025 and 49126081.
OPERATIONS.
(1.) (2.)
6
1205
366025 I 605 Ans. 7
36
6025 14009
6025
49126081 1 7009 Ans.
49
126081
126081
16. Find the square roots of 259081 and 826281.
A71S. 509 and 909.
17. Find the square roots of 49.4209 ; 404.01 ; and
.822649. Ans. 7.03 ; 20.1 ; and .907.
18. Find the square roots of 12321 ; 1234321 ; and
123454321.
19. Find the square roots of 49284 and 4937284.
20. What is the value of l/ 110889— l/ 40376081.
Ans. 312.91.
21. Find the square root of .0011943936.
Ans. .03456.
22. Find the square roots of 99980001 and 9999-
800001.
23. Find the square root of 152399025.
Ans. 12345!
24. Find the square root of 2950771041.
Ans. 54321.
25. Find the square root of 8264446281.
26. Find the square root of 6529932864.
27. Find the square root of 4999479849.
292
EVOLUTION.
28. Find the square root of 2, and also of 3, to 7
decimal places.
OPERATIONS.
(1-) (2.)
1
2.00000011.4142136
1
: 1 3.000000|1.7320508
2.4
1.00
96
400
281
2.7
3.43
3.462
2.00
1.89
2.81
1100
1029
2824
11900
11296
7100
6924
2.8282
) 60400
56564
3.4640
) 17600
17320
3836
2828
280
272
1008
848
160
168
Remark. — After finding four figures of the root as usual, the re-
maining figures are found by simply dividing the last dividend by
the last divisor, except that, instead of annexing ciphers to the
dividend, we take away a figure at the right of the divisor at each
new figure of the root. Attention is of course paid to the rules for
division of decimals. In general,
Having found one more than half the required figures of the root in
the ordinary way, the remaining figures may he found by dividing the last
dividend hy the last divisor, carefully observing the rules in division of
decimals.
In this manner Table II of the Appendix may be verified.
EVOLUTION. 293
285. The square root of the product of two numbers
is the same as the product of the square roots of the
numbers. Thus :
l/4x9 is the same as l/4 Xv'9 ; also V2 xS=V2 xVB,
as may be easily verified.
S8G. To find the square root of a composite number,
Find the product of the roots of its factors, mahing
one of the factors a perfect square if possible.
EXAMPLES.
'1. What is the square root of 8?
■■ We have t/8= 1/4x^ = 1/4 X"l/2 = 2Xl/2.
' N'ow t/2=i1.4142136. Hence, l/8=2X1.4142136.
Ans. 2.8284272.
2. Find the values of l^FS; VM; VM', and 1/72.
Ans. 4.2426408, etc.
3. Find the values of l/'98 ; l/l28 ; VlQ2 ; and 1/200.
Ans. 9.899495, etc.
4. Find the values of Vl2', VVJ; t/48; and VJb.
Ans. i/4x3=2Xt/3=3.464102.
5. Find the values of 1/IO8; l/l47; T^IM; and
1/243. Ans. 10.392305, etc.
6. Find the values of l/20 ; 1^28; 1^99; and 1/8O.
t/20=2Xi/5; t/28=2X>^7; T/99=3X"/li; VW):^4:XV^.
7. Find the values of 1/120 ; l/270 ; 1/8OO ; and
1/450. Ans. 2X1/30=10.954451, etc.
Remark.— In practice reject as many figures on the right of the
product of the tabulated decimal as are found in the root of the
factor which is a perfect square.
294 EVOLUTION.
287. The square root of a fraction is the square root
of the numerator divided bj the square root of the de-
nominator. Thus :
The square root of | is -7^ = 1, because |X|=|.
288. To find the square root of a fraction,
(1.) If necessary, multiply both terms of the fraction
hy the smallest number that will make the denominator a
perfect square.
(2.) Divide the square root of the resulting numerator
by that of the denominator^ for the required root.
EXAMPLES. _^
1. What is the square root of J ? ^ / ' ^
„, , ^ /_ ._ 1/2 1/2 1.4142L36
We have J=f ; then V | = l/f -:^ — ^= -—
=.7071068.
2. What are the square roots of J; J ; 4; ^^1
-^Tis. 1x1/3^.5773502.
3. What are the square roots of f ; | ; f ; i ? (Vide
286, Ex. 6. Ans. ^^^^ =.8944272, etc.
4. What are the square roots of f ; | ; 4 > I '• j
^Tis. ]^=.9258201, etc.
5. What are the square roots of 2| ; 3 J ; 4§ ; 2 J ?
^ns. 2^=1.5811388^ etc.
6. What are the square roots of 30/(j; 11/^; Hi?;
and 6|f ? Jtzs. 5'; sf; 3|.; and 2|.
7. What are the square roots of 2| ; 18 J^; 272^;
51 A? ^ns.li; 41; 5^; 74.
EVOLUTION. 205
CUBE II GOT.
289. The extraction of the cube root of a number is
the operation of finding one of itfe three equal factors.
290. The cube root of any integral number which is
a perfect cube and less tlian 1000, may be found from
memory, or by inspecting the third column of the Table
under 273, Ex. 13.
EI^AMPLES.
1. What are the cube roots of 1 ; 8; 27; 64; 12^?
2. What are the cube roots of 216 ; 343 ; 512 ; and
729? •^
Remark. — All the integral numbers less than 1000, other than
those in the above two examples, have only approximate cube roots,
that is, their cube roots are surds. (Vide 279, (8.) The same is
true of all numbers not perfect cubes.
291. The general method of extracting the cube root
of a number may be drawn from the following opera-
tions. We will cube the number 37 by continuing
Operation (2) of 283.
OPERATION FIRST.
37^=
1369=
30^+2x30x7+72
37-
.37=
30+7
30^+2X30^x7+30x7^
302x7+2x30x7-f7^
37^= 50653= 303+3x30^X7+3x30X7^+7^
In the operation, the quantity 30^+2X30X7+7^ is
first multiplied by 30 and then by 7, and the two indi-
cated products are added together. It is evident that
296
EVOLUTION.
twice 30^X7 added to once 30^X7 is the same as three
times 30^X7, that is, the sum is 3x30'X7; also, that
once 30X7^ added to twice 2>^Y.1^ is three times 30 X7^
that is, the sum is 3X30x7^.
We will now reverse these steps.
1st Col.
30
2X30
3X30+7
OPERATION SECOND.
2d Col.
302
3X30=
3X30^-1-3X30X7+72
Quantity. Root. [30+7
30^+3X302X7+3X30X72+78
30^
3X302X7+3X30X72+73
3X302x7+3X30X72+73
The above operation contracted is as follows :
OPERATION THIRD.
Number, Root,
50653 I 37.
27
1st Col.
2d Col.
3
6
97
9
2700
3379
.
23653
23653
The steps in Operation Second are as folloAvs:
Having arranged the quantity so as to form two
columns on the left, with 30 in the 1st col., its square
in the 2d col., and its cube under the quantity itself.
We now indicate double the 30 of the 1st col., and write
it under 30 of the same column. Next multiply this
2X30 by 30, which gives 2X30^, and add the product to
30^ of the second column, and write the indicated sum
or 3x30^ under 30^ in the same column. Next indicate
the triple of the 30 in the 1st col., and write the indi-
EVOLUTION. 297
oated product, viz., 3x30, under 2X30 of the 1st col.
If we now divide the first term of the remainder, viz.,
3X30^X7, by 3x30-, the second term of the 2d col.,
the quotient will of course be 7, the second term of the
root. This 7 is added to the right of 3x30 of the 1st
col., making 3x30+7, which is then multiplied by 7,
and added to 3x30^ of the 2d col., giving 3x30^+3
X30x7+7^ which is now multiplied by 7, and the pro-
duct is the same as the remainder, after subtracting 30'^
from the quantity itself.
The indicated products and additions of Operation
Second are actually made in Operation Third, omitting
ciphers, which would be of no service if written down.
Hence,
!S9S. To extract the cube root of any integral
number,
(1.) Divide the number into periods of three figures
each, counting from right to left. The left-hand period
may consist of one, two, or three figures. (Vide 278.)
(2.) Find bi/ 290 the largest perfect cube less than the
left-hand period ; place its cube root to the right of the
number J and also to the left, as the first term of the 1st
col. ; square the first term of the 1st col., and write it as
the first term of the 2d col.; cube the first term of the
1st col., and subtract it from the left-hand period of the
number, and annex the next period to the right. The
result is the first dividend. Add the figure in the 1st
col. to itself, and lurite the sum as the second term of the
1st col.; midtiply the second term of the 1st col. by the
figure in the root, and add the product to the first term
of the 2d col.; the sum with two ciphers annexed is the
298 EVOLUTION.
second term of the 2d col. Add the figure in the root
to the second terin of the 1st col., and place the sum under
the second term.
(3.) See hoiv many times the second term of the 2d
col. is contained in the fikst dividend, and write the
quotient as the second figure of the root, and also to the
right of the last number in the 1st col., forming its third
term; multiply the third term of the 1st col. by this
second figure of the root, and add the product to the second
term of the 2d col. as its third ter7n; multiply this third
term by the second figure of the root, and subtract the p)ro-
ducf from the first dividend, and annex the next period of
the number for the second dividend.
Proceed with the second figure of the root precisely
as with the first, and continue the operation till all the
periods have been used.
Remark 1. — It often happens that in making the division for
the figure of a root the quotient will be found too large. In such
a case erase the work as far back as that point, and make the root
figure so small that the subtrahend will be less than the dividend.
(Vide Ex. 13.)
Remauk 2. — If a dividend is too small to contain the divisor,
place a cipher in the root, and also to the right of the last term
of the first column, and two ciphers to the last term of the second
column. (Vide Ex. 19.)
Remark 3. — In the case of decimals, the first three figures on the
right of the decimal point constitute one period, the next three
figures another, and so on.
Remark 4. — This method of taking the cube root is so superior,
it is not a little surprising that it has not been adopted, to the ex-
clusion of all others. Aside from its value in this connection, it
prepares the student for an easy understanding of the best mode
of finding the numerical values of the roots of equations above
the second degree in Algebra.
EVOLUTIOX.
299
EXAMPLES.
1. Find the cube root of 753571.
OPERATION.
9
18
271
81
24300
24571
753571 I 91 Ans.
729
24571
24571
2. Find the cube roots of 531441 and 357911.
Ans. 81 and 71.
3. Find the cube roots of 132651 and 72507.
A71S. ^r^ 43.
4. Find the cube roots of 970299 and 681472.
Ans. 99 and 88.
5. Find the cube roots of 456533 and 287496.
Ans. 77 and 66.
6. Find the cube roots of 6.859 and 24.389.
Ans. 1.9 and 2.9.
7. Find the value of 1^704969+1^421875.
Ans. 164.
8. Find the value of 1^884736+1^314432.
Ans. 164.
9. Find the value of 1^110592—1^^103823.
Ans. 1.
10. Find the value of f" 91125 Xl^4287o.
An.s. 1575.
11. Find the value of 1^456533—1^421875.
Ans. 2.
12. Find the value of 1^4096X1^15625. Am. 400.
300 EVOLUTION.
13. Find the cube root of 2863288.
OPERATION.
1
2
34
38
422
1
300
436
58800
59644
2863288 [142
1
1863
1744
119288
119288
14. Find the cube roots of 2299968 and 2352637.
-^ Am. 132 and 133.
15. Find the cube roots of 10793861 and 14526784.
Am. 221 and 244.
16. Find the cube roots of 36926037 and 63521199.
Am. 333 and 399.
17. Find the cube root of 212776173. Am. 597.
18. Find the cube root of 997002999. Am. 999.
19. Find the cube root of 743677416.
OPERATION.
9
18
2706
81
2430000
2446236
743677416 I 906
729
14677416
14677416
20. What is the cube root of ip30301? Am. 101.
21. What is the cube root of 128787625?
22. What is the cube root of 225866,529 ?
Am. 609.
EVOLUTION.
101
23. Find the cube root of 2 to seven places of
decimals.
OPERATION.
2|1.2599211 An90:,
1
1
2
300
32
364
34
43200
365
45025
370
4687500
3759
4721331
3768
475524300
37779
4758)64311
1.000
.728
272000
225125
46875000
42491979
4383021000
4282778799
10024)2201
9517
507
475
32
47
We proceed in the usual way, and find 1.2599, at
which point divide the dividend 10024 by 4758, reject-
ing other figures on the right as not afi'ecting the result.
In general, having found one more than half the figures
of the decimal in the ordinary way, divide the last divi-
dend hy the last term of the 2d col., rejecting from the
right of the dividend one figure less than from the right
of the divisor. In this manner Table III of the Ap-
pendix may be verified.
24. Find the cube roots of 3, 4, etc., to 20.
302 EVOLUTION.
29S. The cube root of the product of two numbers
is the same as the product of the cube roots of the num-
bers. Thus :
1^8x27-1^8 X1^27, and f'2xE=f2xf'S,
as may be verified from the Table. Hence,
294. To find the cube root of a composite number,
Find the product of the cube roots of its factors, makiiig
one of the factors a perfect cube, if possible.
EXAMPLES.
1. What is the cube root of 16? We have #"^16=
#"8X2=1^8 X#'2 =2X1^2. Now, if 2 -=1.2599211.
Hence, 1^16=2X1.259921. Ans. 2.519842.
2. Find the values of #"24^ 1^321 1^40^ and 1^48.
fSxf'^; 1^8X1^4; 1^8X1^5; 1^8X1^6.
Ans. 2.8845, etc.
3. Find the values of 1^54; 1^128; #'250.
Ans. #^27 X^'g -3.779763.
4. Find the values of ^88^ ^297"; and ^704.
(Vide 286, Rem.) Ans. 2X^11=4.44796.
295. The cube root of q> fraction is the cube root of
the 7iumerator divided by the cube root of the denomi-
nator. Thus, the cube root of /^ is -— ^ = |, because
F 27
fX|X|=3V Hence,
296. To find the cube root of a fraction,
(1.) If necessary, multiply both terms of the fraction
by the smallest number that ivill make the denominator a,
perfffct cube.
EVOLUTION. 303
(2.) Divide the cube root of the resulting numerator hy
that of the denominator for the required root.
EXAMPLES.
1. What is the cube root of i? We have
1=^ Then irT_iri=^4_f4 _]L587401_
.7937005 Ans.
2. What are the cube roots of | ; J ; J ; and I ?
Ans. 1X1^9 =.693361, etc.
3. What are the cube roots of f ; |; |; and li?
Jxrg^ Jxiri2i fxif 9] ixif To:
^ns. .908561, etc.
4. What are the cube roots of | ; | ; y^g ; | ; -J| ?
^?is. -I X 1^18 =.87358.
5. What are the cube roots of ff ; iff; ||f '
Remark. — When the root of a perfect square, cube, fourth power,
etc., contains no more than three figures, it may be taken mentally,
after a little practice, and on observance of the points in sec. 274.
All roots of integral numbers containing not more than six figures,
decimals and fractions, are easily found by a common Table of
Logarithms. (Algebra, 152.)
PROBIiEMS.
29T. The sum of two numbers and their product
being given, to find the numbers.
Find the square root of four times the product
SUBTRACTED FROM the Square of the sum. This root
will be the diiference of the numbers. (Vide 125,
Ex. 10.)
V
304 EVOLUTION.
EXAMPLES.
1. The sum of two numbers is 105 and their product
2666. What are the numbers? l/l052— 4X2666=19.
Ans. 62 and 43.
2. The sum of two numbers is 10 and their product
24. What are the numbers? Ans. 6 and 4.
3. The sum of two numbers is 8j and their product
17 J. What are the numbers? A71S. 5 and 3|.
4. The sum of two numbers is 28 and their product
196. Wliat are the numbers? Ans. 14 and 14.
S98. The difference of two numbers and their pro-
duct being given, to find the numbers,
JFind the square root of four times the product added
TO the square of the difference. This root is their sum.
EXAMPLES.
1. The difference of two numbers is 19 and their pro-
duct 2666. What are the numbers ?
y 192+4x2666=105. Ans. 62 and 43.
2. The difference of two numbers is 3 and their pro-
duct 40. What are the numbers? Ans. 8 and 5.
3. The difference of two numbers is 7 and their pro-
duct 294. What are the numbers ? Ans. 21 and 14.
4. The difference of two numbers is 3^ and their pro-
duct 73 J. What are the numbers? Ans. 10 J and 7.
5. The sum of two numbers is 23j and their product
135. What are the numbers? Ans. 11} and 12.
6. The difference of two numbers is J and their pro-
duct 135. What are the numbers ?
ARITHMIiTICAL PROGRESSION. 305
ARITHMETICAL PROGRESSION.
299. An Arithmetical Progression is a series of
numbers in which any term is found by adding a given
number to the preceding term, or by subtracting a given
number from tJie precediyig term.
(1.) The common difference is the number to be added
or subtracted.
(2.) The progression is increasing when the common
difference is added.
(3.) The progression is decreasing when the common
difference is subtracted.
300. In every progression these five points may be
considered, viz., the first term, the last term, the common
difference, the number of terms, and the sum of all the
terms. Thus, in the arithmetical progression,
1, 3, 5, 7, 9, 11, 13,
the first t'erm is 1, the last term 13, the common differ-
ence 2, the number of terms 7, the sum of all the terms
49, and the progression is increasing.
In the decreasing progression,
22, 19, 16, 13, 10, 7, 4, 1,
the first term is 22, the last term 1, the common differ-
ence 3, the number of terms 8, and the sum of all the
terms 92.
(1.) The first and last terms are sometimes called
extremes, and the intermediate terms the means. The
means must of course be less by 2 than the number of
terms.
20
306 AEITHMETICAL PROGRESSION.
(2.) In a progression of three terms, the middle term
is called the arithmetical mean, and is half the extremes.
301. To find the last term, when the first term, 7mm-
her of terms, and common difference are known.
EXAMPLES.
1. The first term of a progression is 4, common dif-
ference 3, number of terms 50. What is the last term?
ANALYSIS.
If we take a few terms of the progression ; thus,
1st. 2d. 3d. 4th. 5th.
4, 4+3, 4+2x3, 4+3x3, 4+4x3;
that is, 4 7 10 13 16;
we see that any term, as, for instance, the 5th, is found
by adding to the 1st the product of the common differ-
ence into a number less by 1 than that denoting the
term. If we should continue the progression to the
50th term, the quantity standing under 50th Avould
evidently be 4+49x3, and the 50th term would there-
fore be 151. Hence,
When the progres'sion is increasing,
(1.) To the first term add the product obtained by mul-
tiplying the common difference by the number of terms
less 1.
2. In the progression 2, 4, 6, 8, etc., what is the 20th
term? 40th term? 60th? 71st? 100th?
Ans. 40, 80, etc.
3. In the progression 3, 11, 19^ 27, etc., what is the
50th term? 80th? 150th? 200th? 500th?
Ans. 395, 635, etc.
ARITHMETICAL PROGRESSIOX. 307
4. In the progression 7, 11, 15, etc., what is the 10th
term? 45th? 75th? 101st? Ans. 43, 183, etc.
5. In the progression 8, 11, 14, etc., what is the 12th
term? 24th? 47th? 81st? 1000th?
Last Ans. 3005.
6. In the progression 1, 1^, 2, 2|, etc., what is the
900th term? 1200th term? 1500th term?
Ans. 450 1.
7. In the progression 5, 5-|, 5f , 6, etc., w4iat is the
45th term? 90th? 750th? 100th? Ans. 19f, etc.
When the progression is decreasing,
(2.) From the first term subtract the product obtained
by 7nultiplying the common difference by the number of
terms less 1.
8. In the progression 500, 496, 492, etc., what is the
20th term? 25th? 30th? 50th? ^^is. 424, etc.
9. In the series 320, 318, 316, etc., what is the 10th
term? 21st? 31st? 41st? Ans. 302, etc.
10. In the series 412, 409, 406, etc., what is the 15th
term? 85th? 45th? Ans. 370, etc.
11. In the series 100, 95, 90, etc., what is the 12th
term? 15th? 20th? 21st? Last J[?is. 0.
302. To find the common difference^ when the ex-
tremes and the numbers of terms are known.
Divide the difference of the extremes by the number of
terms less 1.
EXAMPLES.
1. The extremes of a progression are 4 and 151;
number of terms 50. What is the common diiference?
What is the series? (151— 4)--(50— 1)=3, the common
difference. The series is then 4, 7, 10, etc.
308 AlHTHMETICAL PROGRESSION.
2. The extremes are 2 and 40 ; number of terms 20.
What is the series? Common difference is 2.
Ans. 2, 4, 6, 8, etc.
3. The extremes are 8 and 395 ; number of terms 50.
What is the series? Ans. 3, 11, 19, etc.
4. Insert 8 means between the extremes 7 and 43.
(Vide 300, (1.)
Ans. 7, 11, 15, 19, 23, 27, 31, 35, 39, 43.
5. Insert 10 means between 8 and 41.
A71S. 8, 11, 14, etc.
6. Insert 18 means between 500 and 424,
A71S. 500, 496, etc.
7. Insert 8 means between 320 and 302.
8. Insert 2 means between 4 and 14.
Ans. 4, 7J, lOf, 14.
9. Insert 3 means between 1 and 2.
Ans. 1, 1J-, 11, If, 2.
10. Insert 5 means between 3 and 4.
Ans. 3, 31, 3|, etc.
11. Insert 7 means between 3 and 6.
Ans. 3, 3|, 3-|, etc.
12. Insert 5 means between 1 and 13.
Ans. 1, 3, 5, 7, 9, 11, 13.
14. What is the arithmetical mean between 4 and 10?
(Vide 300, (2.) Ans. 7.
15. What is the arithmetical mean between 1 and 2?
Ans, li=1.5.
16. What is the arithmetical mean between 7 and 15?
Ans. 11.
17. What is the arithmetical mean between 1.5 and 1 ?
Ans. 1.25
ARITHMETICAL PROGRESSION. 309
303. To find the sum of all the terms, when the
first term, the last term, and the number of terms are
known.
EXAMPLES.
1. The extremes are 1 and 13, number of terms 7.
What is the sum of all the terms ?
ANALYSIS.
We have just found the series to be (Ex. 12 above)
1, 3, 5, 7, 9, 11, 13. This series
reversed is 13, 11, 9, 7, 5, 3, 1. The sum of double
th« series is then 1-1+144-14+14-f 14+14+14:=7Xl^=98, and
7V14
the sum of the series -itself is -^o — =49, as may be verified by
adding the terms. (Vide 300.) ilence.
Take half the product obtained by multiplying the sum
of the extremes by the number of terms.
2. What is the sum of 50 terms of the series 4, 7,
10, etc.? Last term is 151, by 301.
An,. 5^X^^=3875.
3. What is the sum of 20 terms of the series 2, 4, 6,
etc.? 40 terms? 60 terms? 71 terms? Ans. 420, etc.
4. What is the sum of 50 terms of the series 3, 11,
19, etc.? 80 terms? 150 terms? 200 terms?
^?2s. 9950, etc.
5. What is the sum* of IQ terms of the series 7, 11,
15, etc.? 35 terms? 45 terms? 75 terms?
6. What is the sum of 12 t^rms of the series 1, 2, 3, 4,
etc.? 24 terms? 48 terms? 96 terms? Ans. 78, etc.
7. What is the sum of 2 terms of the series 1, 3, 5,
7, etc.? 3 terms? 4 terms? 5 terms? 6 terms? 7 terms?
25 terms? Last Ans. 625.
310 ARITHMETICAL niOGRESSION.
8. What is the sum of 40 terms of the series 1, 4, 7,
etc.? 50 terms? 60 terms? 70 terms?
Ans. 2380, etc.
9. A gentleman started on a journey, traveling 5
miles the first day, 7 miles Jie second day, and so on,
gaining 2 miles each day. Another gentleman, starting
from the same place, travels over the same road at a
uniform rate of 34 miles per day. How far apart were
they at the end of 10 days? 20? 30?
Last Ans. They are together.
10. A gentleman started on a journey, traveling 55
miles the first day, 51 the second day, and so on, losing
4 miles each day. Another gentleman travels uniformly
40 miles per day over the same road and from the same
place. How far apart are they at the end of 10 days?
Ans. 30 miles.
11. A boy buys 12 marbles, giving 1 cent for the first,
2 cts. for the second, 3 cts. for the third, and so on.
How many cents did he give for all? Ans. 78 cts.
12. Suppose the city of Quito to be precisely on the
equator, and that this circle is exactly 25000 miles in
circumference; suppose, furthermore, that the earth
were entirely of land, and that stakes have been set up
at intervals of 1 mile, tho enti|;e distance round the
equator. Now allowing a bag of gold dollars to be at
the bottom of each stake, what distance would a person,
setting out from Quito, h^v^e to travel in order to carry
the bags, one at a time, to that city?
A71S. 312500000 miles.
GEOMETRICAL PROGRESSION. 811
GEOMETRICAL PROGRESSION.
304. A Geometrical Progression is a series of num-
bers, in which any term is found by multiplying the pre-
ceding term by a given number.
(1.) The ratio is the number used as a multiplier.
(2.) The progression is increasing when the ratio is
greater than 1.
(3.) The progression is decreasing when the ratio is
less tha7i 1.
305. In every geometrical progression these five
points may be considered, viz., the first term, the last
term, the ratio, the number of terms, and the sum of all
the terms. Thus, in the series,
1, 3, 9, 27, 81, 243,
the j^rs^ term is 1, last term ^4:^^^atio 3, number of terms
6, sum of all the terms 364, and the series is increasing.
In the decreasing progression,
1 1 j_ _L ^ ^ _ _ 1 _
the f.rst term is 1, last term Y-0^24' '^^^^^ h 'f^umber of
terms 6, and the sum of all the terms i§||.
(1.) The first and last terms are called the extremes,
and the intermediate terms the means.
(2.) In a series of three terms the middle term is
called the geometrical mean, and is the square root of the
product of the extremes.
(3.) The ratio is the quotient of any term divided by
the preceding term.
312 GEOMETRICAL PROGRESSION.
306. To find the last term, when the first term, num-
ber of terms, and ratio are known.
EXAMPLES.
1. If the fi7'st term of a series is 2, ratio 3, and num-
her of terms 10, what is the last term?
ANALYSIS.
If we take a few terms of the progression thus,
1st. • 2d. 3d. 4th. 5tli.
2, 2X3, 2X3^ 2X3\ 2X3^
that is, 2 6 18 54 162
we see that any term, as, for instance, the 5th, is found
by multiplying the first term by that power of the ratio
denoted by the number of terms less 1. If we should
continue the series to the 10th term, the quantity repre-
senting ii would eviden{;|y be 2X3^, and the 10th term
is therefore 39366. (Vj^je ?76, Ex. 1.) Hence,
Multiply that poiver of the ratio denoted by the number
of terms less 1 by the first term.
2. In the series 3, 21, 147, etc., what is the 4th term?
6th term? 9th term? (Vid^e 273, Ex. 13, Table.)
Last Ans. 17294403. '
3. In the series 4, 8, 16, etc., what is the 8th term?
9th term? 16th terra? (Vide 276, Ex. 2.)
Last Ans. 131072.
4. In the series 2, 4, 8, etc., what is the 5th term ?
9th terra? 13th term? Last Aiis, 8192.
5. In the series 1, J, -}, etc., what is the 10th term ?
6th term? 4th term? Ans. ^{.t, etc.
GEOMETRICAL PROGRESSION. 313
SOT. To find the ratio^ when the extremes and the
number of terms are known,
(1.) Divide the last term by the first term.
(2.) Take the root of the quotient denoted by the number
of terms less 1.
EXAMPLES.
1. The extremes of a series are 2 and 39366 ; number
of terms, 10. What is the ratio and the consequent
series ? 39366 -- 2 =-- 19683. Then 1^19683 = 3, the
ratio. (Vide 273, 13, Table ; 274, (7.)
• The series is 2, 6, 18, 54, 162, 486, 1458, 4374,
13122, 39366.
2. Insert 2 means between 3 and 1029. The number
of terms will of course be 4. 1029 -r- 3 --=343; then
#"343=7, the ratio. Ans. 3, 21, 147, 1029.
3. Insert 3 means between 7 and 1792.
Ans. 7, 28, 112, 448, 1792.
4. Insert three means between J and g^^-
Ans. i, J^, Jq, Jq, g*^.
5. What is the geometrical mean between 4 and 16 ?
Ans. 8. (Vide 305, (2.)
6. What is the geometrical mean between 5 and 16 J?
Ans. 9.
7. What is the arithmetical mean between 1 and 0 ?
(Vide 300, (2.) Ans. .5
8. What is the geometrical mean between 10 and 1 ?
Ans. 3.162278.
9. What is the arithmetical mean between 1 and .5 ?
Ans. .75
10. What is the geometrical mean between 10 and
3.162278? Ans. 5.623413.
27
314 GEOMETRICAL PROaRESSION.
11. What is the arithmetical mean between .75 and 1 ?
Ans. .875.
. 12. What is the geometrical mean between 10 and
5.623413 ? Ans. 7.498942.
13. What is the arithmetical mean between 1 and
.875? Ans. .9375.
14. What is the geometrical mean between 10 and
7.498942 ? Ans. 8.659643.
15. Find the values of 1±^ and l/lOX 8.659643.
16. Find the values of
•''"'+•'''' and 1/9.305720x8.659643.
17. Find the values of
.96875+.953125 ^^^ i/9.305720x 8.976871.
18. Find the values of
■960988+.9a3i25 ^^^ ^9.139817x8.976871.
19. Find the values of
.957031+.953125 ^^^ ^-9.057978x8.976871.
20. Find the values of
.955078+.953125 ^^^ t/9.017333x8.976871.
21. Find the values of
.954102+.955078 ^^^ i/8.997079x9.017333.
22. Find the values of
.954590+.954102 ^^^ ^^9.007200X8.99707-9.
28. Find the values of
.954346+.954102
--- and 1^9.002138X8.997079.
and 1/9.000873x8.999608.
GEOMETRICAL PROGRESSION. 3^15
24. Find the values of
■?^^^^!^ and •8.999608X.9002138.
25. Find the values of
.954285+.954224
2
26. Find the values of
^'^^'Y'"^' -d 1^9-000241X8.999608.
27. Find the values of
■954239+.904254 ^^^ ^8.999924x9.000241.
28. Find the values of
.9542474-.954235
and 1^9.000082X8.999924.
Ans. .954243 and 9.000003.
Remark. — Each arithmetical mean in the above examples is the
Logarithm of the corresponding geometrical mean, beginning with
example 7. The logarithm of 9 is .954243. (Vide Algebra, 153.)
308. To find the sum of all the terms, when the ex-
tremes and ratio are given.
EXAMPLES.
1. The extremes are 2 and 162, the ratio 3. What is
the sum of the series?
ANALYSIS.
The series is 2, 2x3, 2X3^, 2x3^ 2x3^ Multiplied
by 3 it is 2X3, 2X3^, 2x3^ 2x3S 2X3^
The first series added would give the sum of all the
terms ; the second series added would give three times
the sum of all the terms.
If, then, the first series be subtracted from the second,
316 GEOMETRICAL PROGRESSION.
the remainder will be twice the sum of all the terms.
The actual subtraction gives 2X3^ — 2, the other terms
canceling each other. Hence, the sum of the series is
2 — 2"^^ -^^^- Hence,
Multiply the last term hy the ratio, and from the pro-
duct subtract the first term; divide the remainder by the
ratio, less 1.
2. In the series 3, 21 . . . 1029, what is the sum of
all the terms? (Yide 305, (3.)
Ans, ^^^^f-^:^1200.
3. In the series 4, 12 . . . 78732, what is the sum of
all the terms? Ans. 118096.
4. In the series 5, 20 . . . 327680, what is the sum
of all the terms? Ans. 436905.
5. In the series 4, 8, 16, etc., what is the sum of 16
terms? (Yide 306, Ex. 3.) J.ws. 262140.
6. In the series 2, 4, 8, etc., what is the sum of 13
terms? Ans. 16382.
7. In the series 5, 50, 500, etc., what is the sum of 8
terms? Ans. 55555555.
309. If the series is decreasing, and is carried on
infinitely, the last term will become 0. Hence, to find
the sum of an infinitely decreasing series.
Divide the first term by the ratio subtracted from 1.
EXAMPLES.
1. What is the sum of the series J, J, ^, etc., to in-
finity? Ans. »^(1— i)=l.
2. What is the sum of J, y^^, ^^5, etc.? Ans. |.
3. What is the sum of ^, i, ^, etc.? Am. \.
PERMUTATIONS. 817
4. What is the sum of 4, 1, J, y^^, etc.? Ans. 5-|.
5. What is the value of .l-f-.Ol+.OOl, etc.?
Ans, ^.
6. What is the value of .1+.05+.025, etc.?
7.
8.
What
What
A
is the value of l+f+g^, etc.?
is the value of |+/u+i3, etc.?
Vns. i-
Ans.
Ans.
=.2.
If.
2f.
9.
What
is the value of
UA+t¥8, etc.
?
Ans*
If
PERMUTATIONS
310. To find the number oi permutations that can be
made with any given number of things, each one differ-
ent from the other,
Multiply all the consecutive integral numbers^ from 1
up to the given number of things, continually together.
The product will be the permutations that can be made.
EXAMPLES.
1. How many permutations can be made of the first
four letters of the alphabet? Ans. 1X2X3X4=24.
2. How many days can 7 persons be placed in differ-
ent positions at table? Ans. 5040.
3. A captain of 26 men told his company that he
should not consider them perfectly drilled till each man
had occupied all possible positions in the arrangements
that might be made of them. Suppose his men drill 12
318 PERMUTATIONS.
hours a day, and make a change every hour, how many
years must they drill before becoming perfect, reckoning
321 days to the year?
Ans. 107716786412020736000000 years.
311. To find how many arrangements may be made
by taking each time a given number of different things
less than all.
Take a series of nwnhers beginning with the number
of things given and decreasing by 1 until the number of
terms equals the number of things to be taken at a time;
then find the product of all the terms.
EXAMPLES.
1. How many sets of 3 letters each may be made out
of 4 letters ? Ans. 4 X 3 X 2==24.
proof:
abc, acb, adb, bac, bca, bda, cab, cba, cda, dab, dba, dca,
abd, acd, adc, bad, bed, bdc, cad, cbd, cdb, dac, dbc, deb.
2. How many integral numbers can be expressed,
each composed of any 5 of the 9 digits ?
Ans. 9X8X7X6X5=15120.
3. How many arrangements can be made out of the
26 letters of the alphabet, 6 being taken at once?
Ans. 165765600.
312. To find the number of combinations that can be
made of any number of things in sets of 2 and 2; 3
and 3, etc.,
(1.) Form a series of numbers^ as in 311, /or a dividend.
(2.) Form a series of numbers^ as in 310, up to the
number of things to be combined at a time, for a divisor.
The quotient will be the number of combinations
sought.
PRACTICAL GEOMETRY. 319
EXAMPLES.
1. How many combinations can be made out of 4
letters, having 3 different letters in each set?
4X3X2
^ns. -^^2x3~'*-
Proof. — abc, abd, acd, bed.
2. How many combinations can be made out of 10
letters, each combination having in it 7 letters, but no
two of them to have all their letters alike ?
Ans. 120.
3. How many different combinations of 3 colors can
be made of the 7 prismatic colors? Ans. 35.
4. How many different combinations of 4 colors can
be made of the 7 prismatic colors? Ans. 35.
5. How many different combinations of 3 may be
made of 10 different things? Ans. 120.
6. How many different combinations of 7 may be
made of 10 different things? Ans. 120.
PRACTICAL GEOMETRY.
DEFINITIONS.
313. Geometry is that branch of Mathematics which
treats of the' relations of extension. Extension has three
dimensions — length, breadth, and thichiess.
(1.) A point is mere position, with no length, breadth,
or thickness.
(2.) A line is length, without breadth or thickness.
320 PRACTICAL GEOMETRY.
(3.) A surface is a figure having length and breadth,
but no thickness.
(4.) A solid is a figure having length, breadth, and
thickness.
LINES.
(5.) A straight line is one, all points of which lie in
the same direction. It is designated bj letters placed
near its extreme points. Thus, the line ab is repre-
sented by A B
(6.) A curved line is one, ail points of which lie in
different directions. Thus, the curved line
A B is represented by
(7.) Parallel straight lines are two or
more straight lines lying in the same c d
direction. Thus, ab, cd, ef, are par- ^ ^
allel straight lines.
(8.) Oblique lines are those which do not lie in the
same direction. Thus, A b and c D are . __ — _— — -^
oblique lines. c d
Remark. — When two oblique lines meet each other, they form an
angle.
(9.) An angle is the divergence of two b
straight lines proceeding from the same ^^ o
'point. Thus, a, or b A c, or c A b is aa
angle. In reading, the letter at the vertex . »
is placed in the middle, or the letter at / ^
the vertex may be used alone to desig- ^
nate the angle when other angles arc not ^^
adjacent. ^^ o
A B
PRACTICAL GEOMETRY. 321
Eemark. — An angle is greater or less^ according as the lines di-
verge more or less.
(10.) A right angle is one of the equal angles which
two straight lines may make
in meeting or intersecting
each other. Thus, A o c, B o c, ^
A 0 D, or B 0 D is a right angle
when each is equal to any other.
An acute angle is one which is less than a right angle.
An obtuse angle is one which is greater • „
than a rio;ht angle. Thus, B o c is acute. ^^
and A 0 c is obtuse. °
(11.) Perpendicular lines are those which form a right
angle with each other. Thus, A o is per-
pendicular to c 0, and c o is perpendicular
to A 0, when A o c is a right angle.
SURFACES.
(12.) A plane is a surface in which if any two points
be assumed and connected by a straight line, that line
ivill lie wholly in the surface. Other surfaces are called
curved surfaces. Thus, the surface of a common slate
represents a plane, and the surface of a slate globe
represents a curved surface.
(13.) A plane figure is a figure bounded by straight
or curved lines.
(14.) A polygon is a plane bounded by straight lines.
(15.) A triangle is a polygon of three sides, and con-
sequently of three angles.
(a.) An equilateral triangle has its three sides equal;
322
PRACTICAL GEOMETRY.
an isosceles triangle has tivo of its sides equal; a scalene
triangle lias its three sides unequal,
(b.) A right-angled triangle has one of its angles a
right angle.
(c.) ArT acute-angled triangle has all of its angles acute.
(e.) An obtuse-angled triangle has one of its angles
an obtuse angle. Thus :
A B c is equilateral and acute-angled ; D E F is isosceles
and right-angled; mno is scalene and obtuse-angled,
p Q R is scalene and acute-angled.
(/.) Any side of a triangle may be assumed as the
base, and its altitude is the perpendicular line extending
from the vertex of the opposite angle to the base.
(g.) In a right-angled triangle the side opposite the
right angle is called the hypothenuse. The other sides
are then designated as the base and perpendicular.
(16.) A quadrilateral is a polygon of four sides.
(17.) A parallelogram is a quadrilateral with its oppo-
site sides parallel.
(18.) A rectangle is a parallelogram
having a right angle.
(19.) A rhomboid is a parallelogram
having an oblique angle.
(20.) A square is a rectangle hav-
ing its sides all equal.
(21.) A rhombus is a rhomboid having
its sides all equal.
S, ,Q
a) |r
Vi IB
PRACTICAL GEOMETRY. 323
(22.) A trapezium is a quadrilateral
with its opposite sides 7iot parallel.
(23.) A trapezoid is a quadrilateral
with two sides parallel. Thus:
A diagonal is a line joining the vertices of two opposite
angles. Thus T E is a diagonal.
Remakk. — A diagonal divides a rectangle into two right-angled
triangles.
(24.) A pentagon is a polygon of five sides; a hexagon,
of six sides; a heptagon, of seven; an octagon, of eight;
a nonagon, of nine ; and a decagon, of ten sides.
(25.) A circle is a plane figure
bounded by a curved line called the
circu7nference, every point of which
is equally distant from a point within
called the center. The diameter is
a straight line drawn through the
center and terminating in the cir-
cumference. A radius is any line draivn from the center
to the circumference. An arc is a part of the circumfer-
ence. Thus, A B is a diameter, o B, o A, and o M are
radii ; A M is an arc.
(26.) Similar figures are such as .are mutually equi-
angular, and have the sides containing the equal angles,
taken in the same order, proportional.
314. Proposition. — The square of the hypothenuse
of a right-angled triangle is equivalent to the sum of the
squares on the other two sides. Hence,
315. To find the hypothenuse, the ttvo sides being known,
(1.) Square the two sides, and add together the results.
(2.) Find the square root of the sum.
324 PEACTICAL GEOMETRY.
EXAMPLES.
1. The sides of a right-angled triangle are 3 and 4
feet. What is the length of the hypothenuse ?
Ans. -1/32 + 42 = 1/25 = 5 feet.
2. The sides of a right-angled triangle are each 1
foot. What is the length of the hypothenuse ?
^Tis. 1/2 = 1.4142136 feet.
3. The sides of a right-angled triangle are each 2 feet
in length ; 3 feet ; 4 feet ; 5 feet in length. What are
the corresponding lengths of the hypothenuse?
Ans. 1^8 = 2.828427, (Vide 286, Ex. 1.)
4. The bottom of a window is 40 feet from the ground,
and I wish to place the foot of a ladder 30 feet from the
bottom of the wall in which the window is situated.
What is the length of a ladder that will reach the win-
dow? Ans. 50 feet.
5. An acre of land is laid out in the form of a square.
What is its distance between opposite corners ?
Ans. 1/320 = 8X1/5 = 17.888544 rods.
6. Suppose a floor to measure 16 feet by 20. What
is the distance between opposite corners ?
Ans. 1/656 =4X l/41 = 25.612497.
7. What is the length of the lower edge of a brace
which touches a post 3^- feet from the corner, and a
beam 4^ feet from the same point ?
Ans. i X 1/130 = 5.7008771 feet.
316. To find the side of a right-angled triangle, when
the hypothenuse and the other side are known.
From the square of the hypothenuse subtract the square
of the given side. The square root of the remainder will
be the other side.
PRACTICAL GEOMETRY. 325
EXAMPLES.
1. The hypothenuse is 5 and a side 4. Wliat is the
other side of the triangle?
2. A ladder 50 feet long is placed at a distance of 30
feet from the bottom of a house, and just reaches the
sill of a window. What is the hight of the sill?
Ans. 40 feet.
3. If a line 144 feet long will reach from the top of a
fort to the opposite side of a river 64 feet wide, on
whose brink it stands, what is the hight of the fort ?
A71S. 129 feet, nearly.
4. In the ruins of Persepolis are left two columns
standing upright ; one is 70 feet high and the other 50
feet ; in a line between them stands a small statue 5 feet
high, the top of which is 100 feet from the summit of
the higher, and 80 feet from that of the lower column.
What is the distance between the tops of the two col-
umns ? Ans. 143 J feet.
317. To find the side of a square when its diagonal
is known.
Multiply half the diagonal by the square root of 2.
EXAMPLES.
1. The diagonal of a square is 1 foot. What is the
side ? Ans. h X V2 =.7071068 feet.
2. The diagonal of a square is 2 feet ; 3 feet ; 4 feet ;
5 feet, etc. What is a side? Ans. 1.4142136, etc.
3. The distance from a corner to an opposite corner
of a square room is 20 feet. What is a side ?
Ans. 14.142136 feet.
326 PRACTICAL GEOMETRY.
SIS. Proposition. — In any triangle, if a perpen-
dicular be drawn from its vertex to its base, then the
whole base is to the sum of the other two sides as the dif-
ference of those sides is to the difference of the parts of
the base made by the perpendicular. c
Thus, suppose c d to be a perpendicular
drawn from the vertex c to the base A B
of the triangle ABC, then,
A B : A c+B c : : A c— b c : A d— b d.
Remark. — The perpendicular must always fall within the tri-
angle.
310. To find the perpendicular, when the th'ee sides
of a triangle are known,
(1.) Make the base the first terin of a proportion; the
sum of the other tivo sides the second term; the difference
of the two sides the third term. The fourth term found
by 258 will be the difference of the parts of the base.
(2.) Find the parts by 125, example 10, (1) and (2).
(3.) Find the perpendicular by 316.
EXAMPLES.
1. In a triangle ABC we have ab=5 rods; AC=4
rods, and B C— 3 rods. What is the length of the per-
pendicular CD?
OPERATIONS.
(1.) 5 : 4+3 : : 4-3 : 1.4.
(2.) (125,Ex.ll.) -^±i:^=3.2-:ADand^::^=1.8=DB.
(3.) 1/42-3.2^=2.4 or 1/32— 1.82=2.4=:D c.
Ans. 2.4 rods.
Why is this triangle right-angled at c'i
PRACTICAL GEOMETET. 327
2. In an isosceles triangle the sides are 5, 5, and 8
rods. What is the length of the perpendicular ?
Ans, 3 rods.
Remark. — Assume the side which is unequal to the other two to
be the base ; then the parts of the base made by the perpendicular
will be each 4 rods.
3. In an equilateral triangle the sides are each 1 rod.
What is the length of the perpendicular ?
Ans. |/r=i = Vi = hxVS = .8660254.
4. The sides of a triangle are 4, 5, and 6 rods. What
is the length of the perpendicular?
Ans. 3.307187 rods.
5. The sides of a triangle are 10, 10, and 16 rods.
What is the length of the perpendicular? Ans. 6 rods.
6. The sides of a triangle are each 25 rods. What is
the length of the perpendicular? Ans. 21.6506 rods.
7. The sides of a triano-le are each 2 rods in leno-th ;
3 rods; 4 rods; 5 rods; 6 rods, etc. What is the
length of the perpendicular?
Ans.l/S; iXV^S; 2X1^3, etc.
MENSURA.TION.
320. Mensuration is the method of finding the
area of surfaces in square units from their known or
implied linear dimensions. By mensuration we also find
the cubical contents of solids.
321. To find the area of a triangle, when its sides
are known,
(1.) Find the perpendicular by 319.
(2.) Multiply the base by the perpendicular, and take
half the product.
328 PRACTICAL GEOMETRY.
EXAMPLES.
1. What is the area of a triangle whose sides are 3,
4, and 5 rods? (Vide 319, Ex. 1.) ?^, or, because
the triangle is right-angled, -^ . Ans. 6 rods.
2. What is the area of a triangle whose sides are 5,
5, and 8 rods? Ans. 12 rods.
3. What is the area of a triangle whose sides are each
1 rod? Ans. .4330127 rods.
4. What is the area of a triangle whose sides are 4,
5, and 6 rods? Ans. 9.9215 rods.
5. What is the area of a triangle whose sides are 10,
10, and 16 rods? Ans. 48 rods.
6. What is the area of a triangle whose sides are each
25 rods? 13 rods? 40 rods? Ans. 270.633 rods, etc.
7. How many square feet of boards will cover the
gable end of a house whose rafters measure 23 feet, the
length of the beam at the ends of the building being
34.8712 feet? Ans. 261.534 feet.
Rule II. — Add together the three sides, and taJce half
their sum. From this half sum subtract each side sepa-
rately; mtdtiply together the half sum and the three re-
mainder. The square root of the product will be the area.
8. What is the area of a triangle whose sides are 10,
15, and 20 rods? 15+10+20--45 ; 45-^2=22.5;
V'22.5X2.5X12.5X7.5==72.62. Ans. 72.62 rods.
333. To find the area of any quadrilateral, two of
whose sides are parallel,
Multiply the sum of the parallel sides hy the perpendic-
ular distance between them, and take half their product.
PRACTICAL GEOMETRY. 329
EXAMPLES.
1. How many yards of carpeting will cover a floor 20
feet long and 16 feet Avide, carpet 1 yard wide? 20 X
16-^-9. Ans. 35 f yards.
2. How many acres in a rectangular piece of land
17 chains long and 5 ch. and 41. wide? (Vide 180,
Rem. 2.) 17x5.04^10 Ans. 8 A. 2 R. 10/o'(j r-
3. How many acres in a piece of land in the form of
a rhombus, each side measuring 70 rods and the width
being 8 rods ? Ans, 3 A. 2 R.
4. How many acres in a square piece of land whose-
sides are each 35 ch. 25 1. ' Ans. 124 A. 1 R. 1 r.
5. A board measures 25 feet in length and 1 ft. 6 in.
in width. What is the area in feet? Ans. 37 J feet.
6. In the trapezoid, 313 (23,) I D is 20 rods; z o,
27 rods; P R, 12 rods. How many acres in the lot?
m^}><}l Ans. 1 A. 3 R. 2 r.
7. In a trapezoid the parallel sides are 25 ch. 13 1.
and 30 ch. 1 1. ; the perpendicular distance between them
is 40 ch. How many acres ? Ans. 110 A. 1 R. 4.8 r.
S23. To find the area of a trapezium, when all its
sides and a diagonal are known,
Find the area of each triangle hy 321, and their sum
will be the area of the trapezium.
EXAMPLES.
1. In a trapezium represented by the figure ZIUM,
813, (22,) z I is 25 rods; mi, 23 rods; zu, 17 rods;
U M, 21 rods ; and z m, 30 rods. How many acres ?
Ans. 2A. 3R. 13.7 r.
28
330 PRACTICAL GEOMETRY.
324. Any iivo similar figures have their areas in pro-
portion to the squares of any ttvo lines similarly situated
in each. Hence, all circles are to each other as the squares
of their radii, or the squares of their diameters.
325. To find the area of a circle,
Multiply half of the diameter by half the circumfer-
ence.
Remark. — The circumference of a circle whose diameter is 1 is
3.141592053589793238462643383279502884197169399375105820974-
94459230781640628620899862803482534211706798214808651327230-
66470938446, etc.
EXAMPLES.
1. What is the area of a circle whose diameter is 1 ?
Ans. .7853981633974483, etc.
Hence, to find the area of a circle, midtiply the square
of the diameter hy enough of the figures composing the
decimal .785398163397, etc., to make the area sufficiently
exact.
2. What is the area of a circle whose diameter is 2
rods? 3 rods? 4 rods? 5. rods? etc.
Ans. 3.1416, etc., rods; 7.06858, etc., rods.
3. What is the area of a circle whose diameter is 20
rods? 30 rods? 40 rods? etc.
4. What is the area of a circle whose radius is 10
rods? 15 rods? 20 rods? etc.
5. What is the area of a circle whose radius is 100
rods? 150 rods? etc.
6. What is the area of a circle whose radius is 5000
rods? Ans. 490873 A. 3 R. 16 r.
7. What is the area of a circle whose radius is 1
mile? 2 miles? Smiles? etc. J.?is. 3.1416 miles, etc.
rRACTICAL GEOMETRY. 331
8. What is the area of a circle whose radius is 5
yards? 6 yards? etc. Ans. 78 J yards, etc.
9. Three men purchase a grind-
stone with a radius of 30 inches. '' / \''~''"-^/\\
How much of the radius must each ,'
grind oiF to secure J of the stone, \
making no allowance for loss at '' ^
the center? We have ""-"
Circ. A 0 :
: circ. c o :
: AO^
: C 0^ ; that is,
1 ;
: 1 :
: 30- ;
:30^Xf.
Circ. A 0 ;
: circ. P 0 :
: AO^
: P 0- : that is,
1
-• i -•
: 30-^ :
; 30^X^
Since c 0^=30^X1, c o=30Xl/|-=10Xl/6--24.4949.
Since p o2=30-Xl, p o=30Xi/1=10XV^3=17.3205.
Hence, the fii?st takes 30—24.4949=5.5051 inches;
the second takes 24.4949—17.3205=7.1744 inches; the
third takes 17.3205 inches.
Remark. — It is only necessary to use the ratio of the .areas in
the proportions.
10. Five men purchased a grindstone, and each man
paid ^ of the price. The radius was 40 inches. How
much of it must each man grind off to secure his
share? Ans, First man 40— 16x/5=4.2229 in.
11. I have a circular garden 25 rods in diameter,
and wish to make a walk around it that shall take up
I of the entire area. What must be the width of the
walk? Ans. 1.0891 rods.
12. I have a circular garden 50 rods in diameter, and
wish to make a walk around the outside of it whose area
shall be J^ the area of the garden. What must be the
width of the walk? Ans. 1.22 rods.
332
PRACTICAL GEOMETRY.
MENSURATION OF SOLIDS.
326. Demnition. — (1.) A cylinder \^ a solid described
by the revolution of a rectangle about one of its sides,
which remains fixed.
(2.) A cone is a solid described by the revolution of
a right-angled triangle about one of its sides^ which re-
mains fixed.
(3.) A sphere is a solid described by the revolution
of a semicircle about its diameter, which remains fixed.
Thus :
0 P B D revolved about o P generates a cylinder ; c 0 B
revolved about c o generates a cone ; and A M B revolved
about A B generates a sphere, o p is the altitude of the
cylinder, o c of the cone, and A B is the diameter of the
sphere.
(4.) A prism is a solid whose bases are j[?<2raZ?eZ and
its sides parallelograms. A right prism has its edges
perpendicular to its bases,
(5.) A parallelopiped is a prism whose bases as well
as sides are parallelograms. A cube has six equal square
faces.
(6.) A pyramid is a solid, having a polygon for a
base and three or more triangles for its sides, whose
PRACTICAL GEOMETllY.
333
vertices meet in a common point called the vertex of
the pyramid. The frustrum of a pyramid is the part
left after cutting off a portion of the top by a plane
parallel to the base. Thus :
/^. \
\
\o
\
F
D
1
li
A B c-D E F is a right triangular prism ; A B c D-E is a
right quadrangular prism, or parallelopiped; A B c-D is
a triangular pyramid ; and A B c-s represents the frus-
trum of a pyramid.
3S7. To find the contents of a cylinder or prism,
Multiply the area of the base by its altitude.
EXAMPLES.
1. Each side of the base of a triangular prism is 1
foot, and the altitude is 3 ft. 2 in. What are the con-
tents in feet? in inches? (Vide 321, Ex. 3.)
.4330127X31. Ans. 1.3712 feet.
2. The sides of a triangular prism are 7, 8, and 9
inches. Its altitude is Ij feet. Find the contents in
inches. Ajis. 482.99 inches.
3. The diameter of each end of a cylinder is 8 feet,
and the hight is 5 J feet. Find the contents in feet.
Alts. 276.46 feet.
334 PRACTICAL GEOMETRl.
4. The diameter of a cylindrical water-pail is 10
inches, and the hight is 1 foot. How many wine gal-
lons does the pail hold? (Vide 182, Rem. 1.)
A71S. 4.08 gallons.
5. How many bushels in a box 15 feet long, 5 feet
wide, and 8 feet deep? (Vide 184, Rem.)
Ans. 482.142 bushels.
6. How many bushels in a box 6 feet long, 1^ feet
wide, and 2 J feet deep? Ans. IS. OS.
7. What are the contents in cubic feet of a wall 24
feet 3 inches long, 10 feet 9 inches high, and 2 feet
thick? Ans. 521|.
3S8. To find the contents of a cone or pyramid,
Multiply the area of the base by the altitude^ and take
I of the product.
EXAMPLES.
1. Each side of the base of a triangular pyramid is 1
foot, and the hight is 14 inches. What are the con-
tents? Ans. 290.9844 in.
2. The sides of the base of a triangular pyramid are
10, 11, and 12 feet, and the hight is 12 feet. What are
the contents? Ans. 206.085 feet.
3. The base of a cone is 10 feet in diameter and the
hight is 5 feet. Find the contents.
Ans. 130.899 feet.
4. A square pyramid, 477 feet high, has each side of
its base 720 feet in length. Find the contents.
Ans. 3052800 cu. yd.
5. The sides of the base of a triangular pyramid,
which is 14J feet high, are 5, 6, and 7 feet. Find the
contents. Ans. 71.0352 feet.
PRACTICAL GEOMETRY. 335
3S9. To find the contents of the frustrum of a cone
or pyramid,
Find the sum of the areas of the two ends and the geo-
metrical mean between them. Multiply this sum hy the
altitude, and take one third of the product
EXAMPLES.
1. A stick of timber is 15 in. square at one end and
6 in. square at the other and 24 feet long. What arQ
the contents? (^^5+B«+«0)X8 ^„,.- 19, feet.
2. A conic frustrum is 18 feet high, 8 feet in diam-
eter at one end and 4 feet at the other. Find the
contents. Ans. 527.7888 feet.
3. A cask, in the form of two equal conic frustrums,
has a bung diameter of 28 inches, a head diameter of 20
inches, and a length of 40 inches. How many gallons
of wine will it hold? Ans. 79.0613 gallons.
7. A cistern is 12 feet in diameter at the top, 10 feet
in diameter at the bottom, and 14 feet deep. What
number of gallons will it hold?
OPERATION.
Area of top = 144 X -7854 = 113.09
Area of bottom .-= 100 X .7854= 78.54
Geomet. mean =t/113.09X78.54 = 94.25
Sum = 285.88 feet.
Then 285.88XV*X1728--231 =9980 gallons, nearly, ^hs.
8. How many Winchester bushels will the above cis-
tern hold? A71S. 1072 bushels.
330. Proposition. — All spheres are to each other as
the cubes of their radii, or diameters.
336 MISCELLANEOUS EXAMPLES.
331. To find the surface and also the solid contents
of a sphere,
(1.) Multiply the square of the diameter by 3.1415^,
eic.^ for the surface.
(2.) ^lultiply the cube of the diameter hy .523598, etc.^
for the solid contents.
Remark. — The latter decimal is i of the former.
EXAMPLES.
1. An artificial globe is 24 inches in diameter; what
is its area, and what the solid contents ?
Arts. Area 1809.556992 sq. in.; solidity 2738.218752 cu. in.
2. A slate globe is 6 feet in diameter; what is its
area, and what its contents ?
Ans. Area 113.0973 sq.ft.; contents 113.0973 cu. ft.
3. A sphere is 40 feet in diameter; what will be the
diameter of one containing | as many cubic feet? J- as
many? i as many? (Vide 296, Ex. 3, 2, and 1.)
Ans. 40X^^1=36.34 ft.; 40X'^ 1=31.75 ft.; 40X#'i=25.19 ft.
332. MISCELLANEOUS EXAMPLES.
1. If a certain number be multiplied by 5, and the
product divided by J, and 3 be added to the quotient,
and 7 taken from the sum, the remainder will be 76.
What is the number ? Ans. 8.
2. If to a certain number 12 be added, and the square
root of the sum taken, the cube of that root will be 64.
What is the number ? Ans, 4.
^
APPENDIX.
353
III.
TABLE OP CUBE ROOTS.
No.
1
Cube Root.
No.
Cube Root.
No.
Cube Root.
No.
Cube Root.
1.000000
Ig"
3.583048
91
4.497941
136
5.142563
2
1.259921
47
3.608826
92
4.514357
137
6.155137
3
1.442250
48
3.634241
93
4.530655
138
5.167649
4
1.587401
49
3.659306
94
4.546836
139
5.180102
5
1.709976
50
3.684031
95
4.562903
140
5.192494
6
1.817121
51
3.708430
96
4.5J8857
141
5.204828
7
1.912931
52
3.732511
97
4.594701
142
5.217103
8
2.000000-
53
3.756286
98
4.610436
143
5.229322
9
2.080084
54
3.779763
99
4.626065
144
5.241483
10
2.154435
55
3.802953
100
4.641589
145
5.253588
11
2.223980
56
3.825862
101
4.657009
146
5.265637
12
2.289429
57
3.848501
102
4.672329
147
5.277632
13
2.351335
58
3.870877
103
4.687548
148
5.289573
14
2.410142
59
3.892996
104
4.702669
149
5.301459
15
2.466212
60
3.914868
105
4.717694
150
5.313293
16
2.519842
61
3.936497
106
4.732624
151
5.325074
17
2.571282
62
3.957892
107
4.747459
152
5.336803
18
2.620742
63
3.979057
108
4.762203
153
5.348481
19
2.668402
64
4.000000
109
4.776856
154
5.360108
■ 20
2.714418
65
4.020726
110
4.791420
155
5.371685
21
2.758924
66
4.041240
111
4.805896
156
5.383213
22
2.802039
67
4.061548
112
4.820285
157
5.394691
23
2.843867
68
4.081655
113
4.834588
158
5.406120
24
2.884499
69
4.101566
114
4.848808
159
6.417502
25
2.924018
70
4.121285
115
4.862944
160
6.428835
2G
2.962496
71
4.140818
116
4.876999
161
5.440122
27
3.000000
72
4.160168
117
4.890973
162
5.451362
28
3.036589
73
4.179339
118
4.904868
163
5.462556
•29
3.072317
74
4.198336
119
4.918685
164
5.473704
30
3.107233
75
4.217163
120
4.932424
165
5.484807
31
3.141381
76
4.235824
121
4.946087
166
5.495865
32
3.174802
77
4.254321
122
4.959676
167
5.506878
33
3.207534
78
4.272659
123
4.973190
168
5.517848
34
3.239612
79
4.290840
124
4.986631
169
5.528775
35
3.271066
80
4.30^70
125
5.000000
170
6.539658
36
3.301927
81
4.326749
126
5.013298
171
5.550499
37
3.332222
82
4.344482
127
5.026526
172
5.561298
38
3.361975
83
4.362071
128
5.039684
173
5.672055
39
3.391211
84
4.379519
129
5.052774
174
5.582770
40
3.419952
85
4.396830
130
5.065797
175
5.593445
41
3.448217
86
4.414005
131
5.078753
176
6.604079
42
3.476027
87
4.431048
132
5.091643
177
6.614672
43
3.503398
88
4.447960
133
5.104469
178
6.625226
44
3.530348
89
4.464745
134
5.117230
179
6.635741
i 45
3.556893
90
4.481405
135
5.129928
180
5.646216
354
APPENDIX.
IV.
TABLE:
Shoiving the ultimate fratisveise strength of a bar 1 foci long and 1 inch
1 inch in diameter, made of either of the materials mentioned. The bar is loaded
in the middle, and lies loose at both ends.
Matebials.
Square Bar
Onk Third.
Round Bar.
One Third.
Oak
800
1137
o69
916
600
2580
4013
269
379
189
305
200
860
1338
628
893
447
719
471
2026
3152
209
298
149
239
157
675
1050
Ash
Elm
Pitch-pine
Pine
Wroui'lit-iron
To find the ultimate transverse strength of any rectangular
beam, supported at both ends and loaded in the middle,
Multiply the strength of an inch square bar 1 foot long, as in the
Table, by the breadth, and by the square of the depth, in inches, and
divide the product by the length, in feet.
The quotient will be the weight in avoirdupois pounds.
Remark 1. — When a beam is supported in the middle and loaded at each end,
it will bear the same weiglit as when supported at both ends and loaded in the
middle ; that is, each end will boar half the weight.
Remark 2. — When the Aveight is applied somewliero between the middle and
the end of the beam, multiply twice the length of the long end by twice the length of
the short end, and divide the product by the whole length of the beam. The quotient^,
is the effective length of the beam.
Remark 3. — If the beam is round, multiply the ultimate strength of the round bar,
in the Table, by the cube of the diameter, in inches, and divide the product by the
length, IN FKET.
Remark 4. — When a beam is fixed at both ends and loaded in the middle, it
will bear one half more than when loose at both ends. If the beam is loose at
both ends, and tho weight is applied uniformfy along its length, it will bear
double; but if fixed at both ends, and tho weight applied uniformly along its
length, it will bear triple tho weight.
EXAMPLES.
1. What weight will break a beam of asli 5 inches broad, 7 inches
deep, and 26 feet deep between the supports?
1137X^X7^
Ans.
:111431b., nearly.
APrENDix. 355
2. What is the ultimate strength of an oak Lcam 20 feet long,
4 inches broad, 8 inches deep, and the weight placed G feet from
the end? ??^ =10.8 feet. (Vide Rem. 2.) Then,
800X4X8^^^,,,^,,^ ^,,.
lb,8
3. What is the ultimate transverse strength of a wrought-iron
solid cylinder, 10 feet long and 5 inches in diameter ?
Ans. "-^ =39400 lb. (Vide Rem. 3.)
ANNUITIES.
An Annuity is an estate which entitles its owner to the pay-
ment of a fixed sum, at regular intervals of time.
The annuity, time, and rate of interest being given, to find the
amount,
Raise the ratio to a power denoted by the time, from which subtract 1;
divide the remainder by the ratio less 1, and multiply the quotient by the
annuity. The product will be the amount.
Remabk 1. — For the powers of the ratio, see Table under 249.
EXAMPLES.
1. What is the amount of a pension of $100 per annum, which
has remained unpaid for 5 years, interest 6 per cent.?
1.06^=1.338226. Then .338226--.06Xl00=$563.71. Ans.
2. What is the amount of an annual rent of $150, in arrears
for 12 years, at 6 per cent, compound interest?
Ans. $2530.489999.
3. Wliat is the amount of a pension of $900, in arrears 17 years,
at 7 per cent, compound interest? Ans. 27756.193.
4. What is the amount of an annual salary of $6000, in arrears
for 8 years, at 5 per cent, compound interest?
Ans. $57294.60.
5. What is the amount of a $100 pension, in arrears 20 years,
at 5 per cent.? 6 per cent.? 7 per cent.?
Ans. $3306.596 ; $3678.558 ; $4099.55.
G. What is the amount of a pension of $1000, in arrears for 12
years, at 7 per cent, compound interest? Ans. $17888.45.
356
APPENDIX.
The annuity, time, and rate of interest being given, to find the
present worth,
Divide the amount^ as found ahove^ hy the ratio raised to a power
denoted hy the time.
EXAMPLES.
1. What is the present worth of a pension of $100, to continue
5 years, at 6 per cent, per annum?
^563.71-f-1.0G'^=$421.24. Ans.
Remauk 2. — In this way the following Table may be constructed :
v. — TABLE:
Shoioing the presetit value of $1.00 for any number of years, from 1 to 25, at 5, G,
7, 8, and 10 per cent.
Yeaes
5 Per Cent.
6 Per Cent.
7 Per Cent.
8 Per Cent.
10 Per Cent.
1
0.952381
0.943396
0.934579
0.925926
0.909091
2
1.859410
1.833393
1.808018
1.783265
1.735537
3
2.723248
2.673012
2.624316
2.577097
2.486852
4
3.545951
3.465106
3.387211
3.312127
3.169865
5
4.329477
4.212364
4.100197
3.992710
3.790787
6
5.075692
4.917324
4.766540
4.622880
4.455261
7
5.786373
5.582381
5.389289
5.206370
4.868419
8
6.463213
6.209794
5.971299
5.746639
5.334926
9
7.107822
6.801692
6.515232
6.246888
5.759024
10
7.721735
7.360087
7.023582
6.710081
6.144557
11
8.306414
7.886875
7.498674
7.138964
6.495061
12
8.863252
8.383844
7.942686
7.536078
6.813692
13
9.393573
8.852683
8.357651
7.903776
7.103356
14
9.898641
9.294984
8.745468
8.244237
7.366687
15
10.379658
9.712249
9.107914
8.559479
7.606080
16
10.837770
10.105895
9.446649
8.851369
7.823701
17
11.274066
10.477260
9.763223
9.121638
8.021553
18
11.689587
10.827603
10.059087
9.371887
8.201412
19
12.085321
11.158116
10.335595
9.603599
8.364920
20
12.462210
11.469921
10.594014
9.818147
8.513564
21
12.821153
11.764077
10.835527
10.016803
8.648694
22
13.W3003
12.041582
11.061241
10.200744
8.771540
23
13.488574
12.303379
11.272187
10.371059
8.883218
24
13.798642
12.550358
11.469334
10.528758
8.984744
25
14.093945
12.783356
11.653583
10.074776
9.077040
2. What is the present worth of a pension of $800, to continue
25 years, at 10 per cent.? Ans. $2723.11.
APPENDIX.
357
MISCELLANEOUS TABLE.
12 units make 1 dozen.
12 dozen " 1 gross.
12 gross " 1 great gross.
20 units " 1 score.
100 years " 1 century.
10 centuries " 1 chiliad.
100 pounds " 1 quintal of fisli.
196 pounds " 1 barrel of flour.
200 pounds " 1 barrel of pork.
14 pounds " 1 stone,
21J stones " 1 pig-
8 pigs " 1 fother.
18 inches " 1 cubit.
6 feet " 1 fathom.
24 sheets " 1 quire.
20 quires r... " • 1 ream.
2 reams " 1 bundle.
5 bundles " 1 bale.
FRENCH WEIGHT.
Frexch weight is that used in the empire of France. Its units
are named milligramme, centigramme, decigramme, gramme, decagramme,
hectogrmme, kilogramme, myriagramme, quintal, millier or bar. The
milligramme is the unit of lowest value, and the bar the highest.
It takes 10 of any order to make one of the next higher order of
units, except that 100 quintals make 1 bar.
The GRAMME is the fundamental unit, and is the weight of a
centimetre of pure water, at the temperature of melting ice, which
is 15.43402 grains Troy.
1 quintal = 1 cwt. 3 gr. 25 lb.; 1 millier or bar = 9 T. 16 cwt.
3 gr. 12 lb.
1 pound Avoirdupois = 453|
1 pound Troy
grammes.
FRENCH LINEAR MEASURE.
The units of this measure are named millimetre, centimetre, deci-
metre, METRES, decametre, hectometre, kilometre, and myriametre; and
it takes 10 units of any lower order to make 1 of the next higher.
358 APPENDIX.
The METRE is the fundamental unit, and one of the ten million
equal parts into which the meridian distance from the equator
to the north pole is divided. This distance is 39.37079 English
inches. It is thence easy to find the value of any French unit in
English measure.
FRENCH SUPERFICIAL MEASURE.
The units of this measure are named milliare, centiare^ deciare,
ARE, decare, hectare, kilare, and myrlare; and it takes 10 units of a
lower order to make 1 of the next higher.
The ARE is the fundamental unit, and is a square decametre^
which is equivalent to 1076.4298 square feet.
FRENCH SOLID MEASURE.
The units of this measure are named centistere, decistere, stere,
and decastere; and it takes 10 units of a lower order to make 1 of
the next higher.
The stere is the fundamental unit, and is a cubic metre, which
is equivalent to 35.3174 cubic feet.
FRENCH MEASURE OF CAPACITY.
The units of this measure are named millitre, centilitre, decilitre,
LITRE, decalitre, hectolitre, kilolitre, and myrialitre.
The LITRE is the fundamental unit, and is a cubic decimetre,
which is 61.027051 cubic inches.
1 litre = 1| English pints; 1 hectolitre = 22 English gallons.
1 decalitre = 2.6414308 wine gallons; 1 hectolitre =2.834 Win-
chester bushels.
RATES OF FOREIGN MONEY OR CURRENCY.
(FIXED BY LAW.)
Ducat of Naples $0 80
Florin of the Netherlands 40
Florin of the Southern States of Germany 40
Florin of Austria and Trieste 48J
Florin of Nuremburg and Frankfort 40
Florin of Bohemia 48^
Guilder of the Netherlands 40
Lira of Lombardo and the Vcuetiuu Kingdom 16
ArpENDix. 3-59
Livre of Leghorn $0 16
Lira of Tuscany 16
Lira of Sardinia 18f
Livre of Geneva 18f
Milrea of Portugal 1 12
Milrea of ]\Ladeira 1 00
Milrea of Azores 83i-
Marc Banco of Hamburg 35
Ounce of Sicily 2 40
Pound Sterling of Jamaica 4 84
Pound Sterling of the British Provinces 4 00
Pagoda of India 1 84
Real Vellon of Spain 05
Real Plate of Spain 10
Rupee of British India 44|
Rix Dollar (or Thaler) of Prussia and North Germany.. 69
Thaler (or Rix Dollar) of Bremen 78|
Thaler (or Rix Dollar; of Berlin, Saxony, and Leipsic... 69
Rouble (Silver) of Russia 75
Specie Dollar of Denmark 1 05
Specie Dollar of Norway 1'06
Specie Dollar of Svreden 1 06
Tale of China 1 48
Banco Rix Dollar of Sweden and Norway 39|
Banco Rix Dollar of Denmark 53
Crown of Tuscany 1 05
Curacoa Guilder 40
Leghorn Dollar or Pezzo ^Oj^iny
Livre of Catalonia 53|
Livre of N\ifchatel 26|
Swiss Livre 27
Scudi of Malta 40
Roman Scudi 99
St. Gall Guilder 403-^^^
Rix Dollar of Batavia 75
Roman Dollar 1 05
Turkish Piastre 05
Current Mark 28
Florin of Prussia..! , 22f
360 APPENDIX.
Florin of Basle $0 41
Genoa Livre 21
Livre Tournois of France. 18|
Rouble (paper) of Russia, varies from 4j^^^ to i^^^jj to the dollar,
BOOKS AND PAPER.
NAMES AND SIZES OF PAPER MADE BY MACHINERY.
Double Imperial 32 by 44 inches.
Double Superroyal 27 by 42 "
Double Medium 23 by 26 "
" " 24 by 37i «
" " 25 by 38 "
Royal and Half 25 by 29 "
Imperial and Half 26 by 32 «
Imperial 22 by 32 «
Superroyal 21 by 27 «
Royal 19 by 24 «
Medium ^. ^ 181 by 23| «
Demy ." 17 by 22 "
Folio Post ,. ♦ 16 by 21 "
Fo?4^5ap.....^...':..:?^^ ,| 14 by 17 «
fo-wn
Cro-wl....c:r:../ ^!V.... 15 by 20 "
^ sheet folded in 2 leaves is called -dT folio.
A sheet '' 4 ' "" " ^^ quarto or 4to,
A sheet " 8 ''■" "an octavo or 8vo.
' A'jshee'f , " 12 " "^ " a 12rao.
A sheet " 18 " " an 18mo.
A sheet " 24 " ** a 24mo.
A sheet « 32 » " a 32mo.
THE END.
s
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M305987
T 5'9
THE UNIVERSITY OF CALIFORNIA LIBRARY
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