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THE
FRANKLIN
WRITTEN ARITHMETIC
'With (gixamples for @ral f nuttte
BY
EDWIN P. SEAVER, A.M.
SUPERINTENDENT OF PUBLIC SCHOOLS, BOSTON
GEORGE A. WALTON, A.M.
AUTHOR OF Walton's arithmetics, arithmetical tables, etc
PHILADELPHIA
PUBLISHED BY J. H. BUTLER
BOSTON
WILLIAM WARE & CO.
„ c • 4 ^o c c.
Copyright
By E. p. SEAVER and G. A. WALTON.
University Press: John Wilson & Son,
Cambridge.
PREFACE.
The Franklin Written Arithmetic contains a full course of arithmetical
instruction and drill for pupils in the common schools. The definitions
and principles are thoroughly illustrated and explained, so that the leanier
may work intelligently ; while the range of applications is broad and varied
enough to afford him good preparation for ordinary business affairs.
Topics of a merely theoretical interest, antiquated or curious matter, and
puzzling problems, are omitted altogether ; while parts of the subject not
very necessary to the greater number of pupils are given in the Appendix.
To avoid a multiplicity of rules, decimals and integers have been treated
together whenever that could easily be done. For the same purpose, the
various problems in Percentage have all been referred to a few fundamental
principles, stated and illustrated at the outset.
The Metric System has been treated in a way to indicate the most prac-
tical course to pursue in teaching it.
The topics that follow Simple Interest '-i this book, as in most arith-
metics, can wisely be deferred till the last years of the common-school
course.
In the arrangement of work it will be noticed that Oral Exercises precede
Examples for the Slate. For convenience in the class-room, the latter are
numbered consecutively through the whole section, with the exception of
four pages of typical examples (pages 17, 25, 35, and 48), which are let-
tered. The Oral Examples are designated by letters. All answers to
Examples for the Slate, except those to Illustrative and Typical Examples,
are omitted from the body of the book. Miscellaneous Examples are given
in great number and variety, and each section is supplemented by a set of
questions for review.
A special feature of the book is the Drill Exercises. In general character
these are like those previously published by Walton and Cogswell in their
Book of Problems ; and they have been introduced in this book by consent
of ]\Ir. Cogswell. They give a large number of miscellaneous examples,
with answers, on all the topics treated in the Arithmetic ; and the teacher
will be spared the trouble of selecting from o-ther books examples for
class drill. The fact that these exercises have been extensively imitated
in books published of late shows the high estimation in which they are
held by teachers and author^-
541853
TABLE OF OONTEKTS.
Page
Reading AND Writing Numbers 1
Addition 12
Subtraction 20
Multiplication 28
Division 38
United States Money 64
Addition, Subtraction, Multi-
plication, and Division .... 65
Coins and Paper Currency. ... 67
Accounts and Bills 68
Factors 75
Symbols of Operation 79
Cancellation 80
Greatest Common Factor 82
Multiples 84
Least Common Multiple 85
Common Fractions 88
Reduction 90
Addition 94
Subtraction 97
Multiplication 99
Division 105
To find the Whole when a
Part is given Ill
To find what Fraction one
Number is of another 113
Aliquot Parts 114
Decimal Fractions 124
Reading and Writing 8, 124, 300
Reduction 125
Addition 15, 17, 19
Subtraction 24, 25, 27
Multiplication 34, 35, 36, 130
Division ...43, 44, 45, 48, 49, 132
Page
Weights and Measures 136
Compound Numbers 146
Reduction 1 47
Fundamental Operations 152
Longitude and Time 158
Mensuration of Surfaces and
Solids.... 160
Metric System 172
Percentage 185
Profit and Loss 192
Commission 194
Stocks, Dividends, and Brok-
erage 197
Insurance 199
Taxes 201
Customs or Duties 204
Simple Interest 209
Accurate Interest 215
Partial Payments 216
Problems in Interest 220
Present Worth and Discount 224
Bank Discount 226
Commercial Discount 230
Compound Interest 231
Average of Payments 234
Average of Accounts 238
Exchange 241
Bonds 245
Ratio and Proportion 254
Partnership 263
Powers and Roots 265
Square Root 267
Cube Root 272
Mensuration 277
Appendix 299
Drill Exercises 57, 59, 63, 73, 123, 135, 171, 253
General Reviews 50, 117, 165, 205, 247,293
Questions for Review 11, 56, 73, 115, 135, 168, 250, 264, 294
Miscellaneous Examples... 27, 37, 52, 74, 118, 119, 166, 206, 229, 248,
295, 815
AKITHMETIC.
SECTION I.
READING AND WRITING NUMBERS.
Article 1. A collection of single things or ones is a
number. By common usage one is also called a number.
2. A knowledge of numbers is Arithmetic.
3. Some numbers have simple names. These are one^
two, three, four, Jive, six, seven, eight, nine, ten ; also a hun-
dred, a thousand, a million, etc. All other numbers have
compound names. (See Appendix, p. 299.)
Exercises.
1. Name the numbers in regular order, or count, from one
to fifty; from fifty to one.
2. Count to a hundred by twos ; by fives ; by tens. Count
from a hundred downward by twos ; by fives ; by tens.
3. Name the number that is made up of two tens and five
ones ; of one ten and seven ones ; of one ten and a one ; of
one ten and two ones ; of six tens and six ; of eight tens and
five ; of nine tens ; of nine tens and nine ; of ten tens.
4. Name the number that is made up of one hundred, one
ten, and a one ; of two hundreds, seven tens, and three ones ;
of six hundreds and three tens ; of five hundreds and four
ones ; of four hundreds, three tens, and three ones ; of nine
hundreds, nine tens, and nine ones.
r'„A-:i r/c,^- ; ; _; UBADtNG AND WRITING
4. From their names we see that small numbers are
reckoned by ones, larger numbers by tens, and still larger
numbers by hundreds, as far as ten hundred. To ten hun-
dred we give the simple name thousand.
Above a thousand numbers are reckoned by thousands,
by tens of thousands, and by hundreds of thousands, up to
ten hundred thousands, or a thousand thousands, which we
call a million.
Above a million numbers are reckoned by millions, by
tens of millions, and by hundreds of millions, up to a
thousand millions, which we call a billion.
Above a billion numbers are reckoned by billions, by tens
of billions, and by hundreds of billions, up to a thousand
billions, which we call a trillion. And so we go on with
higher numbers as far as we choose.
6. One, Ten, a Hundred, a Thousand, Ten-thousand, a
Hundred-thousand, a Million, etc., are called units, because
they are used in reckoning or measuring other numbers.*
6. To distinguish these units, we call one a unit of the
first order, ten a unit of the second order, a hundred a
unit of the third order, a thousand a unit of the fourth
order, ten thousand a unit of the fifth order, and so on.
When we speak of a unit without mentioning the order,
we usually mean a unit of the first order, or one.
7. These units form a scale; and because ten units. of
any order make a unit of the next higher order, the scale
is called a scale of tens, or a decimal scale.
* A unit is a fixed quantity of any kind used to measure other quan-
tities of the same kind. Thus, a foot, a yard, a meter, are units, being
fixed lengths used to measure other lengths ; a pound, an ounce, a dollar,
a cent, an hour, a second, are units, used to reckon or measure weight,
value, or time. The word unit is also much used as a name fo» one, and
units for ones.
SIMPLE NUMBERS. o
8. A system of numbers whose successive units form a
scale of tens is a decimal system of numbers. The sys-
tem of numbers in common use is a decimal system.
9. Table of Units of the Different Orders.
Ten ones (or units) . . . make a Ten,
Ten tens make a Hundred,
Ten hundreds. .... make a Thousand,
Ten thousands .... make a Ten-thousand,
Ten ten- thousands .... make a Hundred-thousand,
Ten hundred-thousands . . make a Million,
and so on.
10. It will be convenient to remember that
A thousand ones (or units) . . are a Thousand,
A thousand thousands . . are a Million,
A thousand millions . . . are a Billion,
A thousand billions . . . are a Trillion,
and so on.
11. Exercises.
5. Count by hundreds to a thousand ; to two thousand ; to
two thousand five hundred.
6. Count by thousands to ten thousand ; by tens of thou-
sands to a hundred thousand ; by hundreds of thousands to a
million.
7. Count by millions to ten million ; by tens of millions to
a hundred million ; by hundreds of millions to a billion.
8. How many units of each order are there in twentj^-'five ?
seventeen? eleven? ninety? ninety-nine? four hundred? five
hundred forty-four ? one thousand eight hundred ?
Note. This kind of exercise may be extended at the discretion of
the teacher.
4 BEADING AND WRITING
Writing Numbers.
12. Besides being expressed in words, numbers are ex-
pressed by writing the signs 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
which are called figures. These signs are also called
Arabic numerals, because they were first made known to
Europeans by the x4rabs.*
13. The first of these signs, 0, is called zero, or cipher
and is used to stand for no number; the others are used to
stand for the first nine numbers, and take their names, thus '
1, 2, 3, 4, 5, 6, 7, 8, 9.
one, two, three, four, five, six, seven, eight, nine
14. Numbers higher than nine have no single signs foi
themselves, but are expressed by writing side by side two
or more of the figures above.
16. Tens are expressed by writing a figure to tell hey
many tens, and then writing a zero at the right of it. The
tens' figure is then said to stand in the second place, the
first or units' place being filled by a zero. Thus, we write
Ten (one ten), 10. Forty (/owr tens), 40. Seventy (seven tens), 70.
Twenty (two tens), 20. Fifty {five tens), 50. Eighty {eight tens), 80.
Thirty {three tens), 30. Sixty {six tens), 60. Ninety {nine tens), 90.
16. Numbers that are made up of tens and ones are
expressed by writing a figure in the second place for the
tens, and a figure in the first place for the ones. Thus,
Eleven (ten and one), 11. Twenty-one (two tens and one), 21.
Twelve (ten and two), 12. Twenty-two (two tens and two), 22.
Let the teacher dictate numhers between ten and a hundred for the
pupil to write.
* For an account of the Roman numerals, which were displaced by the
Arabic, see Appendix, p. 299.
SIMPLE NUMBERS. O
17. Hundreds are expressed by writing a figure in the
third place, the second and first places being filled by
zeros. Thus,
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.
18. Numbers made up of hundreds, tens, and ones are
expressed by writing a figure in the third place for the
hundreds, a figure in the second place for the tens, and a
figure in the first place for the ones. Thus,
Four hundred eighty-three (4 hundreds, 8 tens, 3 ones), 483.
Nine hundred sixty (9 hundreds, 6 tens, no ones), 960.
Nine hundred six (9 hundreds, no tens, 6 ones), 906.
Let the teacher dictate numbers between a hundred and a thousand for
the pupil to write.
19. Thousands, tens of thousands, and hundreds of
thousands are expressed by writing figures in the fourth,
fifth, and sixth places respectively. The figures in these
three places taken together form the Thousands' group;
while the figures in the hundreds', tens', and units' places
taken together form the Units' group. These groups are
usually separated by a comma. Thus,
One thousand, . . . 1,000. Three thousand, . . 3,000.
Ten thousand, . . . 10,000. Twenty thousand, . . 20,000.
A hundred thousand, 100,000. Five hundred thousand, 500,000.
Five hundred twenty-three thousand, 523,000.
Six hundred eight thousand, seven hundred twenty-eight, . 608,728.
20. Exercises.
Write in figures:
9. Four thousand.
10. Four thousand four hundred.
11. Four thousand forty.
12. Four thousand four.
b READING AND WRITING
13. Eight thousand, four hundred twenty-two.
14. Three hundred fifty-six thousand, eight hundred ninety.
15. Sixty thousand, sixty-five.
16. Eighteen hundred seventy-eight.
Let the teacher dictate other numbers, to a miUion, for the pupil to
write.
21. The examples above given illustrate the principle
on which all numbers are written, and which is this :
Units of any order are expressed hy writing a figure in
the place corresponding to that order.
If the units of any orders are wanting in the number,
the corresponding places are filled by zeros.
22. The general method of writing numbers on the
principle above stated is shown by the following
TABLE.
i
a
3
is I § ^ 1 .2 «
S -i^g -0.2. •eiSffl '«§fl -o
•2-?;§ '3'?. 2 'H?a "3^3 'i«a-{S
Hhh Hnm M&h^ Mhh M&Ht?
w^S? 3q?3o Sooic- S5§ «(§iH
480,2 97,034,508, 672 Figures.
ci «h group, 4th group, 3d group, 2d group, ^^*^ ^^'^^^' \ Groum
® Trillions. Billions. Millions. Thousands. Units. '
Note. For the names of higher numbers, see Appendix, page 300.
23. This table shows that the figures used to express a
number fall into groups of three figures each. The first
group expresses simple units, tens, and hundreds ; the sec-
ond, units, tens, and hundreds of thousands; the third,
units, tens, and hundreds of millions ; and so on.
SIMPLE NUMBERS. 7
These groups are called the units^ group, the thousands'
group, the millions' group, etc., from the lowest order of
units which they express.
Note. The units themselves are grouped as the figures are. (Arts. 4
and 10.)
24. In writing large numbers it will be found con-
venient to think chiefly of the groups as above described.
Thus, let it be required to write the number Forty -nine
billion, three hundred seven million, seventy thousand, six
hundred forty-three. The groups are
49 union, 307 million, 70 thousand, 643.
and the number itself is written
49,307,070,643.
25. E:sercises.
I. Beginning with the units' group, repeat the names of the
groups to trillions ; repeat the names from trillions to units.
II. Write the groups of figures required to express the fol-
lowing numbers, with the names of the groups :
17. Forty-six thousand, five hundred twenty.
18. Four hundred six thousand, five hundred two.
19. One million, one thousand, one hundred ten.
26. Exercises.
I. Write in figures the following numbers :
20. Eighty-five million, five hundred three thousand, seven.
21. Nine hundred six million, two hundred eighteen thou-
sand, twenty-eight.
22. Three billion, thirty-seven million, nine hundred thou-
sand, two hundred.
23. Eighteen billion, four.
24. Forty million, seven hundred thousand.
25. Thirty-seven trillion, ninety-nine billion, nine million.
8 READING AND WRITING
II. Write in figures as many of the numbers named on
page 62 as the teacher may indicate.
Reading Numbers.
27. Let it be required to read the number 53869214.
To prepare this expression for reading, we begin at the
right, and point off three figures for the units' group, three
more for the thousands' group, leaving two for the millions'
group, thus :
^ 53,869,214.
Now beginning at the left, we name the number ex-
pressed by each group, adding the name of the group, thus :
Fifty-three million, eight hundred sixty -nine thousand, two hundred
fourteen \units\
Note. The name of the units' group is usually omitted in reading.
28. Exercises.
I. Read the following :
(26.) 361. (30.) 9000200. (34.) 3670980347.
(27.) 3261. (31.) 86320029. (35.) 9008007006.
(28.) 9301. (32.) 81402020. (36.) 7676767676.
(29.) 6,54327. (33.) 89743208. (37.) 90002000.
II. Read across the page as many of the numbers expressed
on page 60 as the teacher may indicate.
29. It is frequently convenient to separate a number
into parts, each part containing only the units of a single
order. Thus, the number 734 may be separated into 7 hun-
dreds, 3 tens, and 4 units. Such parts are called the terms
of a number.
Decimal Fractions.
30. As n hundred is made up of ten equal parts, eacli
of which is a ten, and as a ten is made up of ten equal
DECIMAL FRACTIONS. »
parts, each of whicli is one^ so we may consider one to be
made up of ten equal parts, each of which is a tenth; a
tenth to be made up of ten equal parts, each of which is
a hundredth; a hundredth to be made up of ten equal
parts, each of which is a thousandth; and so on.
Now a hundred is written 100 ; the tenth part of a hun-
dred (ten) is written 10, the figure 1 being moved one place to
the right ; and the tenth part of ten (one) is written 1, tlie
figure 1 being moved one place further to the right; so, fol-
lowing the same plan, the tenth part of one (one tenth) is
written 0.1 ; the tenth part of a tenth (one hundredth) is
written 0.01 ; the tenth part of a hundredth (one thousandth)
is written 0.001 ; and so on.
Tenths, hundredths, thousandths, etc., are fractional units.
or fractions; and, as they form a decimal scale (Art. 7),
collections of such units are called decimal fractions.
31. The dot put at the right of the units' place is called
the decimal point.
32. The relations of these fractional units to the higher
units are shown by the following table, which may be ex-
tended both ways as far as we please :
A thousand 1000.
A hundred 100.
Ten 10.
One , . . 1.
A tenth 0.1
A hundredth 0.01
A thousandth 0.001
33. We see then that decimal fractions may be written
on the principle stated in Art. 21. Thus, we write
Two tenths 0.2 Three thousandths 0.003
Five hundredths . . . 0.05 Thirty-two thousandths .... 0.032
Twenty-five hundredths 0.25 Three hundred sixteen thousandths 0.316
10 READING AND WRITING
34. The method of writing decimal fractions is shown
by the following table, which is merely an extension of the
table given in Art. 22.
TABLE.
p
li
1 1 i 1 1 a I
Names of the Units. •"|§fl§gfl§dg'^
oj "U 'O ■+» '+3 .+3 -f3 +a -t?
PZaces.
f-igurcs. 0.708963432
Note. In writing decimal fractions it is well to fill the units' place with
a zero when there is no other figure to. be written there.
36. To read a decimal fraction, name the number ex-
pressed by the figures, and then add the name of the units
expressed by the right-hand figure.
Thus, 0.0739 is read " seven hundred thirty-nine ten-
thousandths.'" See Appendix, p. 300.
When a whole nuTnber and a decimal fraction are
written together, read first the lohole number and then the
fraction.
Thus, 56076.028 is read "fifty-six thousand seventy-six,
and twenty-eight thousandths."
36.
Exercises.
'.. Read the following :
(38.) 0.7 (43.)
0.072
(48.)
2548.
(39.) 0.03 (44.)
0.0806
(49.)
254.8
(40.) 0.25 (45.)
5.05
(50.)
25.48
(41.) 0.83 (46.)
4.056
(51.)
2.548
(42.) 0.005 (47.)
7.0056
(52.)
0.02548
DECIMAL FRACTIONS. 11
II. Write in figures the following numbers :
(53.) Seven tenths. (58.) 7 units and 5 thousandths.
(54.) Seven hundredths. (59.) 25 units and 49 ten-thou-
(55.) Seven thousandths. sandths.
(p^.) Twenty-five hundredths. (60.) 306 hundred-thousandths.
(57.) Thirty-nine thousandths. (61.) 5047 hundred-thousandths.
Let the teacher dictate other numbers between units and millionths for
the pupil to write.
37. Questions for Review.
What is a number? How are numbers reckoned? (Art. 4.) What
general name do you give to one, a ten, a hundred, a thousand, etc. ?
How do you distinguish the different units ? What kind of a scale
do they form ? What system of numbers is in common use ? Why
is it so called ?
What is the meaning of the word thirteen f eleven ? twelve f twenty f
thirty-seven f (Appendix, page 299.)
How many units make a thousand ? How many thousands make
a million ? How many millions make a billion ?
What is the use of figures ? How are numbers higher than nine
written? On what principle are all numbers written? (Art. 21.)
What is the use of zeros ?
How are the figures used to express a number grouped? Name
the first five groups. How do you write large numbers ? (Art. 24.)
Dlustrate. How do you read a number? (Art. 27.) Illustrate.
What are the terms of a number ? Name the terms of the number
6725.
What is the largest number that can be expressed by one figure ? by
two figures ? by three figures ?
What is the least number of figures that will express units ? thou-
sands? millions?
In 100, how many tens? how many units? In 15000, how many
hundreds? units? tens? In 18462, how many tens, and how many
units remain ? how many hundreds, and how many units remain ?
What is the eff'ect of placing zeros at the right of an expression for
whole numbers ? at the left ?
12 ADDITION,
SEOTIOlSr II.
ADDITION.
38. If you have 5 cents and 3 cents and 2 cents, and
count them together, how many cents do you find there are ?
Counting them together, you find there are 10 cents.
39. The process of counting numbers together is ad-
dition.
40. The result found by addition is the sum or amount
of the numbers added. Thus, the sum of 5 and 3 and 2 is 10.
41. The addition of numbers is indicated by the sign + ,
which is read plus.
The sign = indicates equality, and is read equals, or
is equal to. Thus, the expression 5 + 3 + 2 = 10 means
that the sum of 5 and 3 and 2 is 10, and is xead "five plus
three plus two equals ten."
42. Oral Exercises.
I. Name the sums of the pairs of numbers expressed below
till you can give them rapidly at sight :
a. b. c. d. e. f. g. h. i. j. k. 1.
44645323473 6
233426346232
6
7
8
4
8
7
5
3
6
9
5
2
8
4
8
8
3
8
4
8
7
2
9
7
8
5
3
3
9
6
4
6
6
2
9
7
4
8
5
7
6
4
7
1
6
9
4
5
ORAL EXERCISES. 13
a.
b.
c.
d.
e.
f.
&
h.
i
3-
k.
1.
2
9
3
8
5
7
8
2
9
8
.2
7
8
7
9
6
6
1
7
2
5
9
5
6
7
8
9
1
2
5
4
5
8
6
8
1
7
2
3
8
4
3
9
5
5
9
1
7
3
2
5
7
4
9
7
6
9
9
5
8
2
6
7
3
5
9
9
5
1
8
9
2
II. Count to a hundred or more,
jn.. By twos, beginning with 2 ; with 1.
n. By threes, beginning with 2.
o. By fours, beginning with 3 ; with 2.
P' By fives, beginning with 4 ; with 3 ; with 2 j with 1.
Of' By sixes, beginning with 5 ; with 4.
r. By sevens, beginning with 6.
s^. By eights, beginning with 7 ; with 6.
t. By nines, beginning with 8.
III. Add the numbers expressed in the following columns :
(i.) {2.) is.) (4.) (5.) (6.) (7.) (8.)
2
9
4
4
6
60
600
6000
3
6
2
3
3
30
300
3000
5
3
6
7
8
80
600
9000
7
2
8
4
8
80
800
8000
6
5
8
5
3
30
400
6000
3
8
4
8
5
50
500
5000
9
4
2
9
3
30
300
3000
1
7
8
7
9
90
700
9000
Begin at the bottom and add upward, naming only the
results. Thus, in the first column, say 1, 10, 13, 19, 26, 31,
34, 3G; sum, 36. Now, to see if you are right, begin at the
14 ADDITION.
top and add downward. Thus, 2, 5, 10, IT, 23, 26, 35, tJ6;
sum again, 36. Practise exercises of this kind till you can
add with great rapidity.
For further drill of this sort the teacher is referred to exercises on pages
59 and 61.
Examples for the Slate.
43. Illustrative Example I. What is the sum of
413, 102, and 134?
WRITTEN WORK. Explanation. — To find the sum of large numbers
,-, o like these, we add the units, the tens, and the hun-
. ^„ dreds separately ; hence, for convenience, we write
the numbers so that units of the same order may
be expressed in the same column. (Art. 6.)
Sum, 649 Drawing a line beneath, and adding the units
(thus, 4, 6, 9), we find there are 9 units, which we
write under the line in the units' place. Adding the tens (thus, 3, 4),
we find there are 4 tens, which we write under the line in the tens'
place. Adding the hundreds (thus, 1, 2, 6), we find there are 6
hundreds, which we write under the line in the hundreds' place. The
sum is, then, 6 hundreds 4 tens and 9 units, or 649.
44. Illustrative Example II. What is the sum of
960, 748, 932, and 867 ?
WRITTEN WORK. Explanation. — Writing the numbers as before,
ggQ ■ and adding the units (thus, 7, 9, 17), we find there
«. JO are 17 units, which are equal to 1 ten and 7 units.
QQ9 ^^ write the 7 units in the units' place, but keep
the 1 ten to add with the tens expressed in the next
^^^ column. Adding the tens (thus, 1, 7, 10, 14, 20),
Sum, 3507 we find there are 20 tens, which are equal to 2
hundreds and no tens. We write in the tens'
place, to show there are no tens in the sum, but keep the 2 hundreds
to add with the hundreds expressed in the next column. Adding the
hundreds (thus, 2, 10, 19, 26, 35), we find there are 35 hundreds,
which are equal to 3 thousands and 5 hundreds. We write 5 in the
hundreds' and 3 in the thousands' place. The sum is, then, 3 thou-
sands 5 hundreds tens 7 units, or 3507.
EXAMPLES. 16
Keeping a number and adding it with the numbers
expressed in the next column is called carrying.
In working examples, use as few words as possible. Thus,
in the above example, say merely, " 7, 9, 17 j * 1, 7, 10, 14,
20; 2, 10, 19, 26, 35; sum, 3507/'
1. Add together 6234, 785, and 5861.
2. Add together 582, 2, 49, and 124.
3. How many are 2356, 8004, and 987 ?
4. Find the sum of 70639, 600, and 7000.
5. Add 76, 33, 92, 53, 305, 78, 8, and 19.
6. What is 213 + 819 + 37 + 66 ?
Addition of Decimals.
45. Illustrative Example III. What is the sum of
425.37, 433.126, 0.076, 442.09, 0.6, and 0.319 ?
WRITTEN WORK.
Explanation. — Writing the numbers so that
units of the same order may be expressed in the
4:2,0.61 same column, we begin with the units of the low-
433.126 est order (in this case thousandths) to add, and
0.076 proceed in the manner already explained, briefly
442.09 thus : thousandths, 9, 15, 21 ; write 1, carry 2 ;
OQ hundredths, 2, 3, 12, 19, 21, 28 ; write 8, carry 2 ;
0.319 tenths, 2, 5, 11, 12, 15; write 5, carry 1; units,
- 1, 3, 6, 11 ; write 1, carry 1 ; tens, 1, 5, 8, 10 ;
Sum, 1301.581 write 0, carry 1; hundreds, 1, 5, 9, 13; write 13;
sum, 1301.581.
7. Add together 90.7, 43.68, 0.045, and 0.812.
8. Add together 0.005, 2.864, 0.9, and 0.25.
9. Add together forty-two thousandths, one hundred seven-
teen thousandths, thirteen and twenty-two hundredths, seven
and five hundredths.
* Do not stop to say " write 7 and carry 1," but do it.
16 ADDITION.
46. From the preceding examples we may derive the
following
Rule for Addition.
1. Write the numbers to be added so that units of the same
order may be expressed in the same column. Draw a line
beneath.
2. Add the units of each order separately, beginning with
those of the lowest order.
3. When the sum of the units of any order is less than
ten, write it under the line in its proper place ; when ten or
more, write only the units of the sum, and carry the tens to
the numbers expressed in the next column.
4. Write the whole sum of the last addition.
Proof.
Repeat the work, adding dowmvard instead of upward.
Adding two or more columns at once.
47. Accountants often add at once the numbers ex-
pressed in two, three, or more columns. The- following
example will illustrate the method :
WRITTEN WORK. Explanation. — Beginning with 29, add to it first
OK the 4 tens and then the 2 units of 42 ; then to the
rrn sum the 8 tens and the 7 units of 87 ; and so on.
Naming the results merely, say 29, 69, 71 ; 151,
158; 188, 192; 242, 245; 315,317; 347, 352. Add-
ing downwards, say 35, 105, 107; 157, 160; 190,
^'^ 194; 274, 281; 321, 323; 343, 352.
^ After practice it will he found unnecessary
to name all the results ; and it is by omitting
63
34
29
Sum, 352 to name them that great rapidity is acquired.
Note. The examples on the opposite page embrace the chief varieties
in form of examples in Addition. After performing these, and before taking
the Applications on page 18, pupils will usually need additional practice in
similar work. Examples for such practice will be found on pages 59-63.
EXAMPLES. 1?
48. Examples in Addition.
a. Add 5274, 206, 87, and 428. Ans. 5995.
b. Add 132, 3618, 7, and 53. Ans. 3810.
c. Find the sum of 8972, 980, 5607, and 89. Ans. 15648.
d. What is the sum of 34, 4800, 147, and 675 ? Ans. 5656.
e. How many are 346, 4682, 64, and 798 ? Ans. 5890.
/. What is the amount of 6079, 416, 346, and five thou-
sand one hundred sixty -four ? Ans. 12005.
g. Two thousand eight hundred twenty-one + nine hun-
dred nine + 376 + 43 equals what number ? Ans. 4149.
h. Six thousand two hundred ten + eight thousand eight
+ 4743 + 259 = what number ? Ans. 19220.
i. Five thousand fifty phis 9782 plus seven thousand seven
hundred seventy plus 842 are how many ? Ans. 23444.
j. Six hundred two plus 7524 plus six thousand twenty
plus 78 plus 4 are how many ? Ans. 14228.
k. How many miles are 467 miles, 1349 miles, nine hun-
dred seven miles, and sixty-four miles ? Ans. 2787 miles.
1. How many dollars are 7419 dollars, 864 dollars, four
thousand twenty-five dollars, and ninety dollars ?
Ans. 12398 dollars.
m. Add the numbers expressed by figures in the ex-
amples /, g, and h. Ans. 12262.
n. Add the numbers expressed by words in examples
/, g, and h. Ans. 23112.
E:samples -with Decimals.
o. What is the sum of 7.62, 14.2, 120.5, 9.08, 0.875, and
2.125 ? Ans. 154.4.
p. Find the sum of twenty- three thousandths, five. hun-
dredths, ninety-seven hundredths, seven and eight tenths,
fifteen and forty-one hundredths. Atis. 24.253.
For drill exercises, see pages 59 - 63.
18 ADDITION.
49. Applications.
10. A farmer raised 169 bushels of potatoes in one field,
262 bushels in another, 58 bushels in another, and 1827 in
another. How many bushels of potatoes did he raise in all ?
11. My cow Mabel gave 1388 pounds of milk in April,
1456 pounds in May, 1440 in June, 1317 in July, and 1175
in August. How many pounds did she give in all ?
12. I paid $2400* for my farm, $155 for a horse, %2Q for
a cart, 1 86 for a mowing-machine, $ 10 for a horse-rake, and
1 108 for a yoke of oxen. What did I pay for all ?
13. A merchant buys 5 bales of cloth, the first containing
768 yards ; the second, 754 yards ; the third, 698 yards ; the
fourth, 702 yards; and the fifth, 1003 yards. How many
yards are there in all ?
14. A planter sold 6 bales of cotton, weighing as follows :
the first, 495 pounds ; the second, 509 pounds ; the third, 508
pounds ; the fourth, 498 pounds ; the fifth, 526 pounds ; and
the sixth, 487 pounds. What was the whole weight ?
15. A merchant bought at one time 324 barrels of flour for
$2430; at another, 260 barrels for $2080; at another, 500
barrels for $ 3000 ; and at another, 107 barrels for $ 749.
How many barrels did he buy ? How much did he pay in all ?
16. A steamship sailed 203 miles on Monday, 243 miles on
Tuesday, 214 miles on Wednesday, 226 miles on Thursday,
239 miles on Friday, 241 miles on Saturday, and 238 miles on
Sunday. How many miles did she sail in the week ?
17. In St. Joseph's District, Michigan, there were at one
time 335530 peach-trees on 2953 acres of land, 57519 pear-
trees on 758 acres, 9786 plum-trees on 502 acres, 17654 cher-
ry-trees on 125 acres, 195995 apple-trees on 2958 acres, and
4988 quince-trees on 33 acres. How many acres of land were
occupied by these fruit-trees ? How many fruit-trees were
above in all ?
* $ is tie sign for dollars.
EXAMPLES. 19
18. The distance from Boston to Albany is 202 miles;
from Albany to Buffalo, 297 miles; from Buffalo to Toledo,
296 miles ; and from Toledo to Chicago, 243 miles. What is
the distance from Boston to Chicago ?
19. A man dying left by his will $42000 to his wife;
$ 14000 to his daughter ; 1 3500 in cash, and other property
worth % 13650, to his son ; $ 750 to each of his two nieces ;
and the remainder of his property, worth $2627, to his
brother. What was the value of the whole property?
50. Examples with Decimals.
20. Add together 3.07, 0.096, 8.431, and 0.7.
21. What is the sum of $875.16, $538.12, and $400,875?
22. What is the sum of 0.08 of a mile, 0.39 of a mile, and
4.7 miles ?
23. A man paid $15 for a coat, $8.50 for a hat, $6.75
for a pair of boots, and $ 3.45 for other articles. How much
did he pay in all ?
24. A surveyor measures four fields, and finds in the first
1.625 acres; in the second, 7.316 acres; in the third, 12.776
acres ; and in the fourth, 17.306 acres. How many acres in
all?
25. How far does a man travel who walks 5.5 miles before
breakfast, 17.25 miles between breakfast and dinner, 12 miles
between dinner and supper, and 0.875 of a mile after supper ?
(26.)
(27.)
(28.)
(29.)
(30.)
(31.)
$75.46
$754.60
$2.40
$476.48
$47.84
$780.00
18.72
42.87
74.09
207.42 '
98.76
15.85
9.47
5.30
53.67
77.99
3.69
119.fe
15.08
106.84
184.76
4.44
0.49
45.45
11.80
70.00
66.67
0.85
9.84
99.99
4.55
14.76
407.99
109.98
55.00
710.00
7.67
107.34
31.08
3.17
46.50
84.37
17.38
21.95
213.67
6.51
27.75
76.85
20 SUBTRACTION.
SECTION III.
SUBTRACTION.
51. If Charles has 9 apples and should give 4 of thein
away, how many apples would he have left ?
To find how many he would have left, we take 4, a part
of 9, away, and, by counting or otherwise, find there are 5
left ; thus we know that he would have 5 apples left.
62. The process of taking part of a number away to find
how many are left is subtraction.
53. The number, part of which is to be taken away, is
the minuend,
54. The part of the minuend to be taken away is the
subtrahend.
55. The part of the minuend left after a part has been
taken away is the remainder.
Name the minuend in the example above ; the subtrahend ; the
remainder.
56. The subtraction of numbers is indicated by the
sign — , which is read minus, or less.
Thus the expression 9-4=5 means 9 diminished by
4 equals 5, and is read " nine minus four equals five," or
"nine less four equals five."
57. Oral Exercises.
I.
Give rapidly the remainders in the following examples :
a.
b. c. d. e. f. g. h. i.
3
3 5 7 6 13 11 12 14
1
23424765
ORAL EXERCISES. 21
a.
b.
c.
d.
e.
f.
§•
h.
i
5
4
7
6
13
12
13
15
13
4
3
2
4
8
5
9
8
7
4
5
7-
6
12
17
12
14
16
2
2
6
3
9
9
3
9
7
9
7
9
8
15
14
11
12
16
7
5
4
7
6
7
6
8
9
8
9
8
7
11
11
14
11
12
5
8
2
3
8
4
8
9
4
9
6
9
8
13
16
15
12
11
3
6
6
6
6
8
7
3
3
9
8
8
9
11
11
14
17
15
2
3
4
5
5
2
6
8
9
II. Subtract (that is, count downward)
j. By 2's from 50 ; from 49.
k. By 3's from 50.
1. By 4's from 50 ; from 49.
m. By 5's from 50; from 49; 48; 47; 46.
22. By 6's from 100 ; from 99.
o. By 7's from 100.
p. By 8's from 100; from 99.
q. By 9's from 100.
III. What is 80 - 30 ? 50 - 20 ? 90 - 50 ? 150 - 70 ?
700-400? 1200-300? 1600-900? 1500-800?
IV. From 100 subtract 25, 35, 85, 67, 39, 48, 73, 44, 78,
60, 51, 72, 13, 64, 57, 36, 53, 62, 46, 17, 77, 24, 87, 75.
For swiditional oral drill, see pages 69-63.
22 SUBTRACTION.
Examples for the Slate.
58. Illustrative Example I. If 147 trees are taken
from a nursery of 489 trees, how many trees will be left ?
Written work. Explanation. — To find how many will be left,
489 ^® ^^^^ ^"^^ ^^ ^^^ number 489 away. In subtract-
^ . _ ing a large number like this we take away the
Subtrahend, 147 •, ^u . a ^i. u A ^ ^ i v!
units, the tens, and the hundreds separately ; hence,
Remainder, 342 for convenience, we write the minuend and the sub-
trahend as in the margin, so that units of the same
order shall be expressed in the same column. Drawing a line beneath,
and beginning with the units, we subtract thus : 7 units taken from 9
units leave 2 units, which we write under the line in the units' place ;
4 tens taken from 8 tens leave 4 tens, which we write under the line
in the tens' place ; 1 hundred taken from 4 hundreds leaves 3 hun-
dreds, which we write under the line in the hundreds' place ; and we
have for the whole remainder 3 hundreds 4 tens and 2 units, or 342.
Answer, 342 trees.
1. If a man having 375 oranges in a box should sell 234
of them, how many would be left ?
2. I had a farm of 493 acres, and sold a part containing
172 acres. How many acres had I left ?
69. Illustrative Example IT. If a minuend is 7592
and the subtrahend 3674, what is the remainder ?
WRITTEN WORK. Explanation. — We write these numbers and
(6) (15) (8) (12) subtract as before. As we have but 2 units in the
Min. 7 5 9 2 minuend, we cannot now take the 4 units away.
Sub. 3 6 7 4 so we change one of the 9 tens (leaving 8 tens) to
Rem 3 9 18 ^^i^^- '^^^^ ^ *^^ equals 10 units. We add the
10 units to the 2 units, making 12 units. Sub-
tracting 4 units from the 12 units, we find 8 units left, which we
write as part of the remainder. Subtracting 7 tens from the 8 tens
we now have, we find I ten left, which we write. As we have but 5
hundreds in the minuend, we cannot now take 6 hundreds away, so
we change one of the 7 thousands (leaving 6 thousands) to hundreds,
and add the 10 hundreds thus obtained to the 5 hundreds, making 15
EXAMPLES. 23
hundreds. Subtracting 6 hundreds from 15 hundreds, we find 9
hundreds left, which we write. Subtracting 3 thousands from the 6
thousands we now have, we find 3 thousands left, which we write ;
and we have for the whole remainder, 3 thousands 9 hundreds 1 ten
and 8 units, or 3918.
This explanation maybe given briefly thus: 4 from 12 leaves 8,
7 from 8 leaves 1, 6 from 15 leaves 9, 3 from 6 leaves 3 ; remainder,
3918. In actual work, however, all explanation should be omitted.
Do not stop to say " 4 from 12 leaves 8," etc., but do the work, naming
only results as you write them, thus : " 8, 1, 9, 3 ; remainder, 3918."
In this way you will learn to work rapidly.
What are the remainders in the following examples ?
(3.) (4.) (5.) (6.)
849 321 8642 3089
278 219 370 2435
7. If I had $685 in a bank and withdrew $328, how
many dollars remained?
8. How old was a person in 1876 who was born in 1798 ?
60. Illustrative Example III. If a farm is bought for
$ 965 and sold for $ 2000, how much is gained ?
WRITTEN WORK. Explanation. — To find how much is gained,
(1) (9) (9) (10) we take away a part of $ 2000 equal to $ 965.
$ J As we have no units, no tens, and no hun-
9 6 5 dreds in the minuend, we change one of the
Ans. $ 1 3 5 thousands (leaving 1 thousand) to 10 hundreds ;
then change one of the 10 hundreds (leaving 9
hundreds) to 10 tens ; and one of the 10 tens (leaving 9 tens) to 10
units. 2000 is thus changed to 1 thousand 9 hundreds 9 tens and 10
units, from which taking 9 hundreds 6 tens and 5 units, we have for
the remainder 1035. Ans. % 1035.
9. From 2000 years take 1028 years.
24 ■ SUBTRACTION.
10. From 3000 oxen take 229 oxen.
It. How many more birds are there in a flock of 960 birds
than in one of 487 birds ?
Subtraction of Decimals.
61. Illustrative Example IV. What is the difiference
between 20.69 and 8.745 ?
WRITTEN WORK. Explanation. — Writing these nmnbers bo that
20.69 units of the same order shall be expressed in the
8 745 same column, and beginning with the units of the
lowest order (in this case thousandths) to subtract,
we have for the remainder 11.945.
12. Take 20.5 from 199.
13. From $27.68 take $15.96.
14. Find the difference between one thousand and one
thousandth.
62. From the examples above explained we may derive
the following
Rule for Subtraction.
1. Write the minuend and underneath write the subtra-
hend, so_ that units of the same order may he expressed in the
same column. Draw a line beneath.
2. Begin with the units of the lowest order to subtract, and
proceed to the highest, writing each remainder under the line
in its proper place.
3. If any term of the minuend is less than the correspond-
ing term of the subtrahend, add ten to it and then subtract ;
but consider that the next term of tJie minuend has been
diminished by one.
Proof.
Add the remainder to the subtrahend : the sum ought to
equal the minuend.
EXAMPLES. 26
63. Examples in SubtractioxL
a. From 7282 subtract 4815. Am. 2467.
b. Take 3084 from 6231. Am. 3147.
c. How many are 64037 less 5908 ? Am. 58129.
d. Subtract 807605 from 1740932. Am. 933327.
e. What number taken from 71287 will leave 40089 ?
Am. 31198.
/. How many more than 94736 is 104083 ? Am. 9347.
g. Find the difference between 86045 and 708406.
Am. 622361.
h. 2684753 - 764287 = how many ? Am. 1920466.
i. From four hundred twenty thousand six hundred
eighty-three, take two hundred fifty-nine thousand seventy-
five. Am. 161608.
j. Take eight hundred ten thousand twenty-three from
one million sixty thousand forty-one. Am. 250018.
k. 1001001 minus 909199 equals what ? Am. 91802.
1. Subtract the sum of the numbers in example c from
the sum of the numbers in example d. Ans. 2478592.
m. Find the difference between the amount of the num-
bers in example a and the amount of the numbers in
example b. Am. 2782.
Examples with Decimals.
n. From $ 17.60 take $ 5.25. Am. $ 12.35.
o. From 426.17 take 11.723. Am. 414.447.
p. Subtract three hundred sixty-four thousandths from
one. Am. 0.636.
q. What must be added to 0.0476 to make 1 ?
Am. 0.9524.
Note. The examples on this page embrace the chief varieties in form of
examples in Subtraction, After performing these, and before taking the
Applications on page 26, pupils will usually need additional practice in
similar work. Examples for such practice will be found on pages 59 - 63.
26 SUBTRACTION.
64. Applications.
15. A farmer who raised 948 bushels of corn sold all but 198
bushels. How much did he sell ?
16. The year's earnings of a family were $1172. If their
expenses were $ 875, what was saved ?
17. A and B together own 5740 acres of land. If B owns
2964 acres, how much does A own ?
18. Mount Washington is 6234 feet high, which is 2286 feet
higher than Vesuvius. How high is Vesuvius ?
19. The several items of an account amount to 19867.62;
of this amount $ 7985.75 has been paid. Find the balance.
20. Franklin was born in 1706, and died in 1790. What
was his age at the time of his death ?
21. The difference between A's and B's estates is $1463;
B's, which is the greater, is worth $ 7638. What is A's worth ?
22. In one week a grain elevator received 984560 bushels
of grain ; of this 769386 bushels were delivered. How much
remained in the elevator ?
23. The sailing distance from New York to Queenstown is
2890 miles. If a Cunard steamer has run 1368 miles on her
course from New York, how far has she still to run?
The population of the city of New York was 60489 in
the year 1800; 96373 in 1810; 123706 in 1820; 202589 in
1830 ; 312710 in 1840 ; 515547 in 1850 ; 813669 in 1860 ;
and 942292 in 1870. What was the increase in population
24. From 1800 to 1810 ? 28. From 1840 to 1850 ?
25. From 1810 to 1820 ? 29. From 1850 to 1860 ?
2Q. From 1820 to 1830? 30. From 1860 to 1870 ?
27. From 1830 to 1840 ? 31. From 1800 to 1870 ?
32. The population of London in 1871 was 3266987. How
many times may you subtract from this a population equal
to that of New York in 1870 ?
33. The equatorial diameter of the earth is 41847194 feet,
and the polar diameter 41707308 feet. What is the differei^ce ?
EXAMPLES. 27
65. Examples with Decimals.
34. A person having 205.6 acres of land, sold 10.75 acres.
How many acres had he left ?
35. What is the difference between 0.7 and 0.385 ?
36. How many thousandths must you add to 0.485 to make
1.?
(37.) 86.67 -9.8-? (40.) 641.34-56.345-?
(38.) 7561.2-9.6456 = ? (41.) 101.1-90.014 = ?
(39.) 961.62-54.645-? (42.) 970.2 - 86.37 - ?
66. Miscellaneous Examples.
43. James Fry has in his possession $172; he owes $28
to A, $ 36 to B, and 1 19 to C. After paying his debts, what
will remain ?
44. In a certain mill 2415 persons were employed, of whom
581 were natives, 1119 were foreigners, and the rest unknown.
How many were unknown ?
45. I have $462 in the savings-bank, and $2180 in gov-
ernment bonds. How much more must I have that I may
purchase a house worth $ 4700 ?
46. A man gave to his son $3575, to his daughter $4680,
and to his nephew $2495 less than to his daughter. His
whole property was worth $ 30500 ; what sum remained ?
47. Two persons who are 250 miles apart, travel towards
each other, one 36 miles, the other 52 miles a day. How far
apart will they be at the end of one day ?
48. If the same persons travel away from each other, how
far apart will they be at the end of one day ?
49. From 9460 subtract 5466 ; from the result subtract
1284 ; to this add 3989, and from this subtract 5987.
50. A man bought a lot of land for $ 1296, and built upon
it a house costing $ 7364. If he sold the property for $ 10000,
how much did he make ?
28 MULTIPLIGATIOH.
SECTION IT.
MULTIPLICATION.
67. Unite three 7's into one number.
7 This may be done by adding them together thus : 7, 14, 21.
7 By this process we find that three 7's are 21. In the same way
rr we can find that seven 6's are 42, eight 9's are 72, eight 7's
— are 56, and in fact all the results which we commit to memory
21 when we learn the Multiplication Table.
68. The process of uniting two or luoie equal luinibers
into one number is mvLltiplication.
69. One of the equal numbers to be united is tiie
multiplicand.
70. The number that tells bow many equal numbers are
to be united is the multiplier.
7 1. The result obtained by multiplication is the product.
72. The multiplicand and multiplier are called factors
(makers) of the product.
In the example "three 7's are 21," which is the multiplicand ? the
multiplier? the product ? Name two factors of 21.
73. The multiplication of numbers is indicated by the
sign X . Thus, the expression 50 x 4 = 200 means that
four 50's are 200 ; and is read " 50 multiplied by 4 equals
200."
74. Oral Exercises.
Turn to page 58, and multiply the numbers expressed
a. In column li by 3. e. In column q by 7.
b. In column j by 4. /. In column r by 8.
c. In column k by 5. g. In column v by 9.
d. In column o by 6. h. In column v by 12.
EXAMPLES. 29
76. Compare the product of five 4's with that of four
o's: are they equal or unequal? Compare the products
3x4x6, 4x3x6, and 6x4x3: are they equal or un-
equal ? From examples like these we learn this general
principle :
The ^product of two or more factors is the same, whatever
the order in which the factors are taken.
To multiply mentally Numbers greater than 10.
[At the option of the Teacher.]
76. Illustrative Example. At $34 each, what will
4 cows cost?
Solution. — At $ 34 each, 4 cows will cost 4 times $ 34. Four 30's
are 120, and four 4's are 16, which, added 4o 120, make 136. Ans. $ 136.
i. If 9 men can build a wall in 25 days, how long would it
take 1 man to do it ?
J. How many gallons of water in 5 hogsheads of 67 gallons
each?
k. At $8 a month, what is the amount of a soldier's pen-
sion for 1 year ? for 9 years ?
I, How many are three 27's ? four 16's ? eight times 84 ?
For additional practice, multiply each number expressed in A, page 58,
by such numbers from 1 to 12 as the teacher may select.
See also oral exercises in multiplication, pages 59 and 63.
Examples for the Slate.
77. Illustrative Example I. If a steamship goes 258
miles each day, how far does she go in 6 days ?
Explanation. — If the steamship goes 258 miles
WRITTEN WORK. ^^^^ ^^^^^ -^ g ^^yg ^^^ ^jll g^ q ^^^^^ 258 miks.
Multiplicand, 258 We have then to multiply 258 by 6. Writing the
Multiplier, 6 multiplicand and the multiplier as in the margin,
■iKAQ. we multiply the units, tens, and hundreds sepa-
rately, beginning with the units.
30 MULTIPLICATtOK
Six 8's are 48. The 48 units are equal to 4 tens and 8 units. We
write 8 under the line in the units' place, and carry 4 tens to the
product of tens.
Six 5's are 30. The 30 tens with the 4 tens carried are 34 tens, or
3 hundreds and 4 tens. We write 4 under the line in the tens' place,
and carry 3 hundreds to the product of hundreds.
Six 2's are 12. The 12 hundreds with the 3 hundreds carried are
15 hundreds, or 1 thousand and 5 hundreds. We write, under the
line, 5 in the hundreds' place and 1 in the thousands' place. The
entire product is 1548. Ans. 1548 miles.
For the sake of rapid working, use as few words as possible.
Thus, in the example above say " iorty-ei(/ht ; thirty, thirty-
four; twelve, fifteen " : while saying " forty-eight," write 8 ;
while saying " thirty-four," write 4 ; and while saying " fifteen,"
write 5 and 1.
1. How many pounds of flour are there in 5 barrels, each
containing 196 pounds ?
2. How many pounds of cheese are there in 6 cheeses of 172
pounds each ?
3. If a person earns $ 313 every year for 7 years, how many
dollars does he earn ?
4. What will 9 pianos cost at $ 475 each ?
5. From Chicago to Peoria is 160 miles; how far does a
man travel who goes from Chicago to Peoria and back 8 times ?
6. If a person by working 11 hours a day can do a piece of
work in 37 days, how many days will it take him if he works
1 hour a day ?
7. There are 5280 feet in a mile. How many feet long is a
telegraph-wire that connects Boston with Reading, 12 miles
distant ?
78. Illustrative Example II. If 1 barrel of flour costs
$ 8, what will 427 barrels cost ?
Solution. — If 1 barrel of flour costs $ 8, 427 barrels will cost 427
times $8. But 427 times $8 is the same as 8 times $427 (Art. 75),
which is $ 3416. Ans. ?> .3416.
EXAMPLES. 31
8. What will 732 quarts of milk cost at 7 cents a quart ?
9. What must I pay for 324 sheep at $ 9 apiece ? -
10. When coal is $ 6 a ton, what must I pay for 476 tons ?
11. At 4 cents a mile, what must I pay for riding 1289
miles ?
12. If 294 persons gave $ 8 apiece for a charitable object,
how much did all give ?
13. What must I pay for 626 car-fares at 6 cents apiece,
and for 87 car-fares at 9 cents apiece ?
14. Multiply 267 by 2 ; by 3 ; by 4 ; and add the products.
15. Multiply 628 by 5 ; by 6 ; by 7 ; and add the products.
16. Multiply 3401 by 8 ; by 9 ; and add the products.
17. Multiply 90021 by 10 ; by 11 ; and add the products.
18. Multiply 66285 by 12 ; by 8 ; and add the products.
19. Multiply 89079 by 7 ; by 12 ; and add the products.
For additional examples in multiphcation by one term only, see pages 59
and 63.
79. Illustrative Example III. Multiply 12 by 10;
12 by 100 ; 12 by 1000.
WRITTEN WORK. Explanation. — 10 twelves equal 12
12 12 12 tens (Art. 75), or 120 ; 100 twelves
10 100 1000 equal 12 hundreds, or 1200 ; and 1000
— — • TTTTT" ^r.r.^r. twclvcs equal 12 thousands, or 12000.
120 1200 12000 >
In multiplying by 10, 100,
1000, etc., the written work may be omitted, and the
product immediately found hy annexing to the multiplicand
as many zeros as there are in the multiplier.
20. What will 10 bushels of potatoes cost at 65 cents a
bushel ?
21. At $ 100 a share, what will 100 shares in a whip com-
pany cost ?
22. Multiply % 75 by 10 ; by 100 ; and add the products.
32 MUL TIPLICA TION.
23. Multiply 5872 by 10 ; by 1000 ; and add the products.
24. Multiply 684 by 10; by 100; by 10000 ; by 1000 ; and
add the products.
25. Multiply 3682 by 10000; by 10; by 1000 ; by 100;
and add the products.
80. Illustrative Example IV. Multiply 4520 by 300.
WRITTEN WORK. Explanation. — Here 4520 equals 452 x 10, and
4520 ^^^ equals 3 x 100 ; hence 4520 x 300 is the same
o/^^ as 452 X 10 X 3 X 100, or, since the order of the fac-
tors may be changed (Art. 75), the same as 452 x 3
1356000 X 10 X 100. We shall, therefore, find the product if
we multiply 452 by 3 and annex three zeros (Art. 79).
When the inultip>licand and multiplier, or either of
them, have zeros at the right hand, the zeros may he dis-
regarded in multiplying, hut there must he annexed to the
product as many zeros as were disregarded.
26. I have 600 acres in my farm. What is it worth at
% 250 an acre ?
27. How many strawberry plants are there in 400 rows, if
there are 280 plants in each row ?
28. What is the product of 1870 x 90 ? Of 1870 by 900 ?
Of 1870 by 9000 ?
29. If 268000 is the multiplicand and 80 the multiplier,
what is the product ?
30. Multiply 596 by 3 and by 40, and add the products.
31. Multiply 984 by 8 and by 60, and add the products.
32. Multiply 647 by 9 and by 20, and add the products.
33. Multiply 379 by 5 and by 80, and add the products.
34. Multiply 4837 by 2, by 30, and by 500, and add the
products.
35. Multiply 2802 by 8, by 70, and by 900, and add the
products.
EXAMPLES.
33
81. Illustrative Example Y. Multiply 625 by 39.
WRITTEN WORK.
625
39
5625 = product by 9.
1875 = product by 30.
24375 = product by 39.
Explanation. — We shall find the
product of 625 x 39 if we multiply 625
by 9 and then by 30, and add the re-
sults. We first find the product by 9,
which is 5625, and write it under the
line. The product of 625x30 is the
same as 625 x 3 x 10. To find this we
multiply 625 by 3, obtaining 1875, but
instead of annexing a zero (Art. 79), we write the result as 1875 tens.
We then add the partial products.
KoTE. Compare this process with that of Examples 30 to 33, in the
last Article.
36. How many are 34 x 25 ?
37. Multiply 49 by 98 ; then multiply 98 by 49. Are these
products equal ? Why ?
38. What is the product of 2842 multiplied by 28 ?
39. Multiply 3684 by 36 and by 64, and add the products.
40. Multiply 625 by 339; by 705; by 7005.
WRITTEN WORK.
625
705
WRITTEN WORK.
625
339
5625
1875
1875
= product by 9.
= product by 30.
= product by 30C
3125 = product by 5.
4375 = product by 700.
440625 - product by 705.
211875 = product by 339.
The explanation of this work is left for the pupil. (See Art. 81.)
41. How many are 743 x 657 ?
42. Multiply 237 by 195 ; 195 by 237.
43. Multiply 4387 by 235 ; 235 by 4387.
44. Multiply 7608 by 504 ; 504 by 7608.
45. Multiply 760500 by 307000.
46. Multiply 907200 by 420900.
34 MULTIPLICATION.
Multiplication of Decimals.
82. Illustrative Example VI. Multiply 108.67 by 48.
Explanation. — 10867 hundredths miil-
WRITTEN WORK. . ,. ,, „. ^^^^^i -, -, ,
tiplied by 8 is 86936 hundredths. 10867
1^^-^'^ hundredths x 40 is the same as 10867
48 hundredths x 4 x 10. Now 10867 hun-
QpQQf5 - -n od bv 8 dredths x 4 is 43468 hundredths ; to ex-
4^1fi8 - r1 *>. 10 Pi^sss this product multiplied by 10 we
~ ^ ' '' ' write the figures one place to the left.
5216.16 = prod, bv 48. Adding the partial products we have
521616 hundredths (5216.16) for the en-
tire product. Here, as in the preceding examples, we see that the
'product is of the same order of units as the multiplicand.
47. Multiply 8.648 by 5 ; 432.5 by 21.
48. Multiply 7.0909 by 6 ; 0.0005 by 18.
49. Multiply 0.625 and 0.375 each by 24, and add the results.
83. From the preceding examples may be derived the
following
Rule for Multiplication.
1. Write the multiplicand and underneath write th^ mul-
tiplier. Draw a line beneath.
2. If the multiplier consists of one term only, multiply
each term of the multiplicand hy the multiplier, beginning
with the term of the lowest order, and carrying as in additio7i.
3. If the multiplier consists of more than one term, midti-
ply hy each term of the multiplier separately, writing the
partial products so that units of the same order shall he
expressed in the same column.
4. Add the partial products thus obtained, and the result
will be tlie entire product.
Proof.
Multiply the multiplier by the multiplicand : the two prod-
ucts ought to be equal.
For contractions in multiplication, see Appendix, page 300.
EXAMPLES. 35
84. Examples in Multiplication.
a. Multiply 4687 by 8. Ans. 37496.
b. Find the product of 50875 by 7. Ans. 356125.
c. Multiply 5872 by 10, also by 1000, and add the
products. Ans. 5930720.
d. Multiply 8756 by 300 ; by 500 ; by 7000 ; and add
the results. Ans. 68296800.
e. What is the product of 39700 by 9000 ?
Ans. 357300000.
/. 37406 X 43 = what number? Ans. 1608458.
g. For multiplicand take 46059, for multiplier 76, and
find the product. Ans. 3500484.
h. How many are 309 times 46057 ? Ans. 14231613.
i. Multiply thirty-seven thousand twenty-eight by 508.
Ans. 18810224.
j. The multiplier being 987, the multiplicand six thou-
sand four hundred sixteen, required the product.
Ans. 6332592.
k. What is the product of 908060 by five thousand four
hundred ? Ans. 4903524000.
1. One factor being 718151, the other seven hundred,
what is the product ? Ans. 502705700.
m. At 147 dollars per acre, how much will 385 acres of
land cost? Ans. $56595.
n. There are 24 hours in a day. How many hours in
476 days? Ans. 11424.
Examples "with Decimals.
o. Multiply 40.27 by 87. Ans. 3503.49.
p. Multiply thirty-one thousandths by 25. Ans. 0.775.
Note. The examples on this page embrace the chief varieties in form
of examples in Multiplication. Examples for additional practice will be
fonnd on pages 59 - 6.S.
36 MULTIPLICATION,
85. Applications.
60. At $45 a month for labor, what will a man earn in a
year ? In 5 years ?
51. If a man saves $ 17 a month, what will he save in 25
years ?
52. If a sewing-machine can set 690 stitches in a minute,
how many stitches can it set in 60 minutes or an hour ? In a
day of 12 hours ? In 6 working days or a week ? In ^2
weeks or a year ?
53. The first House of Kepresentatives of the United States
consisted of 65 members ; if each member represented 30000
inhabitants, how many inhabitants were represented ?
54. In a certain mill, material for 65000 dresses is made in
a week. Allowing 18 yards for a dress, how many yards are
made in a week ? In a year ?
55. The cotton crop in Texas in one year was 450000 bales.
Allowing 400 pounds to a bale, how many pounds were raised ?
56. In a day there are 24 hours, in an hour 60 minutes, in
a minute 60 seconds. How many seconds in a day ?
57. Light, according to Foucault, travels at the rate of
185172 miles in a second. If it passes from the sun to the
earth in 8 minutes 13 seconds (or 493 seconds), what is the
distance from the sun to the earth ?
86. Examples with Decimals.
58. It took Mary 3.25 hours to learn a piece of music, and
Olive 5 times as long. How many hours was Olive in learn-
ing it ?
59. Mr. Green has 5.175 acres of land and buys 7 times as
much of his neighbor. How many acres does he buy of his
neighbor ?
60. AVhat will 38 barrels of flour cost at $ 11.75 a barrel ?
61. Mr. Gage sold 175 tons of refined bar-iron at $ 45.50
a ton. What did he receive for it ?
62. Multiply 5.4328 by 62.
EXAMPLES. 37
87. Miscellaneous Examples.
63. I have four bins, containing severally 63 bushels, 54
bushels, 37 bushels, and 29 bushels. If there are 60 pounds of
corn in a bushel, how many pounds of corn will they all hold ?
64. What is the height of an iceberg which is 375 feet above
the surface of the water and 7 times as many feet below ?
65. Myron walks 847 steps of 2 feet each in going to school.
How many more feet must he take to walk a mile, or 5280 feet ?
^Q. What do I save a year, my income being $1600 a j^ear,
and my expenses $ 24 a week, 52 weeks making the year ?
67. Mr. Fiske receives a salary of $ 1500 a year, pays $ 130
for clothing, $ 275 for other expenses, also $ 6 a week for his
board. How much money has he left at the end of the j-ear ?
68. If 768 be one factor, and 861 - 237 the other factor, what
is the product ?
69. Smith & Co. consume 74 tons of coal in a year. How
much more did they pay for their coal in 1864, when coal was
$ 14 a ton, than in 1877, when it was |7 a ton ?
70. If in one yard of cloth there are 580 fibres of warp and
432 of filling, and each fibre of warp contains 32 strands, and
each of filling 48, how many strands are there in the yard ?
71. One house is valued at 16750, and another at three
times as much. How much will pay for both houses ?
72. Mr. Gould had $ 2500 with which he bought 17 acres of
land at % 42 an acre, a house for $ 1500, 2 cows at 1 45 apiece,
and a horse for $ 75. How much money had he left ?
73. Mr. Bod well paid for labor and use of oxen on his land,
the following sums : $ 135, 1 128, and $ 90 ; he also paid % 64
for fertilizers and $ 10 for seed, and raised on the land 23 tons
of hay which he sold at $ 25 a ton. What was his gain above
his expenses ?
74. Add 284, 1752, 45, and 846 ; subtract 2731 from the
sum ; multiply the remainder by 208 ; and find the difference
between the product and 40801.
38 DIVISION,
SEOTIOK" V.
DIVISION.
88. Mr. Eice has 24 bushels of sand to bring from the
beach. If he brings 8 bushels at each load, how many
loads must he bring?
He must bring as many loads as there are 8's in 24. We have
already seen by multiplication that three 8's are 24, so we know that
he must bring 3 loads.
If a cheese weighing 54 pounds be divided equally among
6 persons, how many pounds will each receive ?
Each person will receive one of the 6 equal parts into which the 54
pounds is to be divided. We have seen by multiplication that 6 nines
are 54 ; hence one of the 6 equal parts of 54 is 9, and each person
will receive 9 pounds.
It will be noticed in the first example that we find how many equal
numbers, one of which is given, there are in another number (that is,
how many times one number is contained in another); and in the
second that we find one of the equal parts of a number.
89. The process of finding how many times one number
is contained in another or of finding one of the equal parts
of a number is division.
90. The number to be divided is the dividend,
91. The number by which we divide is the divisor,
92. The result obtained by division is the quotient,
Note I. When the divisor is one of the given equal numbers, the quo-
tient will tell how many such numbers there are in the dividend.
DIVISION. 39
Note II. When the divisor tells how many equal parts the dividend is
to be separated into, the quotient will tell how great one of those equal
parts is.
Note III. By comparing the first process with multiplication (Arts.
69 - 72), we see that the product and multiplicand are given, and the mul-
tiplier is to be found. By comparing the second process with multiplica-
tion, we see that the product and multiplier are given, and the multipli-
cand is to be found.
In either case the product and one of the factors are given, and the other
factor is required.
93. If Mr. Eice has 31 bushels of sand to bring from
the beach, and can bring but 8 bushels at a load, how
many full loads can he bring and how many bushels will
then remain?
The part of the dividend left after the equal numbers
have been taken away is the remainder.
In the example above, which is the dividend ? the divisor ? What is the
remainder ?
04. The division of numbers is indicated by the sign -f-.
Thus, the expression 24 -^ 8 = 3 means that the quotient
obtained by dividing 24 by 8 is 3, and is read " 24 divided
by 8 equals 3." The sign : is also used for division. Thus,
24 : 8 = 3.
Sometimes the dividend is expressed above a line and
the divisor below, in place of the dots. Thus, j = 3. This
expression is called the fractional form of indicating divis-
ion, and is read "24 divided by 8 equals 3," or "1 eighth
of 24 equals 3."
95. When -a thing or a number is divided into 2 equal
parts, the parts are called halves; when divided into 3 equal
parts, the parts are called thirds ; when into 4 equal parts,
the parts are called fourths; and so on.
What is one of the parts called when a number is divided
into 5 equal parts? 6? 7? 8? 10? 20? 100? 1000?
40
DIVISION.
96. Table for Oral Practice in Division.
1.
2.
4
7 2
6
3 8
5
7
11
9
10
12
16
15 21
14
20 13
22 17
23
18
19 24
3.
20
29
35
33
28 31
2&
34
27
30
32
36
4.
5.
6.
7.
39 46 38
42
37 43
47
40
45
41
44
48
58
49
6o
51
54 59
52
57 50
53
m 60
62
71
70
m
61 64
69
63
68
65
67 1 72
76
79
77
81
78 80
75
73
82
74
83
84
8.
90
88
91
87
89 94
85
93
86
95
92
96
9.
10.
11.
99
104
100 98
97 102 106 101 107
103 105 108
110
118 111
109
117 112
115
114
119
113
116 120
124
130
123
125
129 121
128 122 1 131
1 1
126
127 132
12.
134
142
135 140 133 139
136
143 137
141
138 144
' 97. Oral Exercises upon the Table.
Beginning at the left of the table above, divide by 2 each
number expressed in the first two lines, naming quotients
and remainders at sight. In the first line the numbers to
be divided are 4, 7, 2, 6, 3, 8, 5, etc. The results will be
given as follows : " 2 ; 3 and 1 over ; 1 ; 3 ; 1 and 1 over ;
4/' etc.
Divide in the same manner the numbers expressed in either *
a. Of the first 3 lines by 3. /. Of lines 2 to 8 by 8.
b. Of the first 4 lines by 4.
c. Of the first 5 lines by 5.
d. Of the first 6 lines by 6.
e. Of the first 7 lines by 7.
g. Of lines 2 to 9 by 9.
h. Of lines 2 to 10 by 10.
2. Of lines 2 to 11 by 11.
j. Of lines 2 to 12 by 12.
For other oral exercises in division, see pages 61 and 63.
As the teacher may indicate.
EXAMPLES. 41
SHORT DIVISION".
Examples for the Slate.
98. Illustrative Example I. At $ 5 a day for work,
how many days' work can be had for % 4730 ?
WRITTEN WORK. Explanation. — As many days' work can
(2) (3) be had for % 4730 as there are 5's in 4730.
Divisor, 5) 4 7 30 Dividend. For convenience, we write the dividend
and divisor as in the margin, and divide
946 Quotient. ^^^ ^^^^^^ ^^ ^^^ dividend separately, as
Ans. 946 days' work, far as possible, beginning with the highest.
If we divide the four thousands by 5,
we shall have no thousands in the quotient, so we first divide 47
hundreds by 5.
5's in 47 (hundreds), 9 (hundred), and 2 hundreds remain. We
write the 9 hundred under the line in the hundreds' place, and
change the 2 hundreds remaining to 20 tens, which, with the other
3 tens of the dividend, make 23 tens.
5's in 23 (tens), 4 (tens), and 3 tens remain. We write the 4 tens
under the line in the tens' place, and change the 3 tens remaining to
30 units.
5's in 30 (units), 6 (units), which we write iinder the line in the
units' place, and have 946 for the entire quotient. Ans. 946 days'
work.
In dividing, the pupil may simply say, " 5's in 47, 9 and
2 over ; in 23, 4 and 3 over ; in 30, 6." Or, abbreviating
stiU more, " 5's in 47, 9 ; in 23, 4 ; in 30, 6."
1. How many cords of wood at $ 6 a cord can be bought for
$ 522 ? for $ 3804 ? 1st Ans. 87 cords.
2. How many hours will it take to ride 3216 miles at 8
miles an hour ? at 12 miles ? 1st Ans. 402 hours.
3. At 7 cents an hour for work, how many hours must I
work to earn 2835 cents ?
4. How many packages of tea, 9 pounds in a package, can
be made from 8847 pounds ?
42 DIVISION.
99. Illustrative Example II. How many barrels of
flour at $ 8 a barrel can I buy for $ 2597 ?
WRITTEN WORK. Explanation. — Here, after dividing, we have
. p. a remainder of $5: hence, 324 barrels can be
^ ' bought and $5 remain unexpended, which
324 may be expressed as in the margin.
Ans. 324 barrels; $5 remain.
The work may be proved by finding the product of the quo-
tient and divisor (Art. 92, Note III.) and adding the remain-
der. Thus, 324x8 + 5 = 2597.
5. How many weeks are there in 585 days ? in 730 days ?
1st Ans. 83 weeks ; 4 days remain.
6. How many 8 quart cans can be filled with 1865 quarts of
milk ? with 2587 quarts ? 1st Ans. 233 cans ; 1 quart remains.
7. How many years of 12 months each are there in 200
months ?
8. There are in an orchard 1608 trees, 12 in a row. How
many rows of trees are there ?
9. At 11 cents a yard, how many yards of cloth can I buy
for 5972 cents ?
10. At 9 cents apiece, how many oranges can be bought for
29415 cents ?
100. Illustrative Example III. If 8 men buy 9675
acres of land which they are to divide equally among them-
selves, what is each man's share ?
WRITTEN WORK. Explanation. — Each one will have 1 eighth of
8^ Qf\7n ^^"^^ acres. We divide, briefly, thus :
b) ^b75 acres. ^^^ ^^^^^^ ^^ ^ thousand is 1 thousand, and
Ans. 1209 1 acres. 1 thousand (equal to 10 hundreds) remains. One
eighth of 16 hundreds is 2 hundreds ; of 7 tens,
tens and 7 tens (equal to 70 units) remain. One eighth of 75 units
is 9 units, with a remainder of 3 units yet to be divided.
If 1 eighth of each of the 3 acres is taken, we shall have 3 eighths of
an acre. This we express as in the margin, and have 1209f acres for
the entire quotient.
DIVISION OF DECIMALS. 43
11. What is the price of 1 hat if 6 hats cost 375 cents ? it
12 cost 2700 cents ? 1st Ans. 62^ cents.
12. How far must a man travel each day to go 1761 miles
in 4 days ? in 9 days ? 1st Ans. 440^ miles.
13. Mr. Stewart promises to sell me 5 rods of land for
% 1578. What is his price per rod ?
14. At 1 8 a thousand, how many thousands of bricks can
he bought for $ 3287 ?
15. A man left by his will 1 45267 to be divided equally
among his 6 children. What should each child receive ?
16. Eight times a certain number equals 324787. What is
that number ?
17. How many 9's are there in 10000 ?
18. To what number is ^ ^ y ^ equal ?
19. To what number is ^^^^^"^ equal ?
20. How many are 10101019 - 7 ?
21. How many are 98306572 - 5 ?
22. Divide 864024 by 7. 24. Divide 369801 by 9.
23. Divide 164408 by 8. 25. Divide 120087 by 11.
101. Division of Decimals.
Illustrative Example IV. What is 1 twelfth of
109.92?
Explanation. — Briefly thus : 1 twelfth of 109 is
WRITTEN WORK, g^^^^^ ^ remains; of 19 tenths is 1 tenth, and 7 tenths
12) 109.92 remain ; of 72 hundredths is 6 hundredths. Ans. 9.16.
Q ^ ^ In the example above it will be seen that we have
hundredths in the quotient as there are hundredths
in the dividend. In dividing a decimal by a whole number, the quotient
is of the same denomination as the dividend.
In dividing a decimal by a whole number, fix the decimal point in
the quotient as soon as you reach the decimal point in the dividend.
26. What is 1 fifth of 86.4055 ? (28.) $23454-9 = ?
27. What is 1 eighth of 94076.8 ? (29.) $ 907.34 - 7 = ?
44 DIVISION.
102. To Divide, carrying the Division to Decimals.
Illustrative Example Y. Find 1 eighth of 9675 acres.
WRITTEN WORK. Explanation. — We divide as in Illustrative Ex-
g. QA'TK HAA ample III., until we come to the remainder, 3 acres.
^ This we change to 30 tenths. One eighth of 30
1209.375 tenths is 3 tenths, and 6 tenths remain, which are
equal to 60 hundredths. One eighth of 60 hun-
dredths is 7 hundredths, and 4 hundredths remain, which are equal to
40 thousandths. One eighth of 40 thousandths is 6 thousandths. The
entire quotient is 1209.375 acres.
Perform Examples 11 to 16 in Article 100, carrying the division to
decimals.
103. Where the divisor is not greater than 12, it is
customary to divide as shown above without expressing
all the operations. Such a process is short division.
For other examples in short division, see pages 61 and 63.
LONG DIVISION.
104. Illustrative Example VI. Divide 33075 by 82.
WRITTEN WORK. Explanation. —We write the dividend and
divisor as in the margin, and draw a curved
82) 33075 (403f I Une at the right of the expression for the divi-
328 dend.
Zl't Since the divisor 82 is a larger number than
3 or than 33, we first divide 330 hundreds by 82.
Now 330 divided by 82 will give about the
29 same quotient as 33 divided by 8,* which is 4.
The first term of the quotient is then 4 hun-
dreds, which we express by writing a figure 4 at the right of the curved
line. Multiplying 82 by 4 hundreds, and subtracting the product,
we find 2 hundreds remain ; uniting with these 2 hundreds the 7 tens
of the dividend, we have 27 tens.
Dividing the 27 tens by 82, we have no tens in the quotient ; so
we write a zero to show that there are no tens in the quotient, and
unite with the 27 tens the 5 units of the dividend, making 275 units.
♦ So we make 8 our trial divisor.
LONG DIVISION. 45
Dividing the 275 units by 82, using 8 for a trial divisor, we have 3
units in the quotient, which we write. Multiplying and subtracting
as before, 29 units remain.
Dividing each of the 29 units by 82, we have -||, which we write
with the units, and have for the entire quotient 403 1|.
105. When the divisor is larger than 12, it is usually
convenient to express in full, as above, the work of dividing.
The process is then called long division.
To Divide, carrying the Division to Decimals.
106. Illustrative Example VII. Divide 33075 by 82.
WRITTEN WORK. Explanation. — We divide as in the last
82) 33075 (403.35... illustrative example until we reach the re-
328 mainder, 29 units. We now put a decimal
^zzz point in the expression for the quotient,
and, changing the remainder to 290 tenths,
divide as before ; and so we keep on dividing
290 Tenths. as far as desirable, or until there is no re-
246 mainder. In this example we stop dividing
~440 Hundredths ^* hundredths, and indicate that the divis-
- ^ ^ ion is incomplete by Avriting a few dots.
-— 107. Give answers to the following
examples as in Art. 104, or with the
quotient carried to thousandths, as the teacher may direct : *
30. Divide 4684 by 31. 34. Divide 12157 by 23.
31. Divide 9632 by 43. 35. Divide 24898 by 72.
32. Divide 5872 by 54. 36. Divide 36872 by 84.
33. Divide 6748 by 62. 37. Divide 36072 by 91.
108. Illustrative Example VIII. Divide 1849 by 192.
WRITTEN WORK. Explanation. — As 192 is nearly 200, 1849
192) 1849 (9|f ^ divided by 192 will give about the same quo-
1728 tient as 1800 divided by 200, or as 18 divided
~12± ^y 2- ^"^ ^hen make 2 our trial divisor.
The answers in the Key are given in both forms.
46 toivistoist.
38. Divide 26832 by 96. 40. Divide 232848 by 56.
39. Divide 97684 by 79. 41. Divide 682345 by 88.
109. From the preceding examples we derive the fol-
lowing
Rule for Division.
1. Write the dividend; at the left draw a curved liner
and at the left of this line write the divisor.
2. Divide the highest term or terms of the dividend hy the
divisor.
3. Exjpress the result for the first term of the quotient at
the right in long division, beneath in short division.
4. Multiply the divisor hy this term.
5. Take the product thus obtained from the part of the
dividend used.
6. Unite the next term of the dividend with the remainder
for a new partial dividend ; divide, multiply, and subtract
as before; and so continue till all the terms of the dividcTld
are used.*
7. Uxpress the division of the final remainder, should
there be any, in the fractional form. (Or
Change the remarnder to tenths, hundredths, thousandths,
etc., and continue the division as far as desirable^
Proof.
Find the product of the quotient and divisor, and add to it
the remainder, if there is one. The result ought to equal the
dividend.
42. How many are 36247 -189?
43. How many are 53004-398?
44. How many are 932480 - 287 ?
45. How many are 750010 - 677 ?
* If at any time the divisor is not contained in a partial dividend,
write a zero for the next figure of the quotient, and unite with the partial
dividend the next term of the given dividend.
CONTRACTIONS. 47
Contractions in Division.
110. Illustrative Example IX. Divide 12367 by 10;
by 100 ; by 1000.
If the decimal point be moved one place
12367 -^ 10 = 1236.7 to the left, each figure will express a num-
12367 ^ 100 = 123.67 ber 1 tenth as great as before (Art. 30) ;
12367 -^ 1000 = 12.367 therefore, 1 tenth of 12367 is 1236.7.
For a similar reason, 1 hundredth of
12367 is 123.67, and 1 thousandth of 12367 is 12.367. Hence, when
the divisor is 10, 100, 1000, etc., we may find the quotient hy moving
the decimal point of the dividend as many places to the left as there
are zeros in the divisor.
46. There are 100 cents in a dollar ; how many dollars are
there in 2742 cents ? in 12367 cents ? 1st Am. $ 27.42.
47. How many dollars are there in 14863 cents ?
48. There are 1000 mills in a dollar ; how many dollars are
there in 56849 mills ?
49. Divide 25000 by 10, by 100, by 1000, and add the
quotients.
50. Divide 380768 by 100, by 1000, by 10, by 10000, and
add the quotients.
111. Illustrative Example X. Divide 20864 by 6300.
WRITTEN WORK. Explanation. — Since 6300 = 63 x 100, we
63) 208.64 (3.31... may first divide by 100, obtaining 208.64
189 (Art. 110), and then divide this quotient by
^Qg 63, as shown in the margin (Art. 106).
189 Note. In cases where the exact remainder is wanted,
the common form of written work is better. It may
be abbreviated, as in the writ-
74
63 63|00) 208|64 (3^* ^^ ^^^^ ^f ,^.^ ^^,^_
— 189
11 Explanation. — Indicate first,
■*-^"^ by a vertical line, the division
by 100 ; this gives 208 for a quotient, and 64 remain.
Dividing now 208 by 63, we have for a quotient 3, and 19 hundreds re-
main. Uniting the first remainder 64 with the last remainder 19 hundreds,
we have for the entire remainder 1964.
\ For other contractions of division, see Appendix, page 302.
48 DIVISION.
112. Examples in Division.
a. Divide 58643 by 9. Atis. 6515|.
b. At $8 apiece, how many sheep can be bought for
$ 2595 ? Ans. 324 sheep ; $ 3 remain.
c. If the dividend is 86445 and the divisor 51, what is
the quotient? Ans. 1695.
d. What is the quotient of 40076 -^ 98 ? Ans. 408-| f .
e. 48 times a certain number equals 38256. What is
that number ? Ans. 797.
/. What number multiplied by 87 gives a product of
$22446? Ans.%2b^.
g. Divide 759000 by 10, by 1000, by 100, and add the
quotients. Ans. 84249.
h. What is the sum of 93600 divided by 20, and 93600
divided by 7200 ? Ans. 4693.
i. How many are 493689 -^ 47000 ? Ans. lOf |f f f
j. 37884 is 42 times what number ? Ans. 902.
la. What is 1 thirty-eighth of 856406 ? Ans. 22537.
1. The product of two factors is 5063 ; one of them is
83. What is the other? Ans. 61.
m. The product of three factors is 28350 ; two of them
are 42 and 75. What is the third ? Ans. 9.
22. iLSjO^jLSL + one fourth of 2700 = what number ?
Ans. 261554.
Examples ^vith Decimals.
o. Divide 42.8116 by 13. Ans. 3.2932.
p. What is 1 ninth of $ 76.842 ? Ans. % 8.538.
q. What, is the cost of each chair if 25 chairs can be
bought for S 247 ? Ans. % 9.88.
Note. The examples on this page embrace the principal varieties in
form of examples in division. Examples for additional practice will be
found on pages 61 and 63.
EXAMPLES. 49
113. Applications.
51. If I travel 42 miles a day, in how many days can I
travel 273 miles ?
52. How many barrels are required to hold 5488 pounds of
flour, if one flour-barrel holds 196 pounds ?
53. How many days are there in 9684 hours ?
54. How many days will it take a ship to sail 13724 miles,
at the rate of 133 miles a day ?
55. There are 5280 feet in a mile. How many miles high
is Mount Everest, which is 29002 feet high ?
56. A. B. bought a farm for 1 18785 at 1 95 an acre. How
many acres were there in the farm ?
57. A produce dealer packed 19152 eggs in boxes containing
144 eggs each. How many boxes did he fill ?
58. If the dealer would put 19152 eggs into 84 equal-sized
boxes, how many eggs should he put in a box ?
59. In one year, Missouri produced 4164 tons of lead, worth
% 353940. What was the value of a ton ?
60. There was sent to the U. S. Mint in 13 years % 4377984
worth of gold. What was the average value sent a year ? If
gold was worth 16 dollars an ounce, and 12 ounces make a
pound, how many pounds were sent ?
114. Examples Tvith Decimals.
61. A man divided among his three sons 887.625 acres of
land. What was each son's share ?
62. What is the price of 1 comb, when 48 combs can be
bought for $ 53.76 ?
63. When 234 oranges are bought for $ 7.02, what is paid
for 1 orange ?
In the following examples continue dividing to the third order of
decimals :
64. Find 1 ninth of 1.28 acres. (67.) 8.1-21 = ?
65. Find 1 twelfth of 3.75 tons. (68.) 0.5 - 33 = ?
66. Find 1 fifteenth of 128.5 miles. (69.) 1.868- 215 -?
60 MISCELLANEOUS EXERCISES.
SEOTIOl^ YI.
MISCELLANEOUS EXERCISES.
115. General Review, No. 1.
1. 287 + 5 million + 36 thousand + 59481 - ?
2. Add 567 to the sum of the following numbers: 121,
232, 343, 454, m^, 676, 787, and 898.
3. The difference between two numbers is 95478. The
larger number is 148769 ; what is the smaller ?
4. Which of the two numbers 15672 or 10560 is nearer to
13465, and how much ?
5. Take 987 from each of the following numbers, and add
the remainders: 3644; 7573; 2432; 4001.
6. What number must be added to the difference between
68 and 7003 to equal 938000 ?
7. What number taken from the quotient of 1833000-24
leaves 25 ?
8. What number equals the product of the three factors
1785, 394, and 624-48?
9. If 5872 be the multiplicand, and half that number the
multiplier, what is the product ?
10. If 4832796 is the product, and 1208199 the multiplicand,
what is the multiplier ?
11. If 894869 is the minuend, and the sum of the numbers
in the fifth example is the subtrahend, what is the remainder ?
12. If 700150 is a dividend, and 3685 the quotient, what is
the divisor ?
13. If 28936 is the divisor, and 86 is the quotient, what is
the dividend ?
14. Divide 87 million by 15 thousand.
For other questions in review, see pages 59 - 63.
OEAL EXAMPLES. 51
116. Oral Examples for Analysis.
(See Appendix, page SOS.)
a. If a car runs 69 miles in 3 hours, how far can it run
in 5 hours ?
b. If 18 rows of potatoes yield 36 bushels, how many bush-
els will 20 similar rows yield ?
c. If $ 5 pay for 35 quarts of berries, how many quarts will
112 buy?
d. If, when flour is $ 8 a barrel, a ten-cent loaf weighs 25
ounces, what should it weigh when flour is $ 10 a barrel ?
e. If 5 oxen consume 185 pounds of hay in 2 days, how
much will be required for 1 yoke of oxen for the same time ?
/. If 6 cows were bought for $ 224 and sold for $ 260, what
was the gain on each cow ?
g;. If 150 barrels of apples were bought for $ 200 and sold
for $ 350, what would be gained by selling 45 barrels at the
same rate ?
h. I bought a lot of paint for $3.90 and sold it for $5.10,
gaining 12 cents on a pound. How many pounds did I
buy?
i. If a quantity of hay lasts 22 oxen 10 days, how many
days will it last 5 yoke ?
j. A field of wheat was reaped by 10 men in 6 days ; what
length of time would be required for 15 men to reap the same
amount ?
k. A cistern can be emptied in 15 minutes by 7 pipes ; in
what time can it be emptied, if only 5 of the pipes are open ?
1. If 8 operatives can do a piece of work in 12 days, in
what time will 24 operatives perform the same work ?
m. If a certain piece of work can be performed by 50 men
in 4 weeks, how many more must be employed to perform it
in a week ?
22. Ten hunters have provisions to last them 6 weeks ; if 2
men be killed, how long will the previsions last the remainder ?
62 MISCELLANEOUS EXERCISES.
117. Miscellaneous Ezramples.
15. A merchant bought goods for 1 1084, and sold them for
1 594 more than he gave. How much did he receive for them ?
16. From a farm containing 984 acres there were sold at one
time 347 acres, at another time 157 acres. How many acres
remained ?
17. A merchant bought goods for $ 2467, and sold them for
$ 875 less than he gave. How much did he receive for them ?
18. If I take 7642 gallons from 21002 gallons twice, what
will remain ?
19. Of 30070 men who went into battle, 4564 were slain and
1675 were taken prisoners. How many were left ?
20. Bought two horses ; the first cost % 215, the second 1 40
less than the first. How much did the two horses cost ?
21. If $ 19.74 were paid for 14 bushels of wheat, what must
be paid for 25 bushels ?
22. If 19 tons of coal run an engine 798 miles, how far will
14 tons run it ?
23. The area of the New England States is as follows:
Maine, 31766 square miles ; New Hampshire, 9280 ; Ver-
mont, 10212 ; Massachusetts, 7800 ; Connecticut, 4674 ; Ehode
Island, 1306. How many more square miles are there in
Maine than in the three States of Vermont, New Hampshire,
and Massachusetts ?
24. How many States of the size of Ehode Island might be
made out of Massachusetts, and how many square miles would
remain ?
25. How much smaller is Connecticut than Vermont ?
26. Texas contains 237504 square miles. How many States
of the size of New England might be made out of it, and how
many States of the size of New Hampshire out of the remainder ?
27. If 5 bushels of wheat of 60 pounds each are required to
make 1 barrel of flour, how many pounds of wheat are re-
quired to make 100 barrels of flour ?
EXAMPLES. 53
28. In a certain schoolhouse 9 of the rooms will seat 52
pupils each, and 4 will seat 48 pupils each. How many pupils
can be seated in all ?
29. How many feet of fencing will be required to enclose a
lot of land measuring on each of two sides 489 feet, on the
third 548 feet, and on the fourth 596 feet ?
30. In a school there are 7 classes of 54 pupils each j 196 of
these are boys. How many are girls ?
31. A horse cost $ 262, a chaise $ 228, and a hack 3 times
as much as both. What did all cost ?
32. A farmer exchanged 4 cows, worth $ 68 each, for a span
of horses. What were the horses worth apiece ?
33. A merchant bought 45 bales of cotton, each bale con-
taining 42 pieces, and each piece 38 yards, at 9 cents a yard, and
sold the whole at 11 cents a yard. How much did he gain ?
34. A man raised in one year 364 bushels of corn, the next
year twice as much as he did the first year, and the third year
three times as much as the second year. How many bushels
did he raise in all ?
35. A grocer bought 8 chests of tea, each chest containing
48 pounds, at 50 cents a pound. He sold one half of the tea at
Q^ cents^ a pound and the other haK at 72 cents a pound. How
much did he gain ?
36. After $ 158 were taken from a box there remained % 15
more than twice that sum. How many dollars remained ?
37. Mrs. Keyes, having $2000 to invest, bought 10 United
States bonds at $ 112 each, and then as many railroad shares
at $ 92 each as she could pay for. How much money was left ?
38. Mr. Oaks bought a piano for $ 375, paid 1 14 for freight
and cartage, and $ 2 for tuning, then let it 7 quarters at $15
a quarter, and afterwards sold it for $325. Did he gain or
lose, and how much ?
39. A man paid $ 270 ' for a threshing-machine, and hired
help to run it at $ 5 a day. He then let the machine at $ 8 a
day, including the help he hired. How many days must he let
the machine to pay its first cost ?
54 MISCELLANEOUS EXERCISES.
40. How many posts and how many rails will be required
for a fence 156 feet long if the posts are set 12 feet apart
and the fence is 5 rails high ?
41. A man sold three houses; for the first he received
$ 3525, for the second $ 950 less than he received for the first,
and for the third as much as for the other two. How much
did he receive for the three ?
42. A jeweller sold 15 clocks and 22 watches ; for the clocks
he received $ 12 apiece, and for each watch 7 times as much as
for a clock. What did he receive for all ?
43. If 28 men can grade a road in 72 days, how long will it
take 36 men to do half the work ?
44. If a man earns % 180 a month and spends $ 36 for board
and $ 50 for clothes and other expenses, in how many months
can he save 1 1410 ?
45. Mr. Brown bought 18 cords of wood for $ 110. For how
much must he sell it a cord to gain 1 34 on the whole ?
46. Mr. Snow bought some land for $ 13825. He sold 100
acres at $ 55 an acre, and then found, in order not to lose on
his bargain, that he must sell the remainder for 1 62 an acre.
How many acres were there in the remainder ?
47. A had $ 45 ; B twice as much less 1 17 ; and C as much
as A and B togethei ■. How much money ha. ( C ?
48. One half of one number is 1764, and four times another
number is 5876. What is their sum ?
49. A and B, 450 miles apart, travel towards each other.
A travels at the rate of 30 miles a day and B of 35 miles a
day. If B rests the second day, how far apart are they at the
end of the fourth day ?
50. A man bought 163 barrels of flour at I 9 a barrel ; 15
barrels were spoiled, and the remainder he sold at 1 11 a bar-
rel. Did he gain or lose, and how much ?
51. At an election the sum of the votes received by two
opposing candidates was 4324; the successful candidate re-
ceived 218 more votes than his opponent* How many votes
did each receive ?
EXAMPLES. 55
52. On commencing business a merchant had % 7852 in cash,
$ 7919 in real estate, goods valued at % 9728, a lot of lumber
valued at $ 6930, a ship valued at 1 16834 ; during the first
year he was in trade he gained above all his expenses $ 3195.
What was he worth at the end of the year ?
53. The Gulf Stream carries 2787840000000 cubic feet of
water past a given point every hour, which is 1200 times as
much as the hourly discharge of the Mississippi. What is the
hourly discharge of the Mississippi ?
54. There were 1032467 cigars made in Westfield in 1 month.
If these were bought for 5 cents apiece, how many families
would the money thus spent supply with bread for a year (365
days) if each family should consume two 8-cent loaves a day ?
65. The distance from Boston to Albany is 202 miles, from
Albany to Buffalo, 298 miles. How long will it take a train
to pass over the road at the rate of 28 miles an hour, allowing 2
hours for detentions between Boston and Albany, 1 hour at
Albany, and 3 between Albany and Buffalo ?
56. If it takes 5 yards of cloth to make a pair of shirts, what
will 24 pairs, cost at 15 cents per yard for the cloth, 45 cents
apiece for bosoms, wristbands, and buttons, and 95 cents apiece
for making ?
57. In how many days, of 6 hours each, can the president
of a bank sign 90000 bank-notes, if he signs 5 in a minute ?
58. If 8 presses can coin 19200 pieces of money in an hour,
how many pieces can one press coin in a minute, 60 minutes
making an hour ?
Papyrus is said to have been used to write upon 2000 years before
Christ, and parchment to have been invented 1810 years later; from
the invention of parchment to that of paper in China was 20 years ;
to printing by movable types was 1608 years more ; stereotyping was
invented 273 years still later.
59. How many years from the first use of papyrus, as given
above, to. stereotyping?
60. In what year before Christ was paper invented in China ?
61. In what year after Christ was printing invented ?
56 MISCELLANEOUS EXERCISES.
62. There are in a certain school 47 pupils 14 years old ; 96
pupils 12 years old ; 114, 11 years ; 149, 10 years ^ and 168,
9 years old. What is their average age ?
63. If the earth is 92000000 of miles from the sun, and the
moon at its full is 224000 miles farther on, and light travels
at the rate of 185172 miles a second, how many seconds is it
in passing from the sun to the moon and back to the earth ?
118. Questions for Review.
What is Addition ? What is the amount ? Add orally 64 and
87. How do you write numbers to be added? Is this absolutely
necessary ? Add five numbers expressed by four figures each, and ex-
plain. Give the rule ; the proof. Illustrate adding at once numbers
expressed in two or more columns.
What is Subtraction ? What is the minuend ? the subtrahend ?
the remainder? Take orally 28 from 91. Find the difference be-
tween 368 and 7006, and explain. Give the rule; the proof. When
the minuend and difference are given, how can you find the sub-
trahend? When the subtrahend and difference are given, how can
you find the minuend?
What is Multiplication ? What is the multiplicand ? the mul-
tiplier ? the product ? What are factors ? Multiply orally 45 by 6.
Perform and ex])lain an example in which the multiplier has at least
two terms. Give the rule ; the proof. How do you multiply by
10, 100, 1000, etc. ? How do you proceed if there are zeros at the
right of the expression of the multiplicand or the multiplier, or both?
Tens X units = what ? Units x tens ? Thousands x tens ? Tens x
hundreds? Ten-thousands x hundreds?
What is Division? What is the dividend ? the divisor? the quotient ?
the remainder ? Perform and explain an example in short division ;
prove the work. Perform and explain an example in long division.
Give the rule ; the proof. How do you divide by 10, 100, 1000, etc. ?
How do you divide when the expression of the divisor contains zeros
at the right ? When the dividend and quotient are given, how can
you find the divisor ? When the divisor and quotient are given, how/
can you find the dividend ? When the multiplier and product are
given, how can you find the multiplicand ? When the multiplicand
and product are given, how can you find the multiplier ?
DRILL EXERCISES. 57
DRILL EXERCISES.
119. Explanation op the Use of the Drill Tables.
The object of the Drill Tables and Exercises which are
found on the six following pages is to, extend indefinitely
practice in arithmetical operations without additional labor
on the part of the teacher.
The exercises are not to be assigned in order, nor is any
one pupil expected to perform them all ; they may be used,
however, like other examples. (See Notes on pages 16, 25,
35, and 48.)
The following illustration shows how they may be used
for class drill, and each pupil have a different example.
Addition.
1. Let the members of the class number themselves 1, 2,
3, etc., to any given number up to 25 ; and let each member
find his number in the left-hand margin of the table.
2. The teacher then gives a direction in this form : "Add
A, B, and C." (See Exercise 1, page 59.)
3. In obedience to this direction, each pupil will add the
numbers that he finds expressed under the letters A, B, and
C, and in the line of his own number. Thus, pupil No. 1
will add 65, 512, and 7901 ; No. 2 will add 34, 724, and
3053 ; and so on.
Thus a series of examples is given out at a single dicta-
tion, and the pupils are taught to work independently.
4. The key contains answers to all these examples.
Subtraction, Multiplication, and Divisioa
By changing slightly the form of direction described
above, the same table will afford abundant practice in the
other fundamental operations. (See pages 59, 61, and 63.)
68
MISCELLANEOUS EXERCISES.
120. DRILL TABLE No.
Simple Numbers.
D
opq
274
xamples.
1.
A
Q5
B
liij
512
c
k Imn
7 901
2.
34
724
3 053
3.
79
235
5 360
4.
46
941
1 604
5.
98
858
8 029
6.
bQ
467
7 940
7.
32
673
4 809
8.
48
388
6 580
9.
87
747
2 096
10.
76
599
8 920
11.
54
252
2 031
12.
95
381
6 150
13.
^2
817
4 706
14.
71
426
9 059
15.
23
794
4 270
16.
37
261
3 Oil
17.
93
638
5 490
18.
28
372
1 705
19.
62
919
9 630
20.
57
485
5 108
21.
89
591
7 502
22.
45
183
2 610
23.
92
868
3 703
24.
64
655
6 207
25.
25
1
942
2
8 054
3
613
769
133
486
918
675
436
577
814
239
721
544
715
645
978
851
327
163
784
516
297
466
349
922
4
E
r s tu
2 865
3 742
8 604
6 821
4 930
5 439
7 108
4 583
8 057
6 974
9 107
1 580
7 362
2 115
4 276
6 583
2 941
8 724
6 239
4 037
5 482
9 372
6 048
3 659
4 176
5
DRILL EXERCISES.
59
121. Exercises upon the Table.
Addition.
1. Add A,* B, and C.
2. Add B, C, and D.
3. C plus D plus E plus F
equals what number ?
4. A + B + C + D+16042 = ?
5. What is the sum of B, C,
D, E, F, and 61375 ?
6. Find the amount of A, B,
C, D, E, F, and 23456.
In each column indicated by figures at
the bottom of pages 58, 60, and 63,
7. Add the upper six numbers.
8. Add the upper ten numbers.
9. Add the upper fifteen num-
bers.
10. Add all the numbers.
Subtraction.
11. From C take B.
12. Subti-act D from E.
13. Take E from F.
14. Find the difference between
C and E.
15. F minus C equals what
number ?
Multiplication.
16. Multiply B by 6.
17. Multiply C by 7.
18. Multiply D by 8.
19. Multiply E by 9.
20. Multiply B by A.
21. Multiply C by B.
22. Multiply C by D.
23. Multiply E by D.
24. Find the product of F by D.
25. Find the product of F by C.
See the explanation, page 57.
Review.
26. What number added to the amount
of A and B will equal C ?
27. Add together C, E, and the difference
between B and D.
28. Subtract C from 12304, and from the
remainder take B.
29. Multiply D by 1002, and from the
product take F.
30. Multiply C by 6 ; D by 7 ; E by 8 ;
and find the sum of the products.
31. Multiply B by 10; D by 11; and add
the products with C plus E.
32. A man having F dollars paid E dol-
lars to one man and D dollars to
another. How much did he have
left?
33. Bought a house for C dollars ; paid
B dollars for repairs ; then sold
it at a loss of D dollars. How
much did I receive for the house ?
34. A merchant had B barrels of flour.
He sold A barrels at $12 a barrel,
and the remainder at $ 9 a ban-el.
How much did he receive for the
flour?
Oral Practice.
35. How many are 8 + f + g + h, etc. to z ?
36. How many are 27 +h + i, etc. to z ?
37. How many are 55 - f - g - h - i - j ?
38. How many are 100 - o - p, etc. to z ?
39. How many are 7 + f to n less o less
p less q ?
40. How many are h times i less j plus
k toz?
41. How many are 100 less A ?
42. Find the difference between 43 and A.
60
MISCELLANEOUS EXERCISES
122. DRILL TABLE No. 2.
Simple Numbers.
D E
nop qrs t uvw xyz
469 007 8 046 352
743 500 6 530 781
620 085 4 654 380
800 659 7 820 463
389 700 9 068 318
956 800 5 782 630
285 004 3 905 746
500 376 2 849 370
675 400 6 470 856
732 900 5 783 062
486 300 7 560 849
840 015 3 672 083
570 068 • 4 921 608
962 400 9 430 572
435 600 4 187 063
700 843 6 410 342
659 700 1 598 706
800 875 7 421 089
643 007 6 450 713
350 092 2 835 068
800 765 5 306 739
469 007 3 540 687
782 500 7 390 468
946 007 6 083 745
465 300 7 408 653
11 12 13 14
Examples.
1.
A
ef
45
B
ghi
648
C
j klm
6 068
2.
68
473
5 406
3.
74
835
8 049
4.
56
592
7 250
5.
48
726
3 087
6.
64
954
4 503
7.
78
367
2 790
8.
83
289
5 608
9.
54
635
9 160
10.
76
489
6 085
11.
58
375
8 507
12.
47
689
4 130
13.
65
764
7 082
14.
53
865
3 706
15.
49
796
8 154
16.
84
347
4 370
17.
79
586
5 480
18.
63
627
1 094
19.
57
395
8 406
20.
69
738
6 530
21.
75
579
7 209
22.
67
486
8 560
23.
59
942
4 308
24.
46
278
3.960
U.
73
8
587
9
6 805
^0
DRILL EXERCISES.
61
123. Exercises upon the Table.
43.
U-
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
68.
59.
60.
61.
63.
65.
66.
Division.
Divide D by 4 .♦
Divide D by 5.
Divide E by 6.
Divide E by 7.
Divide D by 8.
Divide D by 9.
Divide C by 12.
Divide C by 16.
Divide D by 16.
Divide D by 18.
Divide E by 27.
Divide C by A.
Divide D by A.
Divide E by B.
Divide D by C.
Divide E by C.
Divide D by 800.
Divide E by 4200.
Addition.
How many are
46872 + A to D?
How many are
65478 + A to E?
Subtraction.
From E take D.
Find the diflference '
tween E and D x 1
Multiplication.
Multiply D by C.
Multiply E by D.
Keview.
How many more than C are B times B ?
What number added to ten times the
amount of B and C will equal D ?
A man owns three tracts of land ; the
first is valued at C dollars, the second
at B dollars, and the third is worth
twice as much as the second. How
much is the land worth ?
By selling a house at C dollars I gained
12 times A dollars. What was the
cost?
If a farmer should purchase B acres of
land at A dollars per acre, and pay
down C dollars, how much would he
then owe for the land ?
A man having C dollars spent B dollars
and lost A dollars. How much would
one third of the remainder be ?
73. How many cows, at A dollars apiece, can
be bought for one fifth of ten times B
dollars, and how many doUars will
Oral Practice.
74. How many are 6 + e + f+g, etc. to z?
75. How many are 15 +g + h, etc. to z ?
76. How many are 29+j +k, etc. to z?
77. How many areexf-g-h?
78. How many are g x h -i- i ?
79. How many are h x i -f- j ?
80. Divide A by 2 ; by 3 ; by 4 ; 5 ; 6 ;
70.
71
72.
7;
81. Divide gh (64, 47, 88, etc.) by 3 ; by
4 ; etc.
Other dividends and divisors can be indicated, as
j k by 7 ; no by 8 ; tu by 9; etc.
* Sec page 57, for Explanation of the Use of the Prill Tables.
62 MISCELLANEOUS EXERCISES.
124. DRILL TABLE No. 3.
Simple Numbers.
Examples. A
1, Nine hundred fourteen thousand, forty-one.
One million, forty thousand, fourteen.
Nine hundred seventy-six thousand, sixty-seven.
Sixteen hundred seventy-eight.
Sixty-three million, three hundred six thousand.
Nineteen million, nine hundred thousand, 19.
One hundred seventy million, seven.
Ten million, one thousand, one hundred one.
Three hundred five million, fifty thousand. .
Twelve million, two hundred thousand, two.
One million, eighteen thousand, eight. .
One hillion, six hundred thousand, six.
Nine hundred four million, ninety-four.
Three billion, thirty million, three hundred three.
Two hillion, four hundred twelve thousand, 14.
One hundred one million, one thousand, one.
Six billion, sixty thousand, six hundred.
Four hundred three billion, thirty.
One trillion, seventeen million.
506 billion, sixty-five thousand, five.
Four trillion, four billion, four thousand.
Six hundred eighty-nine billion, six thousand, 89.
Forty-two trillion, forty thousand, two hundred.
Eighteen trillion, 108 million, eighteen.
Four trillion, forty-seven billion, 4700.
DRILL EXERCISES.
63
DRILL TABLE No. 3
(continued).
Ex
amples.
c
r s
63
^2
81
36
64
35
28
68
,51
72
88
48
45
54
57
49
96
34
84
99
91
78
s
D
t uv
819
364
486
324
576
595
728
952
612
648
352
392
912
585
864
594
627
588
715
768
884
672
495
273
624
16
E
w xyz
3 276
4 368
9 234
2 592
7 488
6 545
4 368
7 616
4 896
7 128
8 448
8 232
2 736
2 925
6 912
4 158
8 778
9 996
6 435
5 376
6 188
8 736
7 425
9 282
3 744
17
125. Exercises upon the Table.
82. Express A by figures *
83. Add, in A, the 1st and 2d; 2d and 3d;
etc.
84.
85.
86.
87.
88.
89.
90.
91.
Add, in A, from 1 to 6; 2 to 7; etc.
Find, in A, the difference between the
1st and 2d; the 2d and 3d; etc.
Multiply A by 6.
Multiply A by 7.
Multiply A by 8.
Multiply A by 9.
Divide A by 6. 92. Divide A by 8.
Divide A by 7. 93. Divide A by 9.
Multiply C by B ; add D to the product;
and find the difference between the
amount and E.
Divide D by B ; also divide E by B ;
and find the difference between the
sum of the quotients and D.
Divide E by D; subtract the quotient
from C; and multiply the remainder
byB.
Subtract C from D ; divide the remain-
der by B ; and with the quotient
divide E.
Oral Practice.
98. How many are 9 + q to x less y less z ?
99. How many are 34 + r to v, divided by q?
100. How many are 45 + 1 to z, divided by q?
101. How many are r times s + 1 toz, divided
byq?
102. How many are t times u -!- v to z, di-
vided by q ?
95.
96.
97.
* See Explanation of Table, page 57.
64
UNITED STATES MONEY,
SEOTIOI^ YII.
UNITED STATES MONEY.
126. The picture above represents pieces of metal
weighed and stamped by authority of government, and
used in buying and selling. Such pieces of metal are coins.
Each coin represents a unit of value.
127. Dollars and cents are the units of value chiefly
used in business. Eagles, dimes, and mills are also used,
but there is no coin to represent a mill.
TABLE.
10 mills
10 cents
10 dimes or
100 cents
10 dollars = 1 e
]=
1 cent, marked ct. or f,
1 dime.
1 dollar, marked $.
To read and write numbers in United States Money.
128. The dollar, being the principal unit of United States
money, is expressed at the left of the decimal point ; dimes,
cents, and mills, being tenths, hundredths, and thousandths
of a dollar, are expressed at the right of the decimal point.
EXAMPLES, 65
Thus, 11 dollars, 2 dimes, 3 cents, and 4 mills are written
$11,234;
and the expression is read, "Eleven dollars twenty-three
cents four mills."
For exercises in reading and writing, turn to page 73.
129. Oral Exercises in Reduction.
a. How many mills in 1 cent ? in 18 cents 5 mills ?
By what do you multiply to change cents to mills ? dollars to cents ?
dollars to mills ?
b. Change % 14.08 to cents. e. Change 1 2.625 to mills.
c. Change 1 1.62 to cents. j. Change 1 5.02 to mills.
d. Change % 0.48 to cents. g. Change 1 4 to mills.
h. How many dollars are there in 500 cents ?
By what do you divide to change cents to dollars ? mills to dollars ?
i. How many dollars are there in 170 cents ? in 3689 cents ?
j. How many dollars are there in 1875 mills ? in 4728 mills ?
130. In performing the examples above, you have
changed numbers expressing a certain amount of money
to numbers whose units are larger or smaller, but without
changing the amount itself Such a process is called
reduction.
For other examples in reduction of United States money, see page 73.
131. Examples for the Slate.
ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION,
How do you write dollars, cents, and mills, when you are to add or
subtract them ? (Art. 46.)
1. My deposits in a bank were $ 192.92 and % 155.37 ; of
this I have withdrawn 1 79.48, 1 71.62, and % 78.21. What
is the balance in the bank ?
2. What must I pay for 23 yards of silk at $2.37 a yard,
and 5 yards of lace at $ 1.68 a yard ?
66 UNITED STATES MONEY.
3. What is 1 fifteenth of 1287.40? 1 seventeenth of
1722.50?
In the following examples continue the division to cents and mills.
(Art. 102.) In the answers, reject mills when less than 5, and call 5
mills or more 1 cent.
4. What is 1 fifth of $ 17 ? of $ 83 ? 1st Ans. $ 3.40.
5. What is 1 sixteenth of 1 981 ?
6. Eight men chartered a schooner for $ 295. What was
each man's share of the cost ?
7. When 32 lawn-mowers were bought for 1 696, what was
the price of each ?
8. Mr. Eice paid 1 198.45 for 35 school etesks. What would
168 desks cost at the same price ?
To divide one sum of money by another.
132. Illustrative Example. At $ 2.12 per pair, how
many pairs of slippers can be bought for $ 100 ?
WRITTEN WORK. Explanation. — To divide one sum of money by
212) 10000 (47 another, both dividend and divisor must be expressed
848 '^^ ihe same denomination. Here the divisor being
.p.„^ cents, the dividend must be changed to cents.
118^ (Art. 129.) Dividing 10000 cents by 212 cents,
we have 47 for a quotient, with a remainder of
36 36 cents. Ans. 47 pairs ; 36 cents remain.
9. I paid $80 for turkeys at $2.50 apiece. How many
turkeys did I buy ? Divide 1 42 by 1 1.75. 1st Ans. 32.
10. A conductor took up $ 1224 worth of railroad tickets
from Springfield to New York at 14.25 apiece. How many
tickets did he take ?
11. How many boxes at 33 cents a box can be bought for
$ 20, and how many cents will be left ?
12. How many veils at 92/ each can be bought for 1 30 ?
13. How many dinners at $ 0.625 each will $ 22 pay for ?
For additional examples, see page 73.
UECKONtNG MONEY. 67
Coins and Paper Currency.
133. The legal coins of the United States are
Gold
Silver.
Double-Eagle
=
120.00
Dollar = $1.00
Eagle
=
10.00
Half-dollar = 0.50
Half-Eagle
=
6.00
Quarter-dollar - 0.25
Quarter-Eagle
=
2.50
Twenty-cent piece ~ 0.20
Three-dollar piece
=
3.00
Dime = 0.10
One-dollar piece
=
1.00
Copper and nickel 3-cent and 5-cent pieces and bronze 1-cent piece.
Note. The gold coin is hardened by an alloy of 1 tenth copper and silver
(the silver not to exceed 1 tenth of the whole alloy). The silver coin is
hardened by 1 tenth copper. The bronze cent has 95 parts of copper to 5
parts of tin and zinc. The 3-cent and 5-cent pieces have 75 parts of copper
to 25 parts of nickel.
The silver 5-cent and 3-cent pieces, the bronze 2-cent piece, and the old
copper coins, are no longer issued.
Bank bills and United States Treasury notes (greenbacks) are largely
used in place of coins. These represent the values of $1, $2, $5, $10,
$20, $50, $100, $500, and $1000.
134. Exercises in reckoning Money.
Perform as many as possible of the following examples without
written work :
How much money in
a. . Two 20-dollar bills, three lO's, four 5's, and seven I's ?
h. Eight 5-dollar bills, seven 2's, five lO's, and three I's ?
c. One 50-dollar bill, six 5's, two I's, with 3 half-dollars,
5 quarters and 4 dimes ?
How much more money must you receive to have $ 50 if
you now have
d. Three 5-dollar bills, seven 2's, four I's, with 2 half-dol-
lars, 3 quarters, and four 5-cent pieces ?
e. Two 5-dollar bills, three 2's, and one 10, with 5 quar-
ters, 4 dimes, two 3-cent, and three 5-cent pieces ?
68
UNITED STATES MONEY.
J. How much money shall I have left of six 5-dollar bills,
and 2 quarters, after paying for 6 yards of brilliant at 65/
a yard, for Silesia, 28/', and for buttons $ 1.15 ?
g. What must you pay for 2 dozen eggs, 5 pounds of sugar,
2 gallons of vinegar, and 2 bushels of apples at the present
prices where you live ?
135. Accounts and Bills.
[Extract from the Accodnt-book of T. Smith & Co.]
EDWARD WILLIAMS, J)r.
18
76,
(^ ^ /^. -m^^i^ s^/ou^^ @ ^§
/
40
f
0^^
£
" ^^/^.G^uA^uMa^cn^," /^/
S
7^
Ji^Tte
/
- ir^/^.Ja^a Coffee, " SO iP
4
§o
It
^§
" / c/czy'd- tcw^in, o/ dciM^ man
/
7^
136. Above is a record kept by T. Smith & Co. of ar-
ticles sold and services rendered by them to Mr. Williams.
137. It is customary for persons who buy or sell goods
or services to keep a record of the articles bought or sold,
the kind and amount of services rendered, their value, etc.,
as above. Such a record is an account.
138. The person to whom a debt is owed is a creditor.
139. The person who owes a debt is a debtor.
Who is debtor in the account stated above? Who is creditor?
140. A written statement of an account prepared for the
debtor by the creditor is a bill.
141. When the bill is paid, the creditor, or some one
authorized by him, signs his name to the bill, with the
words "Received payment." The bill is thus receipted.
(See bills Nos. 2 and 3.)
BILLS.
69
Zixamples for the Slate.
142. Find the cost of each article in the following bills,
and their several amounts :
14. (1) New York, Nov. 12, 1877.
^ ^ ¥0ttgl)t of FOWLE, PEATT, & CO.
/^^ (^. (M. c^^^ S^/oui., ex^ia, (^"^ ^^.^O
/J* " Q^7z?ie4o^ S^^ul, " l/0.^3
/4 ^. ^oi^, " §7f
JReceived payment,
15.
(2) Cincinnati, Dec. 18, 1876.
^0txgl)t of J. SMITH.
1876,
Q4i^.
/
/^^.
M,i,t:a ^ue9z, @
S^f
II
S
SOO "
Wdc^cna.
/^
Mec.
7
4^6? "
Wdc^ .^ac/, " .
//^
II
//
S^ "
^/ue.
^Of
Beceived payment.
J. SMITH.
16.
(3) Worcester, Oct. 9, 1877.
STo OTIS LERNED, Dr.t
Received payment,
@
y^.^6?
^f
OTIS LERNED.
By John Waite.
* This sign means "at," and is commonly used in stating prices.
t This means that Mr. Drew is debtor to Mr, Lemed. Dr. is read " debtor."
70
17.
UNITED STATES MONEY.
(4) New York, Mar. 7, 1877.
^0 CHARLES DAT, Dr.
187
7.
o/a /J^ /A S^al^u'c Q^^, @ §^f
It
It
" 7 " M/ue ^auo/, " l/3f
It
It
11
S^ed.
J?
" ^/^^. ^a^Jcz7no77ii^, " £.00
It
It
//
//
Received "payment.
CHARLES DAY.
18.
(5) Bristol, Jan. 1, 1877.
^0 A. E. PEASE, Dr.
18
It
re.
S
fi
It
II
/une
fu/y
It
II
II
@ 7f
" 4^ " ^lead/a^^ o/eu," SOf
Cr,*
MylT^WayfOn. ^2S.OO
^ ^a2^, @ ^S4
^^/. ^O.OO
Mu^Ttce c/ue Q^. M. ^eaae
Received payment.
* This means that Mr. Butler is credited for goods or cash delivered. Cr. is read
creditor."
EXAMPLES. 71
Examples for Bills.
143. Find the amounts due in the following examples,
and make out the bills, supplying dates, etc., when wanting.
19. Charles Miller bought of James Gibbs, Jan. 4, 1877,
1 horse for $ 95.00, 2 cows at $ 50 apiece, 1 wagon for % 62.00,
2 shovels at $ 1.12 apiece, 30 bushels of corn at 65 f per bushel,
and 17 bushels of wheat at 1 1.62 per bushel.
20. Samuel Briggs sold to Alfred Loomis 2 pieces flannel, of
62 yards each, at 49 / per yard ; 38 yards ticking, at 29 /; 86 yards
brown sheeting, at 27/; and 42 yards broadcloth at 13.65.
21. Dr. Holland bought of John Avery 9 pounds oil of pep-
permint at 12.50; 4 pounds oil of cassia at % 1.62; 4 pounds
oil of orange at 1 3 ; 6 pounds oil of lemon at 1 3.25 ; 5 pounds
oxalic acid at 13/ ; and 5 pounds Seneca root at 95/'.
22. Banks & Searles, of Cleveland, bought of Snow & Rising,
Albany, 24 sack coats at % 15.75 ; 36 vests at % 3.50 ; 9 dozen
felt hats at $36 per dozen; 4 dozen pairs suspenders at 42/
per pair ; and 23 dozen pairs gloves at ^S / per pair.
23. J. D. Furber bought of C. O. Clement, Nov. 8, 1876,
2 Dictionaries, at 87/ apiece; 9 Vocal Cultures, at 90/, and
24 Spellers, at 20/. Dec. 2, he bought 2 reams of paper at
12.12, 3 dozen pencils at 50/, and 12 slates at 17/. Dec. 10,
he paid Mr. Clement $ 20.00, and Jan. 1, 1877, Mr. Clement
made out his bill. Required the balance due.
24. Sell to your neighbor 4 pear-trees at $ 1.75 each, 9 to-
mato-plants at 7/ each, 5 geraniums at 30/ each, and make
out the bill.
25. Sell three different articles from a dry-goods store, and
make out the bill.
26. Make out a bill for 3 days' work at 75/ a day, 4 days'
work at $1.50 a day, and 2 bushels of cranberries at $4 u
bushel, crediting the person against whom you make the bill
with 5 hours' work at 35/ an hour.
72
UNITED STATES MONEY.
144. DRILL TABLE No. 4.
A
$18.40
$83.22
$36.41
$30.05
$204.75
$9,208
$5,632
$876.
$100.35
$15,207
$1.36
$20.95
$0,402
$19,005
$63,072
■ $7,645
$419.28
$0,625
$ 500.57
$268.06
$29.70
$11,005
$100.02
$444.44
$100.10
United States Money.
B
Twelve dollars, twenty-five cents.
Seventy-one dollars, ninety cents.
Twenty-five dollars, sixty-two cents.
Eighteen dollars, nine cents.
One hundred thirty dollars, six cents.
Pive dollars, seven cents, five mills.
One dollar, ninety cents, eight mills.
One hundred dollars, twenty cents.
Forty-nine dollars, seventy-two cents.
Six dollars, seven cents, three mills.
Twenty-seven cents, five mills.
Twelve dollars, nineteen cents.
Twenty-five cents, five mills.
Sixteen dollars, six mills.
Forty-nine dollars, twenty-four cents.
Five dollars, sixty-seven cents, five mills.
Ninety-nine dollars, fifty-six cents.
Seventeen cents, eight mills.
Thirty-eight dollars, five mills.
89 dollars, fifty cents, three mills.
Ninety-two cents, five mills.
Seventy-five cents, five mills.
Fifty-four dollars, nine cents.
Four dollars, forty-four cents, four mills.
Nine dollars, nine cents^ nine mills.
DRILL EXERCISES. 73
146. Exercises on Table No. 4.
103. Read as dollars, cents, and mills, the numbers expressed in A. §
104. Read decimally the numbers expressed in A.
105. "Write in figures the numbers expressed in B.
106. Disregarding the mills, change the numbers expressed in A to
cents.
107. Change the numbers expressed in B to mills.
108. Add the numbers from 1 to 8 * in A to each number expressed in B.
109. Add the numbers expressed in A and B, (1st) from 1 to 4 *;
(2d) from 2 to 5 ; (3d) from 3 to 6, etc.
110.
$900-A = ?
114. Ax9 = ?
118.^ A -
^7=?
111.
A-B = ?
ii5.tA-T-10=?
119.x A -
j-$0.25 =?
112.
A X 6 = ?
i/^.t B-v-ll = ?
120.x A -
-$0.16 = ?
113.
Bx7 = ?
ii7.tB-T-12 = ?
121. B^
-$0,005-?
122. If a person saves a sum equal to A in one month, how much wi]?
he save in 13 months ?
123. How many pounds of sugar, at 8 cents a pound, can be bought for
each sum of money expressed in A ? J
146. Questions for Revie-w'.
What are the units of United States money? Give the table.
How are dollars, cents, and mills expressed by figures ? What is con-
sidered the" principal unit ? Give the sign for dollars. How do you
change dollars to cents ? dollars to mills ? cents to mills ? mills to dol-
lars ? cents to dollars ?
How do you add numbers in United States money? How do
you subtract ? When you multiply, where do you put the decimal
point in the product? Divide $185 by 7, continue the division to
mills, and explain. What is necessary in order to divide one sum of
money by another ? Divide $900 by 36 cents.
What are coins ? Why is paper money sometimes used in place of
coins? Name the gold coins ; the silver coins.
What is a creditor ? a debtor ? an account ? a bill ? How is a
bill receipted?
* Inclusive. t See page 66, note. t Reject mills.
§ See page 57, for Explanation of the TJ?o of the Drill Tables.
74 UNITED STATES MONEY,
147. Miscellaneous Examples.
27. A girl bought a pair of boots for 1 2.37, another pair for
$1.65, slippers for $1.25, and shoes for 82/. What was the
whole cost ?
28. I bought a horse for 1 95.00, a wagon for 1 63.00, and
a harness for $15.00; kept them a week, paying $2.50 for
board of the horse, then sold them for 1 175.00. Did I gain
or lose, and how much ?
29. What shculd I pay for 2 dozen pigeons at 85/ pev..
dozen, 2 dozen at $1.10 per dozen, and 1 dozen for 90/.
30. There were sold in one week 8874 sheep at $ 4.13 pe .
head. What did they bring ?
31. There were sold 4778 beeves, averaging 874 poun(?«
apiece, at 7/ per pound. What was received for them?
32. What did I gain by buying 2 pieces of cambric, each
containing 62 yards, for $ 39.68, and selling them for 40 cents
per yard ?
33. A man paid $ 16.25 for 13 days' work. What was that
a day ?
34. Among how many boys must $ 12 be distributed, that
each may receive 75 cents ?
35. I sold 35 barrels Pippins at $ 1.75 per barrel, 17 barrels
Pome Royals at $ 1.80 per barrel, 13 barrels Golden Sweets at
$ 1.25 per barrel, and 25 of Eussets at $ 2.25 per barrel. Paid
17 cents a barrel for picking, and $ 12.00 for freight. What
remained after my expenses were paid ?
36. Paid $ 3.00 for 1 dozen apple-trees, $ 3.36 for 1 dozen
peach-trees, $3.30 for half a dozen pear-trees. What did I
pay for the whole, and how much apiece for each kind ?
37. A carpenter paid for stock and work for a barn, $ 450.75 ;
for mason's work, $ 38.25 ; for digging and stoning cellar,
$47.18; for painting, $40.00; to the plumber, $8.12. He
then sold the barn, and lost, in so doing, $ 14.30 ; how much
did he sell it for ?
FAGTOBS. 75
SECTION YIII.
FACTORS.
148. What numbers multiplied together will produce
10 ? Answer, 2 and 5 ; also 1 and 10 ; thus, 2x5 = 10 and
1x10 = 10.
A number that may be used as multiplicand or as multi-
plier to make another number is a factor of that number.
Name two factors of 15 ; of 16 ; of 18 ; of 24 ; of 36 ; of 45.
Note I. The word factor will be used in this Arithmetic to denote only
such factors as are not fractional.
Note II. If a number be divided by any of its factors there will be no
remainder. Hence a factor of a number is also called a divisor or a tneasure
of that number.
149. Name some factors of 12 besides the number itself
and 1. Has the number 13 factors besides itself and 1 ?
Has the number 14 ? 15 ? 17 ? 18 ? 19 ?
A number that has other factors besides itself and one is
a composite number.
Which of the numbers 12, 13, 14, 15, 17, 18, 19 are com-
posite numbers?
150. A number that has no other factors besides itself
and one is a prime number.
Which of the numbers 12, 13, 14, 15, 17, 18, 19 are prime
numbers ?
Name the composite numbers from 1 to 40.
Name the prime numbers from 1 to 40.
Note. In speaking of the factors of a number, we do not usually include
the number itself and one. Thus, we frequently say that a prime number
has no factors.
76 FACTORS.
151. Name the factors of 12 that are prime numbers.
Name those that are not prime numbers.
A factor that is a prime number is a prime factor.
152. Oral Exercises.
a. What are the prime factors of 6 ? 8 ? 14 ? 24 ? 27 ?
b. What are the prime factors of 22 ? 36 ? 28 ? 20 ? 35 ?
c. What are the prime factors of 16 ? 21 ? 15 ? 33 ? 2Q?
153. In seeking for the factors of a number we may use
certain tests, the more convenient of which are the follow-
ing:
1. A number whose units' figure is 0, 2, 4, 6, or 8, is divisible by 2.
Note. A number that is divisible by 2 is an even numher; a number that
is not divisible by 2 is an odd numher.
2. A number is divisible by 3 if the sum of its digits * is divisible
by 3. Thus, 285 is divisible by 3, for 2 + 8 + 5 = 15 is divisible by 3.
3. A number is divisible by 4 if its tens and units together are
divisible by 4. Thus, 6724 is divisible by 4, while 6731 is not.
4. A number is divisible by 5 if the units' figure is either or 5.
5. A number is divisible by 6 if it is an even number and divisible
by 3..
6. A number is divisible by 8 if its hundreds, tens, and units are
divisible by 8. Thus, 6728 is divisible by 8, while 6724 is not.
7. A number is divisible by 9 if the sum of its digits is divisible
by 9.
8. A number is divisible by 11 if the sums of its alternate digits
are equal, or if their difference is divisible by 11. Thus, 1782 and
1859 are divisible by 11, while 4987 is not.
9. A number is divisible by a composite numher, if it is divisible by
all the factors of the composite number. Thus 3555 is divisible by 15,
for it is divisible by 3 and by 5.
Note. For the reasons of these tests, see Appendix, page 303.
* A digit here means the number denoted by a figure without regard to
its place.
PRIME FACTORS. 77
154. Oral Ezercisea.
Using the tests described above,
a. Name the numbers expressed in B, page 58, that contain
the factor 2 ; 4 ; 5.
b. Name the numbers in C, page 58, that contain the factor
3; 6; 9.
c. Name the numbers in D, page b^, that contain the factor
8; 9; 10; 100.
To find the Prime Factors of a Number.
155. Illustrative Example I. What are the prime
factors of 2205 ?
Explanation. — Applying the tests (Art. 153) to
the given number, we find that 2 is not, but that
3 is, a factor of 2205 ; and, by dividing, see that
2205 = 3 X 735.
Seeking, in the same way, a prime factor of
735, we find that 735 = 3 x 245. Continuing
this process, we find that 245 = 5 x 49, and that
49 = 7 X 7. Therefore, 2205 = 3x3x5x7x7, and
Ans. 3, 3, 5, 7, 7. the prime factors are 3, 3, 5, 7, and 7.
156. Illustrative Example II. What are the prime
factors of 409 ?
WRITTEN WORK. Explanation, - Aipiplying the
19^ 409 r21 23^ 409 n 7 *"'*' ^^''' ^^^' ""' ^^ '^^' ^^^
19} 409 (Jl Z6) 4Uy (17 .^ ^^^ divisible by 2, 3, or 5. We
_ _ then try to divide by the other
29 179 prime numbers in order until we
19 161 reach 23, when we see that the
"77 ~~Z quotient is less than the divisor.
There can then be no prime factor
in 409 gi'eater than 23, for if there were, there would be another factor
(the quotient) less than 23, which we should have found before reach-
ing 23. The number 409 is therefore prime.
157. As we have found in Art. 155 that 2205 equals the
product of all its prime factors, so we shall always find that
A composite number equals the product of all its prime factors.
RIT
TEN WORK.
3
2205
3
735
6
245
7
49
7
%S FACTORS.
158. When a composite number is expressed as a prod-
uct of prime factors, it is said to be separated into its prime
factors.
169. From the above examples may be derived the fol-
lowing
Rule.
To separate a number into its prime factors :
1. Divide the given number hy one of its prime factors.
2. Divide the quotient thus obtained by one of its prime
factors; and so continue dividing until a quotient is ob-
tained that is a prime number.
3. This quotient and the several divisors are the prime
factors sought.
Proof.
Multiply together the prime factors thus found. The
product ought to equal the given number.
Note. If no prime factor is readily found by which to divide, we try to
divide by the several prime numbers in order. If no prime factor is found
before the quotient becomes less than the trial divisor, the given number is
prime. See Illustrative Example II.
160. Examples for the Slate.
Separate into prime factors the following numbers :
(1.) 180. (4) 208. (7.) 329. (10.) 644.
(2.) 192. (5.) 260. (8.) 338. (11.) 684.
(3.) 176. (6.) 169. (9.) 357. (12.) 2500.
Select the prime numbers and find the prime factors of the
composite numbers among the following :
(13.) 341. (18.) 450. (23.) 704. (28.) 945.
(14.) 344. (19.) 590. (24.) 711. (29.) 972.
(15.) 362. (20.) 560. (25.) 762. (30.) 2688.
(16.) 367. (21.) 596. (26.) 808. (31.) 1164.
(17.) 408. (22.) 689. (27.) 836. (32.) 3248.
SYMBOLS OF OPERATION. 79
SYMBOLS OF OPERATION.
161. The signs + , - , x , and -^ , since they indicate
that certain operations (adding, subtracting, multiplying,
and dividing) are to be performed, are called symbols of
operation.
162. In expressing a series of operations by aid of these
signs, it is often necessary to indicate that an operation is
to be performed on two or more numbers combined. This
is done by writing the numbers to be operated upon, with
the proper signs, and enclosing the whole expression in
marks of parenthesis or brackets. The expression so en-
closed is then treated as if it denoted a single number.
Thus,
(7 + 2) X 5 means that the sum of 7 and 2 is to be multiplied
by 5 ; but 7 + 2x5 means that 7 is to be increased by 5 times 2.
(7-2) X 3 means that the difference between 7 and 2 is to be
multiplied by 3 ; but 7-2x3 means 7 diminished by 3 times 2.
7 + 2
(7 + 2) -^ 5, or —^* means that the sum of 7 and 2 is to be
divided by 5.
[(2 + 3) X 5 - 11] X 2 means that the sum of 2 and 3 is to be
multiplied by 5, the product diminished by 11, and the re-
mainder multiplied by 2.
163. In performing a series of operations indicated by
signs.
First, operate on the numbers that are written within
parentheses as indicated by the signs. Next, multiply and
divide as indicated by the signs x and +. Finally, add
and subtract as indicated by the signs + and -.
* The horizontal line here drawn between 7 + 2 and 5 is equivalent to
marks of parenthesis.
80 FACTORS,
164. Oral Exercises.
[The Key contains answers to the following examples.]
a. (6 + 8) X 5= ?
h. 3x8-4x3= ?
b. 6 + 8x5-^ ?
i. 3x8-(4x3)=?
c. (8-3) X 2= ?
d. 8-3x2=?
^ _ 3x4-2x3
J. 14- ^
e. 8 + 12-4=?
/. (8 + 12)-4=?
8+3 8-3 ,
^- 2^ 2 -^
g-. (2 + 1) X (7-2)=?
L [(4 + 6)x4-5x3
= ?
CANCELLATION.
165. Illustrative Example I. If 4 be multiplied by
3 and the product divided by 3, what is the result ?
WRITTEN WORK. From this example we see that
4x3 If cc given number he multiplied by a
^ number, and the product be divided by the
M same number, the result will be the given
number.
In such cases, both the multiplication and the division
may be omitted.
Note. This omission is indicated in the written work above by draw-
ing a mark through the 3 thus, ^.
166. Illustrative Example II. What is the result of
dividing the product of 4 and 6 by 3 ?
Explanation. — As 6 = 2 x 3, the dividend in this
WRITTEN WORK. i • . « o i ..v, v • • o ^i, 4-
- example is 4 x 2 x 3, and the divisor is 3, so that we
. g, may strike out the factor 3 in both dividend and
— = 8 divisor, and multiply by 2 only, thus shortening the
. P work.
The process of shortening work by striking out equal
factors in dividend and divisor is cancellation.
CANCELLATION. 81
167. Examples for the Slate.
All operations upon numbers should first be indicated, as
far as possible, by signs, that the work to be done may be
shortened, if possible, by cancellation.
33. Divide 81 x 42 by 99 x 7.
34. Multiply 75 x 10 by 3 x 6, and divide that product by
15 X 25 X 12.
35. Divide 7 x 8 x 48 by 63 x 4 x 5 x 17, and multiply the
quotient by 51.
36. If 5 sets of chairs, 6 in a set, cost % 75, what did 1 chair
cost ?
37. If it requires 13 bushels of wheat to make 3 barrels of
flour, how many bushels will be required to make 78 barrels
of flour ?
38. If a tree 54 feet high casts a shadow of 90 feet, what
length of shadow will be cast by a flag-staff 105 feet high ?
39. A grocer exchanged 561 pounds of sugar, at 12 cents
per pound, for eggs at 22 cents per dozen. How many dozen
were received?
40. If 12 pieces of cloth, each piece containing 62 yards,
cost $ 372, what do 24 yards cost ?
41. If the work of 7 men is equal to the work of 9 boys,
how many men's work will equal the work of 90 boys ?
42. If 15 men consume a barrel of flour in 6 weeks, how long
would it last 9 men ?
43. If 12 men can build a wall in 42 days, how many days
will be required for 21 men to build it ?
44. If $15 purchase 12 yards of cloth, how many yards
will $48 purchase?
45. A ship has provision for 15 men 12 months. How long
will it last 45 men ?
46. How many overcoats, each containing 4 yards, can be
made from 10 bales of cloth, 12 pieces each, 42 yards in each
piece ?
82 FACTORS.
COMMON FACTORS.
168. Illustrative Example I. What numbers are fac-
tors of both 18 and 24 ?
WRITTEN WORK. Explanation. — Separating 18 and 24 into
1 Q _ 9 5 q their prime factors, we find 2 and 3, and conse-
c " e%^ cy quently 6 (which is the product of 2 and 3), to
^ns. 2 3 and 6. Name any common factor of 12 and 15 ;
of 12 and 18 ; of 30 and 40.
169. Numbers that have no common factors are said to
be prime to each other.
Thus, 14 and 15 are prime to each other, though they are
not prime numbers.
170. The greatest factor which is common to two or
more numbers is their greatest common factor.
What is the greatest factor which is common to 18 and 24 ?
to 40 and 50 ? to 45 and 54 ?
171. We have seen that 6, the greatest common factor of
18 and 24, is the product of 2 and 3, the only prime factors
common to 18 and 24. The greatest common factor of any
two or Tiiore numhers is the product of all the prime fac-
tors which are common to those numhers.
Note. The letters g. c. f. are used for greatest common factor.
To find the Greatest Common Factor.
172. Illustrative Example II. Find the greatest com-
mon factor of 12, 30, and 48.
WRITTEN WORK. Explanation. — The prime factors of 12 are 2,
12 = 2 X 2 X 3 2, and 3. The product of such of these as are
ff. c. f . = 2 X 3 = 6 common to 30 and 48 must be the g. c. f. re-
quired.
We find that 2 is a factor of both 30 and 48 ; therefore 2 is a factor
of the g. c. f. We find that but one 2 is a factor of 30; therefore only
GREATEST COMMON FACTOR. 83
one 2 is used as a factor of the g. c. f. We find that 3 is a factor of
both 30 and 48 ; therefore 3 is a factor of the g, c. f. Thus the g. c. f.
sought is 2 X 3, equal to 6. Hence the following
Rule.
173. To find the greatest common factor of two or more
numbers : Separate one of the numbers into its prime factors,
and find the product of such of them as are common to the
other numbers.
174. Examples for the Slate.
Find the greatest common factor
47. Of 48, 56, and 60. /
48. Of 24, 42, and 54. ^
49. Of 108, 45, 18, and 63.
50. Of 18, 36, 12, 48, and 42.
Note. In Example 60, 18 is a factor of 36, and 12 of 48. The g. e. f.
of 18 and 12 must be the g. c. f. of 18, 12, and their multiples 36 and 48 ;
hence we need only find the g. c. f. of 18, 12, and 42.
Find the greatest common factor
51. Of 42, 28, and 84. 53. Of 32, 18, 108, and 25.
52. Of 26, 52, and 65. 54. Of 114, 102, 78, and 66.
55. What is the width of the widest carpeting that will ex-
actly fit either of two halls, 45 feet and 33 feet wide, respec-
tively ?
56. A' has a piece of ground 90 feet long and 42 feet wide.
What is the length of the longest rails that will exactly suit
both its length and its width ?
57. What is the length of the longest stepping-stones that
will exactly fit across each of three streets, 72, 51, and 87 feet
wide, respectively?
58. What is the length of the longest curb-stones that will
exactly fit each of four strips of sidewalk, the first being 273
feet long, the second 294, the third 567, the fourth 651 ?
84 FACTORS. '
176. When numbers cannot readily be separated into
their factors, the following method for finding the greatest
common factor may be adopted.
Illustrative Example. Find the greatest common fac-
tor of 62 and 91.
WRITTEN WORK. Divide the greater number by the less,
Pi9^ Q1 n ^^^ then divide the less number by the
Ko remainder, if there be any. Continue
— dividing the last divisor by the last re-
Sd) bZ (1 mainder until nothing remains. The last
39
divisor will be the g. c. f. sought.
13) 39 (3 Note. As the explanation of this method is some-
39 what difficult for younger pupils, it is not given
here, but will be found in the Appendix, page 804.
To find the g. c. f. of more than two numbers, find the
g. c. f. of any two of them and then of that common factor and
a third number, and so on till all the numbers are taken.
176. Find the greatest common factor
59. Of 323 and 663. 61. Of 6581 and 1127.
60. Of 147 and 966. 62. Of 187, 442, and 969.
For other examples in factoring, see page 123.
MULTIPLES.
177. Name some numbers which are made by using 3
as a factor. Ans. 3, 6, 9, 12, etc.
Any number made by using another number as a factor
is a multiple of the number thus used.
178. Name the multiples of 4 and of 6 to 36.
^^^ ( Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36.
■ (Multiples of 6 are 6, 12, 18, 24, 30, 36.
Which of these numbers are multiples of both 4 and 6 ?
Numbers which are multiples of two or more numbers
are common multiples of these numbers.
LEAST COMMON MULTIPLE. 85
Thus 12, 24, and 36 are common multiples of 4 and 6.
Name a common multiple of 3 and 5 ; name two more.
179. Oral Exercises.
Name any six multiples of 5. Name three multiples of 12.
Name all the multiples of 11 up to 140. Name any common
multiple of 10 and 6. Of 3, 6, and 5.
Least Common Multiple.
180. Name the least number which is a multiple of both
4 and 6. Ans. 12.
The least number which is a multiple of two or more
numbers, is the least common multiple of those numbers.
Name the least common multiple of 2 and 5 ; of 6 and 9.
Note. The letters 1. c. m. are used for least common multiple.
181. As any number contains all its prime factors, a
multiple of any number must contain all the prime factors
of that number.
A common multiple of two or more numbers must con-
tain all the prime factors of those numbers, and
The least common multiple of two or more numbers is
the least number which contains all the prime factors of
those numbers.
182. Illustrative Example I. What is the least com-
mon multiple of 6, 9, and 15 ?
WRITTEN WORK. Explanation. — The least multiple of
5^2x3 6 is 6, which may be expressed in the
9 = 3x3 form 2x3.
^ ^ _ o K The least multiple of 9 is 9, which may
be expressed in the form 3x3. But in
1. c. m. = 2 X 3 x 3 X 5 = 90 ^ "^^ ha\e already one of the factors (3)
of 9 ; hence if we put with the prime
factors of 6 the remaining factor (3) of 9, we shall have 2x3x3, which
are all the factors necessary to produce the 1. c. m. of 6 and 9.
86 FACTORS,
The least multiple of 15 is 15, which may be expressed in the form
3x5. In the 1. c. m. of 6 and 9 we have one of the prime factors (3)
of 15; hence if we put with the prime factors of 6 and 9 the remain-
ing factor (5) of 15, we shall have 2x3x3x5, which are all the prime
factors necessary to produce the 1. c. m. of 6, 9, and 15.
The product of these factors is 90, which is the 1. c. m. sought.
Note. In finding the least common multiple, the factors of the given
numbers seldom need to be expressed, and the written work may be
greatly reduced. Thus, in this example the written work may be simply
1. c. m. = 2 X 3 X 3 X 5 = 90.
183. From the explanation above may be derived
Rule I.
To find the least common multiple of two or more num-
bers : Take the prime factors of one of the numbers ; with
these take such prime factors of each of the other numbers
in succession as are not contained in any preceding num-
ber ^ and find the product of all these prime factors.
184. Oral Exercises.
What is the least common multiple
a. Of4, 5, andS? c. Of 6, 14, and 21 ?
b. Of 6, 8, and 12 ? d. Of 3, 4, and 5 ?
When several numbers are prime to each other, what must
their least common multiple equal ?
185. Examples for the Slate.
Find the least common multiple
63. Of 8, 18, 20, and 21. 64. Of 3, 5, 12, 36, and 45.
Note. When one of the given numbers is contained in another, the
smaller may be disregarded in tlie operation ; thus, in the preceding ex-
ample, 3, 5, and 12 may be rejected. "Why?
Find the least common multiple
65. Of 18, 36, 60 and 72. 68. Of 18, 32, 48, and 52.
66. Of 12, 42, 56, and 70. 69. Of 16, 28, 35, and 63.
67. Of 13, 28, 39, and 49. 70. Of the nine digits.
LEAST COMMON MULTIPLE. 87
186. The above is a good method for finding the least
common multiple when the numbers are easily separated
into their prime factors. For larger numbers observe the
following method:
Illustrative Example II. Find the 1. c. m. of 18, 56, 38,
and 30.
WRITTEN WORK. Explanation. — Here,
2) 18, 56, 38, 30 ^7 repeated divisions, we
o x Q 90 -iQ IK *^^® ^^* ^^ *^® ^^<^^^^^
O) \J, ^o, i.\j, ±Q ^^^^ ^j.g common to two
3, 28, 19, 5 or more of the given num-
1. c. m. = 2 X 3 X 3 X 28 X 19 X 5 = 47880 JT /^ ^',f'''\ t
these factors (2 and 3)
and those that are not common must be the 1. c. m. sought.
Rule II.
187. To find the least common multiple of two or more
numbers :
1. Write the given numbers in a line as dividends.
Make any prime number which is a factor of two or more
of the given numbers a divisor of those numbers.
2. Write the quotients and undivided numbers beneath
as new dividends, and so continue dividing till the last quo-
tients and undivided numbers are prime to each other.
3. The product of all the divisors, last quotients, and
undivided numbers is the least common multiple required.
188. Examples for the Slate.
Find the least common multiple
71. Of 338, 364, and 448. 75. Of 165, 9500, and 855.
72. Of 184, 390, and 552. 76. Of 1146, 484, and 24.
73. Of 308, 616, and 77. 77. Of 880, 9680, and 176.
74. Of 84, 336, and 472. 78. Of 187, 539, and 8470.
For other examples in multiples, see page 123.
88 COMMON FRACTIONS.
SEOTIOIsr IX.
COMMON FRACTIONS.
189. If a unit, as 1 inch, is divided into two equal parts,
I 1 1 one of the parts is called one half.
If the unit is divided into three equal parts, one of the
I 1 1 1 parts is called one third; two of the
parts are called two thirds.
One of the equal parts of a unit is a traction, or
fractional unit. A collection of fractional units is a
fractional number.
Note I. For the sake of brevity, fractional units and fractional numbers
are both called fractions.
Note II. A number whose units are entire things is an integral num-
ber, or an integer.
Name a fractional unit ; a fractional number ; an integer.
190. The unit of which the fraction is a part is the
unit of the fraction.
191. The number of equal parts into which the unit
of the fraction is divided is the denominator of the
fraction.
Thus, in the fraction two thirds the denominator is three.
192. The number of equal parts taken is the numera-
tor of the fraction.
Thus, in the fraction two thirds the numerator is two.
193. The numerator and denominator are called the
terms of the fraction.
Note. Decimal fractions have been treated of in previous articles. All
fractions except decimal fractions axe called common fractions.
EXAMPLES, 89
"Writing Common Fractions.
194. The terms of a fraction are written, the numera-
mimerator, 2-1 ^^r above and the denominator below a
Denominator, Z "^^ ' ^^^^ ^J^^^g^ ^^^ ^J^^^^^ ^y ^^ ^^^^ ig
written as in the margin.
195. Exercises.
Write in figures the following :
a. One half of a mile. d. Twenty twenty-fifths.
b. One third of a day. e. Twelve thirds.
c. Seven tenths of a dollar. /. Seven sevenths.
g*. Write any fraction you please, having for a denominator
five ; seven ; ten ; seventeen ; one hundred.
h. Write any fraction you please, having for a numerator
six; eight; sixty; one hundred.
2. Where is the denominator of a fraction written ? Where
is the numerator written ?
j. Which is the greater part of a thing, i or J ? ^ or 3^ ?
196. The form of writing fractions as shown above is
the same as the fractional form used to indicate division.
(Art. 94.) Thus the expression f may mean two thirds of
one or one third of two. illustration.
The fact that § of 1 / ^ I h 1 1
equals ^ of 2 may be illus- ' ' ' ' 1 1 1 [
trated as in the margin. ^ ^^ ^ ^q^^l^ ^ ^f 2.
197. Exercises.
a. What is meant by the expression ^ ?
Ans. It means 5 of the 9 equal parts into which a unit is
divided, or it means 1 ninth of 5 units.
h. What is meant by the expression f?T^? f? if? A?
c. Illustrate the fact that | of 1 equals i of 3 ; that § of 1
equals \ of 2.
90 COMMON FRACTIONS.
REDUCTIOlSr.
To change a Fraction to smaller or larger terms.
198. Illustrative Example I. Change ^| to equiva-
lent fractions of smaller terms.
WRITTEN WORK. Explanation. — By dividing both terms of \^ by
ia=:4 = ^ 2, we make the terms half as large, and have the
fraction f. Now dividing both terms of the frac-
tion "I by 3, we make its terms one third as large, and have the fraction
f. If we had divided both terms of ^| by 6, we should have made
the terms one sixth as large, and obtained at once the fraction |.
The illustration shows that
ILLUSTRATION. ^he same part of the unit is
^2 1,1,1,1,1,1,1 expressed by ^|, f, and |. In
' 8 ' ' ' ' ' ' I ' I ' ' ' I ' ' ' ' ' I obtaining | and | from {I the
f I I I I I I I — ] — \ — I number of parts taken has
2. I I I I been diminished as the size of
the parts has been increased.
1/ both terms of a fraction are divided hy the same num-
ber, the value of the fraction will not be changed.
199. Illustrative Example II. Change f to equivalent
fractions of larger terms.
WRITTEN WORK. Explanation.— ^j mviliv^lj-
^ = 4 • ^ = 4-f i^g l^o^h terms of | by 2, we
make the terms twice as large,
ILLUSTRATION. and have the fraction |. By
2 , , multiplying both terms of f
^ ' ' ' by 6, we make the terms six
I I I I I I 1 1 times as large, and have the
\l I I I I I I I I I I I I I . I I ,, I fraction ^f. Here the num-
ber of parts in each case is
increased as the size of the parts is diminished.
If both terms of a fraction are multiplied by the same
number, the value of the fraction will not be changed.
200. When the terms of a fraction have no common fac-
tor, the fraction is said to be expressed in its smallest terms.
EXAMPLES. 91;
201. Oral Exercises.
Perform mentally the examples given below, naming results
merely ; thus, " |^ ; ^- ; \', ^/' and so on.
a. Change to their smallest terms : | ; | ; | ; ^ ; f ; -j^^. ; ^^ ;
t; f ; T^; T^oJ A; f ; f ; 1%; T^^; A; T^; ^.
Z>. Change to their smallest terms : ^^ ; £^ ; y^ ; -jS^ ; ^^^ ;
^\; ^¥5 T^^; -i^\ -i^', ^; /?r; i^^; t^s-; ^^t; ^^; ^3-; /?;
c. Change to their smallest terms: -j^f; ^; ^^; ^|; ^f ;
H; H; M; M; M; M; M; M; t^V; M; M; A%V
d. Change | to equivalent fractions, having 12, 16, 28, 44,
100, and 120 for denominators.
e. Change ^ to equivalent fractions, having 27, 54, 99, and
900 for denominators.
/. Change f , ^, f , -^j^, {^, ^^, each to an equivalent fraction
having 120 for a denominator.
g. How many thirtieths in ^ ? in ^ ? in § ? in f ? in ^ ?
inf? in^?
h. How many 24ths in | ? in f ? in ^ ? in f ? in f ? in | ?
To change a Fraction to its smallest terms.
202. From previous illustrations we may derive the fol-
lowing
^ Rule.
To change a fraction to an equivalent fraction of the
smallest terms : Strike out all the factors which are common
to the numerator and denominator; or divide both terms hy
their greatest common factor.
203. Examples for the Slate-
Change to equivalent fractions of smallest terms :
(1.) ^^\. (4.) Ui. (7.) ^^. (10.) im-
(2.) m- (5-) m- (8.) wwv- (11-) m%-
(3.) JV'j- (6.) ^M- (9-) ^^- (12.) mh
For other examples, see page 123.
92 COMMON FRACTIONS.
To change Improper Fractions to Integers or to Mixed
Numbers.
204. A fractional number, the numerator of which equals
or exceeds the denominator, is called an improper fraction.
205. Illustrative Example III. Change f f and -f|- as
far as possible to integers.
WRITTEN WORK. Explanation. — (1.) Since 12 twelfths make a
(1.) 12) 60 ^^^*> ^^ ^^ twelfths there are as many units as
— there are 12's in 60, which is 5. Ans. 5.
^^*' ^ (2.) In \^ there are as many units as there are
^o\ A^ ^^'^ ^^ '^^J which is 3 and ^. Ans. Z\\.
(2.) 1^^ 47
~T, . 206. The number 3 \^ consists of an
Ans. 0\^ . IP- A
integer and a fraction. A number con-
sisting of an integer and a fraction is a mixed number.
207. Oral Exercises,
a. Change to integral numbers: f ; ^-; J^; -^; ^-; |
¥; ¥; ^; -¥; ¥; ¥; -V-; ff; ¥; fl; ¥-; M-
Z). Change to mixed numbers: -y^; |; J^S-; ^l; J^-j i^
¥; ¥; ¥; ¥; ¥■; ¥; H; M; ¥; -¥; -V-
c. Change to integers or to mixed numbers : f ; %'- ; |
¥; ¥; ¥-; ¥; H; W-; ^F; ¥; ¥•
208. From previous illustrations we may derive the fol-
lowing
Rule.
To change an improper fraction to an integer or a mixed
number : Divide the numerator hy the denominator.
209. Examples for the Slate.
Change to integers or to mixed numbers :
(13.) f f. (15.) W- (17.) i¥*- (19-) W days.
(14.) W- (16.) W- (18.) Wj"-. (20.) W- years.
EXAMPLES. 93
To change an Integer or a Mixed Number to an Improper
Fraction.
210. Illustrative Example IV. Change 23^ to fourths.
WRITTEN WORK. Explanation.— ^ivlcq in 1 there are 4 fourths, in
23^ = ^^ Ans. 23 there are 23 times 4 fourths, or 92 fourths,
4 . which, with 1 fourth added, are 93 fourths.
93 ^^^- ¥•
211. Oral Exercises.
a. Change to improper fractions : 2^ ; 3^ ; 2| ; 5^ ; 2f ;
H) 6^; 5§; 5|; 7f; 7^; 8| ; 8f ; 9i; 9^; 10^.
b. Change to improper fractions : 2| ; 2| ; 3y\ ; 3f ; 4^ ;
4|; 5f; 9f; Gf; 7^ ; Sf; 9^ ; 4f; 4f ; 8^ ; 7^.
c. Change 5 to ninths ; 11 to fiftlis ; 14 to thirds ; 8 to
twelfths ; 15 to fourths ; 1 to sevenths.
d. Among how many persons must 7 melons he divided that
each may receive ^^ of a melon ? i^ ? i^ ?
e. How many persons will 5^ cords of wood supply if each
person receives ^ of a cord ? :^ of a cord ? ^ of a cord ?
212. From previous illustrations may be derived the
following
Rule.
To change an integer or a mixed number to an improper
fraction : Multiply the integer hy the denominator of the
fraction, and to the product add the numerator; the result
will he the numerator of the required fraction.
213. Examples for the Slate.
Change the following to improper fractions :
(21.) 69^. (24.) 76H. (27.) Change 48 to ninths.
(22.) 2721. (25.) IQf^. (28.) Change 567 to tenths.
(23.) 1095\. (26.) mi. (29.) Change 93 to forty-thirds.
For other examples in reduction of fractions, see page 123.
M COMMON FRACTIONS.
ADDITION" OF FRACTIONS.
To add Fractions having a Common Denominator.
214. Illustrative Example I. Add | of an apple,
1 of an apple, and \ of an apple. Ans. | of an apple.
These fractions are like parts (eighths) of the same or
similar units (apples). Such fractions are like fractions.
215. Like fractions have the same denominator, which,
because it belongs to several fractions, is called a common
denominator.
216. Oral Exercises.
a. Add ^s, ^, and ^. e. Add ^^, ^^, and |^.
b. Add tV^, tI ^, and T^^. /. Add |f, f ^, and v^.
c. Add f, %, f, and f . g. Add ^M^, ^■^%^, and ^/^^.
d. Add j7^, ^3^, \\, and ^^. h. Add ^, ^, and ^^j.
How do you add fractions which have a common denominator ?
To add Fractions not having a Common Denominator.
217. Illustrative Example. Add |, |, and -f^.
WRITTEN WORK. Explanation. — To be added, these
2 X 3 X 3 X 5 == 90 i.c. denom. fractions must be changed to hke frac-
tions, or to fractions having a common
^' ~ ^ xi5 = i^ denominator. (Art. 215.) The new
t — t><i8~t^ denominator must be some multiple of
^^ = -^-g^ 6 ~ t^ ^^^ given denominators. A convenient
J-M o 157 _ 1 6 7 multiple is their least common multiple,
Ans.-^,^-m- ^hichis90. (Art. 182.)
To change f to QOths, the denominator 6 must be multiplied by
3x5, or 15 ; hence the numerator 5 must be multiplied by 15. (Art.
199.) Thus, I is found to equal |-^.
In a similar way ^ will be found to equal |^, and -^j to equal f ^.
Adding these fractions, we have -^^-, or If ^, for the sum.
ADDITION. 95
218. Oral Exercises.
a. Add ^, \, and ^. c. Add ^, |, and | .
^715. T»3j = f . d. Add §, I, and /j.
i&. Add f, I, and f . e. Add f, ^, and |.
^7^5. ^t = 2^. /. Add /^, T^, and |.
Note. When the denominators are prime to each other, the new denmni-
nator will he the product of all the denominators^ and the n£W numerators
will he found hy multiplying each numerator by the product of all the de-
nominators except its oum.
g. Add \ and ^ ; ^ and \ ; \ and | ; ^ and ^ ; ^ and ^ ;
\ and \\ ^ and ^1^ ; \ and ^.
h. Add § and I ; ^ and § ; I and f ; f and f ; \ and f.
i. Add i, ^, and i ; §, ^, and f ; i, f , and ^ ; iV, f , and i.
j. If you should spend \ of your time in school, ^ in
practising music, and \ in sewing and studying, what time
would you spend in all ?
k. Owning f of a paper-mill, I bought the shares of two
other persons who owned -^ and f respectively. What part
of the mill did I then own ?
219. From the above examples may be derived the fol-
lowing
Rule.
To change fractions to equivalent fractions having the
least common denominator :
1. For the common denominator, find the least common
multiple of the given denominators.
2. For the new numerators, multiply the numerator of
each fraction hy the number hy which you multiply its de-
nominator to produce the common denominator.
Note. If the number to multiply the numerator hy is not readily seen,
it may be found by dividing the common denominator by the denominator
of the given fraction.
96 COMMON FRACTIONS.
220. From what we have now learned of the addition
of fractions, we may derive the following
Rule.
To add fractions :
1. If they have a common denominator, add their
numerators.
2. If they have not a common denominator, change them
to equivalent fractions that have a common denominator,
and then add their numerators.
221. Examples for the Slate.
(30.) f + ^ + i + *=? (35.) iV + A + Jf = ?
(31.) $4.if* + H + A = ? (36.) M + t'5 + A = ?
(32.) T/v + i?tT + i + f = ? (37.) ? + « + *§ = ?
(33.) ^i + ^ + | = ? (38.) t8j + ^ + ^j = ?
(34.) sV + A + A = ? (39.) A+A + A + H=?
Add the integers and fractions of the following, and similar ex-
amples, separately :
40. In my furnace there were hurned 2f tons of coal in
December, 2f tons in January, and 3| in February. How
many tons were burned in all ?
(41.) 72^ + 161 + 18f + 231 + 37^7^ = ?
42. A horse travelled 43^^ miles in one day, 52| the next,
36^ the third, and 40|^ the fourth. How far did he travel
in all ?
43. A merchant had three barrels of sugar, the first contain-
ing 247^ pounds ; the second, 229j'V pounds ; and the third,
260 J pounds. What was the weight of the whole ?
For other examples in addition of fractions, see page 123.
What operation should first be performed on this fraction ?
SUBTRACTION. 97
SUBTRACTION.
222. Oral Exercises.
a. ^ less f are how many ninths ? Ans. f .
b- H-^T-what? £f. ff-i| = what?
c. M-Zij-^what? e. if-Y\-/T = what?
/. Find the difference between -^-^ of a day and f ^ of a day.
g*. What must be added to ^^ to make ^f ? ^f ?
When the minuend and subtrahend are Hke fractions, how do yoa
subtract ?
«
223. Illustrative Example. If | of a yard of velvet
is cut from a piece containing | of a yard, what part of a
yard will be left ?
WRITTEN WORK. Explanation. — That the subtraction may be
o_o_p_o_j, performed, these fractions must be changed to
% ^ TF" T2 • equivalent fractions having a common denomi-
nator. The least common denominator is 12. f =i% and f = ^*^.
A. i-i? i-l? i-i? i-A? i-H i-i^T?
i. §-f? f-f? f-i? f-i? A-i? i-§?
j. 2-^? 8-i? ll-§? 9-3^? 7-2%? 8-33?
ir. How many yards will be left if from a piece containing
6^^ yards there be taken 1;^ yards ?
1. What is the difference in the height of two boys, one
being 3^ feet, the other 2f feet high ?
222. A pole is standing so that f of it is in the water, 4- in
the mud, and the rest in the air. What part is in the air ?
22. How much will be left of a piece of cloth containing 7
yards, after cutting from it 2 vests and a coat, allowing | of a
yard for a vest and 4^ yards for a coat ?
o. From a bin containing 23^ bushels of wheat there were
taken out 3f bushels at one time and 4^ bushels at another.
How much remained ?
98 COMMON FRACTIONS.
224. From the previous illustrations we may derive the
following
Rule.
To subtract one fraction from another :
1. If they have a common denominator, find the differ-
ence of their numerators.
2. If they have not a common denominator, change them
to equivalent fractions which have a common denorninator,
and then find the difference of their numerators.
225. Examples for the Slate.
(44.) A-A = ? (50.) 12i-f = ?
(45.) t'j-3V = ? (51-) 174-12^ = ?
(46.) f5-/T = ? (52.) 26|-1| = ?
(47.) 36-f=? (53.) 10f-5i| = ?
(48.) 19-24 = ? (54.) 17i-2f = ?
(49.) 75-154 = ? (55.) 181-15,^ = ?
For other examples in subtraction of fractions, see page 123.
226. Examples in Addition and Subtraction.
(56.) !-* + ! = ? (60.) 20-54 + A = ?
(57.) 5 + ^-^ = ? (61.) 8^-2f + 7| = ?
(58.) 1-4 + ^^-^5^ = ? (62.) 7-(/j-^) = ?
(59.) |-4-A-7t5 = ? (63.) 5-(| + A) = ?
64. Two men start at the same place and travel in opposite
directions, one at the rate of 3^^ miles per hour, the other at
the rate of 4^ i miles per hour. How far apart were they at
the end of an hour ?
65. Two boats are 5280 feet apart and rowing towards each
other, one at the rate of 320^''o feet per minute, the other at
the rate 309^^ feet per minute. How far apart are they at
the end of one minute ?
MUL TIPLIOA TION. 99
Q^. From 8 trees I gathered apples as follows : 2\ barrels,
3^ barrels, 5f barrels, 4^ barrels, 3| barrels. If barrels, 3 J bar-
rels, and 2^ barrels. If I sold 15^ barrels of the apples to one
man and 2\ to another, how many had I left ?
67. A lady who had $ 50 received 1 8^ more, spent $ 17f ,
lost % 4y^o, and collected $ 15\ of a debt. How much money in
dollars and cents had she ?
Q^. A man having a sum of money spent ^^ of it for a house,
^ of it for furniture, ^^^ of it for horses and carriages, and f
of it to build a church. What part of his money had he left ?
69. How much more is the sum of 12-^^ and Q^^ than their
difference ?
For other examples in addition and subtraction of fractions, see page 123.
MULTIPLICATION.
To multiply a Fraction by an Integer.
227. Illustrative Example I. If it takes | of a yard
of cloth to make 1 apron, how much will it take to make 2
aprons ?
Solution. — If it takes f of a yard to make 1 apron, to make 2 aprons
it will take 2 times |, or -| of a yard, equal to 1-^ yards.
In multiplying the fraction | by 2, which term of the fraction was
multiplied ? How will you multiply any fraction by an integer ?
228. Oral Exercises.
a. Multiply f by 2; f by 3; f by 7; A by 4; ^ by 5;
^by8; ^by6.
b. How many are 3 times f ? 4 times f ; 5 times ^ ?
8 times t^^ ? 9 times f^ ?
c. How many 2^ve^^x^? t^^x2? |x4? |fx2? -/^xG?
T^^xS? ^^x9? /^x20?
d. If 2| pounds of cane are required to seat 1 chair, how
many pounds will be required to seat 12 chairs ?
Note. Multiply the integer and the fraction separately.
100 COMMON FRACTIONS.
e. At $ 18| a dozen, what is the cost of 5 dozen lamps ?
/. If 7 men can build a dam in 4| days, in what time can 1
man build it ?
g. At $ 10^ each in currency, what is the value of 5 gold
eagles ?
h. In a piece of land 1 foot long and 1 foot wide there is 1
square foot. How many square feet are there in a piece 8|
feet long and 1 foot wide ? in a piece 18| feet long and 5 feet
wide?
J. If a man receives $ | for shoeing a horse and $ ^ for shoe-
ing an ox, how much will he receive for shoeing 4 horses and a
yoke of oxen ?
229. Examples for the Slate.
Illusteative Example II. Multiply ^^g- by 56.
WRITTEN WORK.
7
2
70. If a man can mow /^ of an acre of meadow in 1 hour,
now much can he mow in 38 hours ?
71. How many yards of cloth are required for 6 suits, each
suit requiring 7^^ yards ?
72. What is the width of 18 house lots, each 5| rods wide ?
73. What distance can a vessel sail in 33 hours, going at
the rate of 5f miles an hour ?
74. There are IQ^ feet in a rod. How many feet are there
in 40 rods ? in 320 rods, or 1 mile ?
75. How much ivory worth $ 1 a pound can be bought for the
same sum that will pay for 15| pounds worth |12 a pound ?
76. One quart dry measure contains 67^ cubic inches. How
many cubic inches are there in a bushel, or 32 quarts ?
77. If by working 11 hours a day a piece of work can be
done in 45§ days, in what time can it be done by working 1
hour a day ?
MULTIPLICATION. \ , .V', i , /; i ' i l^,
78. If 17 men can shear a lot of sheep in 9^^ days, in what
time can 1 man shear the lot ?
79. Multiply 14^ hy 9. 82. Multiply 365^ by 39.
80. Multiply 16t\ by 7. 83. Multiply 256 J by 18.
81. Multiply 23^ by 11. 84 Multiply 376^ by 21.
To multiply an Integer by a Fraction.
230. Illustrative Example III. What is -J of 2 inches ?
If -J- of each of the 2 inches is taken,
'3/2 we shall have f of an inch. Ans. ^ of an
•\ i ) inch. (See illustration ; also Art. 196.)
I 1 h
iof2 = §.
231. Oral Exercises.
a. Whatis^of2? ^of8? iof4? iof7? ^ofG?
b. What is tV of 3? iof2? ^oi9? iof5? ^oi5?
232. Illustrative Example IV. What is f of 2 inches ?
Solution. — -^ of 2 inches is f of an inch, and f of 2 inches must be
3 times -f , or ^ of an inch, equal to 1^ inches. Ans. l\ inches.
c. Whatis|of7? f of6? f of4? ^pi5? fof9?
In finding the fractional part of a number, as in the example above,
■what was the first operation ? Ans. Dividing the number by the de-
nominator of the fraction. By what was the result multiplied ? How-
then will you find the fractional part of a number ?
233. The process by which the fractional pait of a num-
ber is found is called multiplying by a fraction.
Illustrative Example Y. Multiply 11 by ^.
Solution. — To multiply 11 by f is to take | of 11. -J- of 11 is \S
and ^ of 11 must be 4 times ^,ov^ = 8f. Ans. 8-f.
d. Multiply 8 by §. /. Multiply 10 by ^.
e. Multiply 6 by |. g. Multiply 12 by f.
•10-2 . , ^ COMMON FRACTIONS.
h. In 1 yard there are 3 feet. How many feet are there in
f of a yard ?
Solution. — If in 1 yard there are 3 feet, in | of a yard there are -f
of 3 feet ; etc.
i. What is the cost of ^^ of a ream of paper at $ 6 a ream ?
j. What is the cost of f of an acre of land at 1 100 an acre ?
E:samples for the Slate.
234. Illustrative Example VI. What must I pay for
^ of an acre of land at $ 165 an acre ?
WRITTEN WORK.
55
^fc^ = ^=68i. ^...1681, or $68.75.
f
85. A man hought a carriage for $ 645, and sold it for f of
what it cost him. How much did he receive for it ?
86. If it requires 9 bushels of apples to make a barrel of
cider, how many bushels will be required to make 13^ bar-
rels of cider ?
Note. Multiply by the integral and the fractional number, separately •,
then add the results.
87. In 1 pound there are 16 ounces. How many ounces in
13/3^ pounds ?
88. 4 quarts equal 1 gallon. How many quarts are there in
65^ gallons ?
89. At $ 3 a bundle, what will 26^ bundles of shingles cost ?
90. If $ 1 will buy 9 pounds of sugar, how much will $ 19|
buy?
91. What must I pay for 63^ yards of flannel at $ 0.54 a
yard ?
(92.) 36 X 5§ - ? (95.) 3681 x 4^^ = ?
(93.) 568 X I = ? (96.) 5432 x 3^1 = ?
(94.) 385 X 3/y = ? (97.) 87036 x ^ = ?
MULTIPLICATION. 103
To multiply a Fraction by a Fraction.
235. Illustrative Example VII. Multiply \ by i
ILLUSTRATION.
Explanation. — To multiply
\ by ^ is to take 1 third of ^.
, To find 1 third of ^, the | must
I be divided into 3 equal parts.
^ ' 1 of i = ^. Since the entire unit -f will con-
tain 5 times 3 or 15 such parts,
one of the parts will be ^ of the unit ; hence ^ of -^ is ^^. (See
illustration.) Ans. ^.
In multiplying -^ by J, how was the new denominator obtained ?
236. Oral Exercisea
a. Multiply i by j ; i by ^ ; i^y tV; i^J^', h^J h
b. Multiply i by i; I by ^; tV by i; J by ^; T^j- by i.
237. Illustrative Example vm. Multiply | by |.
ExplanMtion. — To multiply |- by f is to take |
WRITTEN WORK, ^f |. ^ of ^ is J, (^) ; then ^ of I must be ^\
6^ ^ "A-- (5^), and f of i must be 2 times as much (l||),
or^V ^W5. ^.
In multiplying -^ by f, how was the new numerator obtained ? the
new denominator ? Then how do you multiply one fraction by another ?
c. What isfofi? iof§? foff? foff? foff?
d. Multiply I by I; t V t\; 1^17 ^7 #5 A ^J i? ? ^y f.
238. From the prevdous illustrations may be derived
the following
Rules.
1. To multiply an integer by a fraction, or a fraction by
an integer : Find the product of the integer and the numera-
tor of the fraction for the numerator of the answer, and
take the denominator of the fraction for the denominator
of the answer.
104 COMMON FRACTIONS.
2. To multiply a fraction by a fraction : Find the prod-
uct of the numerators for the numerator of the answer, and
the product of the denominators for the denominator of the
answer.
239. Examples for the Slate.
98. What is -^^ of /^ ? (101.) t^ x ^ = ?
•99. What is xV of ^^ ? (1^2.) if x |f = ?
100. What is T^ of f ? (103.) 4^\ x f = ?
Note. Change mixed numbers to fractional numbers before multiplying.
104. Multiply 8tV by 24f 105. Multiply 4f by 15lf.
106. If a man can walk 4| miles in an hour, how far can he
walk in -^ of an hour ?
107. What is the cost of 22|- thousand bricks at % 111 ^
thousand ?
108. What is the cost of 37f pounds of French carmine at
% 12^ per pound ?
109. A woman who inherited f of a ship divided | of her
part equally between her two daughters. What part of the
ship did each daughter receive ?
Note. Each daughter received | of | of f of the ship. The operation
may be expressed thus : -f^^^.
3
110. If a dress-pattern cost | llf , the trimmings \\ as much,
and the making | as much as the trimmings, what was the
cost of making ?
111. In a school of 840 pupils, f cipher, f as many write as
cipher, ^ as many study grammar as write, f as many study
geography as study grammar. How many study geography ?
240. The following, which are usually called " com-
pound fractions," are simply expressions for multiplication
of fractions, the finding of a part of a part :
112. What is f of T^^^ of f ?
113. Whatisf of T^^yof II?
DIVISION, 105
114. What is /^ of i^oi^?
115. "What is \l of ^ of If of 5| ?
116. What is I oif^oi^t of 6^ ?
117. What is i§ of T^ of f of ^ ?
118. What is ^ X ^3^ of 1^ X 6| ?
119. What is -i^^ of 3fr ^ f of f^f x 8f x 15 ?
For other examples in multipHcation of fractions, see page 123.
DIVISION.
To divide a Fraction by an Integral Number.
241. Illustrative Example I. If -|- of a melon be
divided equally between 2 boys, how many fifths of a melon
will each boy have ?
ILLUSTRATION. "^
\ \ \ \ Explanation. — If | of a melon be
4. . I I I I . equally divided between two boys,
^ [ ' [ ' } ' each boy will have 1 half of f, or f
i i of a melon. (See illustration.)
1 half of I = |. Ans. | melon.
In dividing the fraction ^ by 2, what was done to the numerator ?
242. Illustrative Example II. If -| of a melon be
divided equally between 2 boys, what part of a melon will
each boy have ?
ILLUSTRATION. -^
-^ ^ ^ Explanation. — As the number of
parts, 3, cannot be divided by 2 with-
out a remainder, each one of the parts,
3 3 fifths, may be divided into two equal
1 half of " = A parts ; these parts will be tenths of the
whole melon. We shall then have 6
tenths, of which each boy's share is 3 tenths. (See illustration.)
In dividing | by 2, what was done to the denominator ?
§ I ■ I I I I I I 1 .
106
COMMON FRACTIONS.
243. From the preceding illustrations we learn that to
divide a fraction by an integer, we may divide the numera-
tor or multiply the denominator hy the integer.
Note, In dividing by 2, as above, we find one half of a number ; this
is equivalent to multiplying by ^.
Dividing by 3 is equivalent to multiplying by what ? Dividing by
6 is equivalent to multiplying by what ?
244. Oral Exercises.
a. Divide ffby5; Mbyll; ff by7; M^yl3; Mby25.
b. Divide I by 5; f by 3; /t V 9; ii by 7; "f by 8.
c. Divide f by 3; f by 4; f^ by 7; A V 3; f by 2.
d. If $ 3 will buy i% of a yard of broadcloth, what part of a
yard will $ 1 buy ?
e. If 1 man can do a piece of work in f f of a month, in
what time can 12 men do it ?
/. If $ 2| be paid for 7 pounds of butter, what is the price
of 1 pound ?
Note. Change 1 2|- to fifths before dividing.
g. At $ 2\ per day, what are the wages of a man for 1 third
of a day ?
ii. If by 1 pipe a cistern can be emptied in l|f hours, in
what time can the cistern be emptied by 4 like pipes ?
Examples for the Slate.
245. Illustrative Example III. If 5|- yards of rib-
bon are required for 18 knots of trimming, how much is
required for 1 knot ?
WRITTEN WORK.
6
2
120. If 39|f inches of rain fell in a year, what was the
average fall per week ?
DIVISION. 107
121. If a steamer goes 11\ miles in 5 hours, what is hei-
rate per hour ?
Note. First divide 77 by 5, then change the remainder to thirds, and
divide.
122. If 44| yards of cloth be required to make 8 suits, how
many yards are required for 1 suit ?
123. If land that extends along the street 103^ rods is made
into 18 house-lots of equal width, what is the width of each lot ?
124. What is the length of one side of a square that can be
enclosed by a string 89f feet long ?
125. Divide Hf ^y 32. (130.) 47f-10= ?
126. Divide ft tyl6. (131.) 1211-16= ?
127. Divide jf^ by 72. (132.) 272^-20 = ?
128. Divide i§t by 65. (133.) 5^^-11=?
129. Divide 93/^ by 9. (134.) 1240f - 90 = ?
246. To divide an Integer or a Fraction by a Fraction.
In 1 there are how many fourths ? sixths ? eighths ? ninths ?
In 2 there are how many times ^ ? ^ ? ^^1 ^?
Illustrative Example IV. How many baskets of
peaches at |- of a dollar a basket can be bought for $ 5 ?
WRITTEN WORK. Explanation. — As many baskets can be
5 = J^ bought as there are times f in 5.
jK^o— 15_^2=7A- ^^ ^^^ change 5 to thirds, making •^.
There are as many times f in ^ as there are
2's in 15, or 7^. Ans. 7-J baskets.
Note. For different analysis of this example, see Appendix, page 304.
247. Oral Exercises.
a. Eight are how many times §? f? \^? f? f? l^orf?
b. Twenty are how many times 1^ ? If ? 2^ ? 2^ ? If ? If ?
c. Divide 4 by f; 7 by t; 9 by f; 6 by f; 8 by If
How do you change an integer to divide it by a fraction ? How do
you then divide ? How do you divide by a mixed number ?
108 COMMON FRACTIONS.
d. If a person walks a mile in f of an hour, how many
miles can he walk in 8 hours ?
e. At $1 a pound for coffee, how many pounds can be
bought for $4?
/. How many chairs at $ 1^ each can be bought for $ 10 ?
Note. Change 1^ to fourths.
g. How many tons of coal at $ 6| can be bought for |25 ?
h. The old shilling of New England was worth 16 f cents.
How many shillings made a dollar ?
i. If a boy can write a page in ^ of an hour, how many
pages can he write in ^f of an hour ? in 1 hour ?
j. If a hat can be made from f of a yard of velvet, how
many hats can be made from 3| yards ?
k. Divide ^^ by ^3^ ; if by 3^ ; M by ^ ; il by ^\.
When fractions have a commou denominator, how do you divide ?
248. Illustrative Example V. Divide | by |.*
WRrrxEN WORK. Explanation. — ^ and f changed to frac-
A = 12 • 1 == 4- S- tions having a common denominator are \^
12 io_i9 in_li and If if divided by ^| equals 12 divided
it-xy-J-^-J-^-J-* by 10, or If Ans.ll
1. Divide f by f ; f by ^; ^ by t; f by |; ^ by f.
When fractions have different denominators, how do you prepare
them to divide ?
In the written work of Illustrative Example V., after obtaining a
common denominator we have 12-4-10, or the new numerator of the
dividend divided by the new numerator of the divisor. If, in the place
of these numbers, we put the factors which formed them, we shall
have (4 X 3) -=- (5 X 2) or ±2L^ or | x |, in which the expression for the
divisor, f , is inverted, becoming |, and the answer, found by multi-
plying I by I, is -I, or If as before.
249. To divide one fraction by another, we may then
invert the divisor and proceed as in the multiplication of
* For other explanations of division of fractions, see Appendix, p. 304.
DIVISION. 109
fractions. The written work of Illustrative Example V.
will then be merely fc| = f = 1^.
Perform the following examples hj either of the methods
illustrated above :
222. How many aref-f-^? f^f? f^fr? f^^?
n. Howmany are^-f ? §-f? f-^^j? i^r^f?
260. From the previous illustrations may be derived
the following
Rules.
1. To divide a fraction by an integer, Divide the numera-
tor or multiply the denominator hy the integer.
2. To divide an integer or a fraction by a fraction,
Change the dividend and divisor to fractions having a com-
mon denx)minator, and then divide the numerator of the
dividend hy tJie numerator of the divisor. Or,
2. Invert the divisor, and proceed as in the multiplica-
tion of fraction^.
251. Examples for the Slate.
135. Divide Iff by 6. 138. Divide 181 hy ^.
136. Divide tVV5'- l^^- I^ivide 96 hy ^l,
137. Divide If by 18. 140. Divide 108 hy ^^.
141. At $ -jij per pound, how many pounds of rice can he
bought for $ 11 ?
142. At $ ^ per foot for rubber hose, how many feet can be
bought for $ 41 ?
(143.) 18^1=? (147.) ^^M = ?
= ? (148.) U
(144.) 21
(146.) 54
(145.) 98-./^=? (149.) ^^^=?
if = ?
if
= ?
A=? (150.) il
151. How many bushels of peas at $| a bushel can be
bought for $ 18 ? for % 12^ ?
152. At I -f^ per thousand ems for setting type, how many
thousand ems can be set for $ 75 ?
llO COMMON FRACTIONS.
153. If 1 yard of cloth can be made from || of a pound of
wool, how many yards can be made from 5 tons of 2000 pounds
each ?
154. One rod equals 16^ feet. How many rods in 100
feet ?
155. How many breadths of paper, each §^ of a yard wide,
will reach around a room, the distance being 27^ yards ?
156. At $2^ per yard, how many yards of cloth can be
bought for % 45| ?
157. How many lengths of T^ feet are there in a fence
17061 feet long ?
158. How many square rods^ each containing 30^ square
yards, are there in 75f square yards ?
159. A man had $ 1.50, which he exchanged for francs at
18| cents each. How many francs did he receive ?
(160.) 3f-f-4T^ = ? (162.) 26^-3/t=?
(161.) 51^^-6^ = ? (163.) l-541f = ?
252. Illustrative Example VI. Change the expres-
9ft
sion jry to its simplest form.
WRITTEN WORK.
Expressions like that above are sometimes called com-
plex fractions. But they merely indicate division.
Change the form of the following expressions, and perform
the division indicated :
(164.) 1
(167.) i
(170.) ^ (173.) ?i
(165.) -t
(168.) -t
(171.) g (174.) l|i
(166.) A
(169.) ^
(172.) g (175.) i;|
For other examples in division of fractions, see page 123.
ORAL EXAMPLES. Ill
TO FIND THE WHOLE WHEN A PART IS GIVEN.
Oral Exercises.
263. Illustrative Example I. If | of a ton of hay-
costs $16, what will -^ of a ton cost ? what will 1 ton cost ?
a. If f of a certain number is 28, what is the entire number ?
b. 81 is -j^y of what number ?
c. A man bought a harness for $ 75, which was ^ of what
he paid for his carriage. What did he pay for his carriage ?
d. I paid $ 6 a week for board in Albany, which was f of
what I paid in Buffalo ; this was f of what I paid in Chicago ;
and this was f of what I paid in San Francisco. What did I
pay in San Francisco ?
e. An exploring party having lost \ of their bread, are
obliged to subsist on 14 ounces a day. What were they allowed
at first ?
Note. If \ is lost, | remain.
/. If f of a piece of work be performed in 24 days, how
many days will it take to do the remainder ?
g. A vessel, having lost ^ of her cable, has 200 feet remain-
ing. How many feet had she at first ?
h. Mary is 24 years old, and her age is equal to once and \
the age of her brother. How old is her brother ?
Note. Mary's age is f of that of her brother.
L A mother and her son have $ 45 in a purse ; the son's
part is § as great as the mother's. What is each one's part ?
Solution. — The mother's part must be f of itself, and her son's part
added to her part must be | of her part. But the two together have
$ 45 ; then $ 45 is f of the mother's part.
j. If I sell an article for 1 80, and thereby gain a sum equal
to ^ of the cost, what is the cost ?
k. If I seU an article for $ 80, and thereby lose a sum equal
to ^ of the cost, what is the cost ?
112 COMMON FRACTIONS.
254. Examples for the Slate.
(176.) 16^ is ^^ of what number ?
(177.) 25f is f of what number?
(178.) f of M is f of what number?
(179.) I of 6| is I of what number ?
(180.) ■f\ of f^ is I of what number ?
(181.) 2^ X 7^ is 3|^ times what number (or 1^ of what num-
ber) ?
(182.) 182 -f- (12 X 2\) is 3^ times what number ?
183. An author's copyright on a book was 1 54.57. If this
was -^Q of the whole profit, what was the whole profit ?
184. Mr. Smith owns f f of an acre of land ; his neighbor
Mr. French owns % as much, which is f of what Mr. Brown
owns. What does Mr. Brown own ?
185. If ^ of my property is in real estate, f in trade, and
the balance, which is $33000, is in stocks: what is the value
of my property ?
186. A man sold a lot of land for % 1440, which was 2^ times
what it cost him. What did it cost him ?
187. Having lost ^S" ^^ ^7 inoney in trade, I now have
$ 2476.50. What had I at first ?
188. A person against whom I had an account has failed,
and I have lost f of what he owed me. If I receive $1584.72,
how much did he owe me ?
189. A body of 4800 troops had \ as many cavalry as
infantry. What was the number of each ?
190. A lot of land yielded 4140 bushels of grain in two
years, yielding f as much the second year as the first. What
was the yield each year ?
191. What number is that to which if f of itself be added
the sum will equal 275 ?
192. In counting his fowls, a farmer finds that he has 396
in all, which is \ more than he had the previous year. How
many had he then ?
EXAMPLES, 113
TO FIND WHAT FRACTION ONE NUMBER IS OF
ANOTHER.
Oral Exercises.
255. Illustrative Example I. 1 is what part of 5 ?
Answer. 1 is |^ of 5 because it is one of the five equal parts
into which 5 may be divided.
a. 1 is what part of 7? of 9? of 10? Why?
In comparing 1 with any number to see what fraction it is of that
number, what do you take as the numerator ? as the denominator ?
b. 1 is what part of 7 ? 2 is what part of 7 ? Why ?
c. What part of 9 is 2 ? What part of 10 is 7 ?
d. What part of 200 is 20 ? 60 ? 2^ ? 40 ?
e. 1 peach is what part of 7 peaches ? 3 pears of 13 pears ?
/. ^ is what part of ^ ? ^^ is what part of J§ ?
g^. ^ is what part of ^. Note. Change ^ and J to sixths.
h. What part of 10 is 3i? is2i?
256. From the foregoing illustrations we may derive the
following
Rule.
To find what fraction one number is of another, Make
the number which is the part the numerator of a fraction,
and the number with which it is compared the denominator.
i. If a piece of work can be performed in 9 days, what part
of the work can be performed in 7 days ?
j. If Mr. Chase has $ 54 and spends $ 18 for a coat, what
part of his money does he spend ?
k. Stock originally worth 1 50 a share now sells for % 40.
What part of the original value does it bring ?
1. When goods which cost 75 cents sell for 1 1, what is the
gain ? What part of the cost is the gain ?
222. A and B hired a pasture together ; A pastured 12 cows
in it and B 13 cows. What part of the price should each pay ?
114
COMMON FRACTIONS.
257. Examples for the Slate.
193. A man owing % 316, paid $ 84 of the debt. What part
of the debt did he pay ?
194. What part of 272^ square feet is 9 square feet ?
195. I bought a house for 13000 and sold it for $4500.
What part of the original cost was the gain ?
196. Four men were hired to work on a farm. A worked 7
days, B worked 5 days, C 8 days, and D 4 days. They received
% 72. What was each man's share ?
What part (201.) 12f is what part of 19 ?
(197.) Of 75 is 30? (202.) 1 is what part of 2^ ?
(198.) Of 267 is 89? (203.) 1 is what part of 1^^^ ?
(204.) \^ is what part of ^^ ?
(205.) 2f is what part of 3f ?
206. What part of 100 is 33^? 66f ? 87^? 37^? 12^?
62^? 61? 56^?
(199.) Of 8 is i?
(200.) Of 11 is If ?
To solve Examples by using Aliquot Parts of Numbers.
258. What is one of the three equal parts of 9 ? of 10 ?
One of the equal parts of a number is an aliquot part
of the number. Thus, 3^ is an aliquot part of 10.
259. Oral Esiercises.
Find such aliquot parts of the following numbers as are
indicated below:
a. Of the number 30 find ^ ; J ; | ; J ; ^ ; ^
b. Of the number 60 find ^ ; ^; J ; ^; ^; ^^
c. Of the number 100 find I; ^', i
d. Of the number 100 find § ; £ ; §
e. Of the number 144 find | ; ^ ; |
/. Of the number 200 find I; ^', I
i; i
g:. Of the number 1000 find i ; ^; i; ^; ij i; ,1^; §; I
ALIQUOT PARTS OF NUMBERS. 115
260. By using the aliquot parts of numbers, the work
of multiplying and dividing may often be shortened, thus :
Illustrative Example. What is the cost of 25350 ft.
of gas at 3 J mills per foot ?
Operation. — 3 Jm. = |^ of 10 mills, or ^ of a cent. 25350 ft. at 1 cent
a foot costs $253.50, and at -^ of a cent a foot it must cost ^ of $253.50,
or $84.50. ^ns. $84.50.
Oral Exercises.
Find the cost
a. Of 1872 lbs. of butter at $ 0.33^ per lb. ?
h. Of 64 bu. potatoes at $ 0.87^ per bu. ?
c. Of 44 yds. of silk at $ 1.12^ per yd. ?
d. Of fencing 50 rods of road, both sides, at 1 3.75 per rod ?
e. Of insuring a house 5 years at % 6.66f per year ?
/. Of 750 feet of boards at $ 12 per thousand ?
g. Of 80 pounds of butter at 37^^per pound ?
h. How many pounds of cheese at 16 1/ a pound can be
bought for $ 10 ?
i. For % 20 how many yards of cloth can be bought at 1 1
a yard ? at 12^/ ? at 16f / ? at 2b f ? at 50/ ? at 37^/ ?
261. Examples for the Slate,
(207.) 987--16§=? (210.) 496-^12i=?
(208.) 864 -33^= ? (211.) 684 x 66|= ?
(209.) 572 X 62^= ? (212.) 487 x 37^= ?
262. Questions for Review.
What is a factor of a number ? What is a composite number ? a
prime number ? a prime factor ?
What is an even number ? an odd number ? What numbers are
divisible by 2? 3?4? 5? 6? 8? 9? 11?
How can you find the prime factors of a number? A composite
number equals what product ? Find the prime factors of 180 and
explain the process. How ca:i you make sure that a number is prime ?
What is CANCELLATION ? Why should arithmetical processes first
be indicated by signs ? Explain the use of the parenthesis.
116 COMMON FRACTIONS.
When are numbers prime to each other ? What is a common fac-
tor of two or more numbers ? the greatest common factor ? Find
the g. c. f. of three numbers by the first method given ; explain and
give the rule. Find the g. c. f. of two numbers by the second method
given ; give the rule. In what cases would you find the g. c. f. by the
second method ? When do we make use of the g. c. f. of numbers ?
What is a multiple ? a common multiple of two or more numbers ?
the LEAST common multiple ? When do we make use of the 1. c. m. ?
Explain the first method of finding it ; the second. What does the
1. c. m. of numbers prime to each other equal ?
What is a fractional unit? a fractional number? What name
is applied to both ? Name and define the terms of a fraction. Ex-
plain the expression |. How do you change fractions to smaller
terms ? to larger terms ? When is a fraction expressed in its smallest
terms ? How do you change improper fractions to integers or mixed
numbers ? How do you change integers or mixed numbers to fractions ?
When are fractions said to have a common denominator? For
what operations upon fractions do we first change them to others
having a common denominator ? Change -^j -^, and -^j to fractions
having a common denominator, and explain.
How do you add fractions? Take three fractions of different
denominators, add and explain. How do yoii add mixed numbers ?
How do you subtract fractions ? Give a general rule for the addi-
tion of fractions. Give a general rule for the subtraction of fractions.
Let 4^ be the minuend and ly^ the subtrahend ; subtract and explain.
How do you multiply a fraction by an integer ? a mixed num<
ber by an integer ? Explain, by an example, the method of multi-
plying an integer by a fraction. Multiply a fraction by a fraction ;
explain and give the rule. How do you multiply a mixed number by
a mixed number or a fraction ? How can you simplify the expres-
sions called compound fractions?
How do you divide a fraction by an integer ? a mixed number
by an integer ? an integer by a fraction ? Explain, by an example, the
method of dividing a fraction by a fraction, and give the rule. How
can you simplify the expressions called complex fractions ? How do
you find what fraction one number is of another? What is an aliquot
part of a number ?
What effect does multiplying both terms of a fraction by the same
number have upon it? Why? What effect does dividing both terms
GENERAL REVIEW. 117
of a fraction have upon it 1 Why ? What effect does multiplying
the numerator of a fraction have upon the fraction ? Why 1 In
what other way could you produce the same effect, and why 1 ' What
effect does dividing the numerator have upon a fraction ? Why 1
In what other way could you produce the same effect, and why 1
263. General Review, No. 2.
213. What are the prime factors of 420 ?
214. Divide 18 x 7 x 15 x 6 by 28 x 10 x 3 x 4.
215. Find the greatest common factor of 35, 84, and 56.
216. Find the least common multiple of 63, 18, 14, and 28.
217. Change ^f-f and ^^f to their smallest terms.
218. Change 465| and 84 to improper fractions, having 8 for
their denominator ?
219. Change ^^^ and ^^^ to mixed or integral numbers.
220. Change |, -^j, and § to fractions having a common de-
nominator.
221. Change ■^■^, yV> ^^^ ^h ^^ fractions having the least
common denominator.
222. Add t of |, §, and ^. Add 25|, 6f, and 46J.
223. From 24 take 12f Subtract f^ from ^^ of /^.
224. Multiply 7f by 4 ; 7f by 5 ; -^% by 8f .
225. Simplify the expression ^^ of ^ of f J of 2f .
226. Divide /^ byf ; | of /« ^J 1/^-
i -A- 4
227. Simplify the expressions -J-, -I^, and ^.
228. What part of 4f is 3^ ?
229. Two trains which are 75 miles apart are running
toward* each other, one 30| miles an hour, the other 40|
miles an hour. How far apart will they be in half an hour ?
230. A man paid $ 18| for a load of hay weighing 1^ tons.
At the same rate what should he pay for f of a ton ?
231. Having spent f of his money, Fred has 1 13^. How
much had he at first ?
232. Make out a bill of sale for three barrels of sugar,
weighing respectively 235 pounds, 241 pounds, and 254
pounds, at llf/ a pound.
118 COMMON FRACTIONS.
264. Miscellaneous Oral E:samples.
a. If ^ of a pound of candles cost 35 cents, what is the price
of 1 pound ? of \^ of a pound ?
b. In I of an acre of land there are 120 square rods. How
many square rods are there in ^ of an acre ?
c. When 1^^ bushels of oats will feed 10 horses for a cer-
tain time, how many horses will 2\ bushels feed for the same
time ?
d. At $ I each, how many cedar posts can be bought for
$12? for 17^?
e. At 1 2^ per day, how many days' work can be paid for
with 1 20? with|37|?
/. At $2 per day, how many days' work can be paid for
with |7|? with %%?
g. If it requires 12 yards of carpeting f of a yard wide to
carpet a hall, how much will be required of that which is 1^
yards wide ?
h. What number is that, \ of which exceeds ^ of it by 2 ?
i. If % of the distance from Springfield to Albany is 80
miles, what is f of the distance ?
j. If % h\ pays for the lodging and breakfast of 7 persons,
for how many persons will 1 11;^ pay ?
k. What is that number to which if f of itself be added the
sum will equal 64 ?
1. I sold my watch for % 72, which was \ more than I gave
for it. What did it cost me ?
222. Bought a horse and saddle for % 75, giving f as much for
the saddle as for the horse. What was the cost of each ?
22. A can build a wall in 3 days, and B can do the same
work in 4 days. What part of the work can each do in one
day ? What part can both do in one day ? In how many
days can both do it working together ?
o. C can do a piece of work in 5 days, and D in 8 days.
What time will be required for both to do it ?
MISCELLANEOUS EXAMPLES. 119
265. Miscellaneous Examples for the Slate.
233. What will 16^ yards of cloth cost at 53 f a. yard ?
234 What will 9^ bushels of corn cost at 87^/ a bushel ?
. 235. What will 271f acres of land cost at $ 31| per acre ?
236. I paid 65/ for 2 boxes of strawberries. What will be
the cost of 45 boxes at the same rate ?
237. What is my bill for 7 pear-trees at 87^ cents apiece for
the trees, and $ 2 a dozen for setting ?
238. What do I receive per pound by selling 15 pounds of
coffee for $3.75?
239. If ^ of a man's property is in land, valued at $ 2324f ,
what is the value of his whole property ?
240. Boughtf of ashipfor $4075. What would the whole
ship cost at the same rate ?
241. What is the cost of 3 pieces of calico, 37^ yards in a
piece, at 19^ cents per yard ?
242. Sold my house and farm of 47f acres for $ 6150. Allow-
ing $ 3500 for the house, what did I receive per acre for the
land?
243. How long will a quantity of flour last a family of 8
persons if it lasts 3 persons 14^ months ?
244. If in 32^ years a man saved $1694, what was his
average saving per year ?
245. What number is that which diminished by 1^ will
leave a remainder of 1^ ?
246. What number is that to which if you add 9| the sum
will be 124| ?
247. What is that number to which if you add | of 26^ the
sum will be 147^ ?
248. If you buy 7^ yards of silk at $ 5 a yard, 14^ yards of
cashmere at $ 1.25 per yard, 4| yards of silk at 75 cents per
yard, and | of a yard of velvet at $ 4.50 per yard, giving in
payment a $ 100 bill, what balance will be your due ?
249. What wiU 50 oranges cost at 62^/ a dozen ?
120 COMMON FRACTIONS.
250. How long will 200 pounds of meat last 9 persons at
the rate of | of a pound a day for each person ?
251. A farmer has sold his eggs at an average of 23| cents
per dozen, which is ^ higher than they averaged the previous
year. What did they average then ?
252. He is paid for grain $ 1,80 per bag, which is ^ less
than he was paid last year. What was he paid last year ?
253. Mr. Stevens, dying, left $ 75000 to his wife and two
sons. To his wife he left $ 30000 ; to his oldest son just as
large a part of the remainder as his wife's portion was of the
entire property ; and to his youngest son the rest. What was
each son's share ?
254. A man sold 54| yards of cloth at the rate of 3 yards
for 2 dollars. What did he receive for it ?
255. Mr. Day bought a house and barn for % 4050, giving ^ as
much for the barn as for the house. What did he pay for each ?
256. If a body falls 16yV f®^^ ^^ the first second of time, 3
times 16y\j^ feet in the next second, and 5 times 16^ feet in the
third second, how far will it fall in the three seconds ?
257. What length of time would a man require to travel
around the earth if the distance is 25000 miles and he travels
at the rate of 21\ miles per day ?
258. If a man can build 2f rods of waU in a day, how much
can he build in 6|^ days ?
259. What number is that f of which exceeds \ of it by llf ?
260. If I buy 1250 bushels of corn at 41 cents per bushel,
and sell it at b2\ cents per bushel, how much do I gain ?
261. What number divided by f equals 125f ?
262. What are the contents of 3 floors measuring as follows :
13| square yards, 32^(5- square yards, and 49f:| square yards ?
263. The product of three numbers is 63|^; two of them are
8^ and 6^^- What is the third ?
264. I exchanged 42 tubs of butter, averaging 48f pounds, at
21^ cents per pound, for 42 barrels of flour, at 1 9f per barrel;
and received the balance in cash. What was the balance ?
MISCELLANEOUS EXAMPLES. 121
265. Owing a man in Paris 1325 francs, I have shipped to
him $ 375 worth of rice. If the franc is worth 19^ cents, how
much have I overpaid him in United States money ? in francs ?
266. I have three boxes, each containing 12 pieces of cloth,
each piece 4f yards in length, and weighing 3| pounds to the
yard. What is the weight of the whole ?
267. What will 42^ quires of paper weigh at | pound per
quire ?
268. Owning f of a flour-mill, I sold | of my share for
$ 1750. What is the value of the whole mill at the same rate ?
269. When hay was % 15 per ton, I gave | of a ton for If
tons of coal. What was the coal worth per ton ?
270. If a man walks 9^ miles in 2^ hours, how far will he
walk in 4f hours ?
271. At the rate of 4^ miles an hour, what time will be
required to walk 122 miles ?
272. In 1860 I purchased cotton at 8^ cents a pound, which
I sold in 1862 at 90| cents. What did I gain on 1000 lbs. ?
273. If a man can earn $2.30 per day, how many days'
work will he have to give for a suit of clothes, of which the
coat costs $ 25^, the trousers $ 8, and the vest 1 5^ ?
274. If I of I of a ship cost $42000, what is § of it worth ?
275. In a certain manufactory ^ of the operatives are Ger-
mans, \ French, ^ Scotch, \ English, ^^ Swedes, and the
remainder, 140, native Americans. What is the whole num-
ber, and the number of each nationality ?
276. If \ of my money is in gold, ^ of the remainder in
silver, and the balance, $ 360, in bank-notes, how much money
have I in all ?
277. A certain piece of work can be performed by A in 8
days, by B in 10 days, and by C in 16 days. In what time
can all do it working together ?
278. In what time can A and B do it together ?
279. In what time can A and C do it together ?
280. In what time can B and C do it together ?
122
COMMON FBAGTI0N8.
Examples.
1.
2.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
If
tV
if
f
i
f
266. DRILL TABLE No. 5.
B
A
\
Iff
f
c
D
E
F
Q
f
4M
n
9
121
*
m
8|
15
24f
H
j?^A
4|
6
18|
«
Mi
9|
12
28f
«
t's'^
6|
14
m
A
A'sV
5t
8
32t
11
III
4f
10
15f
i
m
If
7
21f
il
m
2i
23
54i
fj
m
5|
20
m
4
\m
n
21
35|
^j
f/A
M
24
36f
il
^fh
2i
5
45f
A
im
6?
16
56i
A
111
5|
18
64f
11
gti
n
22
55i
f
Ml
3f
17
51J
A
iff
7J
13
40J
A
III
8?
19
38J
t
§ti
4f
11
44f
A
*M
H
24
36?
f
%n
3*
12
18f
A
^^6
8|
28
63^
1*
iWff
6t
15
334
A
A^
7i
26
39J
DRILL EXERCISES.
123
267. Exercises upon the Table.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144:
145.
161.
162.
163.
Find the prime factors of each
numerator in D. t
Find the prime factors of each
denominator in D.
Find the g. c. f. of the terms
of each fraction in D.
Find the 1. c. m. of F, G,* and
H.
Change D to lowest terms.
Change the mixed numbers in
G to improper fractions.
Find the sum of A and C.
■E.
A + B + C.
C + D + E.
E + F + G
+ H.
A-B.
H-G.
G-E.
H + A-G.
A-Bof C.
DiflFerence
of C and D.
Cof E
-AofB.
Simplify
AofBofC.
Simplify
A of B of
Cof E.
AxB.
CxF.
CxE.
If G is B of some number, what
is C of the same number ?
H is C of how many times E ?
C of E is B of how many times A?
146.
147.
14s.
149.
150.
151.
152.
153.
154.
155.
157.
GxF.
GxE.
A-^B.
C^E.
C-^-F.
H-^A.
E^F.
G-^E.
AofF-
A of C
4- B of E.
156. (A + B)
-r-(B X C).
(A-B)
-(B-C).
A + C-B.
ExF + G.
158.
159.
160.
AofE
A-B.
B + C G-j-F.
164. What number is that from which
if you take A the remainder
will be B ?
165. What number is that to which
if you add C of F the sum will
beG?
166. What number multiplied by F
will give G for a product ?
167. What number divided by E will
give D for a quotient ?
168. What divisor will give E for a
quotient, H being the divi-
dend?
169. What number is that to which
if A of itself be added the sum
will equal H ?
170. What number is that from which
if B of itself be subtracted, the
remainder will be F ?
171. Divide H into three such parts
that the 2d shall be twice the
1st, and the 3d F more than
the 2d. What is the 3d part ?
172. At E dollars a yard, what will
F yards of cloth cost ?
173. At E dollars a yard, how many
yards of cloth can be bought
for F dollars ?
174. If B pounds of tea cost H cents,
what will E pounds cost ?
John can do a piece of work in
E days, and James can do the
same work in F days. In what
time can both together do it ?
If George and Albert can do a
piece of work in E days, and
Albert can do it alone in F
days, in what time can George
do it alone ?
175.
176.
* Omitting fractions.
t See page 57, for Explanation of the Use of the Drill Tables.
124 DECIMAL FRACTIONS.
SECTION" X.
DECIMAL FRACTIONS.
268. Articles 30 to 36 treat of a series of fractions, —
tenths, hundredths, thousandths, etc., — each of which has
for a denominator 10, or a number made by using lO's
only as factors. Such fractions are decimal fractions.
Note. Decimal fractions are usually called decimals.
To read and -write Decimals.
269. The method of reading and of writing decimals
has been explained in Articles 34 to 36. These the pupil
may review.
270. Exercises.
a. Bead 5.368; 0.406; 2.007; 0.039; 105.105.
b. Kead 0.4721; 7.0497; 10.010; 15.0015.
Read the following :
c. 30.0094 e. 120.250049 g. 200.005
d. 17.01845 /. 1.001025 h. 0.205
Note. To distinguish 200. 005 (Example g) from 0. 205 (Example h), use
the word decimal before reading the decimal part. Thus, 200. 005 may be
read "two hundred and the decimal five thousandths"; while 0.205 may
be read " decimal two hundred five thousandths."
Read the following :
i. 0.315 222. 500.0074 q. 1000.00001
j. 300.015 22. 4700.0065 r. 14.00375
k. 36000.00018 o. 430.06 s. 0.0000027
1. 0.36018 p. 43000.06 t. 0.1000012
REDUCTION OF DECIMALS. 125
271. To write a decimal : Write the number as an
integer, and place the decimal point so that the right-hand
figure shall stand in the place required by the denomina-
tion of the decimal.
Note. "When the given number does not fill all the decimal places, sup-
ply the deficiency with zeros.
For other exercises in reading and for exercises in writing decimals, sou
page 135,
The pupil may now review addition and subtraction of deci-
mals (Articles 46, 60, 61, and 66).
REDUCTION OF DECIMALS.
To change the Denomination of a Decimal Fraction.
272. Exercises.
a. What is the denominator of the fraction 0.5? 0.25?
0.075? 7.3? 4.86?
b. What is the numerator of the fraction 0.4 ? 0.04 ? 0.075 ?
0.0101 ? 0.000007 ? 0.25 ? 0.1125 ?
c. Write as a common fraction 0.3; 0.08; 0.375; 0.0204.
273. Illustrative Example. Change 0.5 to thousandths.
WRITTEN WORK. Explanation. — Multiplying both numerator and
5 = 500 denominator of -^ by 100, we have ^^V, which is
expressed decimally by writing 0.500.
274. From the written work above we derive the fol-
lowing
Rule.
To express a decimal fraction in any lower denomina-
tion : Annex zeros to the given expression until the place
of the required denomination is filled,
126 DECIMAL FRACTIONS.
275. Examples for the Slate.
1. Change 0.07 to thousandths.
2. Change 0.4, 0.75, 2.5, and 1.06 to thousandths.
3. Express 0.003, 1.75, and 0.006 as ten-thousandths.
4. Express 3 as tenths ; as hundredths ; as thousandths ; as
ten-thousandths; etc. Answers. 3.0; 3.00; etc.
Note. Read the above answers: ** Thirty tenths; three hundred hun-
dredths"; etc.
5. Express 7, 40, and 37 as tenths ; as hundredths ; as ten-
thousandths.
To change a Decimal Fraction to a Common Fraction.
276. Illustrative Examples. Change 0.25 and 0.33J
to common fractions in their simplest forms.
WRITTEN WORK. Explanation. — After writing these frac-
0.25 = \^^ = i tions with their denominators, we find that
33i = ^7^ ^ = -i-^^ft — i *^^ ^^^^ ^^^ ^^ changed to smaller terms
(Art. 198), and that the second may be
changed to a simple fraction (Art. 252) and then to its smallest terms.
277. From the examples above we derive the following
Rule.
To change a decimal fraction to a common fraction :
Write the decimal in the form of a common fraction, and
then change the result, if necessary, to its simplest form.
278. Examples for the Slate.
Change the following to common fractions in their simplest
forms :
(6.) 0.4
(11.) 0.3i
(16.) 0.750
(21.) 0.0625
(7.) 0.80
(12.) 0.37^
(17.) 0.368
(22.) 0.0333
(8.) 0.35
(13.) 0.62^
(18.) 0.66f
(23.) 0.14^
(9.) 0.75
(14.) 0.87i
(19.) 0.666f
(24.) 7.5
(10.) 0.7i
(15.) 0.875
(20.) 0.072
(25.) 1.16J
REDUCTION OF DECIMALS. 127
To change Common Fractdona to Decimal Fractions.
279. Illustrative Example. Change | to a decimal
fraction.
WRITTEN WORK. Explanation. — The fraction | is the same as \ of
8) 3.000 ^' °^ i °^ 3.000 (3000 thousandths), which is found by
-— — dividing 3.000 by 8 in the usual way (Art. 102).
280. From the example above we derive the following
Rule.
To change a common fraction to a decimal fraction :
Express the numerator as tenths, hundredths, thoitsandths,
etc., by annexing as many zeros as may he required, and
then divide it hy the denominator.
281. Examples for the Slate.
Change to decimals :
(26.) f. (29.) ff. (32.) 1,^. (35.) 1.06,^.
(27.) irs- (30.) 5i. (33.) 8J. (36.) O.Oij.
(28.) tIit- (31-) W. (34-) 17t%. (37.) 0.03i.
Change to decimals and add (Art. 45) the follow^ing :
(38.) f , h I, and /^. (40.) f, |, ^, and ^.
(39.) \, I, li, and ^^. (41.) \^, 15f , and 1^.
42. A carpenter paid for a mantel-piece |27f, for a grate
$ 22f , and for a hearth $ 4^^^. How much did he pay in all ?
(43.) 2^ + 3^ + 87^ + 18| = what ?
44. A drover bought a cow and a calf for $ 38.85, and sold
the cow for 1 32|, and the calf for % lOf. How much did he
gain?
45. A man owning 17.635 acres of land, sold 1^ acres to
one person, and -^ of an acre to another. How much had he
left?
46. Change to seven decimal places, and add 1.82^,
0.009/^, and O.IO^VV-
128 DECIMAL FRACTIONS.
282. Illusteative Example. What is the sum of 5J
yards, 2| yards, and 7Jf yards ?
WRITTEN WORK. Explanation. — In this example there are
51 =5 125 fractions which cannot be completely ex-
25- — 2 CCC^ pressed as decimals ; for, however far the
_ _ „ AOAs division be carried, there will still be a
*^ '- ^^ remainder.
Exact sum, 15.215^f If we choose to stop dividing at thou-
sandths, the quotients are expressed accu-
o^ = O.lzo rately by writing |- of a thousandth and -^
2f = 2.667 of a thousandth, as in the margin. But
7^f = 7.424 these results are no more convenient to add
Approximate sum, lK216 ^^^^ *^^ ^^g^^^l numbers ; hence nothing
has been gained by changing the latter to
the decimal form if our object was to find the exact sum.
There are, however, many cases in which the error arising from
the neglect of such small fractions as parts of a thousandth is of no
importance. For such cases the second form of written work given in
the margin is to be adopted. Here the decimal values are expressed
to the nearest thousandth. This is done by increasing the last term of
the decimal by 1 whenever the neglected fraction is ^ or more.
Greater accuracy would be attained by carrying out the decimal to
the nearest ten-thousandth, or to a. still lower denomination.
283. Examples for the Slate.
Note. Unless some other direction is given, the pupil will hereafter
understand that decimal values are to be expressed to the nearest ten-
thousandth.
47. Find the decimal values of §, ^^, f , and add the results.
48. Change to ten thousandths, and add 9|, 16|, and So^\.
49. Change -^ and 0.68 to ten thousandths, and find their
difference.
50. Mr. Carpenter has worked for Mr. Bates 2f hours, 3^ hours,
and 5.5 hours. How many hours has he worked for him in all ?
51. How many rods are there in 25f rods, 0.48^ rods,
105^ rods, and 8.62^^ rods ?
Other examples in addition and subtraction may be found on page 135.
ClRCULATma DECIMALS. 129
Circulating Decimals.
284. We have seen (Art. 282) that in expressing |- deci-
mally (0.666 . . .) the figure 6 is repeated again and again.
So in expressing JJ decimally (0.4242 . . .) the figures 4 and
2 are repeated again and again.
Decimal fractions that are expressed by the same figures
repeated again and again are called repeating or circulat-
ing decimals.
Note. Circulating decimals arise from the reduction of common frac-
tions whose denominators contain prime factors other than 2 and 5.
285. The repeating figures of a circulating decimal are
called a re pet end.
A repetend is marked by placing dots over the first and
last of the figures that repeat.
Thus, l\ = 0.297297 . . . = 0.297 ;
XL = 0.4242 . . . = 0.42 ; 3^ = 3.166 . . . = 3.16.
286. Change the following fractions to decimals tiU the
figures repeat, and mark the repetends :
(52.) J. (55.) $. (58.) t'j. (61.) ^.
(63.) i. (56.) H. (59.) ^. (62.) 1^.
(54.) f (57.) f. (60.) A- (63.) 3^.
To change a Circulating Decimal to a Common Fraction.
287. Illustrative Example I. Change 0.63 to a com-
mon fraction.
To chancre a circulating decimal to n
WRITTEN WORK. '^ ^
A«Q_fi.3_jr common fraction: Take the repetend /or
^ * the figures of the numerator, and for the
figures of the denominator as mamj 9's as there are figures
in the repetend. Change the fraction thus expressed to its
smallest terms.
For an explanation of this rule, see Appendix, page 305.
130 DECIMAL FRACTIONS.
Change to common fractions in their smallest terms :
(64.) 0.3 (67.) 0.39 (70.) 0.016 (73.) O.iSSi
{m:} 0.6 (68.) 0.27 (71.) 0.62i (74.) 0.428571
Im.) 0.42 (69.) 0.648 (72.) 0.108 (75.) 0.571428
To change a Mixed Circulate to a Common Fraction.
288. Illustrative Example II. Change 0.263 to a
common fraction.
To change a mixed circulate to a com-
mon fraction : Take for the numerator the
9§o-tTff difference between the mixed circulate and
the part which does not repeat, hath regarded
as integers, and take for the figures of the
denominator as many 9's as there are figures in the repetend,
with as many zeros annexed as there are figures in that part
of the circulate which does not repeat. (See Appendix, p. 305.)
Change to common fractions in their smallest terms :
(76.) 1.86 (78.) 0.033 (80.) 0.016 (82.) 2.07671
(77.) 2.73 (79.) 0.027 (81.) 0.042 (83.) 7.16i88i
MULTIPLICATION.
In Articles 82 and 86 the multiplication of decimals hy
integers has heen taught. These the pupil may now review.
289. Illustrative Example I. Multiply 175 by 0.01.
Multiply 175 by 0.5.
Explanation. — (1.) To multiply 175 by
WRITTEN WORK. 0.01 is to take 1 hundredth of it, which we
(l^ 175 01— 17^^ ®xpi'6ss by placing the decimal point so that
"^ . — . ^^^ figures 175 may express hundredths ;
(2) 175 thus, 1.75.
rir^K (2.) To multiply 175 by 0.05 is to take 5
_! hundredths of it. One hundredth of 175 is
8.75 1-75, and 5 hundredths is 5 times 1.75, which
equals 8.75. Ans. 8,75.
MUL TIPLICA TION. 131
290. Illustrative Example II. Multiply 0.4 by 0.9.
WRITTEN WORK.
Explanation. — To multiply 0.4 by 0.9 is to take 9
tenths of 4 tenths. One tenth of 0.4 is 4 hundredths,
^•^ and 9 tenths of 4 tenths is 9 times 0.04, which equals
0.36 0.36. Ans. 0.36.
291. From the written work above may be derived the
following
Rule.
To multiply by decimals : Multiply as in integers, and
pcint off as many places for decimals in the product as
there are decimal places in the multiplicand and the mul-
tiplier counted together.
Note. If there are not figures enough in the product, prefix zeros.
292. Examples for the Slate.
Multiply
(84.) 0.048 by 9. (93.) 40.5 by 0.016
(85.) 0.027 by 34. - (94.) 1842 by 0.07
(86.) 0.075 by 20. (95.) 0.0758 by 20.
(87.) 84 by 0.056 (96.) Q.Q by 33^
(88.) 600 by 0.07 (97.) 10.75 by 8f
(89.) 8.4 by 0.56 (98.) 18| by 0.054
(90.) 4.65 by 2.2 (99.) 56^ by 2.73
(91.) 0.8 by 0.0206 (100.) 1.7 by 272^
(92.) 7.06 by 0.053 (101.) m.^ by 5.7
102. What is the sum of 75 x 100 and 0.001 x 1000 ?
103. What is the sum of 7.5 x 1000 and 0.0001 x 0.001 ?
104. How many are 56.8 x 0.01 + 5.29 x 1000 + 0.7 x 0.001 ?
105. How many are 48.125 x 8.33^ + 8169.5 x 0.09 ?
106. What is the cost of paving 146.74 squares at $ 16.84
per square ?
For other examples in multiplication of decimals, see page 135.
132 DECIMAL FRACTIONS.
DIVISION.
In Articles 101, 102, and 114 the division of decimals by
integers has been taught. These the pupil may review.
To divide an Integer or a Decimal by a Decimal.
293. Illusteative Example I. Divide 72 by 1.2
Explanation. — Before dividing by a fraction, the
WRITTEN WORK. ■,.•-,-, , , j • x? i •
dividend must be expressed m the same denomma-
1.2) 72.0 tion as the divisor. The divisor is a number of
60 tenths ; the dividend expressed in tenths is 72.0
(720 tenths).
720 tenths divided by 12 tenths gives the same quotient as 720
divided by 12, which is 60. Ans. 60.
294. Illustrative Example II. Divide 1.935 by 0.45
WRITTEN WORK. ^ Explanation. — Here^ the divisor is a num-
ber of hundredths ; the dividend expressed as
0.45) 1.93^5 (4.d hundredths is 193.5 hundredths (the denomi-
^^^ nation may be indicated in the written work
135 by a caret).
135 193.5 hundredths divided by 45 hundredths
— — gives the same quotient as 193.5 divided by 45,
which is 4.3. Ans. 4.3.
296. From the preceding examples may be derived the
following
Rule.
1. To divide by decimals : Express the dividend in the
came denomination as the divisor by putting a mark as
many places to the right of the decimal point as there are
decimal places in the divisor.
2. Divide as if the divisor were an integer, making a
decimal point in the quotient, when the terms of the divi-
dend have been used as far as the mark.
Note. When there is a remainder after all the terms of the dividend
have been used, the division may be continued, as in Articles 102 and 106.
DIVISION. 133
296. Examples for the Slate.
107. How many books at % 0.08 each can be bought for
$3.84?
108. At $2.80 per yard, how many yards of muslin can be
bought for 155.30?
109. How many rods, each 16.5 feet, are there in 99 feet ?
110. One quart dry measure equals 67.2 cubic inches ; one
quart liquid measure equals 57.75 cubic inches : in a keg whose
capacity is 5040 cubic inches, how many quarts dry measure ?
How many quarts liquid measure ?
111. If coal is % 6.67 per ton, how much coal can be bought
for $3,335?
112. At $ 0.125 per yard for cotton cloth, how many yards
can be bought for $25?
Divide Divide
(113.) 14.91 by 7. (126.) 56.28 by 0.0056
(114.) 8.25 by 1.5 (127.) 0.417196 by 68.76
(115.) 3.24 by 0.81 (128.) 0.08 by 1.611
(116.) 0.00468 by 0.013 (129.) 1.3 by 197.59
(117.) 180.375 by 1.625 (130.) 1203.488 by 28.6
(118.) 579 by 0.075 (131.) 49.2654756 by 0.0759
(119.) 6.9705 by 0.45 (132.) 2464.176 by 57.2
(120.) 0.0033 by 0.011 (133.) 164.6156 by 1334.
(121.) 0.705 by 7.5 (134.) 0.789 by 0.03|
(122.) 3 by 29.9 Note. First multiply both divi-
(123.) 20 by 0.013 dend and divisor by 7.
(124.) 4066.2 by 0.648 (135.) 1.36 by 5.807f
(125.) 68077 by 71.66 (136.) 43.2^ by 0.58J
137. Multiply 0.648 by 100; divide the product by ^J^ ;
divide this quotient by 0.001 ; and multiply the result by ^ of
0.0362.
138. Find the sum of the following: 756.02 x ^-^ 18.3 x 100;
0.7 - 0.001 ; 8.16 - jf^ ; and 0.24 - 16.
For other examples iii division of decimals, see page 135.
134
DECIMAL FRACTIONS.
E
0.025
25.75
0.3504
1.01
250.
0.0008
16.005
8000.
5.6
0.708
19.364
0.0516
1.732
8016.
4.95
0.012
12.007
45.9
8.621
0.00562
1002.
1.87^
0.12i
1.015
8.33|
297. DRILL TABLE No. 6.
Decimals.
F
Twenty-five, and two thousandths.
Two hundred six ten-thommndths.
Seven hundred, and eight tenths.
404 hundred-thousandths.
505050 ten-thousandths.
Eight, and ninety-six hundredths.
Five thousand two, and 5 hundredths.
Sixteen hundred-thousandths.
One hundred twelve millionths.
Two, and twenty-five hundredths.
Seven hundred one, and six tenths.
Two, and 206 thousandths.
936 ten-thousandths.
54, and 54 thousandths.
806, and 1047 milliontlis.
5 hundred, and 2Qf hundredths.
One thousand millionths.
Twenty-nine millionths.
846291 hundred-thousandths.
Five hundred eleven thousandths.
4271, and 4271 ten-millionths.
68 thousand, and 4^ tenths.
One hundred twenty-two thousandths.
Eight, and 4f hundredths.
Five hundred, and f tenths.
DRILL EXERCISES, 135
298. Exercises upon the Table.
177. Read the numbers expressed in E.*
178. Read the numbers expressed in G.
179. Write in figures the numbers expressed in F.
180. Change E to equiv. common fractions in lowest terms.
181. Change G to equiv. common fractions in lowest terms.
182. Change H to equiv. decimals (4 places).
183. Add E and F.
184. Add F and G.
185. AddEFandG.
186. Find the difference of E and F.
187. Find the difference of E and G.
188. Find the difference of F and G.
189. Multiply E by F. 192. Divide E by F.
190. Multiply E by G. 193. Divide G by E.
191. Multiply F by G. 194- Divide G by F.
195. Multiply E by 10 ; divide F by 100 ; add the results to G.
299. Questions for Review.
What are Decimal Fractions ? How are their written
expressions distinguished from those of integral numbers?
What indicates the denomination of the decimal ? How do
you read a decimal expression ? Read 7.05 as a mixed num-
ber ; as a fraction. Read 0.504 and 500.004 so that they may
be distinguished. How do you write decimals ? What is
the effect of annexing ciphers to a decimal expression ?
How do you change decimals to common fractions ? Com-
mon fractions to decimals ? What fractions cannot be changed
wholly to a decimal form ? What are they called when ex-
pressed decimally ? How do you change a circulate to a com-
mon fraction ? How do you add and subtract decimals ?
Perform an example in multiplication by a decimal, explain
and give the rule. Perform an example in division by a
decimal, explain and give the rule.
How do you express the multiplication of a decimal by
10 ; 100 ; 1000 ; 0.1 ; 0.01 ; 0.001 ? How do you express the
division of a decimal by 10; 100; 1000; 0.1 ; 0.01 ; 0.001 ?
* See page 57, for Explanation of the Use of the Drill Tables,
136 WEIGHTS AND MEASURES.
SEOTIOIT XI.
WEIGHTS AND MEASURES.
MEASURES OF LENGTH.
300. We measure the length of anything by applying
to it a line of known length, as 1 foot, 1 yard, and finding
how many such lengths it contains. The line of known
length so used is a linear unit.
301. The length of a line is reckoned in linear units.
302. In measuring length we employ the mile (m.),
rod (rd.), yard (yd.), foot (ft.), and inch (in.). These are
the units of
Long Measure.
12 inches = 1 foot.
3 feet = 1 yaxd.
5^ yards or 16J feet = 1 rod,
320 rods or 5280 feet = 1 mile.
Note L The standard unit of length is the yard. From this the other
units of length are derived.
Note II. For surveyors' and mariners' measures, see Appendix, page 306.
303. Oral Exercises.
a. How many inches are there in 1 yard ? in 2 yards ? in
half a yard ? in a quarter ? in an eighth ? in a sixteenth ?
b. What will it cost to grade a mile of road at $ 1 a rod ?
c. What is the length in feet of a hall that is 15 yards 2
feet long ?
d. How many feet in the length of a fence 5 rods long ?
e. One eighth of a mile is sometimes called a furlong.
How many rods in a furlong ?
MEASURES OF SURFACE.
137
MEASURES OF SURFACE.
304. Two lilies meeting at a point form an angle. Thus
the lines a b and b c form the anixle ab c.
An Angle.
305. The lines are the sides of the
angle, and the point where they meet is
the vertex.
306. The size of an angle is the amount by which one
side is turned away from the other. Thus
the angle d e f is greater than the angle
a be, for the side ef is turned away from e d
more than 5 c is turned away from b a. «^ <^
An Angle.
307. When one line meets another so
as to form two equal angles, each of these angles is a
light angle, and the lines are per-
pendicular to each other. Thus,
the line & c is turned away equally
from b a and b d, making the
angles abc and cbd equal to each ^ b
other. I^ig^* Angles.
308. A flat surface, as the surface of a slate or the top
of a table, is a plane surface.
309. A rectangle is a plane surface
bounded by four straight lines, and hav-
ing all its angles right angles.
310. A square is a rectangle all of whose sides are
equal.
311. A square each of whose sides is
1 inch long is a square inch; a square
each of whose sides is 1 foot long is a
square foot, etc.
312. The area of a surface is its
contents reckoned in square units. a Square.
A Eectangle.
138
WEIGHTS AND MEASURES.
ILLUSTRATION.
To find the Area of a Rectangle.
313. Illustrative Example. If the length of a rec-
tangle is 4 inches and its breadth is 3 inches, how many-
square inches does it contain ?
Explanation. — A rectangle which is 4 in.
long and 1 in. wide will contain 4 square
inches, and a rectangle of the same length
and 3 in. wide must contain 3 times 4, or
12 square inches. (See illustration.)
In the same way it can be shown that
the area of any rectangle is found by multi-
plying the number of units in the length by
the nwmher of like units in the breadth. This is expressed, for brevity,
as multiplying the length by the breadth.
314. Oral Exercises.
a. How many square inches are there in a rectangle 8 in.
long and 5 in. wide ? 11 in. long and 10 in. wide ? 12 in. (1
foot) long and 12 in. (1 foot) wide ? Then how many square
inches are there in a square foot ?
b. How many square feet are there in a rectangle 8 ft. long
and 7 ft. wide ? How many square feet are there in a square
whose sides are each 3 ft. (1 yard) long ?
c. How would you find the number of square yards in f
square whose sides are each 5|^ yards or 1 rod long ?
315. In measuring surface we employ the square mile
(sq. m.), acre (A.), square rod (sq. rd.), square yard (sq. yd.),
square foot (sq. ft.), and square inch (sq. in.). These are the
units of
Square Measure.
144 square inches - 1 square foot.
9 square feet = 1 square yard.
30| square yards, or 272^ square feet = 1 square rod.
160 square rods = 1 acre.
640 acres = 1 square mile.
MEASURES OF VOLUME.
139
MEASURES OF VOLUME.
316. A rectangular solid is a solid bounded by six
rectangles.
A Rectangular Solid.
317. The rectangles are the
faces of the solid, and, together,
make its surface. The bounding
lines of the solid are its edges.
318. A cube is a rectangular solid bounded by six
equal squares.
A cube each of whose edges
is 1 inch long is a cubic inch.
A cube each of whose edges
is 1 foot long is a cubic foot,
etc.
319. The volume of a solid
is its contents reckoned in cubic
units. A Cube.
To find the Volume of a Rectangular Solid.
320. Illustrative Example. What is the volume of a
block of marble 4 ft. long, 2 ft. wide, and 3 ft. thick ?
Explanation. — If the block is 4 feet
long and 2 feet wide, its lower base
must contain 4X2, or 8 square feet
(Art. 313). A solid 1 foot thick upon
these 8 square feet will contain 8 cubic
feet, and a solid 3 feet thick will contain
3 times 8 or 24 cubic feet.
In the same way it can be shown that
the volume of any rectangular solid is
found by multiplying the number of units
in the length by the number of like units in the breadth, and this product
by the number of like units in the thickness. This is expressed, for
brevity, as multiplying together the length, breadth, and thickness.
ILLUSTRATION.
^
^
^
y"
^^ y y
y\
y^ y y y
y
y
\}
140
WEIGHTS AND MEASURES.
321. Oral Exercises.
a. How would you find the number of cubic inches in a
cube 12 inches long, 12 inches wide, and 12 inches thick, or in
1 cubic foot ?
b. How many cubic feet in a cube 3 feet long, 3 feet wide,
and 3 feet thick, or in 1 cubic yard ?
322. In measuring solids we employ the cubic yard
(cu. yd.), cubic foot (cu. ft.), and cubic inch (cu. in.). These
are the units of
Cubic Measure.
1728 cubic inches = 1 cubic foot.
27 cubic feet = 1 cubic yard.
128 cubic feet = 1 cord (cd.), vsed in measuring wood.
I CORD FOOT
I CORD
A<»Hsr'p*i'^
Wood is generally cut for the market into sticks 4 feet long, and
laid in piles, so that the length of the sticks becomes the width of the
pile. A pile 4 feet wide, 4 feet high, and 8 feet long, contains 1 cord.
One eighth of a cord is called 1 cord foot. 1 cord foot contains
16 cubic feet. (See illustration above.)
c. How many cords are there in a pile of wood 4 feet wide,
4 feet high, and 20 feet long ? 32 feet long ? 90 feet long ?
d. What is the cost of a pile of wood 2 feet wide, 4 feet
high, and 10 feet long, at $8 a cord?
e. What must I pay for 3 cords of hard wood at $9.50 a
cord, and J of a cord of pine at $ 5 a cord ?
MEASURES OF WEIGHT.
141
MEASURES OF WEIGHT.
In weighing grocer-
323.
ies and most other common
goods, we use the ton (T.),
pound (lb.), and ounce (oz.).
These are the units of
Avoirdupois Weight.
16 oz. = 1 lb.
2000 lb. = 1 T.
Note I. In weighing some articles,
as iron and coal at the mines, and
goods on which duties are paid at
the United States custom-houses, the
l(mg ton of 2240 lbs. is used. In this
weight
28 lb. =1 quarter (qr.),
4 qr. =1 houdredweiglit (cwt).
20 cwt. = 1 T.
Note II.
324. In weighing silver,
gold, precious stones, etc., we
use the -pound, ounce, penny-
weight (pwt.), and grain (gr.).
These are the units of
Troy "Weight
24 gr. =1 pwt.
20 pwt. = 1 oz.
12 oz. = 1 lb.
325. Comparison of Weights.
175 lb. Troy = 144 lb. av.
170 oz. " = 192 oz. av,
7000 gr. " = 1 lb. av.
Which is heavier, a pound Troy
or a pound avoirdupois ? an ounce
Troy or an ounce avoirdupois ?
For apothecaries' weight, see Appendix, page 307.
Note III. The standard unit of weight is the Troy pound. From this
the other units of weight are derived.
Note IV. A cubic foot of water weighs 62^ lbs., or 1000 oz. avoirdupois.
326. Oral Exercises.
a. How many ounces in 1 lb. avoirdupois ? in 2 lb. 1 oz. ?
h. How many ounces in 1 lb. Troy ? in 4 lb. 5 oz. ?
c. Change 50 gr. to pennyweights ; 90 pwt. to ounces.
d. How many ounces in 3 pounds of silver ?
e. What is the value of a gold chain weighing 1^ ounces at
90/ a pwt.?
/. At 80/ a pound for camphor, what is the cost of an
ounce ?
g. Kow many more pounds in a long ton than in a common
ton ?
142
WEIGHTS AND MEASURES,
MEASURES OF CAPACITY.
327. In measuring liquids
we use the gallon (gal), quart
(qt.), pint (pt), and gill (gi.).
These are the units of
328. In measuring dry
articles, as grain, small fruits,
seeds, etc., we use the husJiel
(bu.), jpeck (pk.), quart, pint,
and gill. These are the units
of
Dry Measure.
4 gi. = 1 pt.
2 pt. = 1 qt.
8 qt. = 1 pk.
4 pk. — 1 bu.
Liquid Measure.,
4 gi. = 1 pt.
2 pt. = 1 qt.
4 qt. = 1 gal.
Note I. A pint of water weighs
about a pound avoirdupois.
Note II. The standard unit for liquid measure is the gallon.
Note III. The standard unit for dry measure is the bushel.
Note IV. In buying and selling grain and many other kinds of produce,
the bushel is reckoned at a certain number of pounds. Thus, potatoes have
60 pounds to a bushel and com has 66 pounds to a bushel.
329. Comparison of Liquid and Dry Measures.
Liquid Measure. Cu. In. | Dry Measure. Cu. In.
1 quart
1 gallon
671
231
1 quart
1 bushel
67^
2150.42
330. Oral Exercises.
a. How many half-pint tumblers can a person fill with 1
gallon of jelly?
h. How many quart measures can he filled with 1 bushel
of cranberries ?
c. What does a vender receive for 1 peck of peanuts which
he sells at 5 cents a pint ?
d. Which is larger, 1 quart of milk, or 1 quart of berries ?
e. How many pints in a bushel ? How many gills in a
gallon ?
/. I bought 3 bushels of pears for $ 2 a bushel, and sold
them at 10 cents a quart ; what did I gain by the sale ?
CIRCULAR AND ANGULAR MEASURES. 143
CIRCULAR AND ANGULAR MEASURES. .
331. A plane surface bounded by a line every point of
which is equally distant from a point
within, called the centre, is a circle.
332. The bounding line of a circle is
the circumference. Any part of the
circumference is an arc.
333. The circumference of a circle
is divided into 360 equal arcs, called ^ Circle.
degrees (°), each degree into 60 minutes ('),
and each minute into 60 seconds ("). These are the units of
Circular Measure.
60 seconds = 1 minute.
60 minutes = 1 degree.
360 degrees = 1 circumference.
Note. As the circumference of every circle has 360 degrees, the length
of the degree differs in different circles.
334. A degree of the circumference of the earth at the
equator is about 69.16 common miles in length.
335. A minute of the circumference of the earth at the
equator is a geographical or nautical mile, and equals
about 1.15 common miles.
336. If the centre of a
circle is placed at the vertex
of an angle, the arc included
between the sides is the meas-
ure of the angle. Thus, if the
arc contains 30 degrees, the angle is called an angle of 30
degrees. (See illustration.)
Note. An angle of one degree has always the same size, but the arc
that measures it differs in different circles.
144 WEIGHTS AND MEASURES.
337. Oral Exercises.
a. How many degrees are there in a semi-circumference ?
in ^ of a circumference, or a quadrant ? in ^ of a circumference,
or a sextant ?
b. The torrid zone is 47° wide. How would you find its
width in nautical miles ? in common miles ?
c. Through how many degrees does the hour hand of a
clock move in 3 hours ? in 1 hour ?. in 2 hours ?
d. Through how many degrees does the minute hand of a
clock move in 5 minutes of time ? in 1 minute ? in a quarter
of an hour ? in half an hour ?
e. How many degrees in a right angle ?
/. How long does it take the hour hand of a clock to move
through a right angle ? How long does it take the minute
hand?
g. The hour and minute hand of a clock form an angle of
how many degrees at 3 o'clock ? at 4 o'clock ? at 10 o'clock ?
at 7 o'clock ? at 12 o'clock ?
MEASURES OF TIME.
338. In measuring time we employ the century, year,
month (mo.), week (w.), day (d.), hour (h.), minute (m.),
and second (s.). These are the units of
Time Measure.
60 seconds = 1 minute.
60 minutes = 1 hour.
24 hours = 1 day.
7 days = 1 week.
or 52 weeks
365 days ) ^ , .
>eksldayi=^''°'"'^'"'y"^<''-y->-
366 days = 1 leap year (1. y.).
100 years = 1 century (C).
MEASURES OF TIME, ■ 145
339. Any year is a leap-year when the number denot-
ing the year is divisible by ^ ^'^d not by 100, and when
it is divisible by JfiO. (See Appendix, page 307.)
Which of the following named years are leap-years :
1878? 1892? 1888? 1900? 2000? 1864? 1880?
340. The year begins with the first of January, and is
divided into four seasons of three months each, as follows :
The winter months are December, January, and February.
The spring months are March, April, and May.
The summer months are June, July, and August.
The autumn months are September, October, and November.
341. April, June, September, and November have 30
days each. February has 28 days, in leap year 29. The
other months have 31 days each.
342. Oral Exercises.
a. What date is three months from Jan. 5 ? July 10 ?
b. What date is 6 months from May 2 ? Feb. 11 ? Nov. 1 ?
c. What months contain 30 days each ? 31 days each ?
d. At 10 cents an hour for 5 hours of every working day,
how much can you earn in 4 weeks ?
e. 8 years and 9 months are how many months ?
/. How many years are there in 100 mo. ? in 200 mo. ?
g*. What date is 30 days from May 5 ? from Apr. 4 ?
h. How many days from May 3 to June 5 ?
Miscellaneous Measures.
343.
"Sxaahers,
344.
Paper.
12 units
= 1 dozen.
24 sheets
= 1 quire.
12 dozen
= 1 gross.
20 quires
= 1 ream.
12 gross
= 1 great gross.
2 reams
= 1 bundle.
20 units
= 1 score.
5 bundles
= 1 bale.
Note. For other measures sometimes used, see Appendix, page 307.
146 COMPOUND NUMBERS.
SEOTIOlsr XII.
COMPOUND NUMBERS.
345. In 2 feet 7 inches, how many inches ? Arts. 31
inches.
The number 31 inches expresses a quantity by reference
to a single integral unit. Such a number is a simple
number.
346. The number 2 feet 7 inches expresses a quantity
by reference to two units of different denominations. A
number expressing a quantity by reference to two or more
units of different denominations is a compound number.
The compound number 2 feet 7 inches expresses the
same quantity that the simple number 31 inches does.
347. When the name of the units is given, the number
is a denominate number. Thus, 31 inches and 2 feet 7
inches are both denominate numbers.
348. When the name of the unit is not given, the num-
ber is a general number. Thus, 31 is a general number.
Note. Denominate numbers are sometimes called concrete numbers, and
general numbers are called abstract numbers.
Name a simple number ; a compound number ; a denominate num-
ber ; a general number. Is 5 feet 2 inches a denominate or general
number ? a simple or compound number ? Is 25 a denominate or a
general number ? a simple or a compound number ?
349. "Written Exercises.
Write from memory the table for Long Measure, Square
Measure, Cubic Measure, Liquid Measure, Dry Measure,
Avoirdupois Weight, Troy Weight, Circular or Angular
Measure, Numbers, Paper.
REDUCTION. 147
REDUCTION.
To change a Compound Number to a Simple Number.
350. Illustrative Example. Change 2 bu. 3 pk. 4 qt.
to quarts.
Explanation. — Since in 1 bushel there are
WRITTEN WORK. 4 pecks, in 2 bushels there are 2 times 4, or 8
2 bu. 3 pk. 4 qt. pecks, which with 3 pecks added are 11 pecks.
4 Since in 1 peck there are 8 quarts, in 11
— pecks there are 11 times 8 quarts, etc.
11 pk.
8
92 qt.
Rule.
351. To change a compound number
to a simple number of a lower denomi-
nation : Multiply the number of the highest denomination hy
the number of units it takes of the next lower denomination
to make one of that higher, and to the product add the given
number of the next lower denomination. Multiply this sum
in like manner, and so proceed till the given number is
changed to units of the required denomination,
352. Examples for the Slate.
1. Change 4 T. 350 lb. 8 oz. to ounces.
2. What is the value of 2 lb. 8 oz. of gold at $ 20 an ounce ?
3. Change 3 rd. 4 yd. 1 ft. to feet.
4. What will it cost to fence both sides of a road 26 rd. 6 ft.
long, at 22/' afoot?
5. How many square feet are there in an acre ?
6. Change 3 sq. m. 35 A. to acres.
7. In a cubic yard, how many cubic inches ?
8. What shall I receive for 25 gal. 3 qt. of milk at 7 cents
a quart ?
9. Mr. Eussell sold 4 bu. 1 pk. 2 qt. of cherries at 12 cents
a qt. ; what did he receive for them ?
148 COMPOUND NUMBERS.
10. If the pulse beats 80 times in 1 minute, how many-
times will it beat in a common year ?
11. If a child sleeps ^ of his time, how many hours will he
sleep in 5 years, allowing for 1 leap year ?
12. How many minutes were there in the first century ?
13. The Tropic of Cancer is 23° 30' north of the equator.
What is the distance in geographical miles ? in common miles ?
363. Changing numbers to numbers of lower denomi-
nations is called reduction descending.
For other examples in reduction descending, see page 171.
To change a Simple Number to a Compound Number.
364. Illustrative Example. Change 4354 feet to rods,
yards, etc.
Explanation. — Since 3 ft. make
WRITTEN WORK. ^. . ^^^. r^ ^.
a yard, m 4354 it. there are as
3) 4354 — 1 ft. Rem. many yards as there are 3's in
6J) 1451 4354, which are 1451, and 1 ft. re-
2 2 mains.
77n 77~?7^ ^ A ^ Since 5|- yards make a rod, in
11) 2902 - I yd. = 4iyd. Ren.. ^^^^ ^^^^^ ^^^^^ ^^^ ^^ ^^^^ ^^^^
263 as there are times 5-^ in 1451,
which are 263, and f yd., or 4^ yd.,
remain, etc.
Ans. 263 rd. 4 J yd. 1 ft., Or
263 rd. 4 yd. 2 ft. 6 in
Rule.
355. To change a simple number to a compound num-
ber of higher denominations : Divide the given number by
the number of units it takes of its denomination to make one
of the next higher. Set aside the remainder, and divide, as
before, the quotient thus obtained; and so proceed till the
required denomination is reached. The last quotient with
the several remainders is the number sought.
REDUCTION. 149
356. Examples for the Slate.
Change to compound numbers :
(14.) 3268 yards. (20.) 9328 lb. of soap.
(15.) 4687 feet. (21.) 19547 oz. of salt.
(16.) 9687 sq. rd. (22.) 9321 pwt. of silver.
(17.) 5692 sq. yd. (23.) 2089 gr. of gold.
(18.) 4791 sq. in. (24.) 5087 qt. of berries.
(19.) 53684^' (25.) 1127793 minutes.
26. What will 20 old silver dollars weigh in oz. pwt. etc.,
each dollar weighing 412^ grains ?
27. The trade-dollar weighs 420 grains. What will 20
trade-dollars weigh ?
28. How many miles is it through the earth from pole to
pole, the distance being 41707308 feet ?
29. In a certain pasture 973 quarts of berries were picked
in one week. How many bushels were picked ?
357. Changing numbers to numbers of higher denomi-
nations is called reduction ascending.
For other examples in reduction ascending, see page 171.
REDUCTION- OF DENOMINATE FRACTIONS.
To change a Denominate Fraction to Integers of Lower
Denominations.
358. Illustrative Example I. Change f rd. to yards,
feet, and inches.
WRITTEN WORK. Explanation. — We first change | of a
^ of -JgL yd. = 4^Tj yd. rod to yards, and have 4^ yards for the
-^ of I? ft = 1^ ft result. We then change ^"^ of a yard to
4 feet, and have If feet for the result, f of
I of 12 in. = 9 in. a foot is 9 inches. Am. 4 yd. 1 ft. 9 in.
Ans. 4 yd. 1 ft. 9 in.
150 COMPOUND NUMBERS.
369. Illustrative Example II. Change 0.62 rd. to
yards and feet.
WRITTEN WORK. Explanation. — We first change
0.62 rd. or 0.62 rd. 0.62 of a rod to yards, and have
5^ 5.5 3.41 yards. We next change 0.41
Q-jA ~310 °^ ^ ^^^^ ^^ ^^^*' ^^^ ^^^® ^''^^
o-i 3;|^Q feet. Ans. 3 yd. 1.23 ft.
3.41yd. 3.410 yd.
3 3
1.23 ft. 1.23 ft.
Ans. 3 yd. 1.23 ft. lowing
360. From the preceding
operations we derive the fol-
Rule.
To change a fraction of one denomination to integers of
lower denominations : Change the fraction, as far as pos-
sible, to an integer of the next lower denomination. If a
fraction occurs in the result, proceed with it as with the
first fraction, and so continue as far as required.
361. Examples for the Slate.
Change to units of lower denominations :
(30.)
(31.)
(32.)
(33.)
(34.)
(35.)
42. At 20/ a foot, what is the cost of | of an acre of land ?
43. In f of an ounce of Dover's powder, how many doses of
5 grains each ?
44. How many planks 8 inches wide will cover the roadway
of a hridge | of a mile long, each plank reaching from side to
side ?
§ of a rod.
(36.) -^s of a cu. yard.
/y of a mile.
(37.) 0.15625 of a gal.
^^ of a sq. mile.
(38.) 0.6 of a bushel.
^ of f of an acre.
(39.) f of a degree.
1 of a ton.
(40.) f of a c. year.
0.875 of a lb. Troy.
(41.) 0.75 of a 1. year.
REDUCTION. 151
To change Integers of LoTver Denominations to a Fraction
of a Higher.
362. Illustrative Examples. (I.) Change 5 oz. 6 pwt.
16 gr. to the fraction of a pound. (II.) Change 2 pk. 6 qt.
to the decimal of a bushel.
WRITTEN WORK.
(I.)
16 gr. = ^1 pwt. = f pwt.
3
(11.)
8) 6 qt.
4) 2.75 pk.
0.6875 bu. Ans.
Explanation (I.). — Since 24 grains make a pennyweight, 16 gr. are
■^ pwt., or I pwt., which, added to the 6 pwt. given, are 6| pwt.
6| pwt. = ^ pwt. Since 20 pwt. make an ounce, ^ pwt. equals
^ as large a part of an ounce, or ^ oz., etc.
Explanation (II.). — Since 8 qts. make a peck, 6 qts. are equal to 0.75
pk., which, added to the 2 pecks given, are 2.75 pecks. Since 4 pks.
make a bushel, 2.75 pk. are equal to 0.6875 bu. Ans. 0.6875 bu.
363. From the preceding operations we derive the fol-
lowinor
° Rule.
To change integers of lower denominations to a fraction
of a higher denomination : Clmnge the mimher of the low-
est given denomination to a fraction of the next higher.
Unite this fraction with the number of that higher de-
nomination. Change, in like manner, the number thus
formed, and so continue as far as required.
45. Change 1 qt. pt. 1 gi. to the fraction of a gallon.
46. Change 242 rd. 2 yd. to the fraction of a mile.
47. What part of a rod is 4 yd. ft. 4^ in. ?
48. What part of an acre is 81 sq. rd. 24 sq. ft ?
49. What part of a cu. yd. is 13 cu. ft. 864 cu. in. ?
50. What part of a year are the three winter months ?
51. Change to the decimal of a mile 87 rd. 10 ft.
152 COMPOUND NUMBERS.
52. Change to the decimal of an acre 135 sq. rd. 54 sq. ft.
53. E-egarding a year as 12 months of 30 days each, what
decimal of a year is 6 mo. 18 d. ? 8 mo. 24 d. ? 5 mo. 27 d. ?
For other examples in reduction of denominate fractions, see page 171.
ADDITION.
364. The operations upon compound numbers are simi-
lar to those upon simple numbers, the principal difference
being that in operations upon compound numbers we use
irregular scales, instead of the scale of tens. No special
rules, therefore, are necessary for addition, subtraction, mul-
tiplication, and division.
365. Illustrative Examples. (I.) What is the sum
of 11° 4' 58", 37° 30' 27", and 27° 24' 54" ?
Explanation. — (I.) We write these numbers
so that units of the same denomination shall
be expressed in the same column. Adding the
seconds, we have 139". Dividing 139" by 60
(60" = 1'), we have 2' 19". We write the 19"
under the line in the seconds' place. Adding the
Ans. 76° 0' 19'' 2' with the minutes of the given numbers, and
dividing the sum by 60 (60' = 1°), we have TO'.
We write 0' under the line in the minutes' place. Adding the T with
the degrees of the given number, we have 76°. Ans. 76° 0' 19".
(in.)
'
(I.)
WRITTEN WORK.
11°
' 4'
58"
37°
30'
27"
27°
24' 54"
(11.)
bu.
pk.
qt.
85
3
7
9
2
5
•98
6
2
3
1
m.
rd.
yd.
ft.
3
192
4
2
316
1
5
76
4
2
Am. 9 m. 265 rd. 3|-yd. 2 ft.
Ans. 196 bu. 2 pk. 3 qt. OT 9 m. 265 rd. 4 yd. ft. 6 in.
KoTE. Change any denominate fraction which occurs in an answer, or
in an example, to units 6f the lower denominations given. (See examples
56, 57, and 58.)
SUBTRACTION. 153
366. Examples for the Slate.
54. What are the contents of three barrels which contain
respectively, 45 gal. 2 qt., 42 gal. 3 qt., and 47 gal. 1 qt. ?
b5. How much land in four lots which contain as follows :
7 A. 83 sq. rd. 31 sq. ft., 15 A. 146 sq. rd., 22 A. 52 sq. rd.
13 sq. ft., and 5 A. 9 sq. rd. ?
6^. What is the length of three roads measuring respec-
tively 15 m. 87 rd., 28 m. 40 rd., and 35^ miles ?
57. To 8° 17' 32" add 4.735° and f .
58. Add together 7 d. 6 h., ^ d., and 0.375 of a week.
367. Illustrative Example II. To -^^ of a gallon add
f of a quart.
Explanation. — That these fractions
may be added they must first be ex-
^^ gal. = ^Tj of 4 qt. = 1^ qt. pressed in the same denomination.
They may be so expressed by chang-
ing ^ gal. to quarts, etc.
69. Add § of a quart to ^^ of a busheL
60. Add 45 1 rods to ^ of a mile.
61. Add 54^ pounds to :^ of a ton.
Perform such examples in exercises 206-208, page 171, as the
teacher may indicate.
SUBTRACTION.
368. Illustrative Example. What is the difference
between 5 rd. 3 yd. 1 ft. and 1 rd. 4 yd. 2 ft. ?
Explanation. — We write these num-
WRITTEN WORK. ^^^.g ^g ^^ simple subtraction, and subtract
5 rd. 3 yd. 1 ft. fi^s^ the 2 feet of the subtrahend. As we
14 2 have but 1 foot in the minuend, we can-
. not now take 2 feet away. So we change
Ans. 3 rd. 6^ yd. J ft. ^ ^^ ^^^ 3 ^^^^^ (leaving 2 yards) to feet.
or 3rd. 4yd. Oft Gin. rpj^jg ^ y^rd equals 3 feet. We add the
WRITT
EN WOEK
I-=S'5
of4qt.=
= liqt.
fqt.
Jtu.
IJqt.
154 COMPOUND NUMBERS.
3 feet to the 1 foot, making 4 feet. Subtracting 2 feet from 4 feet we
have 2 feet left, which we write as part of the remainder.
As we have but 2 yards left in the minuend, we cannot now take
4 yards away, so we change 1 of the 5 rods to yards. This equals
5^ yards, which, added to 2 yards, make 7^ yards. Subtracting 4
yards from 7^ yards, we have 3^ yards left, etc.
369. E2:amples for the Slate.
(62.) (63.) (64.)
bu. pk. qt. oz. pwt. gr. a i ii
5 3 2 6 10 13 35 47 28
2 17 3 15 18 19 54 48
m. 1 m. 80 rd. 2 yd. less 315 rd. 3 yd. equals what ?
^^. What is the difference between 5 ft. 6 in. and f rd. ?
67. A man who had f of a square mile of woodland sold 5^
square rods. How much had he left ?
68. A man having § of a pound of silver ore, gave away Z\
pennyweights. How much had he left ?
69. What is the difference between 0.378 of a day and 44.55
of a minute ?
70. Cape Horn is in h^"" 58^ 4'' south latitude, and the Cape
of Good Hope is in 34° 22' south latitude. Which is farther
south, and how much ?
The difference of latitude between places on opposite sides of the equator
is found by adding the latitudes. The difference of longitude between
places on opposite sides of the first meridian is found by adding the longi-
tudes. If their sum exceeds 180% the difference of longitude equals 360°
minus that sum.
For a table of longitudes, see page 159.
What is the difference of longitude between
71. Albany and Chicago ? 73. Eome and New York ?
72. Berlin and Paris ? 74. San Francisco and Calcutta ?
75. What is the difference in latitude between Philadelphia
39° 5? north latitude, and Buenos Ayres 34° 3' south latitude ?
SUBTRACTION. 155
To find the Number of If ears, Months, and Days from one
Date to another.
N'oTE. The following method of finding the time is generally used in
computing interest.
370. Illustrative Example I. What is the time in
years, mouths, and days from Jan. 11, 1877, to May 5,
1881?
Explanation. — From Jan. 11, 1877, to Jan. 11, 1881, is 4 years ;
from Jan. 11, 1881, to April 11, 1881, is 3 months ; from April 11 to
April 30 is 19 days, and from April 30 to May 5 is 5 days more. An^.
4 y. 3 m. 24 d.
Rule.
371. To find the difference in time between two dates :
First find the number of entire years betvjeen the two dates,
then the number of calendar months remaining, and lastly, the
remaining days.
372. Oral Exercises.
a. How many years, months, and days are there from Feb.
3, 1875, to Oct. 17, 1878 ?
b. How many years, months, and days are there from Sept.
25, 1874, to Jan. 4, 1882 ?
c. Mozart was born Jan. 27, 1756, and died Dec. 5, 1791 ;
at what age did he die ?
d. Goethe died March 22, 1832, and Bryant was born Nov.
3, 1794 ; what was Bryant's age when Goethe died ?
373. Illustrative Example II. How many days are
there from Nov. 12, 1875, to March 10, 1876 ?
Explanation. — There are 18 days remaining in November, 31 days
in December, 31 in January, 29 in February, and 10 in March.
18 + 31 + 31 + 29 + 10 = 119. Ans. 119 days.
e. How many days from March 7 to July 1, 1878 ?
/. How many days from Oct. 9, 1876, to Feb. 11, 1877?
g*. How many days from January 15 to August 7, 1875 ?
156
COMPOUND NUMBERS.
374. A Table showing the Number of Days
From any-
Day of
To the corresponding Day of the following
Jan.
Feb.
Mar.
Apr.
May.
June.
July.
Aug.
Sept.
Oct
Nov.
Dec.
January . .
365
31
59
90
120
151
181
212
243
273
304
334
Februaiy.
334
365
28
59
89
120
150
181
212
242
273
303
March....
306
337
365
31
61
92
122
153
184
214
245
275
April
275
306
334
365
30
61
91
122
153
183
214
244
May
245
276
304
335
365
31
61
92
123
153
184
214
June
214
245
273
304
334
365
30
61
92
122
153
183
July
184
.215
243
274
304
335
365
31
62
92
123
153
August . . .
153
184
212
243
273
304
334
365
31
61
92
122
September
122
153
181
212
242
273
303
334
365
30
61
91
October ..
92
123
151
182
212
243
273
304
335
365
31
61
November
61
92
120
151
181
212
242
273
304
334
365
30
December
31
62
90
121
151
182
212
243
274
304
335
365
Note. In leap years, if the last day of February is included in the time,
a day must be added to the number obtained from the table.
Find from the table above the number of days
h. From April 19 to June 19. j. From Dec. 5 to Feb. 5.
i. From Jan. 1 to March 4. k. From Oct. 12 to Feb. 15.
Perform such examples of exercises 209 and 210, page 171, as the
teacher may indicate.
MULTIPLICATION".
375. Illustrative Example. How much land is there
in 4 gardens, each containing 13 sq. rd. 72 sq. ft. ?
Explanation. — Multiplying 72 sq. ft. by 4,
we have 288 sq. ft. for a product, which equals
1 sq. rd. and 15f sq. ft. We write the 15|
sq. ft. and carry the 1 sq. rd. to the square
53 sq. rd. 15| sq. ft Ans. ^.q^^ {^i the product. 13 sq. rd. multiplied
by 4 are 52 sq. rd., which, with the 1 sq. rd.
carried, are 53 sq. rd. Ans. 53 sq. rd. 15| sq. ft.
WRITTEN WORK.
ISsq.rd. 72sq. ft
4
DIVISION. 157
376. Examples for the Slate.
76. How much syrup will 7 jars contain if each jar holds
1 pt. 3 gi. ?
77. How much wheat is contained in 5 bins if each bin con-
tains 7 bu. 4 pk. 3 qt. ?
78. If a car runs 18 m. 149 rd. in half an hour, how far will
it run in 7 hours ?
Perform such examples in exercises 211 and 212, page 171, as the
teacher may indicate.
DIVISION.
377. Illustrative Example. Divide 47° 18' 36'' by 11-
WRITTEN WORK. Explanation. — Dividing 47° by 11, we
1 1 ^ A7° 1 »/ ^a/f haye 4° for a quotient, with a remaindei of
^ 3°. We write the 4° under the line, and
4° 18' S^j" Ans. change the 3° remaining to minutes, ob-
taining 180'. Adding ISO' to the 18' in
the dividend, we have 198'. Dividing 198' by 11, we have 18' for a
quotient, etc.
378. Examples for the Slate.
79. A farmer brought 5 bu. 3 pk. of corn to mill. How
much corn did the miller take as toll, if he took ^^g- part ?
80. If 65 A. 125 sq. rd. be divided into 50 house-lots, what
is the size of each ?
81. How long will it take to travel 1 mile, at the rate of 75
miles in 10 h. 18 min. ?
82. Among how many men may 624 gal. 3 qt. be divided,
that each man may receive 12 gal. 3 qt. ?
Note. Change both numbers to quarts before dividing.
83. How many bins, each containing 5 bu. 3 pk., will be re-
quired to hold 885 bu. 2 pk. of potatoes ?
84. If a man walks 3 m. 264 rd. in one hour, how long will
it take him to walk 23 m. 273 rd. ?
Perform such examples in exercises 213 to 215, page 171, as the
teacher may indicate.
158 dOMPomb NUMBERS.
LONGITUDE AND TIME.
379. As the earth turns upon its axis once in 24 hours,
it follows that ^ of 360°, or 15° of longitude, must pass
under the sun in 1 hour, and -^ of 15°, or 15', must pass
under the sun in 1 min. of time, and -^^ of 15', or 15", must
pass under the sun in 1 sec. of time. Hence the following
TABLE.
A difference of 15° ) (A difference of 1 hour
in longitude ) ^^ ^^ ( in time.
A difference of 15' ) (A difference of 1 minute
in longitude ] "^^ ^'^ \ in time.
A difference of 15" ) (A difference of 1 second
in longitude ] "^^ ^® ( in time.
380. From the table above we derive the following
Rule.
To find the difference of longitude between any two
places when the difference of time is known : Multiply the
difference of time between the two places, expressed in hours,
minutes, and seconds, hy 15. The product will express the
number of degrees, minutes, and seconds required.
Note. As. the earth turns from west to east, midday occurs sooner in
places east and later in places west of any given point. Hence the time
shown by a clock is later in all places east, and earlier in all places west,
of any given point than it is at that point.
381. Examples for the Slate.
What is the difference in longitude between two places, the
difference in their time being
(85.) 4 h. 17 m. ? (87.) 6 h. 12 m. 10 s. ?
(86.) 2 h. 9 m.? (88.) 1 h. 5 m. 25 s. ?
In what longitude from Greenwich is a place whose time
compared with that of Greenwich is
(89.) 3 hours earlier ? (91.) 1 hour 12 minutes later ?
(90.) 5 minutes later ? (92.) 4 hours 8 minutes earlier ?
LONGITUDE AND TIME.
159
93, The time in St. Louis is 1 b. 5 min. ^f 's. slower thaii
in New York; what is the difference in longitude between
these places, and what is the longitude of St. Louis, that of
New York being 74° 0' 3" west ?
94. A and B sailed together from San Francisco. A kept
his watch by San Francisco time, and B set his by the sun
every day. After 10 days, A's watch was 4 hours 39 minutes
faster than B's : in what longitude were they then, the longi-
tude of San Francisco being 122° 2& W west ?
382. From Art. 379 we may also derive the following
Rule.
To find the difference in time between any two places
when the difference in longitude is known : Divide the
difference in hngitvde, expressed in degrees, minutes, and
seconds, hy 15. The quotient will express the number of
hours, minutes, and seconds reqicired.
383. The names of a few important cities are given
below, with the longitude of each from Greenwich.
Places.
Longitudes.
Places.
Longitudes.
Albany
Boston
Berlin
Calcutta
Chicago
London
Montreal
73° 44' 53" W.
71° ,3' 30" W.
13° 23' 43" E.
88° 19' 2" E.
87° 35' W.
0° 5' 38" W.
73° 25' W.
New Orleans
New York...
Paris
Philadelphia
Rome (Italy)
San Francisco
Washington
90° 7' W.
74° 0' 3"W.
2° 20' 22" E.
75° 10' W.
12° 27' 14" E.
122° 26' 15" W.
77° 2' 48" W.
384. Using the longitudes given above, find the difference
in time between
95. Albany and Boston. 97. Montreal and New Orleans.
96. London and New York. 98. Philadelphia and Chicago.
160 COMPOUND NUMBERS.
When it is noon in "Washington, what is the time
99. In Philadelphia ? 103. In Kome ?
100. In New Orleans ? 104. In Berlin ?
101. In Chicago ? 105. In Paris?
102. In San Francisco ? 106. In Calcutta ?
3VIENSURATI0N OF SURFACES AND SOLIDS.
386. Oral Exercises.
a. How many square feet are there in the top of a table
that is 7 feet long and 3 feet wide ? (Art. 313.)
b. How many square yards are there in a concrete walk
16 J feet long and 4 feet wide ?
c. How do you find the area of any rectangle or square ?
386. From Art. 313, it follows that when the area and
one dimension of a rectangle or a square ar^ given, the other
dimension is found hy dividiTig the number of units of area
by the number of units in the given dimension.
d. There are 15 square yards in a piece of carpeting 5 yards
long ; what is its width ?
e. What must he the length of a walk 2^ feet wide to con-
tain 17 square feet ?
/. How many cubic feet will a box contain that measures
on the inside 7 feet in length, 3 feet in width, and 2 feet in
height ? (Art. 320.)
ff. How do you find the volume of any rectangular solid ?
387. From Art. 320, it follows that when the volume
and two dimensions of a rectangular solid are given, the
other dimension is found by dividing the number of units
of volume by the product of the number of units in each
of two cfiven dimensions.
SURFACES AND SOLIDS. 161
h. What must be the depth of a cistern 5 feet long and 4
feet wide to contain 80 cubic feet ?
i. What must be the height of a room 6 yards long and 5
yards wide to contain 90 cubic yards ?
j. A box 4 inches square must be how deep to contain a
quart dry measure ?
Examples for the Slate.
388. Squares and Rectangles.
107. How many yards of carpeting 1 yard wide will cover
a floor 17 feet long and 15 feet wide ?
108. How many yards of carpeting 27 inches wide will be
required to cover the same floor ?
109. What must I pay for laying a sidewalk 5 rods long
and 5 feet wide at 90/ per square yard ?
110. If one side of a square field is 4 rd. 8 ft. long, how
many square feet are there in the field ?
111. What must I pay for a building lot in St. Louis, 90
feet long and 2 rods wide, at $ 1.75 per square foot ?
112. My building lot contains 1 quarter of an acre, is rec-
tangular, and measures on the street 90 feet, how far back
does it extend?
113. What must I pay for a quarter of an acre of land at
20/ per square foot ?
114. How many more square rods are there in a field 42
rods square than in a 10-acre lot ?
115. My neighbor's garden is 2 rods square, and mine con«
tains 2 square rods ; what is their difference in size ?
116. How many acres were covered by the main Centennial
building in Philadelphia, which was 1880 feet long and 464
feet wide ?
117. What would it cost to make the floor of the above-
named building, the boards costing $37 per thousand feet,
square measure, and the work costing 25/ per hundred, square
measure ?
162
COMPOUND NUMBERS.
W
6
7
18
19
30
31
5
8
17
20
29
32
4
9
16
21
28
33
3
10
15
22
27
34
2
11
14
23
26
35
1
12
13
24
25
36
389. Government Lands.
Before being brought into market, the public lands of
the United States are usually divided by parallels and
^ meridians into townships,
each being as nearly as pos-
sible six miles square. Each
township is divided in the
same way into 36 sections,
and each section into 4 quar-
ter-sections. The township
and sections are numbered
and referred to special me-
s ridians and base lines, so as
A Township. to be easily designated and
pointed out on government maps.
a. How many square miles in a township ? in a section ?
"b. How many acres in a section ? in a quarter-section ?
118. What must I pay for the N. W. quarter of section No.
9 of township 5 North, 20 West, meridian Michigan, at $2.50
an acre ?
390. Rectangular Solids.
119. How many cubic inches are there in a he^n 4 ft. 5 in.
long, 8 in. wide, and 4 in. thick ?
120. What is the weight of a block of Quincy granite 15 ft.
long, 1^ ft. wide, and 6 in. high, if 1 cubic foot weighs 165
pounds ?
121. How high must a block of freestone be to contain 84
cubic feet, if its length is 4^ ft. and its width 3 ft. ?
122. If a bin contains llf cubic yards, and its height is 2
ft., what is the area of its base ?
123. There being 112^ cubic feet in a shaft of marble which
is 27 in. square at each end, what is its length ?
TFOOD MEASURE. 163
391. "Wood Measure.
124. If a pile of wood is 3 ft. 8 in. high and 4 ft. wide, how
long must it be to contain 1 cord ?
125. At $ 6 a cord, what is the cost of a pile of wood 33 ft.
long, 8 ft. 10 in. high, and 4 ft. wide ?
126. If wood is cut in lengths of 3^ ft. and piled to a height
of 4 ft., how long must the pile be to contain 1 cord ?
127. On measuring what I bought for a cord of wood, I
found it 8 feet long, 4 feet wide, and only 3 feet 8 inches high.
At $ 5 a cord, how much money should be deducted from the
original price ?
392. Lumber and Boards.
Sawed timber and boards, when 1 inch or less in thickness, are
generally reckoned by the square foot of surface measure. When more
than 1 inch in thickness, they are reckoned in proportion to their
thickness. Thus,
2000 sq. ft., 1 inch or less in thickness, = 2000 ft., board measure,
2000 sq. ft., IJ inches thick, = 3000 ft., hoard measure,
2000 sq. ft., 2 inches thick, = 4000 ft., board measure,
and so on.
128. How many feet of boards f of an inch thick will be
required to make a fence 2 rods long and 3 feet high ?
129. How many feet, board measure, are there in a piece of
square timber 10 in. wide, 6 in. thick, and 9 ft. long ?
130. How many feet, board measure, are there in 200 pieces
of scantling, each 18 ft. long, 4 in. wide, and 2 in. thick ?
131. How many feet in 8 boards, each 15 ft. long, 8 in. wide,
and 1^ in. thick ?
132. How many feet, board measure, in a plank 24 ft. long,
3 in. thick, 11 in. wide at one end, and 16 in. wide at the
other ?
Note. First find the average width, which equals one half the sura of
the widths at the ends.
164 COMPOUND NUMBERS.
133. At $ 22 a thousand, what is the cost of 20 boards, each
18 ft. long, 1 in. thick, 20 in. wide at one end, and 17 in. wide
(^ at the other ?
\ 134. Arthur bought wood for Sorrento carving, each piece
being 2 feet long and \ of an inch thick, as follows : white
holly, 12 inches wide at 8/ a foot, board measure ; black wal-
nut, 18 inches wide at 6/ ; ebony, 9 inches wide at 25/' ; red
cedar, 14 inches wide at 10/. What was the cost of the
whole ?
393. Capacity of Cisterns, Bins, etc.
135. I have a cask that contains 2 cu. ft. ; how many quarts
of berries will it hold ? (See Art. 329.)
136. How many gallons of water will a cistern hold that is
3 ft. long, 3 ft. wide, and 2^ ft. deep ?
137. If a jar weighs 10 pounds when empty and 74 pounds
when full of water, what is its capacity in cubic feet ? How
many gallons will it hold ? (See page 141, Note lY.)
138. How many bushels of wheat can be put into a bin 8 ft.
long, 3 ft, 2 in. wide, and 2 ft. 3 in. deep ?
In measuring bulky fruits and vegetables, as apples and potatoes, the
' measures are heaped. Heaped measures fill about ^ more space than the
even measures.
139. If 24 bushels of wheat can be put into a certain bin,
how many bushels of apples might be put into the same bin ?
140. How many bushels of beets can be put into a barrel
that holds 47 gallons ?
141. What is the difference in inches between f of a bushel
and 1 cubic foot ?
Note. The difference being so slight, for rough estimates of the con-
tents of bins, etc., it is sufficiently accurate to call eveiy cubic foot f of a
bushel, even measure, or ^f of a bushel, heaped measure.
142.. A box whose capacity is 50 cubic feet, will contain
how many bushels of rye ? how many bushels of pears ?
• GENERAL REVIEW. 165
394. General Review, No. 3.
143. Change 5ra. 42 rd. 8 ft. to feet.
144. Change 4865 gr. to Troy pounds, ounces, etc.
145. Change f cu. yd. to feet and inches.
146. What cost 12 bu. 2 pk. of plums at 6/ a pint ?
147. What cost 2 qt. 1^ pt. of oil at $ 1.12 a gallon ?
148. Change 41' 42^' to the decimal of a degree.
149. Change |§ yd. to the decimal of a rod.
150. What part of anA. is 116sq.r. 88f sq.ft. + ITsq.r. 2 sq.ft.?
151. Add 0.44 c. y. to 2^ d. 5 h. 4 m.
152. Dividel2A. lessTA. 16r. by9.
153. Change 2 lb. av. to integers of Troy weight.
154. How many square feet in a garden 4 rd. long and 1 rd.
15ft. wide?
155. How many cu. ft. of space in a cellar measuring on the
inside of the wall 5 yd. lit. in length, 4 yards in width, and 10
feet in depth ?
156. What must be the depth of a cistern to contain 420
gallons of water, the base being a square covering 12^ sq. ft. ?
157. When a cistern 4 feet high is full of water, what weight
is supported by every square inch of the base ? (See page 141,
Note lY.)
158. How many bricks 4 in. by 8 in. will be required to pave
a court 20 ft. long and 10 ft. wide ? Find the cost at $ 9 per M.
159. Divide 0.006 by 0.06, multiply the quotient by 0.05;
and divide that product by 0.005.
160. Change 0.0625 to a common fraction in smallest terms ?
<^ 161. How many yards of carpeting f of a yard wide must
be bought to cover a floor 13 feet square, no allowance being
made for matching and no breadth to be divided ?
162. What is the difference of time in two places whose
longitudes differ 7° 8' 4" ?
163. When the difference of time between two places is
3 h. 4 m. 6 s., what is the difference of longitude ?
164. How many days from Jan. 5, 1876, to March 3, 1877 ?
166 COMPOUND NUMBERS.
395. Miscellaneous Examples.
165. If eggs are worth 30 cents a dozen, and 10 weigh a
pound, what are eggs worth by the pound ?
166. If I burn 30 lbs. of coal a day, and buy my coal by the
long ton, at $ 7 a ton, what is the cost of my coal for December ?
167. How many furrows, each 20 inches wide, will be made
in ploughing lengthwise a lot of land which is 6 rd. 1 ft. wide ?
168. A quantity of silver weighed 4 lb. 10 oz. 3 pwt. before
refining, and 3 lb. 11 oz. 2 pwt. 9 gr. afterwards ; what weight
was lost in the process ?
169. How many square feet on the top and sides of a box
that is 3 ft. long, 2 ft. wide, and 2 ft. 6 in. high ?
170. What will be the cost of fencing a lot of land 20 rods
by 26 rods at 25 cents a foot ?
171. Change f of a great gross to units of lower denomina-
tions.
172. Divide an angle of 20° 4' ^" by 9.
173. Dr. Smith's wagon-wheel, which is 3 ft. 4 in. in circum-
ference, turns round 200 times in going from his house to the
post-office ; how far does he live from the post-office ?
174. If a bird can fly 1° in 1 h. 8 m. 15 s., in what time can
it fly around the world at the same rate ?
175. What is the cost of 137 gal. 2 qt. of vinegar at 50 cts.
per gal. ?
176. How many bushels of grain will a bin contain which
is 10 ft. long, 8 ft. wide, and 5 ft. deep ?
177. What is the cost of oil-cloth to cover a floor 12 feet by
16^ feet, at 75 cents per square yard ?
178. A farmer divided one half of his estate of 350 A. 140 rd.
equally between his two daughters, and the balance, after set-
ting off 17| A., equally between his two sons. What was the
share of each son and daughter ?
179. How many yards of carpeting % yd. wide will cover a
floor 18 ft. sq. ?
MISCELLANEOUS EXAMPLES. 167
180. If a cotton-mill can make 1200 yds. of cloth per hour,
how many yards could be made by working 10 hours a day
from July 7th to January 4th, allowing for 2Q Sundays ?
181. Change 15 ib. 8 oz. Av. to pounds and ounces Troy.
182. How many sq. ft. does the surface of a block contain,
which is 3 ft. long, 2 ft. wide, and 6 ft. thick ?
183. When 2 dozen grape-vines can be bought for $ 6.50,
what is the cost of each vine ?
184. From a pile of wood b^ ft. long, 4 ft. high, and 4 ft.
wide, was sold at one time 3f cords, at another 2\ cords.
What is the remainder worth at $ 4 a cord ?
185. I have a shed which measures on the inside 18 ft. 7 in.
by 8 ft. by 10 ft. in height. How many cords of wood can be
put in it ?
186. A man purchased 75 cords of wood for $ 360 ; he sold
the following lots, 10^ cd., 15 cd., and llf cd., all at $ 5 per
cord. What did he gain on what he sold ?
187. What would be the cost of sawing the remainder of
the 75 cords, at $ 1 a cord ?
188. How many gals, of water will be contained in a tank
3 ft. square, if the water is 4 ft. 3 in. deep ?
189. At 15 cents per pound, what was the cost for lead,
5 lbs. to the sq. ft., to line the above tank, it being 5 feet deep ?
190. What must I pay for a dozen silver spoons, each weigh-
ing 2 oz. 9^ pwt., at % 1.50 per ounce ?
191. Add I of the month of February, 1876, to f of th^
days from March 21st to June 17th, 1877.
192. How much carpeting f jdi. wide will cover the top and
sides of a block 3 ft. long, 8 inches wide, and 6 inches high ?
193. Estimate the cost of feeding a pair of oxen through
the winter of 1879 and 1880, if 1 ox weighed 1772 lbs. and
the other 1431 lbs., and hay was $ 13.75 per ton, and the oxen
were allowed -^ of their weight in hay each day.
194. How many paving-stones 6 in. by 8 in. will be required
to pave a street 27 rods long by 50 ft. wide ?
168 COMPOUND NUMBERS.
195. At 9 o'clock p. m. in Boston, what is the time in
Paris ?
196. If a druggist sells 1 gross 2 doz. powders a day, how
many will he sell from the 19th of Dec, 1877, to 15th Mar.,
1878, deducting 12 Sundays ?
197. In what time will a vessel go through a strait 2 miles
long, if she is carried ahead by tide 30 feet a minute, by wind
25 feet a minute, and by steam 100 feet a minute ? In what
time can she go through the strait against wind and tide ?
396. Questions for Revie"wr.
Repeat the table of Long Measure. Draw a line an inch long.
Hold your hands a foot apart. What do you think the height of
your school-room to be? In some convenient place mark off and
walk 100 feet, counting your steps as you walk, and find their
average length. By counting your steps, find how far you live from
school. What is the standard unit of length ?
How is an angle formed ? Upon what does its size depend ? What
is a right angle ? a rectangle ? a square ? area ? How do you find the
area of a rectangle or a square? Illustrate. Repeat the table of
Square Measures. From what are the units of square measure
derived ? What is the principal unit of land measure ?
What is a rectangular solid ? a cube ? How many faces has a cube ?
how many edges ? How do you find the volume of any rectangular
solid? Illustrate. Repeat the table of Cubic Measure. From
what are the units of cubic measure derived ?
Repeat the table of Liquid Measures ; of Dry Measures. Which
is larger, 1 quart liquid measure, or 1 quart dry measure ? What is
the standard unit of liquid measure ? of dry measure ? How many
cubic inches are there in a gallon ? in a bushel ? How do we ascer-
tain the WEIGHT of anything? Repeat the table of Avoirdupois
weights ; of Troy weights. By which would you buy iron ? silver ?
salt? emeralds? flour? What is a long ton? How many grains.
Troy make a pound Avoirdupois ? Which is heavier, 1 lb. Avoirdu-
pois, or 1 lb. Troy ? 1 oz. Avoirdupois, or 1 oz. Troy ? What is the
standard unit of weight ?
What is a circle ? the circumference ? an arc ? Repeat the table
of Circular Measures. Are all degrees of the same length ? What
QUESTIONS FOR REVIEW. 169
is the length of a degree of the circumference of the earth at the
equator ? What is a nautical mile ? What is its length in English
miles ?
How is an angle measured ? Are all angles of one degree of the
same size ?
Repeat the table of Time. How do you know what years are leap
years ? Name the months which contain 30 days each ; name the
months which contain 31 days each.
What is a compound number ? a denominate number ? >a general
number ? How do the units of different denominations in compound
numbers increase ?
Give the rule for Reduction Descending; for Reduction Ascend-
ing. How do you change a denominate fraction to integers of lower
denominations? How do you change integers of lower denomina-
tions to the fraction of a higher ?
How are compound numbers added, subtracted, multiplied,
and divided?
How do you find the number of years, months, and days between
two dates ? (Art. 371.) How do you find the time in days between
two dates ? When the difference in time between two places is given,
how do you find their difi'erence in longitude ? When the difference
in longitude is given, how do you find the difference in time ?
Find the area of the top of your desk. Draw a square 1 inch each
way ; \ inch each way. What part of the first square is the second ?
Difference between 5 square inches and 5 inches square ? When the
length of one side of a rectangle is given in feet, and the other in
rods, how do you find the surface ? When the area and one dimen-
sion are given, how do you find the other ?
How are the public lands of the United States divided ?
When the volume and two dimensions of a rectangular solid are
given, how do you find the third ?
How is WOOD generally cut for market ? How many cubic feet
are there in 1 cord ? How would you estimate the contents of sawed
timber and boards ? How do you find the average width of a board
that decreases regularly in width from end to end ?
How are bulky fruits and vegetables measured? How does a
heaped measure compare in bulk with an even measure ? A cubic
foot is equal to what part of a bushel, even measure ? What part of
a bushel, heaped measure ?
170
DRILL TABLE.
397. DRILL TABLE No. 7.
A
B
C
D
1.
T.
lb.
4'r- 625'^-
12°-
2"^- 1428'^-
go.
2.
l.T.
lb.
5I.T. ;|L2^"*-
3qr.
Igcwt.
2^-
71b.
3.
lb.*
pwt.
91b. goz.
5 pwt.
31b.
70Z.
10 pwt.
4.
m.
ft.
10™ 200'-^-
4 yd.
34 rd.
3 yd.
2''-
5.
sq.. m.
sq. rd.
-I^sq.m.5gQA.
4sq.rd.
48^-
Q sq. rd.
6.
cu. yd.
cu. in.
Q cu. yd. A cu. ft.
^QQcu.in.
g cu. yd.
15cu.ft.
1506cu.ia
7.
cd.
cu. ft.
"LQQ cd. 2''^-^'-
14cu.ft.
92'"-
gcd.ft.
12 cu. ft.
8.
gal.
gi.
258^^1- S^'
Qpt-
4.al.
2qt-
ipt
9.
bu.
pt.
gbu. 3pk.
iqt.
5bu.
iPk.
2qt
10.
circ.
(')
1 '''"'■ 90°
40'
280°
2'
28"
11.
c.y.
hours
3c.y. 4d.
12^-
2c.y.
7d.
18^-
12.
l.y.
min.
Vy 65^
18 '^
7d.
20'^-
5. .in.
13.
rd.
in.
gm. 310 rd.
2 yd-
15 rd.
lift.
3 in.
14.
A.
sq. yd.
5 A. 29 ^'^■'•'^
gg sq.ft.
Igsq.rd.
206sqft-
gsq.in.
15.
pk.
pt.
7 pk. 2 1'-
ipt.
6qt.
iPt.
2^'-
16.
D
(")
54° 51'
45"
18°
36'
64"
n.
oz.*
gr.
51b. IQoz.
2 pwt.
11 oz.
Igpwt.
20^'--
18.
yd.
in.
3 m. 4;^'^'^
2y<*-
5 yd.
1ft.
4in.
19.
sq. rd.
ft.
5sq.rd. 29sq-yd
4 sq. ft.
40 sq. yd
8 sq.ft.
9 sq.m
20.
qt.t
gi-
3 qt P*
28^'-
2qt-
ipt.
22'-
21.
w.
min.
28"- 3^-
12^
3w.
5^-
18'^-
22.
sq. yd.
sq. in.
21 sq. yd. Q sq. ft.
12 ^'i- '"
4sq.yd.
3 sq.ft.
3(5sq.m
23.
d.
sec.
284^- 13^-
9 min.
169^-
19'"
42 '"'"•
24.
sq, ft.
sq. in.
W sq. rd. 4 sq. ft.
90 sq. in.
3 sq. rd.
204 sq.ft.
lose in
25.
gross
units
Troy.
-| gross ^7 doz.
2
g gross
3 doz.
t Liquid.
9
DRILL EXERCISES.
171
DRILL TABLE No. 7
{continued).
n\h.
E
goz.
4cwt.
3qr. 111b.
11°^
14pwt
5 yd.
1ft
2 sq. rd.
4 sq. yd.
14- ft.
329cu.in.
13 edit.
14cu.ft.
3qt.
1 P*- (liquid)
4Pk.
7 qt. 1 pt.
98'
14//
348 '^
3^^
21*^-
IQmin.
2^^-
gin.
Qsq yd.
110 ^'i- "•
ipt.
3^- (dry)
68'
58''
*
22 P"^*-
23^'^-
2^'-
3 in.
9 sq. yd
8 sq.ft.
ipt
3 Si- (liquid)
23'^-
41 min.
27 sq.ft.
28 sq. in.
48 'nin.
18 sec.
178 sq.ft.
108 ^^- '"•
1 gross
gdoz. 10
398. Exercises upon the Table.
196. Change five A to B.*
197. Change E to units of the lowest
denomination in the example.
198. Change D to units of the lowest
denomination in the example.
199. Change 3284 B to A.
200. Change 132687 B to units of
higher denominations.
201. Change A A to B.
202. Change 0.4627 A to B.
203. Change the numbers of lower de-
nominations in D to a fraction
of the highest.
204' Change the numbers of lower de-
nominations in C to a decimal
of the highest. (4 places.)
205. What part of A is E ?
206. Add C, D, and E.
207. Add ? A to D.
208. Add 0.5784 A to E.
209.- Take E from D.
210. Take D from C.
211. Multiply C by 6.
212. Multiply E by 15.
213. Divide C by 10.
214- Divide D by 7.
215. Divide D by 4 of the lowest de-
nomination in the example.
* See page 57, for Explanation of the Use of the Drill Tables.
172 THE KETRIQ SYSTEM.
SECTIOI^ XIII.
THE METRIC SYSTEM OF W^EIGHTS
AND MEASURES.
399. The metric system of weights and measures, now
used in the greater part of Europe and coming into use
in the United States, is derived from the standard meter.
Note. The word meter means a measure. The standard meter is a
certain bar of platinum carefully preserved at Paris. Copies of this bar,
made with the utmost precision, have been procured and are carefully pre-
served by the nations that have adopted the Metric System. The standard
meter of the United States is such a copy, and it is kept at Washington.
The meter-sticks made for ordinary use are copies of the standard meter.
MEASURES OF LENGTH.
400. The standard unit of length in the metric system
is the meter.
Note. The teacher should show the pupil a meter and its subdivisions.
If none can readily be obtained, one can easily be made from the decimeter
represented on the next page. This meter may be divided into decimeters
and centimeters. From this measure the pupils can easily make their own
of paper or wood.
401. One tenth of a meter is a de&i-meter.
Note. The prefix deci- means one tenth of.
402. One hundredth of a meter is a c^nfti-meter.
Note. The prefix centi- means one hundredth of.
403. One thousandth of a meter is a miTli-meter.
Note. The prefix milli- means one thousandth of.
MEASURES OF LENGTH.
173
404. Exercises on the Meter and its subdivisions.
a. How many meters long is the room ? How many meters
wide?
b. How many decimeters long is the
table ?
c. How many decimeters wide is the
door?
d. How many centimeters long and wide
is your slate ? the window-pane ? etc.
e. How many millimeters apart are two
lines on a sheet of writing-paper?
/. How many millimeters thick is your
slate-frame? your ruler? etc. §
g. How many millimeters are there in one |
centimeter ? §
h. How many centimeters are there in one ^
decimeter ? ^
i. How many decimeters are there in one
metier?
j. How many millimeters are there in one
decimeter ? in one meter ?
k. How many centimeters are there in
37 millimeters, and how many millimeters
remain ?
1. How many decimeters are there in 84
centimeters, and how many centimeters re-
main ?
222. How many meters are there in 347 centimeters, and how
many centimeters remain ?
n. In measuring the length of the room, did you find it to
be an exact number of meters long ?
o. If not, how many decimeters do you find in the remain-
der ? Do you find an exact number of decimeters ?
p. If there is still a remainder, how many centimeters do
you find in it ?
174 THE METRIC SYSTEM.
To "write Numbers in the Metric System.
405. To express a length in meters and parts of a
meter, we write whole meters in the units' place, deci-
meters in the tenths' place, centimeters in the hundredths'
place, and millimeters in the thousandths' place.
Thus, if a room is found to be 8 meters 6 decimeters
9 centimeters long, we write :
Length of the room = 8.69 meters.
2 decimeters 3 centimeters 5 millimeters is written :
0.235 meters.
406. The abbreviations used in writing expressions of
length are : For meters, m ; for decimeters, dm ; for centi-
meters, cm ; and for millimeters, mm.
407. Lengths may be expressed in other denominations
as well as in meters, hy putting the decimal point at the
right of the place of the required denomination, and writing
the proper name or abbreviation after the figures.
Thus, 0.235™ may be written 2.35^'", 23.5'^'", or 235"™.
So also 728'"-" may be written 72.8 ^"^ 7.28^'", or 0.728"".
408. Exercises in reading Numbers.
Read the following :
a. 5-
e. 5.926™
i. 6.58^™
b. 47'-
/. 36^™
J. 3.4^™
c. 3.9-
g, 428^™
k. 43.7^™
d. 4.21™
h. 23™"
1. 2.5 ™™
409. Examples for the Slate.
Change the following to meters :
(L) l'^™ (4.) 1^™ (7.) 1'
(2.) 13 '^™ (5.) 38^™ (8.) 48™™
(3.) 214^™ (6.) 529^™ (9.) 3675'
mm
MEASURES OF LENGTH. 175
Multiples of the Meter.
410. Besides the meter and its subdivisions, there are
longer measures, which are multiples of the meter.
411. The dekfa-meter is ten times as long as the meter.
Note. The prefix deka- means tenfold.
412. The heltto-meter is a hundred times as long as
the meter.
Note. The prefix hekto- means a hundredfold.
413. The kil'o-meter is a thousand times as long as
the meter.
Note. The prefix kilo- means a thousandfold.
414. The my^ria-meter is ten thousand times as long
as the meter.
Note. The prefix myria- means ten thousandfold.
Note. Of these longer measures, the kilometer is used in measuring
distances on roads, canals, rivers, etc. The other measures are much less
frequently used ; the myriameter hardly ever.
415. Exercises on the Multiples of the Meter.
a. Measure off a string ten meters long. What name is
given to the length of this string?
Note. The string may be used in measuring distances. For this pur-
pose it will be well to make knots at the end of each meter.
b. Measure in dekameters and meters the length and
breadth of the school-yard ; of a garden ; of a field, etc.
c. Measure off in the street, or other convenient place, a
distance of 10 dekameters. What name is given to this distance?
d. Walk from the beginning to the end of the distance thus
measured off, and count your paces. How many of your paces
make a hektometer ?
e. How many of your paces would make a kilometer ?
/. How many kilometers from your home to the school-house?
176 THE METRIC SYSTEM.
g. How long does it take you to walk a kilometer ?
h. How many kilometers can you walk in an hour ?
i. If 1500 of your paces make a kilometer, how many make
a dekameter ?
416. To express distances in meters and multiples of a
meter, we write meters in the units' place, dekameters in
the tens' place, hektometers in the hundreds' place, and so on.
417. To express a distance in kilometers, we write
kilometers in the units' place, and then hektometers,
dekameters, and meters will be written in the tenths', hun-
dredths', and thousandths' places respectively.
Thus, if the distance from one town to another is found
to be 9780 meters, the usual form of writing would be
9.78 kilometers.
Note. The greatest distances are usually expressed in kilometers. Thus,
the distance of the earth from the sun is about 149000000 kilometers.
418. The abbreviations used in writing are : For the
dekameter. Dm-, for the hektometer, Hm\ and for the
kilometer Km.
419. Table of Long Measure.
10 millimeters (mm) = 1 centimeter (cm).
10 centimeters
= 1 decimeter (dm).
10 decimeters
= 1 meter (m).
10 meters
= 1 dekameter (Dm).
10 dekameters
= 1 hektometer (Hm).
10 hektometers
= 1 kilometer (Km).
10 kilometers
= 1 myriameter (Mm).
420.
Oral Exercises.
Read the following :
a. 123- d.
42 Dm g 49 Km
b. 497.6" e.
36.7°'" h. 593.7^"
c. 346"'" /.
57,5""* h 6000^""
MEASURES OF LENGTH. 177
421. Examples for the Slate.
Change the following to meters : " •
(10.) 425°™ (13.) 94.6"- (16.) 0.72 k--
(11.) 35 »™ (14.) 9.24^™ (17.) 0.073"-
(12.) 23.5^- (15.) 39.7°- (18.) 0.05^-
Addition, Subtraction, Multiplication, and Division of Metric
Numbers.
422. Illustrative Example. Chaoge to meters and
add 14.83°", 75.6""', and 948"°.
WRITTEN WORK. Explanation. — When these expressions
14.83°-= 148.3- have been changed to meters, they are all
75.6 "- = 7560. of the same denomination, and the sum is
948 *=- = 9.48 found in the same way as in the addition
7717 78 - ^^ simple numbei^.
423. Numhers expressiTig metric measures and weights
are added, subtracted, multiplied, and divided hy the same
rules as apply to simple numbers.
424. Examples for the Slate.
19. Add 5.6- 24.07-, 30.5- and 7.508 -.
20. Express as meters and add 582"=- 6428^- and 495-".
21. Express as meters and add 369 °- 4073 »- and 5 '^-.
22. Add 48.06- 709.63- 3708.9- 800.9- and express the
answer in kilometers.
23. If 7 ^- be taken from 42 ^- how many meters remain ?
24. From 87.04 - take 42 ^-.
25. The distance round a certain park is 2.58 kilometers.
How many meters will a man go who rides around it six times ?
2Q. A school-boy walked one third around the above park in
12 minutes. How many meters did he walk in 1 minute ?
27. How many kilometers in 36.68 - x 2004 ?
28. Divide 38.07 - by 4 and by 3, and add the answers.
29. Ellen's hoop is 3.6 - around. How many times will it
turn in rolling a distance of 1.08 ^- ?
178 TME METRIC SYSTEM,
MEASUEES OF SURFACE.
425. The units used in measuring surfaces are squares,
each having sides equal to a unit of long measure.
Thus, a square meter is a square having sides one meter
long; a square decimeter is a square having sides one
decimeter long ; etc.
426. Exercises.
a. How many square decimeters in a square meter ? Illus-
trate by drawing a square meter on the blackboard or on the
floor and dividing it into square decimeters.
b. How many square centimeters in a square decimeter ?
Illustrate by drawing a square decimeter on your slate and
dividing it into square centimeters.
c. How many square meters in a square dekameter ?
427. The square dekameter, when used as a unit of
land measure, takes a special name, and is called an ar.
One hundredth of an ar, which is one square meter, is
called a centar. A hundred ars, equal to one square
hektometer, is called a hektar.
428. Square Measure.
100 square millimeters (sq mm) = 1 square centimeter (sq cm).
100 square centimeters = 1 square decimeter (sq dm).
100 square decimeters = 1 square meter (sq m) =1 centar (ca?.
100 square meters = 1 square dekameter = 1 ar (a).
100 square dekameters = 1 square hektometer = 1 hektar (Ha).
100 square hektometers = 1 square kilometer (sq Km).
429. As the units of square measure form a scale of
hundreds, in writing numbers expressing surface two deci-
mal places must be allowed for each denomination.
Thus, 45^^"^ 4^^*^™ 86^^*='" are written 45.0486 ^*i"; and 7"^
6*6^^ are written 706.05 ^
MEASURES OF VOLUME. 179
430. Examples for the Slate.
30. How many square meters of carpeting will be requirec?
to carpet a room 5.3 "" long and 4.5 "" wide ?
31. How many meters of carpeting 0.7"" wide will be re-
quired to carpet a room 4™ long and 3.5"" wide ?
32. What is the cost of polishing the surface of a rectangu-
lar piece of marble 2.8 meters long and 1.2 meters wide, at
$ 2.50 per sq. meter ?
33. In a piece of land 15 "" long and 14.5 '" wide are how
many square meters or centars ? how many ars ?
34. Express the following in ars and add them : 1.3 hektars,
155.5 ars, 43 hektars, 26 centars.
35. A had 6 hektars, 7 ars, 9 centars of land, and sold 0.2 of
it at $ 54 an ar. How much did he receive for what he sold ?
MEASURES OF VOLUME.
431. The units used in measuring cubic contents, or
volume, are cubes, each having its edges equal to a unit
of long measure.
Thus, the cubic meter is a cube having edges one meter
long; a cubic decimeter is a cube having its edges one
decimeter long ; etc.
432. Exercises.
a. How many cubic decimeters in a cubic meter ?
b. How many cubic centimeters in a cubic decimeter?
Illustrate by means of a cubical block having edges one deci-
meter long, marked off into centimeters.
433. The cubic meter, when used as a unit of measure
for wood and stone, takes a special name, and is called a
ster.
434. The cubic decimeter, when used as a unit of liquid
or dry measure, is called a liter.
180 THE METRIO SYSTEM,
435. Cubic Measure.
1000 cubic millimeters (cu mm) = 1 cubic centimeter (cu cm).
1000 cubic centimeters = 1 cubic decimeter (cu dm) = 1 liter.
1000 cubic decimeters = 1 cubic meter (cu m) =1 ster.
436. "Wood Measure.
10 decisters (ds) = 1 ster (s).
10 sters = 1 dekaster (Ps).
437. As the units of cubic measure form a scale of
thousands, in writing numbers expressing volume three
decimal places must be allowed for each denomination.
Thus, 427 "" ™ 29 *=" '^ 3 *=" *="" are written 427.029003 "" ™.
438. As the units of wood measure form a scale of tens,
only one decimal place is needed for each denomination.
Thus, 7 dekasters 5 sters 6 decisters are written 75.6 sters.
439. Examples for the Slate.
36. Express the following in cubic meters and add them :
7 cu. meters 40 cu. decimeters ; 5 cu. meters 3 cu. decimeters
19 cu. centimeters ; 25 cu. centimeters 49 cu. millimeters.
37. How many cubic meters of earth must be removed to
dig a cellar 14.5'" long, 4.6"" wide, and 2.3'" deep ?
38. At $ 1.25 a cubic meter, what will it cost to dig a trench
75.5 "" long, 2.2 "^ wide, and 1.8 ™ deep ?
39. How many loads of earth, each filling 2.25^"'", will fill
a space 15.4 "' long, 12 "" wide, and 4.5 " deep ?
40. If a cubic centimeter of gold is worth $ 12.50, what is
the value of a brick of gold 2.4^™ long, 1.3^" wide, and 0.75""
thick?
41. If I burn 27 sters of wood in the three winter months,
what must be the length of a pile 1 meter wide and | meter
high to last a month, and what will it cost at $ 2.25 a ster ?
MEASURES OF CAPACITY. 181
MEASURES OF CAPACITY.
440. The primary unit of measure for all substances
that can be poured into a dish or box is the liter.
441. A liter is equal in volume to one cubic decimeter.
442. Larger and smaller measures are derived from the
liter in the same way that longer and shorter measures
are derived from the meter, that is, by taking decimal
multiples and subdivisions.
443. Liquid and Dry Measures.
1 milliliter (ml) = 1 cu cm.
10 milliliters — 1 centiliter (cl).
10 centiliters = 1 deciliter (dl).
10 deciliters = 1 liter (1) = 1 cu dm.
10 liters = 1 dekaliter (Dl).
10 dekaliters = 1 hektoliter (HI).
10 hektoliters = 1 kiloliter (Kl). = 1 cu m.
Note. The milliliter is employed in computations, but rarely, if ever,
in actual measurements. Chemists and druggists use cubic centimeters
instead of milliliters.
444. Examples for the Slate.
42. If one hektoliter of kerosene costs $20, what is the
price of a liter ?
43. What must he paid for 2.5 liters of milk each day for a
week, at 7 cents a liter ?
44. From a vessel containing 1 hektoliter of syrup, 25 liters
were drawn out. How many liters remained ?
45. How many hektoliters of oats can he put into a bin that
is 2™ long, 1.3™ wide, and 1.5"* deep?
46. What must he the length of a bin 1 meter wide and
1 meter deep, to contain 4500 liters of grain ?
182 THE METRIC SYSTEM.
WEIGHTS.
445. The primary unit of weight is the gram.
446. A gram is the weight of one cubic centimeter
of pure water at the temperature of 4 degrees centigrade
( = 39.2 degrees Fahrenheit), at which temperature water
has its greatest density.
447. Larger and smaller weights are derived from the
gram by taking decimal multiples and subdivisions.
448. "Weights.
10 milligrams (mg) = 1 centigram (eg).
10 centigrams = 1 decigram (dg).
10 decigrams = 1 gram (g) = wt. of 1 cu cm of water.
10 grams = 1 dekagram (Dg).
10 dekagrams = 1 hektogram (Hg).
• 10 hektograms = 1 kilogram (K) =wt. of 1 cu dm of water.
10 kilograms = 1 myriagram (Mg).
10 myriagrams = 1 quintal (Q).
10 quintals = 1 metric ton (T.) = wt. of 1 cu m of water.
Note I. The gram, kilogram, and metric ton are the only units used in
actual weighing, except by jewellers, druggists, and those who weigh very
small or very expensive articles, like gold or powerful medicines.
Note II. The kilogram is generally called the kilo. The kilo is the
unit of weight for weighing common articles, such as sugar, tea, etc.
Note III. The metric ton is used to weigh very heavy articles, like
hay, coal, etc.
449. Examples for the Slate.
47. At $ 0.60 a kilo for honey, what is the cost of 5.1.5 kilos ?
48. At $ 11 per T. for coal, what will the coal cost to keep
a fire one week if 30 kilos are burnt each day ?
49. What weight of mercury will a vessel contain whoso
capacity is 10 ''" ^, mercury being 13.5 times as heavy as water ?
50. If marble is 2.7 times as heavy as water, what is the wei ght
of a pedestal 1 meter square at each end and 2 meters high ?
EQUIVALENTS.
183
450. Table of Equivalents.
The equivalents here given agree with those that have been established
by Act of Congress for use in legal proceedings and in the interpretation of
contracts.
1 inch = 2.540 centimeters.
1 foot = 3,048 decimeters.
1 yard = 0.9144 meters.
1 rod = 0. 5029 dekameters.
i mile = 1.6093 kilometers.
1 centimeter = 0.3937 inch.
1 decimeter = 0.328 foot.
1 meter = 1.0936 yds. = 39.37 in.
1 dekameter = 1.9884 rods.
1 kilometer = 0.62137 mile.
1 sq. inch = 6.452 sq. centimeters. 1 sq. centimeter = 0.1550 sq. inch;
1 sq. foot = 9.2903 sq. decimeters. 1 sq. decimeter = 0.1076 sq. foot.
1 sq. yard = 0.8361 sq. meter. 1 sq. meter = 1.196 sq. yards.
1 sq. rod = 25.293 sq. meters. 1 ar = 3.954 sq. rods.
1 acre = 0.4047 hektar. 1 hektar = 2.471 acres.
1 sq. mile = 2.590 sq. kilometers. 1 sq. kilometer = 0.3861 sq. mile.
1 cu.inch = 16.387 cu. centimeters. 1 cu. centimeter = 0.0610 cu. inch.
1 cu. foot = 28.317 cu. decimeters. 1 cu. decimeter = 0.0353 cu. foot.
1 cu. yard = 0.7645 cu. meter. 1 cu. meter = 1.308 cu. yards.
1 cord = 3.624 sters.
1 liquid quart = 0.9463 liter.
1 gallon = 0.3785 dekaliters.
1 dry quart = 1.101 liters.
1 peck = 0.881 dekaliter.
1 bushel = 3.524 dekaliters.
\ ounce av. = 28.35 grams.
1 pound av. = 0.4536 kilogram.
1 ster = 0.2759 cord.
1 liter = 1.0567 liquid quarts.
1 dekaliter = 2.6417 gallons.
1 liter = 0.908 dry quart.
1 dekaliter = 1. 135 pecks.
1 hektoliter = 2.8375 bushels.
1 gram = 0.03527 ounce av.
1 kilogram = 2.2046 pounds av.
1 ton (2000 lbs.) = 0.9072 met. ton. 1 metric ton = 1.1023 tons.
1 grain Troy = 0.0648 gram. 1 gram = 15.432 grains Troy.
1 ounce Troy = 31.1035 grams. 1 gram = 0.03215 ounce Troy.
1 pound Troy = 0.3732 kilogram. 1 kilogram = 2.679 pounds Troy.
184 THE METRIC SYSTEM.
451. To change numbers in the metric system to equiva-
lents of the old system : [Use preceding table.]
Examples.
51. In 48 meters how many feet ?
b2. If you travel 50 kilometers in a day, how many miles
do you travel ?
53. Change 18 hektars of land to acres.
54. How many inches long is an insect that is 5.2 centi-
meters long?
66. How many pounds av. are there in 85.6 kilos of salt ?
6Qt. How many gallons are there in 24 kiloliters ?
57. In 20 metric tons how many tons ?
462. To change numbers in the old system to equiva-
lents of the metric system : [Use preceding table.]
Examples.
58. Change 25 miles to kilometers.
59. In 200 acres are how many hektars ?
60. How many liters will a cistern hold that measures on the
inside 5 feet in length, 4 feet in width, and 4 feet in height ?
61. In 3 rods how many meters ?
62. Change 18 qt. 1 pt. to liters.
63. In 1 lb. 7 oz. 18 pwt. of gold, how many grams ?
64. What is the weight of a barrel of flour (196 lbs.) in
kilograms ?
463. Approximate Equivalents.
The equivalents here given are accurate enough for most purposes, and
are easy to remember.
A decimeter
= 4 inches.
A ster
= ^ of a cord.
A meter
_(3 ft. 3f in.,
i or 1-h yards.
A liter
_( 1.06 liquid qt.,
~ (ori^o-ofadryqi
A dekameter
= 2 rods.
A dekaliter
= 1 peck and 1 qt.
A kilometer
= t of a mile.
A hektoliter
= 2§ bushels.
(4 sq. rods,
"" ( or -lo of an acre.
A gram
= 15^ grains.
An ar
A kilogram
= 2\ pounds av.
A hektar
= 2i acres.
A metric ton
= 2200 pounds av
FERGENTAGE. 185
SEOTIOIsr XIV.
PERCENTAGE.
454. rind -^-^ of 500 men. Ans. 45 men.
A number obtained by finding a number of hundredths
of another number is a percentage of that number.
455. The number of which the percentage is found is
the base of that percentage.
In the above example what number is the percentage ? the base ?
456. If a person having $2000 should gain a sum
equal to -^\ of it, how much would he then have ?
2000 x^jj = 200 ; 2000 + 200 = 2200. Ans. $ 2200.
The sum of the base and percentage is the amount.
457. If a person having $ 2000 should lose -^ of it,
how much would he have left ?
2000 X ^0^ = 200 ; 2000 - 200 = 1800. Ans. $ 1800.
The part of the base left after a percentage is taken
away is the remainder.
In the above examples what number is the amount? the remainder?
458. The number of hundredths which the percentage
is of the base is the rate per cent, generally called the
per cent. Thus, j^ of anything is 7 per cent of it.
Note. Per cent is a contraction of the Latin per centum, and means by
the hundred.
459. Oral Exercises.
a. Find 1 per cent of 600 ; 7 per cent ; 20 per cent.
b. Find 10 per cent of 8250 5 5 per cent; 50 per cent.
186 PERCENTAGE.
To express a given Per Cent.
460. The sign % is used for the words per cent. Thus,
5 % means 5 per cent.
461. Any per cent may be expressed as a common frac-
tion, as a decimal, or with the sign for per cent, %. Thus,
1 per cent may be expressed y^^, 0.01, or 1%.
6 per cent "
a
jU, 0.06, or 6%.
7^ per cent "
a
^/h 0.071, or 7i%.
100 per cent "
ii
igg, 1.00, or 100%.
120 per cent "
a
ifg, 1.20, or 120%.
\ per cent "
a
^U O.OOi, or i %.
Exercises.
462. Express the following in the three forms given above :
a. 2 per cent. c. 7f\ per cent. e. 175 per cent.
b. 5 per cent. d. 200 per cent. /. ^ per cent.
463. Express the following as common fractions, and
change them to their smallest terms :
g.5fo. k. 50%. 0.4:%. s. 61%.
h. 10%. 1. 100%. p. 75%. t. 8J%.
i. 20 %. 222. 12^ %. q. 90 %. u. 83J %.
J. 25%. 22. 16|%. r. Sl\%. V. 125%.
To find the Complement of a given Per Cent.
464. What is the difference between 100 % and 25 % ?
The difference between 100 % and any given per cent
less than 100 % is the complement of the given per cent.
465. Oral Exercises.
a. What is the complement of 75%? 40%? 60%? 33j%?
6i%? 15%?
b. What is the complement of 62^%o'? 16%? 37^%? 18%?
87 J%? 72%?
EXAMPLES. 187
To change a Common Fraction to a Per Cent.
466. Illustrative Example. What per cent of a num-
ber is \ of it ?
WRITTEN WORK. Explanation. — Since any number equals
^\ -j^QQ 100% of itself, \ of the number must equal \
— - ^^^ of 100%,or25/o. Ans.^b%.
25 Ans, 25%.
467. Oral Exercises.
a. What per cent of a number is ^ of it? \? ^7 ^^ ?
12^? 3V? 3?V? 1^? *? H V I? tV?
b. What per cent is | ? |? f? f? f? f? f? f? t?
I? ^? §? i? T^CF? /^? ^U? A? A? T^? T^? H?
c. What per cent is ^^? /^? ^^ ? ^^ ? ^j? H? ^(i? /u?
The Base and Rate per cent being given, to find the Per-
centage, Amount, or Remainder.
468. Oral Exercises.
a. What is 7% of $300?
Solution. — 7 per cent of $ 300 is yj^ of $ 300, or $ 21. Ans. $21.
b. What is 20% of 80 trees ? of 300 words ? of 90 ? 60 ?
240?
c. What is 10% of 80 days? 12^% of 80 days? 25%?
40%? 50%? 90%?
d. What is 6% of $ 100 ? of $200 ? of $1.50 ? of $2.50 ?
of $ 500 ? of $ 12.50 ?
e. What is the amount of $ 40 + 5% of $ 40 ? $16 + 25%
of $16?
/. What is the amount of $100 + 7% of $100? $60 +
50% of $60?
g*. What remains of an income of $ 500 after 40% of it is
spent? after 25% is spent? 10%? 15%? 30%? $0%?
188
PERCENTAGE.
469. Illustrative Ex-
ample I. A had S 500. If
by trading he gained a sum
equal to 25 fo of his money,
what was his gain? How
much money did he then
have ?
WRITTEN WORK.
Base, $500
Per cent, 0.25
Percentage, $ 125 A's gain.
$ 625 Amount.
470. Illustrative Ex-
ample II. B had $800. If
by trading he lost 12 % of
his money, what was his
loss ? How much money
did he then have ?
WRITTEN WORK.
Base, 1 800
Per cent, 0.1^
Percentage, $ 96 • B's loss.
$ 704 Reinainder.
In the examples above, A's amount and B's remainder might have
been considered as a percentage of the base and obtained directly thus :
(I.)
Base, $500
Percent, 1.25
$ 625 A's amount.
(11.)
Base, $800
Per cent, 0.88
$ 704 B's remainder.
Explanation. — (I.) The money A had in trade was 100 fo of itself.
Adding to this the 25 % gain, his amount was 125 % or 125 hun-
dredths of $500, equal to $625.
(II.) The money B had in trade was 100 % of itself Having lost
12 fo oi this, he had remaining 88 % or 88 hundredths of $800, equal
to $ 704.
471. From the operations above we derive the following
Rules.
1. To find the percentage : Multiply the base hy the rate
per cent.
2. To find the amount : Add the percentage to the base,
or multiply the base by 1 plus the rate per cent.
3. To find the remainder : Subtract the percentage from
the base, or multiply the base by 1 minus the rate per cent.
EXAMPLES. 189
472. Examples for the Slate.
What is
(1.) 12 % of $ 940 ? (6.) 85 % of l^ pounds ?
(2.) 25 % of $250.60 ? (7.) 75 f^ of 120 % of 486800?
(3.) 62 % of 2000 men ? (8 ) 37^ % of 4000 feet.
(4.) I % of $ 28.80 ? (9.) 100% of 16000 + 75% of 16000 ?
(5.) 120% of 75 days ? (10.) 100% of 10800 - 13% of 10800 ?
11. If 5% of the price of goods is deducted for cash, what
deduction is made from a bill of $ 25.40 ?
IS. If a piece of rubber hose 146 feet long shrinks 10%
when wet, what is its length when wet ?
13. What is 25% of 125% of 75% of 50% of 384 inches ?
14. A farmer paid for shearing 104 sheep 4% of what he
received for the wool ; the fleeces averaged 5 pounds each, and
sold at 40/ a pound. What did he pay for shearing ?
The Percentage and Rate per cent being given, to find the Base.
473. Illustrative Example. 750 bushels is 25 % of
what number ?
Explanation. — Since 25 %
WRITTEN WORK. of the number sought is 750,
^^^ ^ IQQ _ or^cio . Qnnn k,, ^ ^ ^*' *^^ number sought is
25 tE of 7^0, and 100 % of the
number sought, or the number
itself, is 100 times ^ of 750, or 3000. Ans. 3000 bu. Hence the fol-
lowing
Rule.
To find the base when the percentage and rate per cent
are given : Divide the percentage hy the mimerator of the
rate per cent, and multiply the quotient ly 100.
Note. Since 750 divided by 25 and multiplied by 100 equals 750 divided
by 0. 25 (which is the rate per cent), the form of written work given below
may be used instead of that above :
WRITTEN vroRK. — — = 3000. Ans. 3000 bu.
0.25
Here the work is done by dividing the percentage by the rate per cent.
This rule agrees with formula 4, page 192.
190 PERCENTAGE.
474. Examples for the Slate.
(15.) 148 is 3^ of what number ? 10% of what numher ?
(16.) 436 days is 24% of what number? 8% of what
number ?
(17.) $ 31.35 is 5% of what number ? 15% of what number ?
(18.) $300 is 1^% of what number? 5% of what number?
(19.) $220.50 is 105% of what number? 75% of what
number ?
(20.) f is 25% of what number? \% of what number? ^
21. The number of children of school age in a certain town
is 1275 ; if this is 20% of the whole population, what is the
whole population ?
22. I drew out 25% of my deposits in a bank ; of this I
have spent $468.72, which is 9% of what I drew out. What
did I draw out ? What remains in the bank ?
23. If $ 240 is 20% more than some number, what is that
number ?
Note. Since $ 240 is 20 % more than the number sought, it must he
120 % of the numher sought, etc. Hence when the amount is given instead
of the percentage, divide by 100 plus the numerator of the rate per cent, and
multiply by 100.
24. $ 1860 is 25% more than what number ?
25. A sold a horse for $225, which was 5% more than he
paid for it. What did he pay for it ?
26. A grocer sold tea for 115% of its cost, and made 9 cents
per pound. What did it cost a pound ?
27. If $450 is 10% less than some number, what is that
number ?
Note. Since 1 450 is 10 % less than the numher sought, it must he 90 %
of the numher sought, etc. Hence when the remainder is given instead of
the percentage, divide by 100 minus the numerator of the rate per cent, and
multiply by 100.
28. $ 1000 is 4 % less than what number ?
29. Having lost 40% of my money, I have $ 750 left. How
much had I at first ?
EXAMPLES. 191
30. A son is 15 years old, which is 62^^ less than his
father's age. What is his father's age ?
31. The daily attendance upon a school is 558, which is 7%
below the number belonging. What is the number belonging ?
32. After the wages of a workman were reduced 1^7o, he
received % 3.70 a day. What were his wages before they were
reduced ?
33. By assessing a tax of f % on the valuation, a town
raised $ 75000. What was the valuation ?
The Percentage and Base being given, to find the Rate per
cent.
476. Illustrative Example. If a pupil is absent from
school 6 days in a term of 75 days, what per cent of the
time is he absent ?
WRITTEN WORK. Explanation. — If he is absent 6 days in 75
75) 6.00 days, he is absent ^ of the time. -^ changed
— — ^ ^ to hundredths is 0.08, or 8 %. Ans. 8 %.
0.08, or S% Ans. ' ^
476. From the example above may be derived the fol-
lowing
Rule.
To find the rate per cent when the base and percentage
are given : Divide the percentage hy the base, carrying the
division to hundredths.
477. Examples for the Slate.
34. What per cent of % 104 is $ 26 ? is $ 52 ? is $ 18.20 ?
35. What per cent of $ 3 is 12/ ? is $ 3.75 ? is 1/ ?
36. What per cent of a dozen is a score ?
37. Out of 300 words, Charles spelled 280 correctly, Mary
284, Sarah 268, and Dwight 272. What per cent of the words
did each spell correctly ?
192 PERCENTAGE.
38. The surface of the earth contains ahout 144 million
square miles of water, and about 53 million square miles of
land. What per cent of the entire surface of the earth is
water ?
39. From a cask containing 120 gal. of oil, 6 gal. 2 qt. leaked
out. What % was lost ?
478. The operations in percentage, illustrated above,
may be expressed by the following formulas :
1. Percentage = Base x Rate.
2. Amount = Base x (1 + Rate).
3. Remainder = Base x (1 - Rate.)
4. Base = Percentage -i- Rate.
5. Rate = Percentage -^ Base.
For additional examples in percentage, see page 253.
PEOriT AND LOSS.
479. Oral Exercises,
a. How much money is gained by selling goods at 25%
above cost, the cost being 1 8 ? 1 10 ? $ 1.60 ?
b. How much money is lost on goods which cost $ 24, by
selling them at a loss of "25%? 50%? 12^%?
c. At what price must paper which cost $ 2 a ream be sold
to gain 10%? 20%? 25%? 50%? 100%?*
d. At what price must hats which cost 80/ be sold to lose
10%? 5%? 25%"^ 50%? 12^%?
e. What must have been paid a pound for nutmegs if by
selling them at $1.00, there is a gain of 25%? 33^%? 10%?
/. What was the cost of gloves which sold for $ 1.00 a pair
at a loss of 20%? 50%? 33j%? 25%?
g. What per cent is gained if goods costing 10/ a yard are
sold for 11/? 12^? 15^? 20/?
PROFIT AND LOSS. 193
h. What per cent would be lost if goods costing 15 f a yard
were sold for 12/ ? 10/ ? 9/ ?
i. A drover bought cows at $ 25 a head, and paid $ 7 each
to get them to market. If he sold them at I 40 a head, what
per cent did he gain ?
J. What is the cost of goods when a gain of 20/ a yard in
selling is 10% of the cost? 5%? 8%? 50%? 12^%?
k. What was the length of a piece of cloth before shrinking,
if when shrunk 6 inches, it was shortened 1%? 2%? 3%? 4%?
480. The difference between the cost of goods and the
price at which they are sold is a profit or a loss.
481. Profit and loss may be reckoned as percentage, the
cost being taken as the base. Hence the rules of percentage
already illustrated apply to profit and loss.
482. Examples for the Slate.
40. A farm which cost $ 6842 was sold at a gain of 16%.
What was received for it ?
41. A lot of coal was bought for $ 750. For what must it
be sold to gain33L%?
42. If 2000 reams of paper were bought for $ 1500, at what
price per ream must it be sold to gain 40%?
43. A merchant sold a cargo of wheat at 12^% profit, and
gained $ 746.25. What was the cost ?
44. By selling a farm for $ 2760, a man gained on the cost
15%. What was the cost ?
45. What was my property worth 5 years ago, if it has
increased 150%, and is now worth $17500 ?
46. A man sold a picture for % 275 at a loss of 16| %. What
did he pay for it ?
47. If I pay 45/ a pound for tea, and sell it at bQ>f', what
per cent do I gain ?
48. What was the original value of a share in a bridge,
which, selling at an advance of 35%, brings t 780 ?
194 PERCENTAGE.
49. What is the per cent of gain if goods which cost % 750C
sell at a gain of $ 1875 ?
50. A grocer sold 280 barrels of apples for % 708.40. If he
paid $1.40 per barrel for the apples, and 44/ a barrel for
transportation, what per cent did he gain ?
51. A merchant bought carpetings at 85/, 1 1.20, and % 1.50
a yard. At what prices must he sell them to make 20 % profit ?
52. If % 1000 be paid for goods of which one half sells for
$ 640, and the remainder for % 300, what is the per cent of loss ?
53. Bought paper at $1.75 per ream, and sold it at 20 cents
per quire. What per cent did I gain ?
54. A dealer bought 10 gross of combs at $ 12.50 a gross.
If he sold 50 of the combs at 20 cents apiece and the rest at
18 cents apiece, what per cent did he gain ?
55. If 150 beeves are bought at the rate of $ 42.50 each, and
30 at the rate of $ 45.00 each, and the lot is sold for $ 10300,
what per cent is gained ?
COMMISSION.
483. One person is sometimes employed to buy goods
or collect money for another, and is allowed for the service
a percentage on the amount he lays out or collects. This
percentage is called commission.
484. A person employed to transact business for another
is an agent or factor.
485. A person who sends goods to another for sale is a
consignor, and the person to whom the goods are sent is
a consignee.
486. The remainder, after the commission and other
charges of a sale are deducted, is the net proceeds.
487. Commission being a percentage, of which the
money expended or received is the base, the rules of per-
centage already illttstrated apply to commission.
COMMISSION. 195
488. Examples for the Slate.
56. At 1 % commission, what is the commission on the sale
of 4750 pounds of sugar at 7^ cents per pound ?
57. A factor in Mobile purchased for the Pacific Mills
% 90000 worth of cotton at If % commission. What was the
bill for cotton and commission ?
b^. If an auctioneer sells on a commission of 8%, 14 chairs
at $1.25 each, 1 table for $10, and a miscellaneous lot for'
$ 53.70, what is his commission, and what sum will be due the
person for whom he makes the sale ?
59. A lawyer collected 25 % of an account of % 680, charging
5% commission. What was his commission, and what sum
should he pay over ?
60. What is the commission on the sale of 200 yards of
cloth at $4.80 per yard, 6% being paid for selling, and 2\%
for guaranteeing payment ?
61. What are the net proceeds from the sale of 1250 barrels
of flour at $5.50 per barrel, charges for freight and storage
being 40/^ per barrel, commission for selling being 2%, and for
guaranteeing payment 1\%?
Q2. An architect charged $139.75 for plans and for super-
intending the building of a house. If his commission was 2^%,
what was the cost of the house, including his commission ?
63. What is the per cent of commission when an agent
reserves to himself $ 270.00 of $ 9270, sent him to invest ?
.489. Illustrative Example. What part of a remittance
of $328.25 wiU remain to be invested after 1% of the in-
vestment has been deducted ?
Solution. — The reioaittance contains both the investment and the
commission upon it. The commission being 1 % of the investment,
the remittance must be 101% of the investment. Hence $328.25
-M.Ol = $ 325, the investment.
64. I send to my agent at Havana $ 1224. What part of
this sum will remain to invest in sugars, after deducting his
commission of 2% on what he lays out?
196
PERCENTAGE.
65. How many barrels of flour at $ 5 each can a factor pur-
chase with a remittance of $ 2575, after deducting his commis-
sion of 3% ?
(5^. A real estate broker received $ 2593.75 for the purchase
of land.' Reserving 3f % commission on the purchase, what
number of acres of land could he purchase at $ 125 per acre ?
67. If $ 109.65 is sent to an agent to purchase 2000 pounds
of sugar at 5§ cents per pound, and to pay his commission on
the purchase, what % is the commission ?
68. An agent sold 62 lawn-mowers at 1 20 each, and 18 at
$15 each. If, after deducting his commission, he remitted
% 1057 to the manufacturer, what was the % of his commission ?
69. Find the balance of the following account of sales :
?i2 COON, BEO., & CO.
" /^. ^ " ... SS^ §4 44^ ^^^
CHARGES:
Paid Freight and Cartage ^£t?. §p
Commission and Guarantee, 4%---
Philadelphia, April 15, 1877. Balance
490. Written Exercises.
a. Supplying names and dates, write an account of the sales
given in Example 58.
b. In the same way write an account of the sales given in
Example 61.
* Gross weitjht.
t Weight of tubs.
X Net weight.
STOCKS, DIVIDENDS, AND BROKERAGE. 197
STOCKS, DIVIDENDS, MD BEOKEEAaE.
491. An association of individuals formed for the pur-
pose of transacting business is a company or partnership.
492. An association of individuals authorized by law to
transact business under a company name, to hold property
and be liable for debts in that name as an individual would
be, is a corporation.
493. When a corporation is formed for transacting busi-
ness, the persons forming the corporation subscribe what
money is needed for conducting the business. This money
is called capital stock. This stock is divided into shares,
usually of S 100 each.
494. The owners of the stock are stockholders. As
evidence of their ownership, they hold papers called cer-
tificates of stock. The stockholders form the corporation
and elect directors, who are responsible for the business
transacted.
495. A sum levied upon a stockholder to help meet the
expenses or losses of the business is an assessment.
496. The gain upon the capital of a corporation is di-
vided among the stockholders. Gain thus divided is called
a dividend.
Each stockholder's part of the dividend is the same per
cent of his stock that the whole dividend is of the capital.
497. Stocks may be bought and sold like other prop-
erty. Persons who make a business of buying and selling
stocks are called stock- brokers. The commission paid to
a broker is called brokerage.
Note I. When a share of stock will sell at its nominal value, it is at
par; when for more than its nominal value, it is above par, or at a premium ;
when for less than its nominal value, it is below par, or at a discount.
198 PERCENTAGE.
Note II. The market values of stocks are ** quoted " daily in the prin-
cipal newspapers, at given per cents of their values. When a stock is quoted
at 90, it is worth 90% of its face or nominal value ; it is then 10% below
par. When quoted at 105, stock is worth 105% of its face or nominal
value ; it is then 5 % above par.
498. Tlu rides of percentage already illustrated apply
to stocks, dividends, and brokerage.
Examples for the Slate.
499. The following quotations are taken from a daily
paper :
Sales of Stock this day at the Brokers' Board.
70 Chicago, Burlington, & Quincy R. R 103f
150 Burlington & Mo. R. R. in Neb 43^
$ 5000 Atchison, Topeka, & Santa Fe 7's, 1st mortgage . . 88;^
AT AUCTION.
15 Bates Manufacturing Co. 80f
12 Neptune Insurance Co.. 122|
5 Maveric-k Bank 150|
10 N. England Bank 135^
8 American Watch Co 90|
5 Metropolitan Bank 92|
40 Boston & Albany R. R.... 125
36 Nashua & Lowell R. R.... 94^
At the ahove quotations, what is the cost
70. Of 3 shares in the Maverick bank, and 7 in the Metro-
politan ?
71. Of $ 2000 Atchison, Topeka, and Santa Fe 7's ?
72. Of 8 shares in the Bates Manufacturing Co., and 7 in
the Neptune ?
73. Of 75 shares in the Burlington and Missouri, including
I % brokerage on the par value ?
Note. Brokerage is usually \ % , and reckoned on the par value. It is
thus reckoned in this book, unless otherwise specified.
At the above quotations, what is the cost, with brokerage,
74. Of 10 shares Boston and Albany E. E., and 25 Nashua
and Lowell ?
75. Of 15 shares in the Chicago, Burlington, and Quincy
E. E., 5 shares in the American Watch Co., 40 shares in the
New England Bank, and 12 shares in the Neptune Insurance
Co.?
INSURANCE. 199
76. What is the value of 7 shares in a gold company's stock
at 4§ % above par, the original value being 1 200 per share ?
77. A dividend of 3% having been declared by a gas com-
pany, what should a stockholder receive who owns 700 shares,
the par value of each share being % 100 ?
78. A broker sold a lot of stock for $2250, which was 10%
below par. What was the par value ?
79. When stock, originally worth $ 30 per share, sells for
$ 45, at what % above par does it sell ?
600. At present, 1878, paper currency is below par. The
value of gold as compared with it is given from day to day
in the newspapers.
80. When gold is quoted at 102^, how much paper currency
can be bought for $ 200 in gold, no allowance being made for
brokerage ?
81. If the passage to Liverpool is % 125 in gold, and gold is
at 103^, what shall I pay in "greenbacks" for two tickets ?
82. Wishing to send to Ireland 6 pounds sterling, valued
at $4.86 each in gold, what shall I pay for them in "green-
backs," gold being at 102|, and brokerage \%?
83. When gold is quoted at 103, what per cent of a gold
dollar is the value of a 1-dollar bill ?
INSUEANOE.
601. A, owning a house, agrees to pay B a certain per-
centage on its value, B on his part agreeing to pay A the
whole value of the house in case it should within a limited
time be destroyed by fire. Such a contract is a contract of
insurance: and A's house is said to be insured.
502. Insurance is security against loss.
603. Fire insurance is security against loss of build-
ings or goods by fire ; marine insurance is security against
loss of ships or cargoes at sea ; accident insurance against
200 PERCENTAGE.
loss by accident in travelling or otherwise ; health insur-
ance secures a stated allowance during sickness, and life
insurance secures a certain sum to one's heirs or assigns
in case of death.
604. The parties that insure are called insurers or
underwriters.
505. The written contract that binds the parties is the
policy.
506. The sum paid for insurance is the premium.
Note I. When property is insured, the valuation or amount insured is
generally made less than the value of the property.
Note II. Policies are renewed yearly, or at stated periods, and the
premium is paid in advance.
507. The premium is a percentage of which the sum
insured is the base. Hence the rules of percentage already
illustrated apply to insurance.
508. Examples for the Slate.
84. What is the insurance on $1500 worth of goods at f %,
including 1 1 for the policy ?
85. What amount is paid for insurance on § of a store
valued at $15600 at f%, including |1 for the policy?
S6. A merchant insured a cargo from Liverpool worth 2000
pounds at a premium of 1^%. What was the premium, the
pound being valued at $ 4.86 ?
87. A merchant insured $ 3600 worth of goods in one com-
pan}?- at lj% premium, and $2500 worth in another at 1^%
premium. What was the cost, including $ 1 for each policy ?
88. A druggist paid $ 125 for the insurance of a lot of goods
in transportation. If the face of the policy was $ 10000, what
was the rate of insurance ?
89. Jan. 1, 1876, a person took out a health policy, paying
$ 1.50 on the first day of each month. March 2, 1877, he was
disabled by sickness, and received $ 12 a week for 3 weeks.
How much did he receive more than he paid out for premiums ?
TAXES. ' 201
The yearly rates of life insurance depend upon the age of the per-
son when he begins to insure, younger persons paying less per year
than older persons, because they are likely to live longer. Thus A,
being 35 years old, pays $ 109.50 a year for a policy of $ 5000, while
B, who is 40 years old, pays $ 131.50 a year for a policy of the same
amount. The number of years that a person of a given age is likely
to live is called his expectation of life.
90. At the age of 38, I secured a policy upon my life for
$5000, paying the first year $122.55, including $1 for the
policy. What was the premium' paid upon $ 1000 ?
91. Jan. 1, 1868, a man took out a policy on his life for
$3000, in favor of his wife, paying $21.30 on $1000 yearly.
If the man died Feb. 15, 1878, how much did the widow re-
ceive more than had been paid in premiums ?
TAXES.
609. The citizens of a town or city or the members of a
society usually meet the expenses of their government or
society by a sum assessed on their property, their income,
their business, or their persons. Such a sum is called a tax.
610. A tax on the person of a citizen is called a poll
tax. A tax on property is called a property tax. A tax
on annual income is called an income tax.
611. Movable property, such as money, stocks, cattle,
ships, etc., is called personal property. Immovable prop-
erty, as lands, houses, etc., is called real estate.
612. Officers appointed to estimate the value of prop-
erty and to apportion the sum to be raised among the indi-
viduals are called assessors.
613. A property tax is reckoned at a certain per cent
on the estimated value of each person's property, or at a
given number of miUs or cents on $1, |100, or $1000.
202 PERCENTAGE,
614. An income tax is reckoned at a fixed per cent on
the net income of a person after certain deductions have
been made.
515. Illustrative Example. The whole amount to
be raised for State, county, and town taxes in a certain
town is $ 10600. The property of the town is valued at
$ 1250000, and there are 300 polls, each taxed $2. What
is the tax on $ 1 ? What is the tax of E. Stiles, who
has $4000 worth of real estate and $1000 worth of per-
sonal property, and who pays 1 poll tax ?
WRITTEN WORK. Explanation.— I^IOQOO less
110600 the amount of poll taxes
600 leaves % 10000 to be levied on
12510000) 110000 $1250000, which is 8 mills
' ^-^— — - on 11. If E. Stiles pays 8
^•^^^ mills on $ 1, on $ 5000 he will
^^^^ pay 5000 times 8 mills, or
$40. $40. $ 40 plus his poll tax
$40 + $2 = $42 of $2 is $42. Ans. 8 mills
Ans. $ 0.008 ; $ 42. on $ 1 ; $ 42 tax.
616. From the above may be derived the following rules
for assessment of taxes :
I. To find the rate of the property tax : Deduct from the
ivJiole amount to he raised the amount of the poll taxes, and
divide the remainder hy the amount of taxable pt^operty.
II. To find each person's tax : Multiply each person's tax-
able property by the rate, and to the product add his poll
tax.
517. Examples for the Slate.
92. The tax levied by a certain town is % 46800 ; the valua-
tion of the town is $ 3600000, and there are 1800 polls, at $ 1
each. What is the tax on $ 1 ? What is the tax of A, who
has $ 15000, and who pays a poll tax of $ 1 ?
TAXES.
203
93. The valuation of a school district is 1 48000. A tax of
$ 120 is levied for the repairs upon a school-house. What is
the tax on 1 1 ? What is assessed upon a person having
$3500 of taxable property?
94. What is the net tax in a town whose taxable property
is $430000, the rate 12 mills on the dollar, 5% of the tax '
assessed being paid for collecting ?
95. The school-tax of a certain town being $5625, at the
rate of 3| mills on the dollar of taxable property, what is the
taxable property ?
96. The amount of money to be raised by taxes in the town
of H is $212093.20; the taxable property is $ 11522400 j
there are 3350 polls, each taxed $ 1.40. Find the tax on $ 1.
Note. Assessors commonly constrnct a table giving the tax on con-
venient amounts of property at the determined rate.
518. TAX TABLE.
Showing the tax on various sums, at the rate of 18 mills on $ 1
Prop. Tax.
Prop.
Tax.
Prop.
Tax.
Prop.
Tax.
Prop.
Tax.
$1 $0,018
$10 $0.18
$100
$1.80
$1000
$18
$10000
$180
2 0.036
20
0.36
200
3.60
2000
36
20000
360
3 0.054
30
0.54
300
5.40
3000
54
30000
540
4 0.072
40
0.72
400
7.20
4000
72
40000
720
5 0.090
50
0.90
500
9.00
5000
90
50000
900
6 0.108
60
1.08
600
10.80
6000
108
60000
1080
7 0.126
70
1.26
700
12.60
7000
126
70000
1260
8 0.144
80
1.44
800
14.40
8000
144
80000
1440
9 0.162
90
1.62
900
16.20
9000
162
90000
1620
97. Find by the above table the tax on $ 4250.
Note. Find the tax on $ 4000, $ 200, and $ 50 separately, and add the results
Find by the above table the tax
98. Of A on $3000. 102. Of Eon $9068.
99. Of B on $ 2800. 103. Of F on $ Q565.
100. Of C on $7850. 104. Of G- on $ 5687.
101. Of Don $1565. 105. Of Hon $10793.
204 PERCENTAGE.
CUSTOMS OR DUTIES.
619. The expenses of the national government are met
in part by taxes npon imported goods ; these taxes are
called customs or duties.
Note I. A tax called tonnage is laid upon a vessel according to the
weight she is estimated to carry.
Note II. Places are established by government for the collection of cus-
toms or duties ; these places are called ports of entry. Each port of entry
has a custom house, which is in charge of an officer who collects the customs ;
this officer is called the collector of customs.
620. A duty proportioned to the quantity of goods im-
ported, is a specific duty. Thus a duty of 30 f a pound
on yarn is a specific duty. .
Note. In estimating specific duties, an allowance is made (1) for waste,
or impurities, called draft ; (2) for the weight of boxes, casks, etc., called
tare ; (3) for the waste of liquids, called leakage ; (4) for the breaking of
bottles, called breakage.
621. The weight of goods, before allowances are made,
is called gross weight; and the weight, after all allow-
ances are made, is called net weight.
622. A duty proportioned to the cost of goods in the
country from whence they are imported, is an ad valorem
duty. Thus a duty of 15% on iron castings is an ad valorem
duty.
Note. A list of a ship's cargo containing a description of each package
of goods imported, with the price in the currency of the country from
whence imported, must be exhibited to the collector. Such a list is called
an invoice or manifest. When no invoice is received, the value of the goods
is determined by appraisement.
623. Examples for the Slate.
106. What is the duty at 5/ a gallon, on 238 hogsheads of
molasses, 60 gallons in a hogshead ?
107. What is the duty at 30 cents a gallon on 25 barrels of
spirits of turpentine, 32 gallons in a barrel, leakage 2% ?
QUESTIONS FOR REVIEW, ^05
108. At 15%, what is the duty on 75 boxes of tin, 112 lbs.
in each box, invoiced at 7/ a pound, tare 10 pounds a box ?
109. What is the duty at 2^ cents a pound on 13 boxes of
raisins, 24 lbs. in a box, tare 6^ lbs. a box ?
110. At 25%, what is the duty on 100 dozen watch-crystals
invoiced at $ 1.50 a dozen, breakage 3 % ?
111. At 36 %, what is the duty on 200 tons of bar-iron
(2240 lbs. to a ton), invoiced at 2^/ a pound, tare 5 % ?
112. At 3^f a pound and 10% ad valorem, what is the duty
on 7147 lbs. of steel, invoiced at 20 cents a pound, damage
being 8% ?
113. What is the cost at the store of 2556 lbs. of sugar
bought in Havana for $ 148.92, on which is paid 1 35.75 for
freight and carting, and 2^/ a pound for duties, after deduct-
ing 15% for tare?
524. General Review^, No. 4.
114. Change -^ to a per cent.
115. Kepresent 1-^% decimally.
116. Change 106^% to a common fraction in its smallest
terms.
117. What is \ per cent of % 56.49 ?
118. $700 is 140% of what number ?
119. If a percentage is $540 and the rate 3%, what is the
base?
120. 25% of a certain number exceeds 10% of it by $75.
What is that number ?
121. A schooner formerly valued at $ 7500 has depreciated
20%. What is her present value ?
122. Find the cost of goods which sell for $ 120 at a gain of
25%.
123. What per cent is 125 of 1200 ?
124. What commission must be paid for collecting $ 17380
at 3^ per cent ? , _
125. What amount of stock can be bought for $9682, allow-
ing \ per cent brokerage ?
^06 PERCENTAGE
126. What is the value of 20 shares bank stock, at 8^ per
cent discount, tlie par value of each share being 1 150 ?
127. How many shares of stock at 35% advance on a par
value of 1100 can be bought for 11215?*
128. Insurance was effected on the ship Susan, to Cadiz and
back, for % 10000 at 2%, and on her return cargo, worth % 7500,
at 1\%. What was the amount of premium, including $1 for
policy ?
129. What insurance may be covered by a premium of % 28
ati%?
130. What is the insurance premium at ^% on f of a house
worth 1 6000?
131. What is the duty, at 12 '^ a lb. and 10% ad valorem, on
20 bags of wool, each containing 115 lbs., valued at 42 cts. per lb.?
525. Miscellaneous Examples.
132. A man paid for a house $4500, for repairs $157.50,
and then sold it for 18% above the entire cost. What did he
receive for it ?
133. I bought 100 railroad shares at 116|^ and sold them at
120^. What did I gain, the par value being 1 100 ?
134. A mason sold 75 barrels of lime at 27% profit, and
gained 1 40.50. What was the cost per barrel ?
135. A broker bought 48 shares of |50-stockat 9|^% dis-
count and sold them at 2\% premium. How much did he
make ?
136. What amount of current money will be given in ex-
change for % 450 of that which is at 5 % discount ?
137. If I buy 10 shares of stock originally worth $ 100 each
at 18% above par, and sell it at 7% below par, what do I lose?
138. A cotton-mill valued at 1 175000 is insured for | of its
value by two companies, the first taking f of the risk at 0.9%,
the second the remainder at ^ % . What is the total cost of the
premium ?
* See Art. 499, note.
MISCELLANEOUS EXAMPLES. 207
139. A school-house was insured for 115500 at 2f %, $1.50
being paid for the policy and survey. What was the entire
expense for insurance ?
140. If the school-house named above was lost by fire, what
was the net loss to the insurance company ?
141. Suppose I buy 20 shares of stock originall}'^ worth $ 50
a share, at 10% discount, and sell at a premium of 8%, what
do I make ?
142. A merchant sold some iron for 1 278, and made 15 % .
What should he have sold it for to make 25 % ?
143. When 75 shares of stock originally worth $ 100 a share
sell for % 7556.25, at what per cent above par does it sell ?
144. If a company takes an accident risk of $ 8000 at 1^%,
and reinsures one half of it in another company at 1^%, what
will the first company gain if no accident occurs ?
145. After losing 11 % of his apples, a dealer has 133.5 bbls.
of apples left; if they cost him $2.50 per bbl., for what must
they be sold per bbl. that he may lose nothing upon his pur-
chase ?
146. A broker bought insurance stock at 80, and sold it at
112. What per cent did he make upon his investment ?
147. A broker sold 19 shares of stock for $ 1389.85, which
was at 4^ % above par. What was the brokerage at ^ % on the
par value ?
148. A factory is insured at the rate of $ 2 on $ 100. If the
premium, with $ 1 for the policy, is $ 241, and the insurance
is upon f of the value of the property, what is the value of the
property ?
149. When an insurance stock, originally $ 100 per share,
is quoted at 102|, how many shares can be bought for $ 8815,
brokerage \%?
150. If a watch sells for $ 60 at a loss of 22%, what should
it have sold for to gain 30 % ?
151. The capital of a gas company is $ 200000, and the net
earnings are $10746. What rate of dividend can the com-
208 PERCENTAGE.
pany declare, reserving a surplus of $2746 to meet future
demands ?
152. A vessel brought into port 12000 melons. 8 % proved
worthless, 10% of the remainder sold for 18/ apiece, and the
rest for 12^/ apiece. What was received for the whole?
153. At the sale of a piano, 20% was deducted from the
retail price, and 5% of the balance for cash payment. If the
retail prioe was % 750, and the wholesale price 1 475, for what
per cent advance upon the wholesale price was it then sold ?
154. A regiment of 1000 men was reduced to 850 by sick-
ness and battle, the loss by sickness being 50 % as great as by
battle. What was the entire per cent of loss ? what by sick-
ness ? by battle ?
155. I sold 250 lbs. of fish, gaining thereby % 3.75, which
was 42f % of the cost. What was the cost ? For how much
a pound was the fish sold ?
156. A grain dealer's sales amounted in one year to % 75000 ;
f of his receipts were for wheat, on which he made 10 % profit,
and the balance for other grains, on which he made 20% profit
What was the cost of the whole stock ?
157. A broker bought stock, at 8 % premium, and sold it at
9% discount, and lost $510. How many shares originally
worth 1 100 each did he buy ?
158. Two horses were sold for % 144 each ; on one there was
a gain of 20%, and on the other a loss of 20%. How much
was the gain or loss on both ?
159. What is the cost of 5 hhds. of molasses containing in
all 2074 gallons, which was bought in Porto Eico at 42/ a
gallon, and on which is paid $45.75 for freight and carting,
and 5/ a gallon for duty, after deducting 12% for leakage ?
160. A certain corporation wishing to increase its stock
without multiplying the number of its shares, assessed the
stockholders 40% on the par value of their stock, which was
$ 500 per share. What was the par value of the stock after
the assessment was made ?
SIMPLE INTEREST. 209
SEOTIOH" XV.
SIMPLE INTEREST.
626. A had the use of $ 300 of B's money for a year.
At the end of the year he paid B for its use a sum equal to
7 % of the money borrowed. What did he pay for its use ?
Ans. $21.00.
527. Money paid for the use of money is interest.
628. The money for the use of which interest is paid is
the piincipal.
629. The sum of the principal and interest is the amount.
In the above example, what is the interest? the principal? the
amount ?
630. Interest is reckoned at a certain per cent of the
principal. It is, therefore, a percentage of which the base
is the principal.
631. The number of hundredths of the principal taken
in finding the interest for one year is the rate per cent
per annum, usually called the rate.
Note. When a rate of interest is given, it is understood to be the rate
per year, unless a different time is stated.
632. The rate of interest established by law is the legal
rate. Interest at a rate higher than the legal rate is usury.
Note. Debts of all kinds draw interest from the time they become due,,
but not before, unless it is so specified. Interest on interest unpaid when
due is sometimes, though not usually, allowed.
633. Interest on the principal alone is simple interest.
Note. The laws regulating rates of interest are frequently changed,
but the following is a table compiled from official sources in 1877.
210
SIMPLE INTEREST.
534. Table of Legal Rates of Interest.
When two rates are given in this table, any rate not exceeding the highest is allowed,
if agreed upon in writing.
States.
Rate %.
States.
Rate %.
States.
Rate %.
States.
Rate%.
Ala
8
Ill
6
Mo
6
10
S. C
7
Any
Ark
6
Ind
6
10
Montana..
10
Tenn. .
6
10
Arizona.
12
Anv
Iowa . . .
6
10
N. H
6
Texas.
8
10
Oal
10
Kan. . . .
7
12
N. J
7
Utah..
10
Any
Conn
(t
Ky
La
6
«
N. Y
«
Vt. . . .
6
Colo
10
Any
6
N. C
6
8
Va. . . .
6
Dak....
12
Any
Maine . .
6
Any
Neb
10
12
W.Va.
6
Del
6
Md
6
Nev
10
Any
W. T..
10
Any
D. C. . . .
6
10
Mass....
6
Any
Ohio
6
8
Wis. . .
7
10
Fla
8
Any
Mich....
7
10
Or
10
12
Wy. ..
12
Any
Ga
7
12
Minn. . .
7
12
Penn
6
Idaho. . .
10
Miss....
6
10
6
Any
Note I. In this book, when no rate is mentioned or implied, 6% is
understood.
Note II. In reckoning interest, it is customary to consider a year to be
12 mouths, and a month 30 days.
635. In reckoning the months and days between two
dates, take the entire calendar months as months, and then
the exact number of days remaining. (See Art. 371.)
Note. In computing interest for short periods of time, it is customary
to take the exact number of days.
536. Oral Exercises.
What is the interest
a. Of $ 100 for 1 year at 7 % ? for 2 years at 3 % ?
b. Of 1300 for 2 years at 6%? at 8%? at 11%? at 12%?
c. Of 1400 for 3^ years at 4%? at 10%? at 7%? at 8%?
d. Of $40 for 3 years at 10%? at 5%? at 7%? at 6%?
e. What part of a year's interest is the interest on any sum
of money for 6 mo. ? 2 mo. ? 3 mo. ? 4 mo. ? 1 mo. ?
/. At 5 %, what is the interest of 1 600 for 1 year ? for 6 mo. ?
3mo.? 4mo. ? 2 mo. ?
g. At 9%, what is the interest of $100 for 1 year? for
1 mo. or 30 days ? for 6 days ? for 1 day ? for 5 days ?
h. What is the amount of $ 100 for 4 years 6 months at 8 % ?
i. What is the amount of $100 for 1 year 4 months at 5%?
J. What is the amount of $ 200 for 3 years 3 months at 10 % ?
METHODS OF COMPUTING INTEREST. 211
METHODS OF COMPUTING INTEREST.
To THE Teacher. Two methods of computing interest are given in
the following pages ; but the teacher is advised to have pupils use but one.
The method by aliquot parts will be found on page 308 of the Appendix.
GENERAL METHOD.
537. Illustrative Example. Find the interest of $840
for 4y. 3mo..5d. at 8%.
WRITTEN WORK Explanation. — The interest of $ 840
$840x0.08x4 = $268.80 ^^ 1 year at 8% is $840x0.08. The
interest tor 4 years is 4 times as much,
*»Ia nns an or $268.80.
1^0x0.08x95 ^ 17.73 3 mo. 5 d. equal 95 days. The
^^^ Ans. $286.53 interest of $840 for 1 year being
$840 X 0.08, the interest for I day is ^^^
of this (Art. 534, Note II.), and for 95 days it is 95 times as much, or
$ 17.73, which, added to $268.80, makes $286.53, the entire interest.
638. From the example above may be derived the fol-
lowing „ ,
® Rule.
1. To find the interest at any per cent for any number
of years : Multiply the principal hy the rate for 1 year,
and that product hy the number of years.
2. To find the interest for months and days : Change the
months to days (Art. 635) and take as many 360ths of a
years interest as there are days in the given time.
639. This rule may be expressed by the formula :
Interest = Principal x Rate x Number of years.
640. Examples for the Slate.
1. What is the interest of 1 720 for 3 y. 7 mo. 6 d. at 8% ?
2. Of $472.30 for 2 y. 2 mo. 12 d. at 4% ?
3. Of $ 400.50 for 3 y. 10 mo. 24 d. at 10% ?
4. Of $84.80 for 5 y. 3 mo. 20 d. at 6% ?
5. Of $ 116.20 for 2 y. 10 mo. 10 d. at 7% ?
212 SIMPLE INTEREST.
SIX PER CENT METHOD.
641. Oral Exercises.
a. At 6 %, what part of the principal is the interest for 1
vear ? for 2 months ?
b. If the interest for 2 months is 0.01 of the principal, what
part of the principal is the interest for any number of months ?
Ans. One half as many hundredths of the principal as there
are months.
c. At 6%, what is the interest of $600 for 2 mo. ? for
4mo. ? 6mo. ? 8mo. ? 10 mo. ? 5 mo. ? 7mo. ? 15mo. ?
d. If the interest for 2 months, or 60 days, is 0.01 of the
principal, what part of the principal is the interest for 6 days ?
e. If the interest for 6 days is 0.001 of the principal, what
part of the principal is the interest for any number of days ?
Ans. One sixth as many thousandths of the principal as there
are days.
/. At 6% what is the interest of % 500 for 6 days ? 1 day ?
2 days ? 3 days ? 12 days ? 18 days ? 24 days ?
642. Illustrative Example. What is the interest of
$480 for 1 y. 3 mo. 7d. at 6 % ? at 7 % ? What is the
amount at 7 % ?
WRITTEN WORK. Explanation. — 1 y. 3 mo. equals
A Aor. A QrrK 15 mo. The interest for 15 mo. at
07^1 0011 ^^^ '^ ^-^^i' °'' ^'^^^ ""^ *^^ P'"'"'''''
^:!:XH^ __^ pal. The interest for 7 days is 0.001^
2880 0.076^ of the principal. Hence the interest
3360 for ly. 3 mo. 7d. at 6 % is 0.076^
80 of the principal. 0.076^ of the prin-
6) $36:560 Int. at 6 % . ^^P^^ '' ^/f^-^^' ^ ^ ^ ^ ^
^ To find the mterest at 7 %, we add
1 to the interest at 6 % | of itself, and
$ 42.65 Int. at 7%. have for the sum $42.65.
480. $480 + $42.65 =$522.65, the
$522.65 Amt. at 7%. ^°^°"^* ^* '^^^•
Ans. $36.66 ; $42.65 ; $522.65.
SIX PER CENT METHOD. 213
643. From the foregoing may be derived the following
Rule.
1. To compute interest at 6 % : Take 6 times as many
hundredths as there are years, 1 half as many hundredths as
there are months, and \ as many thousandths as there are
iays, and hy this decimal multiply the principal.
2. To find the interest at any rate other than 6 % : Hav-
ing found the interest at 6%, increase or diminish that in-
terest hy adding or subtracting such part of itself as will
give the interest at the required rate.
3. To find the amount : Add the principal to the interest.
Note I. Observe that l%=iof6%; 2% -J of 6%; 3%=iof6%;
4%= 6% -2%; 5%=6%-l%; 7%=6%+l%; 7i%=6% + (i of
6%), etc.
Note II. It will often be more convenient to increase or diminish the
principal before taking the interest instead of increasing or diminishing the
interest. Thus, in the foregoing illustrative example we might add to $480
J of itself and then take 6 % interest on $ 560. This would be the samei as
the interest at 7 % on $ 480, which is $ 42. 65.
544. Examples for the Slate.*
Find the interest on $ 1 at 6 %
6. For 1 y. 3 mo. 6 d. 10. For 1 y. 1 mo. 10 d.
7. For 4y. 16 d. ^ 11. For 1 y. 8 mo.
8. For 4 mo. 5 d. * 12. For 16 y. 8 mo.
9. For Imo. 25 d. 13. For 7y. 10 mo. 18 d.
At 6% what is the interest
14. Of $300 for 2 y. 5 mo. ?
15. Of 136.18 for 3 y. 7d.?
16. Of % 872.32 for 6 y. 2 mo. 16 d. ?
17. Of $ 130.50 for 2 y. 9 mo. 13 d. ?
18. Of $ 800.20 for 3 y. 4 mo. 12 d. ?
19. Of $ 1000 for 3 y. 10 mo. 2d.?
20. Of % 25.50 for 1 y. 1 mo. 1 d. ?
21. Of $ 400.37 for 2 y. 5 mo. 2Q d. ?
214 SIMPLE INTEREST,
What is the interest
22. Of 1 837.36 for 3 y. 2 mo. at 7 % ?
23. Of 1 187.50 for 2 mo. 12 d. at 10 % ?
24. Of 1 1000 from Nov. 11, 1874, to Aug. 15, 1880, at 7%?
25. Of 1130.16 from Feb. 7, 1874, to Dec. 1, 1878, at 8%?
26. Of $ 19.80 from Oct. 15, 1875, to April 19, 1876, at 5%?
27. Of 1 62.50 from Aug. 3, 1874, to April 11, 1875, at 1\ % ?
Find the amount
28. Of $540 for 3y. 6mo. at 6%.
29. Of 1 495.60 for 2 y. 2 mo. at 12 % .
30. Of 1 830 for 5 y. 4 mo. at 8%.
31. Of|110.10for3y. 5mo. at9%.
32. Of % 896 for 2 y. 6 mo. 15 d. at 6§%.
33. Of $416for3y. 16d. at7%.
34. Of $ 720 for 3 y. 9 mo. 19 d. at 8 % .
35. A note for 1 150, dated July 5, 1872, was paid Mar. 17,
1874, with interest at 6 % . What was the amount ?
36. I gave my note to a person, Jan. 1, 1877, for % 387.20,
with interest at 7% from date. What should I pay to dis-
charge this note Oct. 20, 1877 ?
37. Chase and Fowle bought goods to the following amounts,
agreeing to pay 7% interest from the date of purchase : July 8,
1876, % 470 ; July 28, $ 235 ; Oct. 2, % 206. What will be the
amount due Jan. 1, 1877 ?
Short Method for Days ; Application of 6 per cent Method.
545. Illustrative Example. What is the interest of
f^ 126.80 for 93 days at 6%?
Explanation. — The interest at
6% for 60 days, or 2 months, is
0.01 of the principal. 0.01 of
$ 126.80 may be expressed by mov-
ing the decimal point two places
towards the left; this gives $ 1.268-
The interest for 1 month, or 30
WRITTEN
WORK.
$126.80
1.268 Int.
for 60 d.
0.634
u
" 30 d.
0.063
a
" 3d.
ins. 11.965
a
" 93 d.
ACCURATE INTEREST. 215
days, is \ of 1 1.268, or $0,634, and for 3 days it is ^ of $0,634, or
$ 0.063. Adding these interests, $ 1.268 + $0,634 + $ 0.063 = $ 1.965.
Ans. $ 1.97.
646. From the foregoing may be derived the following
Rule.
1. Find tlie interest for 60 days at 6% hy taking 0.01 of
the principal.
2. For other periods of time, Take convenient multiples
or aliquot parts of the interest for 60 days.
547. Examples for the Slate.
Find the interest of
(38.) $ 300 for 93 d. at 6 % . (40.) 1 1000 for 33 d. at 10 % .
(39.) 1 250 for 95 d. at 7 % . (41.) 1 280 for 127 d. at 12 % .
(42.) % 270.80 from Aug. 20 to Oct. 30 at 8 %. [Exact days.]
(43.) $416.60 from Nov. 12, 1875, to Feb. 5, 1876, at 5%.
(44.) $1560.50 from Mar. 27, 1875, to June 7, 1875, at 9%.
(45.) $6000 from Nov. 15, 1875, to March 7, 1876, at 6%.
ACCURATE INTEREST.
KoTE. The above methods of performing examples in interest heinf^
based upon the supposition that a year equals 12 months of 30 days each,
or 360 days, though in common use, are not exact. The government of
the United States and that of Great Britain pay accurate interest.
548. To obtain accurate interest for months and days :
Find the exact number of days between the given dates, and
take as many S65ths of a years interest as there are days.
649. Examples for the Slate.
46. Find the accurate interest of $2000 from Mar. 1 to
Aug. 10 at 5%.
What is the accurate interest
47. Of 1 700 from May 7 to July 9 at 7^%?
48. Of $ 20000 from April 4 to July 7 at 7%?
49. Of $1000 from Nov. 15, 1875, to April 1, 1876, at 5%?
For additional examples in interest, see page 253.
216 SIMPLE INTEREST.
PARTIAL PAYMENTS.
650. [demand note.]
'^^ai/ed. ^/ea^o?i, S^oui, (^dc?ic/iec/ (Seventy ^ ^a//ai4^,
07i aemanc/, wcm' cn^ieat at u ^e^ cent.
S^mcccTTz <^un,t.
651. The above is the written promise of one person,
Flint, to pay another person, Gleason, or any one to
whom Gleason may order it paid, a certain sum of money,
$ 470.60, for value received. Such a promise is called a
promissory note, or simply a note.
552. The sum named in the note (as $470.60 above)
is the face of the note.
To discharge the interest and in part pay the above note
a payment of $ 94.13 was made Nov. 1, 1874. What balance
then remained due ? Ans. % 400.
Suppose the above balance of $400 to remain on interest
from Nov. 1, 1874, to Nov. 1, 1875, when a payment of I 224
was made, what sum then remained due ? Ans. $ 200.
653. Payments in part of a note or other debt, as the
payments described above, are partial payments.
554. A record of the sum paid, with the date of the
payment, is made upon the back of the note ; such a record
is an indorsement.
The method adopted by the Supreme Court of the United States,
and by most of the States, for computing interest in case of partial pay-
ments, requires (1.) That a paymeni be applied first to discharge accrued
interest, and then, if the payment is large enough, to reduce the principal.
(2.) That no unpaid interest be added to the principal to draw interest.
PARTIAL PAYMENTS. 217
655. Illustrative Example. A note for $600, dated
June 20, 1874, had payments indorsed upon it as follows :
Oct. 2, 1874, $110.20. May 23, 1876, $125.25.
Feb. 29, 1876, 24.00. Dec. 11, 1876, 113.20.
Find the balance due Jan. 21, 1877 ; interest 6%.
WRITTEN WORK.
Principal from June 20, 1874 .... $600.00
Interest to Oct. 2, 1874 (3 mo. 12 d.) . . . 10.20
Amount 610.20
First payment, Oct. 2, 1874 . . . . 11 0.20
New principal from Oct. 2, 1874 . . . . 500.00
Interest on $500 to Feb. 9, 1876 (1 y. 4 mo. 27 d.)
f 42.25.
Second payment, $ 24 will not discharge interest.
Interest on $500 from Oct. 2, 1874, to May 23,
1876 (1 y. 7 mo. 21 d.) 49.25
Amount . 549.25
Second and third payments, $24 +$125.25 . . 149.25
New principal from May 23, 1876 . . . 400.00
Interest to Dec. 11, 1876 (6 mo. 18 d.) . . . 13.20
Amount . 413.20
Fourth payment 113.20
New principal from Dec. 11, 1876 . . . 300.00
Interest to Jan. 21, 1877 (1 mo. 10 d.) . . . 2.00
Balance due Jan. 21, 1877 . . . (^tis.) $302.00
&6Q. The above is in accordance with
The United States Rule for Partial Payments.
1. Find the amount of the principal to the time when
the payment or the sum of the payments equals or exceeds
the interest; take from this amount a sum equal to the pay-
ment or payments.
2. With the remainder as a new principal, proceed as
hefore, to the time of settlement.
218 SIMPLE INTEREST.
557. Examples for the Slate.
50. Oct. 12, 1873, I gave my note on demand, with interest
at 6%, for 1480; Feb. 6, 1874, I paid 1120. What remained
due Aug. 24, 1874 ?
51. I held a note for $500, which bore interest at 6% from
May 10, 1869 ; Sept. 16, 1870, 1 received 1 140 ; July 28, 1872,
I received % 50. What remained due Sept. 4, 1872 ?
52. June 15, 1873, George Kich borrowed of John Jones
1 2000, and gave his note for the same, with interest at 8 % .
Aug. 27, 1874, a payment of 1 1450 was made, and a new note
given for the balance. For what sum was the new note given ?
Write the new note in proper form, dating it at Boston.
53. A note for 1 1000, dated Oct. 5, 1874, was indorsed as
follows: Dec. 8, 1874, $125; May 12, 1875, $316; Sept. 2,
1875, $ 417. What balance was due March 9, 1876 ; interest
6% ?
54. What balance will be due July 1, 1881, on a note of
$935 on interest from Sept. 1, 1875, and indorsed $125.75,
Jan. 15, 1876; $250, March 25, 1877; $300, May 10, 1877;
interest being 6%.
(55.) $425. l^^VfYoRK, July 13, 1869.
Six months after date I promise to pay A. Hyde «& Co.
Four Hundred Twenty-five Dollars, with interest at 6%; value
received. Stewart E. French.
Indorsements: Aug. 9, 1871, $50; Nov. 17, 1872, $150.
What was due July 12, 1873 ?
(pQ.) $800. ^T.-Lom^, July 15,1870.
For value received. We jointly and severally promise to pay
H. Hooker, or order. Eight Hundred Dollars on demand, with
interest at 7 %. James Holland.
Henry Holland.
Indorsements: April 18, 1871, $100; Dec. 31, 1872, $70;
June 14, 1874, $62.50.
What was due J uly 14, 1875 ?
PARTIAL PAYMENTS. 219
658. Illustrative Example.
Indorsements.: Aug. 16, 1876, $200 ; Oct. 8, 1876, $480 ; Feb. 20,
1877, $ 49.92. What balance was due July 1, 1877 ?
559. When partial payments are made upon notes on
interest for short periods of time, as upon the above, inter-
est is often computed by the following, called
The Merchants' Rule.
1. Compute interest on the principal from the time it begins to draw
interest to the time of settlement, and also on each payment from
the time it is made
WRITTEN WORK OP EXAMPLE ABOVE.
Principal on interest from July 7, '76 $800.00 ^ ^^^ *^^ ^^ ^^^'
Interest to July 1, '77 (11 mo. 24 d.) . 47.20 '^^'^^•
Amount of note .... 847.20 2. Tale the differ -
Payment, Aug. 16, '76 . 200.00 ence between the sum
Interest to July 1, '77 (lo mo. isd.) 10.50 of the principal and
Payment, Oct. 8, '76 . . 480.00 its interest and the
Interest to July 1, '77 (Smo. 23 d.) 21.04 ,^^ of the payments
Payment, Feb. 20, '77 . 49.92 ^ ^
Interest to July 1, '77 (4mo. iid.) 1.09
762.55
and their interests;
this difference will
be the balance due.
Balance due .... Ans. $ 84.65
560. Examples for the Slate.
(57.) I lOOOOy^^V Washington, Oct. 3, 1875.
In two months from date I promise to pay to the order of
Cyrus Parsons, at Suffolk Bank, Boston, Ten Thousand ^^^
Dollars, with interest at 6%; value received, j n^
Indorsements: Nov. 5, 1875, $672.41; Nov. 15, 1875, $7682.42;
Nov. 16, 1875, $437.98; Nov. 19, 1875, $833.42.
What was the balance due on the above when it became due ?
220 SIMPLE WTEREST.
(58.) $ 1200. Baltimore, A'pril 1, 1875.
One year from date, for value received, I promise to pay
B. F. Bryant, or order, Twelve Hundred Dollars, with interest
at 7%. Isaac C. Fellows.
Indorsements: April 12, 1875, $161.08; July 19, 1875, $224.14;
July 28, 1875, $17.90; Jan. 29, 1876, $100.25.
What was due on the above note April 1, 1876 ?
For annual interest, also for Vermont, New Hampshire, and Connecticut
rules for partial payments, with annual interest, see Appendix, pages 309
and 310.
PROBLEMS IN INTEREST.
To find the Time, having the Interest, Principal, and Rate given.
661. Illustrative Example. In what time will $480
on interest at 5% yield $ 36 of interest ?
WRITTEN WORK. Explaiiation. — The interest of $ 480 for 1
$ 480 X 0.05 = $ 24. year at 5 % is $ 24.
$ 36 -^ 1 24 = 1^. Since $ 480 at 5 % yields $24 of interest in
li- yr. = 1 yr. 6 mo. ^ ^^^^' *° ^^^^^ ^ ^^ ^^ ^^^^^ require as many
years as there are times $ 24 in $ 36, which is
1^. Ans. 1 yr. 6 mo.
662. From the above may be derived the following
Rule.
To find the time, having the principal, interest, and rate
given : Divide the given interest hy the interest of the prin-
cipal at the given rate for 1 year ; the quotient will he the
number of years.
This rule may be expressed by the formula :
Interest
1. Number of yeaxs
Principal x Rate
Note. It will often be found more convenient to divide by the interest
for 1 month or 1 day, in which case the answer will be in months or in days.
PROBLEMS IN INTEREST. 221
563. Examples for the Slate.
In what time will
(59.) $ 400 gain 1 20 at 6 % ? (62.) $ 3000 gain % 205 at 5 % ?
(60.) 1 500 gain $ 60 at 4 % ? (63.) $ 408 gain $ 170 at 7^ % ?
(61.) 1 640 gain $ 67.20 at 7 % ? (64.) $450 gain $192.30 at 8 % ?
Q6. In what time will $280 amount to $301 at 5%?
Note. To find interest, subtract $ 280 from % 301.
&Q. How long must a note of 1 7500 run to amount to $ 7800
at8%?
67. In what time will $500 double itseK at 1%? at 2%?
at3%? at6%? at 10%?
To find the Rate, having the Interest, Principal, and Time
given.
664. Illustrative Example. The interest on $200
for 10 mo. 24 d. was $ 14.40 ; what was the rate % ?
WRITTEN WORK. Explanation. — The interest of $ 200 for
$ 200 X 0.009 = $ 1.80. 10 i»o- 24 d. at 1 % is $ 1.80.
^ 14 40 — ^ 1 80 = 8 Since the interest at 1 % on $ 200 for
~J 10 mo. 24 d. is $1.80, to yield $14.40 the
rate must he as many times 1 % as there
are times $ 1.80 in $ 14.40, which is 8. Ans. 8%.
665. From the above may be derived the following
Rule.
To find the rate, having the interest, principal, and time
given: Divide the given interest hy the interest of the prin-
cipal for the given time at 1% ; the quotient will he the
number of the per cent.
The above rule may be expressed by the formula :
« „ , Interest
2. Rate = •
Principal x Number of years
666. Examples for the Slate.
68. At what rate % will $ 360 gain $ 40.80 in 1 y. 5 mo. ?
69. At what rate % will $ 100 gain $ 33^ in 12 y. 6 mo. ?
222 SIMPLE INTEREST.
At what rate %
70. Will $250 gain $3.75 in 4 mo. ?
71. Will 1 25 gain $ 7.87^ in 3 y. 6 mo. ?
72. Will 1 100 gain $ 25 in 7i y.?
73. The amount of $75 for 2y. 6 mo. was $78.75; what
was the rate % ?
Note. To find the interest, deduct $ 75 from % 78.75.
74. A note of $ 50 on interest from Feb. 29, 1872, to Feb.
28, 1874, amounted to $ 55.25 ; what was the rate % ?
75. When a note of $1000 amounts to $1058.33 J in 7 mo.,
what is the rate % ?
To find the Principal, having the Interest or Amount, the
Time, and the Rate given.
567. Illustrative Example I. What principal on in-
terest at 6 % for 3 y. 4 mo. will yield $ 80 of interest ?
WRITTEN WORK. Explanation. — The interest of $1 at
1 X 0.06 X 3^ - 0.20 6 % for 3 y. 4 mo. is 1 0.20.
$ 80.00 H- $ 0.20 = 400 ^i^^^ 1 ^oll^^ of principal at 6 % in 3 y.
A ^400 4 mo. yields 20 cents of interest, to yield
$ 80 of interest will require as many dol-
lars of principal as there are times 20 cents in $ 80, which is 400.
Ans. $ 400.
668. Illustrative Example II. What principal on
interest at 10% for 2 y. 6 mo. will amount to $478.50 ?
WRITTEN WORK. Explanation. — The interest of $1
1 X 0.10 X 2^ = 0.25 for 2 y. 6 mo. at 10 % is 1 0.25, and the
$1.25) $478.50 (382.8 amount of $1 is $1.25.
375 Since $ 1 of principal at 10 % in 2 y.
6 mo. amounts to $ 1.25, to amount to
$478.50 will require as many dollars
of principal as there are times $ 1.25
350 in $478.50, which is 382.8.
250 Ans. $ 382.80
1035
1000
1000 etc.
PROBLEMS IN INTEREST. 223
669. From the foregoing may be derived the following
Rules.
I. To find the principal, having the interest, the time,
and the rate given: Divide the given interest hy the interest
of $1 for the given time and rate.
II. To find the principal, having the amount, the time,
and the rate given: Divide the given amount hy the amount
of S 1 for the given time and rate.
The above rules may be expressed by the formulas :
Interest
3. Principal
4. Principal =
Rate X Number of years
Amount
1 + Rate X Number of years
670. Examples for the Slate.
What principal on interest
76. At 6% will gain 1 15 in 2 years ?
77. At 5% will gain 1 20 in 4 years ?
78. At 3% will gain $ 76.50 in 2 y. 6 mo. ?
79. At 4% will gain 1 1.705 in 7 mo. 15 d. ?
80. At 6% will gain $4,128 in 11 mo. 14 d. ?
Note. 4.128-h 0.057 J (both changed to thhds of thousandths) equals
12.384 -^ 0.172.
81. At 2% a month will gain $ 24 in 60 days ?
. 82. At 6% will amount to $870 in 7 y. 6 mo. ?
83. At 5% will amount to $2072.25 in 30 d. ?
84. At 1 % a month will amount to $ 412 in 90 d. ?
85. What sum on interest 3^yrs. at 5^7? vrill amount to
1100?
86. What sum put upon interest Jan. 1, 1875, at 7% will
amount to $ 343.75, Feb. 1, 1877 ?
87. What principal put upon interest to-day at 5% will
amount to $ 206.25 in 7 mo. 15 d. ?
224 SIMPLE INTEREST.
PRESENT WORTH AISTD DISCOUNT.
671. Illustrative Example. If one person owes
another $ 214, to be paid 1 year hence, without interest,
what sum should be paid to-day to discharge the debt, the
current rate of interest being 7 per cent ?
WRITTEN WORK. Explanation. — In justice to both parties,
1.07) 214.00 (200 ^^^^ ^ ^^^ should be paid to-day as would,
2\^ if put at interest at 7 %, in 1 year amount to
$214.
A S200 Since $1 in 1 year at 7 % amounts to
$ 1.07, it would require as many dollars to
amount to $214 as there are times $1.07 in
$ 214, which is 200. Ans. $ 200.
672. A sum which will without loss to either party
discharge a debt at a given time before the debt is due
is the present worth of the debt.
673. A sum deducted from a debt or from a price is
discount. The difference between the face of a debt and
the present worth is the true discount.
What is the present worth in the example above ? What is the
true discount 1
Note. — It will be seen that the present worth is the principal, the true
discount is the interest, and the sum due at a future time is the amount.
This subject is then an application of that illustrated in Art. 568.
674. From the illustrative example above may be de-
rived the following
Rules.
I. To find the present worth : Divide the given debt hy
the amount of $1 for the given time and rate.
II. To find the true discount: Subtract the present worth
from the face of the debt.
PRESENT WORTH AND DISCOUNT. 225
676. Examples for the Slate.
The current rate of interest being Q%, what is the present
worth and what is the true discount
88. Of 1 27.50, due 1 year 8 months hence ?
89. Of 1 100.96, due 8 months hence ?
90. Of $200, due in 3 months?
91. Of $ 175.80, due in 9 months 20 days ?
92. Of 1 661.371, due in 3 months 15 days ?
93. What is the present worth and true discount of $ 1609.30,
due in 10 months 24 days, current rate 5 % ?
94. If a bill of % 600 is payable in 3 months after May 1,
without interest, what sum will <^ischarge it June 1, current
rate of interest being 10 % ?
95. Macomber & Earle sold goods to the amount of $ 138.48
on 6 months' credit. For how much ready money could they
afford to sell the same goods, the use of the money being worth
to them 2 % a month ?
96. A merchant bought goods to the amount of 1 1574, one
half payable in 3 months and the rest in 6 months, without
interest. What sum would pay the debt at the time of pur-
chase, rate 7%?
97. A dealer bought $1500 worth of grain on 6 months'
credit, and sold it immediately for 10 % advance. If with the
proceeds he paid the present worth of the $1500, rate 8%,
what sum remained ?
98. A bookseller bought $ 240 worth of books at a discount
of 33 J % on the amount of his bill, and 5% on the balance for
present payment. He then sold the books on 3 months' time for
the price at which they were billed to him. Money being worth
7%, and the purchaser discounting his own bill by true present
worth at the time of purchase, what was the bookseller's gain ?
For other examples in present worth, see page 253.
226 SIMPLE INTEREST.
BANK DISCOUNT.
576. Holding a note against James Peak for $500,
dated April 1, and given for 4 months, without interest, and
desiring the money April 1, I transfer the note to a bank,
and allowing the bank to take interest on the sum named in
the note for 4 months, and 3 days ($ 10.25), receive from
the bank the balance ($ 489.75) in cash. The note is then
said to be discounted.
The sum named in the note is called the face of the
note.
Before transferring the above note, I endorsed it by writ-
ing my name across the back and thus became responsible
for the payment of the note when due.
677. The three days for which interest is taken beyond
the specified time for paying a note are called days of grace.
Note I. A note is nominally due at the expiration of the time specified
in the note, but it is not legally due till the expiration of the 3 days of
grace. A note is said to mature when it is legally due.
678. The interest upon the face of a note from the time
it is discounted to the time it matures is hank dit: count.
What is the bank discount in the example given ?
679. The face of a note, less the discount, is the pro-
ceeds, avails, or cash value of the note.
What are the proceeds in the example given ?
Note II. The time when a note is nominally and when legally due is
usually written with a line between the dates ; thus, Aug'ist II A.
Note III. When a note is given for months, calendar months are under-
stood, and the note is nominally due on the day corresponding with its
date ; if the month in which it falls due has no corresponding day it is due
on the last day of that month.
Note IV. Notes maturing on Sunday or on a legal holiday must be
paid on the business day next preceding.
Note V. In computing bank discount, the more general custom is to
reckon the time in days ; hence, in the examples in bank discount which
follow, the time is so reckoned, when dates are given.
BANK DISCOUNT. 227
680. Illustrative Example. What is the bank dis-
count of a note for $400, payable in 90 days, dis.count at
7%? What are the proceeds ?
WRITTEN WORK. Explanation. — Bank discount is in-
1 400 terest for the specified time and 3 days
0.0155 ^^^g"^"^-
The interest of 1 400 for 93 days at
6) 6.2000 y^^ .g ^7 23^ ^^^ discount. |400 less
1.0333 $7.23 equals $392.77, the proceeds of
$7^333 ^^e ^ote.
% 400 - S 7 23 = S 392 77 ^^^' ^ ^'^^ discount; % 392.77 proceeds.
681. From the above may be derived the following
Rules.
I. To find bank discount on a note due at a future
time, without interest : Compute interest on the face of the
note from the time of discount to maturity {including the
three days of grace).
II. To find the proceeds of the note : Subtract the dis-
count from the face of the note.
Note. When a note drawing interest is discounted, the discount is
computed upon the amount of the note at the time of its maturity.
682. Examples for the Slate.
99. What is the hank discount of a note for $ 750, payable
in 30 days, discount 6 % ? What are the avails ?
Find the hank discount and proceeds of a note
100. For $1000, payable in 90 d., discount 7%.
101. For $300, payable in 4 mo., discount 8%.
102. For $ 700, dated Dec. 10, payable in 69 days, and dis-
counted at date at 10%.
103. For $ 500, dated Aug. 20, payable in 3 mo., and dis-
counted at date at 7^%.
228 SIMPLE INTEREST.
Find the bank discount and proceeds of a note
104. For $290, dated Dec. 30, 1877, payable in 2 mo., and
discounted at date at 9%.
105. For $ 500, dated May 10, payable in 90 days, and dis-
counted June 9 at 6%.
106. For $256.84, dated Oct. 28, payable in 60 days, and
discounted Nov. 12 at 12%.
107. For $ 1200, dated Jan. 31, payable in 3 months, and
discounted March 8 at 5%.
108. I bought a horse and carriage for $ 324, for which I
gave my note Nov. 5, payable in 1 year, with interest at 6%.
What would be the avails of this note at a bank, Aug. 1, dis-
count 7%?=*
109. Find the bank discount and avails of the following
note, discounted Feb. 12, 1876, at 10%.
$4000. San Francisco, Nov. 7, 1875.
Six months from date, with interest at 10%, I promise to
pay F. Egleston & Co., or order. Four Thousand Dollars ;
value received. James Noble.
683. Illustkative Example. For what sum must a
note be drawn, payable in 60 days, without interest, that
the avails may equal $591.60 when the note is discounted
at a bank at 8 % ?
Explanation. — Tha
bank discount of $ 1 for
63 days at 8% is $0,014;
hence, the avails of 1 1
discounted will be $ 1
minus $0,014, which
equals $0,986. Since the avails of $1 are $ 0.986, that the avails may
be $591-60 the note must be drawn for as many dollars as there are
times $ 0.986 in $ 591.60, which is 600. Ans. $ 600.
* See Art. 581, note.
WRITTEN WORK.
Bank discount of 1 1 for 63 d.
= $0,014
Avails of $ 1 for 63 d.
= 0.986
$591.60-10.986
= 600
Ans. $ 600.
BANK DISCOUNT. 229
584. From the foregoing may be derived the following
Rule.
To find the face of a note which discounted at a bank
will yield given proceeds : Divide the given proceeds hy
the proceeds of 1 dollar for the given rate and time, with 3
days of grace.
Note. To find the face of the note when the discount is given : Divide
the given discount hy the discount of $ 1 for the given rate and time,
with 3 days of grace.
585. Examples for the Slate.
110. For what sum must a 30 days' note, without interest,
be drawn that the avails at 6 % discount may be $ 80 ?
111. For what must a 4 months' note, without interest, be
drawn that when discounted at a bank it may yield $489.75
at 6 % discount ?
112. What must be the face of a note given for 90 days,
without interest, that the avails at a bank may be $ 1469, dis-
count being 8%?
113. What was the face of a note given for 45 days, not
bearing interest, on which the bank discount at 9 % was $ 11.40 ?
586. Miscellaneous.
114. What difference does it make in the avails of a note
for % 200, payable without interest in 18 months, whether it be
reckoned by true or by bank discount, rate 8 % ?
115. What will be the difference between the true and the
bank discount of a note for $ 9171, payable May 9, 1878, and
discounted Jan. 15, 1878, at 6 % ?
$500. Richmond, Oct. 5, 1876.
For value received, I promise to pay Charles Towle, or order,
Five Hundred Dollars in three months. James Allen.
116. What cash must be paid to discharge the above note at
its date by true present worth, rate of interest 6 % ?
117. What would be the avails of it at a bank, Dec. 5, 1876 ?
230 SIMPLE INTEREST.
118. What would be the amount of it, March 17, 1877 ?
119. What would be the true discount of it, Nov. 5, 1876 ?
120. What would be the bank discount of it, Nov. 5, 1876 ?
For other examples in bank discount, see page 253.
COMMERCIAL DISCOUNT.
587. Business men are usually allowed a deduction for
making cash payment for goods purchased on time. Notes
also not bearing interest are discounted by the deduction of
a certain per cent, not wholly depending upon the time.
Such a deduction is called business or commercial dis-
count.
588. Examples for the Slate.
121. A merchant bought a lot of goods amounting to $ 124,
on 30 days' credit ; 5 % discount on the price was allowed for
making payment at the time of purchase. What was paid ?
122. A man having bought a bill of goods amounting to
$468.20 on 6 months' time, cashed the bill for 10% off. What
did he pay ?
123. What is the cash value of a bill of cloth amounting to
1347.20, on the face of which a discount of 6% is made, and
on the balance another of 5 % ?
124. What is the difference between discounting a bill of
$1000 at 33^% and taking 10% off from the remainder, and
discounting the whole bill at 43^%?
125. A person paid 1 1.14 per yard for goods after a dis-
count of 5 % had been made upon the invoice price. What was
the invoice price ?
Note. Since 5 % had been deducted, 95 % remained.
126. What was the invoice price of a lot of French plate-
glass for which I paid $39 per pane after a discount of 40%
had been made ?
127. If from the retail price of a book 20% is deducted, and
a discount of 10 % is made upon the balance, and then the book
sells for $ 1.33, what is the retail price ?
COMPOUND INTEREST, 231
COMPOUND INTEREST.
689. A sum of $ 500 was loaned at 7%, interest payable
annually. At the end of the first year the interest for that
year was added to the principal, and upon the amount as a
new principal the interest was reckoned for the second year.
The amount for the second year formed a new principal,
upon which interest was reckoned for the next six months,
at the end of which time the note, with interest, was paid.
What was the amount then due ? What was the interest
gained ?
WRITTEN WORK.
Principal . . ... . . . |500.
Interest for Ist year 35.
Amount, or 2d principal .... 535.
Interest for 2d year 37.45
Amount, or 3d principal .... 572.45
Interest for 6 months .... 20.0357
' Amount % 592.49 1st Ans.
1st principal 500.
Interest $ 92.49 2d Am.
690. Interest upon both interest and principal, the sum
of the two forming a new principal for specified periods of
time, is compound interest.
In the example above the interest is compounded annually. It may
be compounded semi-annually, or for any period of time agreed upon.
691. From the operation above may be derived the fol-
lowing
Rule.
To compute compound interest:
1. FiTid the amount of the given principal for the first
period of time. With this as a new principal, find the
232
COMPOUND INTEREST.
amount for the second period of time, and so continue
for the whole time. The last amount is the amount re-
quired.
2. The last amount minus tJie given principal is the
compound interest.
692. Examples for the Slate.
At compound interest, what is the amount
128. Of $200 for 3 years at 6% ?
129. Of $350.50 for 4 years at 5% ?
130. Of $2000 for 3 years 11 months at 6% ?
131. Of $2000 for ly. 6 mo. at 7%, interest compounded
semi-annually ?
Note. Take interest at 3^ % for three intervals of time.
132. What is the compound interest of $ 40 for 1 y. 2 mo. at
6%, interest compounded semi-annually ?
133. What is the compound interest of $ 900 for 1 y. 1 mo.
at 6%, interest compounded quarterly?
693. The work of computing compound interest may
be shortened by the use of the following
TABLE,
Showing the amount of $1 at compound interest from 1 year to 10 years, at
3, 4, 4^, 5, 6, and 7 per cent.
Years.
3 per cent.
4 per cent.
4J per cent.
5 per cent.
6 per cent.
7 per cent.
1.
1.030000
1.040000
1.045000
1 .050000
1.060000
1 .070000
2.
1.0G0900
1.081600
1.092025
1.102500
1.123600
1.144900
3.
1.092727
1.124864
1.141166
1.157625
1.191016
1.225043
4.
1.125509
1 .169859
1.192519
1.215506
1.262477
1.310796
5.
1.159274
1.216653
1.246182
1.276282
1.338226
1.402552
6.
1.194052
1.265319
1 .302260
1.340096
1.418519
1.500730
7.
1 .229874
1.315932
1 .360862
1.407100
1.503630
1.605781
8.
1.266770
1.368569
1.422101
1 .477455
1 ..593848
1.718186
a
1.304773
1.423312
1.486095
1.551328
1.689479
1.838459
10.
1.343916
1 .480244
1.552969
1.628895
1.790848
1.967151
EXAMPLES. 233
594. Illustrative Example. What is the compound
interest of % 1000 for 2 y. 4 mo. at 7 % ?
WRITTEN WORK.
Amount of $ 1 at 7 % for 2 years .
. $1.1449
1000
Amount of $ 1000 for 2 years
. 1144.90
Amount of $ 1 144.90 for 4 mo. ^
A mount of $ 1000 for 2 y. 4 mo. )
1.02i
1171.6143
1000.
Compound interest ....
. $ 171.61 ^na.
Note. In the above operation, the amount of $1000 for 2 years is first
found, and the amount for the months is then obtained by multiplying by
1.02^. It would be equally well to find the amount of $1 for the entire
time, and then multiply that amount by 1000.
695. Examples for the Slate.
Using the preceding tahle, find the amount at compound
interest
134. Of 1 200 for 2 y. 4 mo. at 7 % .
135. Of -1 580 for 7 y. 10 mo. at 6 % .
136. What is the compound interest of % 300 for 3 y. 2 mo.
6 d. at 8 % , interest payable semi-annually ?
137. What is the compound interest of $ 380 for 1 y. 10 mo.
22 d. at 6%, interest payable semi-annually?
138. If at the age of 25 years, a person puts $ 1000 on in-
terest, compounding annually at 6%, what will be the amount
due him when he is 40 years old ?
Note. First find by the table the amount for 10 years, then find the
amount of that amount for 5 years more.
For additional examples in compoimd interest, see page 253.
WRITTEN
WORK.
Due.
Iteins.
Days. Interest.
Oct. 1,
$262
« 10,
220
9 10.66
Nov. 6,
250
36 3.00
234 AVERAGE OR EQUATION OF PAYMENTS.
AVERAGE OR EQUATION OF PAYMENTS.
696. Illustrative Example. A debtor owes to one per-
son the following sums at the dates specified : Oct. 1, $262;
Oct. 10, $ 220 ; Nov. 6, $ 250. At what date may he pay
the total of these items without loss of interest to either
party ?
Interest Method.
Explanation. — To do this
example, we may suppose all
the items to be paid at the
earliest date at which any
item becomes due, viz. Oct. 1.
This would involve a loss
1 day's int. of 732 - 0.244) 3.66 (15 to the debtor of interest on
I 220 from Oct. 1 to Oct. 10
Oct. 1 + 15 d. - Oct. 16. Ans. (9 days), and on 1 250 from
Oct. 1 to Nov. 6 (36 days).
The interest of 1 220 for 9 days at 12 % * is $ 0. 66
" " " 250 "36 ** «* 12% is 3.00
• ' Total interest . . . $3.66
That no loss may result, the total of the items, $732, should be paid
as many days after Oct. 1 as will be required for 1 732 at 12 % to gain
$ 3.66 of interest. To find this time, we divide $ 3.66 by the interest
of $ 732 for 1 day at 12 % (Art. 562, note), and have for a quotient 15.
15 days after Oct. 1 is Oct. 16. Ans, Oct. 16.
597. The process of finding the time when the payment
of several items, due at different times, may be made at
once, without loss of interest to either party, is average,
or equation of payments.
598. The date at which several sunis due at different
times may be paid at once is the average date or equated
time of payment.
* Any per cent may be taken, but 12 per cent (1% a month) is taken for convenience,
the interest then being for every month 0. 01 of the principal, and for every 3 days 0. 001
of the principal.
AVERAGE OR EQUATION OF PAYMENTS. 235
699. From the foregoing operation may be derived
Rule L
To find the average time for the payment of several sums
due at different times :
1. Select some convenient date ; for example, tJie earliest
date at which any item matures.
2. Compute the interest on each item from the selected
date to the date of its maturity.
3. Add the interests thus found; divide their sum hy
the interest of the sum of the items for one day ; the quo-
tient will express the member of days from the selected date
to the average date of payment.
4. Add this number to the selected date; the result will
be the average date required.
600. The foregoing illustrative example performed by
The Product Method.
WRITTEN WORK. Explanation. — To do this example
Days. Products. hj the product method, we select some
X 262 — 00 date, for example the earliest date at
9 X 220 = 1980 which any item becomes due, and sup-
36 X 250 =: 9000 V^^^ ^^^ ^^^ items to be paid at this
date. This would involve a loss to the
732) 10980 (15 ^^^^^^ ^f interest on 1 220 for 9 days,
Oct. 1 + 15 d. = Oct. 16. Ans. ^^^ «^ ^ 250 for 36 days.
The interest on | 220 for 9 days
equals the interest on $ 1 for 1980 days ; the interest on $ 250 for 36
days equals the interest on $ 1 for 9000 days, which together equals
the interest on $1 for 10980 days, but $732 is the sum to be paid,
and the time required for the interest on this sum to equal the
interest on $1 for 10980 days will be y^ of 10980 days, which is
15 days.
15 days after Oct. 1 is Oct. 16. Ans. Oct. 16.
236 AVERAGE OR EQUATION OF PAYMENTS,
601. From the preceding operation may be derived
Rule II.
To find the average date for the payment of several sums
due at different dates :
1. Select some convenient date ; for example , the earliest
date at which any item matures.
2. Multiply the time each item has to run hy the num-
ber of dollars in the item.
3. Divide the sum of the products thus obtained by the
number of dollars in the sum of the items; the quotient
will express the time from the selected date to the average
date of payment.
4. Add this time to the selected date ; the result will be
the average date required.
602. Proof.
Find the sum of the interests on all items due before the
average date, from the date at which they are severally
due to the average date; also find the sum of the interests
on all items due after the average date from that date to
the dates at which they are severally due. If these sums
are equal, or differ by less than half a day's interest on the
sum of all the items, the result is correct.
Note I. The examples in this book are performed by the interest
method, which has the advantage of brevity when the accountant uses
interest tables. The pupil will perform the work by either or by both
methods, as directed by the teacher.
Note II. Any date may be selected from whicb to average an account.
The last day of the month previous to the earliest day at which any item
becomes due is a convenient date.
Note III. When any item contains cents, if less than 50, disregard them,
if 50 or more, increase the units of dollars by $ 1.
Note IV. When a quotient contains a fraction of a day, if less than \,
disregard it; if ^ or more, call it 1 day.
AVERAGE OR EQUATION OF PAYMENTS.
2Zl
603. Examples.
139. What is the average date for paying three bills due as
follows: March 31, 1 400 ; April 30, 1300; May 30, $200?
140. What is the average date of maturity of three notes of
$ 800 each, due respectively Nov. 5, Dec. 8, and Feb. 3 ?
141. What is the average date of maturity of the following
items of account, viz., $ 900 due Sept. 10 ; % 2250.48 due Oct.
21 ; and % 1049.65 due Oct. 2d, ?
142. Find the equated time for paying $430 due in 5
months ; $ 270 due in 9 months ; and $ 300 due in 8 months ?
143. Average the above, having the first item due in 3
months, the others in 9 months each.
144. A gentleman purchased a farm for $ 3600, agreeing to
pay $ 600 down, and the remainder in five equal semi-annual
instalments. At what time may the whole be paid at once ?
145. When shall a note to settle the following account be
made payable ?
J. R. INGERSOL To E. PISH & CO., Dr.
1876,
April 10
To Mdse on 30 days' credit
May 16
il {i il gQ <( ((
June S
« li (( QQ H H
July 18
" Cash
200
300
520
250
Note. First find at what time each item falls due by adding the time of
credit to the date of the item.
146. What is the equated date of maturity of the following ?
V. M. HURON To COLTON IRON CO., Bt.
1876,
Mar.
11
((
29
Feb.
29
May
8
June
12
To Mdse on 30 days' credit.
a II II QQ (( II
(( IC li QQ il il
II li II QQ U II
II II li gQ U il
254.
145
300
159
238
AVERAGE OF ACCOUNTS.
AVERAGE OF ACCOUNTS.
Note. Younger pupils may omit this subject.
604. Illustkative Example. What is the average date
of maturity of the following account ?
Dr. PHILIP AEOHEE in Acct. with E. GRANGEE. Cr.
1877,
$
1877,
far. 18
To Mdse
250
Apr. 1
" SO
11 i(
600
« 20
By Cash
" Real Estate
700
300
WRITTEN WORK.
Dr.
Or.
Due.
Items. Days.
Interest.
Due.
Items.
Days.
Interest.
1877.
1877.
March 18.
$250
April 1.
$700
14
$3.27
« 30.
600 12
$2.40
« 20.
300
33
3.30
860
2.40
1000
850
6.57
2.40
1 day's int. of 150=0.05 0.05) 4.17
~83
March 18 + 83 d. = June 9. Ans. June 9, 1877.
Explanation. — The payment of all these items at the earHest date,
March 18, would involve a loss, at 1 % a month, to the debtor of $2.40
of interest, and to the creditor of $ 6.57, or $4.17 more to the creditor
than to the debtor.
Now, as the balance of the account, $ 150, is due from the creditor,
he may, to avoid loss of interest, defer payment of the balance as many
(lays after March 18 as will be required for $ 150, at 1 % a month, to
giin $4.17 of interest, which is 83 days. 83 days after March 18 is
June 9. Ans. June 9, 1877.
In this case it will be seen that the balance of the account and of the
interest are on the same side of the account.
606. Suppose, on the contrary, the first item in the fore-
going account to be $500, instead of $250, what would
then be the average date of its maturity ?
Explanation. — In this case the loss of interest to both debtor and
creditor is the same as before, but the balance of the account, $ 100,
AVERAGE OF ACCOUNTS. 239
is due from the debtor, who, to cancel the excess of interest lost by the
creditor, should pay the balance of the account as many days before
March 18 as will be required for $ 100, the balance, to gaiii f 4.17 of
interest, which is 125 days. 125 days before March 18, 1877, is Nov.
13, 1876. Ans. Nov. 13, 1876.
In this case the balance of the account and of the interest are on
opposite sides of the account.
606. From the above illustrations we derive the fol-
lowing
Rule.
To find the average or equated time for the settlement
of an account when there are both debit and credit items :
1. Find the interest on the several items of the account
from the earliest date at which any item becomes due to
their several maturities.
2. Find the balance of interest of the debit and credit
sides of the account, also the balance of the items.
3. Divide the balance of interest by the interest of the
balance of the items for one day. The quotient will be the
time in days between the selected date aTid the average time
of settlement.
4. Count thvi time forward from the selected date, when
the balance of the account and of the interest are on the
same side of the account, and BACK when on opposite sides.
The result will be the date of settlement.
Note I. When settlement takes place after the equated time of payment,
interest on the balance is charged ; when before the equated time, discount
IS allowed.
Note 11. The balance due on an account at. any day selected for settle-
ment may be, and usually is, found without averaging the account, by com-
puting the interest of the items on each side of the account from their
several dates of maturity to the day of settlement. The interests so found
on each side of the account are then added to that side. If any item matures
after the day of settlement, the discount is computed and added to the
opposite side of the account (which is equivalent to subtracting it from
the side on which it occurs). When the two sides of the account have
been so increased, their difference is the balance due.
240
AVERAGE OF ACCOUNTS.
607. Examples.
147. At what date can the balance of the following ledger
account be paid without loss to either party ?
Dr. EDWIN 0. OASTLETON. Cr.
1877.
April 1
July 8
$
f
1877,
ToMdse...
1000
00
April 14
" Cash...
118
98
Aug. 10
By Mdse
" Eeal Estate
1393
94
f
00
33
148. What is the average date of maturity for the following
account ?
Bt. rude, ALDEIOH, & 00. Ct.
1876,
$
f
1876,
$
f
Apr. 5
To Sundries on 2 mo. . . .
400
00
June 1
Mdse on 60 d.
250
00
Aug. 5
" Mdse " 1 " ...
600
00
July 8
Mdse '' SOd.
700
00
" 15
" Mdse " 1 " ...
200
00
Aug. 13
Cash
200
00
149. Find the average date of maturity of the following
account :
Dr. EARL INQALLS. Cr.
1878,
Jan. 6
Feb. 7
To Mdse. on 30 d.
t( it a QQ ««
$
f
1878,
600
00
Jan. 1
420
00
Mar. 16
\mi 90 d.
$
By Real Estate
500
''Cash
300
150. Average the following :
Dr. CHARLES RAYMOND.
Cr.
1876,
Aug. 20
Oct.
14
((
18
((
30
To Mdse, 60 d.
" Cash
" Cash
** Mdse, 1 mo.
$
f
1876,
173
15
Aug. 25
314
68
Sept. 12
230
00
81
25
By Mdse, 30 d.
" Mdse, 30 d.
500
102
151. Average the following account :
Dr. WILLIOI SMITa
Cr.
1877,
Jan. 6
" 25
Feb. 21
May29
To Mdse, 3 mo.
" Mdse, 30 d..
" Mdse, 3 mo.
$
f
1877,
339
92
Jan. 1
582
20
" 15
85
12
2200
00
Feb. 7
By Bal. of acct.
*• Real Estate,
3 mo
** Mdse, 2 7m.
361
4000
580
EXCHANGE. 24.1
SEOTIOISr XVI.
EXCHANGE.
608. To avoid the risk and expense of sending money
to make payments in distant places, merchants and others
make use of drafts or bills of exchange. What these are,
and how they are used, will best be shown by an example.
Suppose that J. G. Ames, in Boston, wishes to pay $200 to William
Smith, in New Orleans. He may pay the money to a banker, James
A. Dupee, in Boston, who will write an order on his correspondent,
George Flint, a banker in New Orleans, in the following form :
^
^
^
A
/i'^^.
Mo^^o.i,/u/y /cS /c^//.
■o-
1 (£//ia^^
1 al oic^l.
i
\ To George Flint, Esq.,
1 New Orleans.
609. Such a written order for the payment of money
is a draft, or hill of exchange. The method of making
payments by drafts or bills of exchange is exchange.
Ames will take this draft and send it to Smith, who, when he re-
ceives it, will ])resent it to Flint for acceptance. If Flint is willing to
obey the ordei and pay the money, he writes the word " Accepted "
across the face of the draft, adds the date, and signs his name. In due
time Smith gets the money from Flint, gives up the draft, and the
transaction is complete.
242 EXCHANGE.
610. The person who makes and signs a draft is the
drawer. The person to whom it is addressed is the
drawee. The drawee when he accepts the draft becomes
the acceptor. The person to whom the draft is payable
is the payee.
In the case described above, who is the drawer ? the drawee ? the
acceptor ? the payee ?
611. If the payee wishes to transfer the draft to
another person, he writes his own name across the back
of the paper; this is called an indorsement, and the
payee then becomes an indorser. The person to whom
the draft is so transferred is an indorsee. If the indorsee
wishes to transfer the draft to a third person, he also writes
his name under that of the former indorser. He thus be-
comes a second indorser ; and there may be a third in-
dorser, a fourth, and so on indefinitely.
612. The person who holds the draft at any time (the
payee or the last indorsee) is called the holder.
The holder, looks for payment first to the acceptor, and then to the
indorsers in their order. Each indorser is liable to pay the draft when
the acceptor and previous indorsers have failed to do so. To avoid
becoming liable, an indorser may write over his name the words
" Without recourse."
Drafts may be " at sight " or " on time " ; bankers charge less for
the latter than for the former, the difference in price being equivalent
to a discount for the given time.
When our exports to another country, England for example, exceed
in value our imports from that country, more money is due to us from
the English merchants than is due to them from our merchants. The
larger sum due us in England will make it easy for us to buy bills of
exchange on England. They will be plenty here, and the price of
them will fall. If they can be bought for less than their face, they
are at a discount, or helow far.
On the contrary, when the value of the goods imported from Eng-
land exceeds the value of those sent to England, more money is due
EXAMPLES. ^ 243
to the English merchants from us than is due to us from them. The
smaller sums due us in England will then make it difficult for us to
buy bills of exchange on England, and the price of them will rise. If
they cost more than their face, they are at a premium, or ahove par.
613. Bills of exchange are either foreign bills or inland
bills. Foreign bills are those which are drawn or are pay-
able in a foreign country ; and for this purpose each of the
United States is foreign to the others. Inland bills are
drawn and payable in the same State.
614. Examples for the Slate.
1. What is the cost in Philadelphia of a draft on San Fran-
cisco for $ 800 at 1 % premium ?
2. What is the cost of a draft on Detroit for $2500 at
^% premium?
3. What is the cost of a draft on New York for $ 700 at
12 days after sight, premium ^ % ?
4. What is the cost of a sixty days' draft on New Yorlc
for $2000 at 2% discount ?
6. I bought a bill on Chicago for $ 700 at a discount of
f%. What did I ^ay ?
615. Illustrative Example. What is the face of a
draft on New York bought in St. Louis for $ 8820, when
the discount is 2%?
WRITTEN WORK.
$1-2% of $1 = $0.98, cost of $1. -
$ 8820 - $ 0.98 = 9000. Ans. $ 9000.
6. What is the face of a draft that may be bought for
.$500 at a discount of 1^%?
7. A merchant in New York bought a draft on Cincin-
nati at ^% premium for $275. What was the face of the
draft ?
244
EXCHANGE,
Exchange with Europe.
616. Exchange with Europe is effected chiefly through
large business centres, as London, Paris, Hamburg, etc.
• In computing foreign exchange, it is necessary to change the vahies
expressed in the currency of one country to equivalent values ex-
pressed in the currency of another country.
On page 311 of the Appendix will be found a list of the monetary
units of foreign countries, with their values in United States money,
as proclaimed by the Secretary of the Treasury, Jan. 1, 1878; also on
page 312, tables of English, French, and German money.
The rates of exchange between this country and the principal busi-
ness centres are given from day to day in the newspapers. The follow-
ing is an extract showing the exchange value of the pound sterling
in United States money ; the number of francs and centimes which
equal a dollar ; and the exchange value of 4 marks in cents :
"We quote bankers' 60-day bills on London at $ 4. 84 @ 4. 84 J, and short-
sight bills at $4.86, both in gold. On Paris, francs 5.15 per dollar for
short sight, and 5.18f for 60-day bills, Gossler & Co.'s rates on Hamburg
for 60-day bills are 95, and short-sight bills 95|. "
In making bills on foreign countries, it is customary to write two
or more of the same tenor and date, the payment of either one of
which cancels the other one or two. And to provide against accident
in their transmission, it is customary to send two, at least, of a set, at
different times, or by different modes of conveyance.
617. Examples for the Slate.
8. What was the cost of the following bill in U. S. money,
the rate of exchange being $4.86 ?
c^cz^e (^Cu7icAeG^ ,£^u?2^ a^i^una, z>^auie i^ececuee/, a7za
To M^srs. McCalmont Bros. <Sc Co., ^^^^^ Mcz/^o^y Jf ^a.
3 Crown Court, London,
UNITED STATES BONDS 245
9. T. Van Horn, of New York, bought of E. J. Birney &
Co. a set of exchange payable at sight for £1000 sterling on
Brown, Shipley, & Co., of Liverpool, at 1 4.84 What was the
cost in gold ?
10. What is the cost in gold of a set of exchange on Paris
for 550 francs, exchange being 5.15 per dollar ?
11. What is the cost of the above-named sum in currency,
gold being quoted at 102f ?
12. What is the cost of a draft on Hamburg for 200 marks
when the quotation is 95 ?
13. What is the cost in New Orleans of a bill on London
for £ 75 10 s, when exchange is there quoted at % 4.85^ ? (See
Appendix, page 312.)
14. When exchange is 1 4.86, what is the face of a bill on
London which can be bought for $ 9720 ?
15. What is the face of a draft on London which can be
bought for $ 1938.42, the rate of exchange being % 4.84 ?
United States Bonds.
618. Governments and corporations sometimes borrow
money, giving, as evidence of the loans, certificates or notes
payable at or within some definite time, with interest at
stated periods. Such certificates or notes are called bonds.
619. Bonds sometimes have certificates attached, prom-
ising the holder certain sums of interest as they become
due upon the bonds. These interest certificates are called
coupons.
Note I. When the interest is paid, the coupons are cut off by the holder
and given up as receipts.
Note II. The extraordinary expenses of the government of the United
States during the civil war were met in part by the sale of bonds.
620. United States coupon bonds are issued in the de-
nominations of $ 50, $ 100, $ 500, and $ 1000. Eegistered
246
UNITED STATES BONDS.
bonds are issued in the same denominations, and also in
denominations of $ 5000 and $ 10000.
Bonds are usually named according to the rate of interest they bear.
621. The following is a list of the more important United
States bonds not redeemed in 1878 : \
Names of Bonds.
Redeemable.
Payable.
Bate
of
Int.
Int. payable in
6'sof 1881
5-20's
1881 ..
1881
6%
6%
5%
5%
^%
4%
Semi-annually.
(<
((
Quarterly.
((
1 In 5 years from \
\ date of issue... \
1874
In 20 years
1904 .
10-40's
5'sofl881
4i's..
1881..
1881
1891..
1891
4's
1907
1907
Bonds are bought and sold as other stocks. Their prices from day
to day are quoted in the newspapers.
622. The rules of percentage already illustrated apply
to bonds.
623. Examples for the Slate.
16. When U. S. 5-20's are sold at 108|^, what is received for
eight $ 500 bonds ?
17. When U. S. 5's are worth 106^, what will $850 in
bonds cost ?
18. What amount in bonds shall I receive for $ 2675 in-
vested in U. S. 5's at 107 ?
19. What shall I pay a broker for a $ 1000 U. S. 5-20 bond
at 110^, and two $ 1000 U. S. 6's of 1881 at 113|, with his
brokerage oi l%?
20. How much money must I remit to a broker that he
may purchase for me three U. S. 10-40 bonds of $ 1000 each,
the bonds selling at 109, and his commission being J % ?
GENERAL REVIEW. 247
21. When gold was at 102f, what was my semi-annual
income in currency from 12 U. S. 6's of % 1000 each ?
Note. First find the semi-annual interest in gold.
22. If the premium on gold is 3 %, what per cent do I get
semi-annually in currency on a U. S. 5-20 bond purchased at
106 ?
23. Which yielded the greater per >cent semi-annually, and
how much, U. S. 6's at 110, gold at 102^, or a mortgage on
real estate paying 3^% semi-annually ?
24. How much money must be invested in U. S. 4|^'s to yield
a quarterly income of $ 225 in gold, bonds selling at 105^, gold
at par ?
624. General Review, No. 5.
25. By losing 3 cents a pound, I lose 12\% of the cost of
butter. If I had lost 4 cents a pound, what % should I have
lost ?
2Q. What is the simple interest of $ 300 from May 5, 1876,
to Feb. 2, 1878, at 1^% a month ?
27. I hold a note for % 500, which bore interest at 7 % from
May 10, 1875. Sept. 16, 1875, received % 140 ; July 28, 1877,
received $ 50. What remained due Sept. 3, 1877 ?
28. If I pay % 45 interest for the use of $ 500 for 3 years,
what is the rate per cent ?
29. How long must $ 204 be on interest at 7 per cent to
amount to $ 217.09 ?
30. What principal will gain % 9.20 in 4 mo. 18 d., at 4 per
cent?
31. What sum, at 7 per cent, will amount to $ 221.075 in
3 yrs. 4 mo. ?
32. At 6%, what is the compound interest of $ 600 for 1 yr.
4 mo., interest payable semi-annually ?
33. What is the present worth of a note for % 488.50, due in
2 yrs. 5 mo. 15 d.-, at 9 per cent ?
34. What is the true discount of % 105.71, due 4 yrs. hence,
rate6%?
248 MISCELLANEOUS EXAMPLES.
35. What is the bank discount of $ 450 for 80 days and
grace at 5% ?
36. What are the avails of a note of $100 given for 27
days, and discounted at a bank at 6 % ?
37. For what must a GO-daj^s' note be given, which, dis-
counted at a bank at 6%, will yield $1295?
38. A debtor owes $ 200, | due in 2 months, \ in 3 months,
and the remainder in 5 months. What is the equated time
for paying the whole ?
39. A man about to travel in England bought a bill for
250 pounds sterling. Exchange being $4.85| in gold, and
gold being quoted at 103, what amount of currency did he
pay for the bill ?
625. Miscellaneous Examples.
40. What is 124% of 5 T. 300 lbs. ?
41. What is the amount at 6%, simple interest, of $38.75,
from Aug. 5 to Nov. 10 ?
42. What is the amount of $380.25, at 6% compound in-
terest, for 2 yrs. 5 mo. ?
43. How much ought a broker to charge me for 5 shares of
stock purchased for me at 7 % advance, shares having originally
been $ 500, his brokerage at | % included ?
44. What will be the length and breadth of a piece of cloth,
originally 2\ yards long by 1 yard wide, after sponging, if in
that operation it shrinks 4% in length and 6% in width ?
45. A commission merchant receives $ 544 ; of this he is to
invest such a portion as remains after deducting his commission
of 2^% on the investment. What is his commission, and what
will remain ?
46. What is the cost of insuring $ 2500 at $ 17.50 on $ 1000 ?
47. If ^ of a sum of money be due in 2 months, \ in 4
months, ^ in 3 months, and the remainder in 4 months, at
what time might the whole be paid without loss to the debtor ?
48. A dealer has 18 barrels of sound apples remaining in a
lot of which 10% have decayed. If his lot cost him $ 1.50 per
MISCELLANEOUS EXAMPLES. 249
bbl., would he gain or lose on the lot, and what %, by selling
the remainder at $ 1.75 per bbl. ?
49. What will be the net loss to an insurance company in
case of the loss by fire of a property insured for % 4500, on
which the company had received 3% premium, no allowance
for interest?
50. What must be paid for a policy to cover $2575 at a
premium of 1;]^%?
51. What per cent of 1 bushel is 1 peck 2 quarts ?
52. In what time will a sum of money double at 2 % simple
interest ?
53. A person lent a certain sum for 1 yr. 6 mo. at 5 % . The
interest being 1 9.30, what was the sum ?
54. What principal will amount to $ 63.25 in 1 yr. 3 mo. at
8%?
55. A merchant imports from Hamburg a bale of cloth, con-
taining 12 pieces of 40 yards each; the cloth, with charges
there, cost him % 480 ; he pays a duty here of 35 cts. per yd.,
freight $28.50, and other charges $7.11. At what must he
sell the cloth per yd. to gain 10 % above all charges ?
56. In the year 1872 the town of B voted to raise, by taxes,
$ 97290 ; ^ of this was levied upon the polls ; the valuation of
the town was $ 10134375. What was the tax on $ 1, and what
was the tax of a non-resident who owned a house in town
valued at $2000?
57. What must be the face of a note, which, discounted at
a bank for 30 days and grace, would yield $ 500 ?
58. On a note for $2500, dated Sept. 5, 1875, were paid
$ 50 January 29, 1876, and $ 500 July 1, 1877. The note
being on interest at 6% from its date, what was due Sept. 5,
1877?
59. Paid % 18.77 for insuring my schooner at a premium of
\ % . What was the sum covered ?
60. What is the par value of stock, which, selling at 25%
above par, brings $ 500 ?
250 QUESTIONS FOR REVIEW.
1 150.25. Chicago, Jan. 5, 1876.
On the fifteenth of May, 1876, I promise to pay to the order
of B. F. Archer, One Hundred and Fifty 3^% Dollars ; value
received. D. T>. Cokwin.
61. If the holder of the above note has it discounted at a
bank Feb. 15, 1876, at 6%, what will he receive ?
62. What is the true present worth of the above note at its
date, rate 7%?
63. If the note above was unpaid when due, and drew in-
terest at 6% from the time it became due, what would settle
it Oct. 27, 1876?
64. Find the amount of the above note at compound interest
at 5% from the time it became due till Oct. 27, 1881 ?
626. Questions for RevieTv.
"What is PERCENTAGE ? What is the base ? the amount ? the re-
mainder ? rate per cent? Express -f and its complement decimally.
How do you change a fraction to a per cent ? a per cent to its lowest
terms ? How do you find a percentage of a number ? the amount ?
the remainder ? the base ? the rate per cent ? Give the formula for
finding the percentage ; the amount ; remainder ; base ; rate per cent.
Upon what is the percentage of profit or loss reckoned ? If goods
cost 24 cents, for what must they be sold to gain 8^%? to lose 16|%?
What per cent would be gained or lost by selling goods that cost 24 cents
for 30 cents ? for 21 cents ? If 24 cents is 20% less than the value of
goods, what is the value ? If 24 cents is 33|-% more than the value of
goods, what is the value ? If 18 cents is 10% less than cost, for what
would you sell goods to gain 10%? to lose 25%? If 10% of what
you receive for goods is gain, what is your gain per cent?
What is COMMISSION ? Who is the factor ? the consignor ? the con-
signee ? What is meant by net proceeds ?
What is a company ? a corporation ? capital stock ? a certificate of
stock? Who are stockholders ? What is an assessment ? a dividend ?
a stockbroker ? brokerage ? When are stocks above par? below par ?
Upon what is the per cent of commission or brokerage estimated ?
How do you find what sum is to be expended when a remittance
contains that sum together with the commission?
QUESTIONS FOR REVIEW. 251
What is INSURANCE ? a policy ? a premium ? Who are under-
writers ? What is expectation of life ?
What is a tax ? a poll tax ? real estate ? personal property ? Who
are assessors ? How do you find the tax to be assessed on a dollar
in any town ?
What are customs or duties ? What is a specific duty ? an ad va-
lorem duty ? gross weight ? net weight ? In estimating specific duties,
what allowances are made ?
What is INTEREST ? what is the principal ? the amount ? What is
meant by the rate, in interest ? What is legal rate ? usury ? simple
interest? Give your method of computing interest. How do you
find accurate interest?
What is a promissory note ? What is the face of a note ? What
are partial payments ? Where is the record of payments made ?
What is the United States rule for partial payments ? What is the
merchant's rule ?
What three factors are used to find interest ? The interest, princi-
pal, and rate being known, how do you find the time ? The interest,
principal, and time being known, how do you find the rate V The in-
terest, rate, and time being known, how do you find the principal ?
What is the dividend in each case ? The amount, rate, and time
being known, how do you find the principal ?
What is the present worth of a debt ? What is discount ? Give
a rule for finding present worth. How do you find discount ? How
can you prove the work ?
What is BANK DISCOUNT ? What are days of grace ? avails of a note ?
Which is the larger, true or bank present worth ? true or bank dis-
count ? Describe the process of getting a note discounted at a bank.
What is indorsing a note ? How do you find the face of a note, which,
discounted at a bank, will yield a certain sum ?
What is COMPOUND interest ? How often may interest be com-
pounded ? For how many periods of time will interest be compounded
in 2 y. 9 mo., if it is compounded semi-annually ? quarterly ?
How do you find the average time for paying several bills due at
different times ?
What is EXCHANGE ? what are drafts or bills of exchange ? Who
are the parties to a draft? who is the holder? When are drafts at a
premium ? at a discount ? Where can you find rates of exchange ?
What are bonds ? What is a coupon ?
252
DRILL TABLE.
627. DRILL TABLE No. 8.
A
Principal.
$640.08
$305.40
$90,508
$705.38
$4000.
$240.08
$9,034
$80.50
$3050.
$560.08
$150.20
$5400.
$690.40
$60.75
$850.06
$6,508
$700.01
$38.20
$590.04
$11.80
$809.06
$654.09
$10000.
$3600.
$908.70
B
Interest.
$16,305
$28.14
$17,083
$78.90
$100.
$50.40
$15.08
$7,005
$430.20
$6,095
$30.75
$175.60
$290.14
$5,872
$25,642
$11.75
$10.90
$3,956
$105.20
$5,769
$340.50
$75.80
$500.
$1640.
$64.37
C
Time.
ly. 6m. 24d.
ly. 2m. 6d.
4y. 11m.
3y. 7m. 27 d.
2y. 3m. 20 d.
4 y. 9 m. 5 d.
7y. 5m. 18 d.
2y. 11m. 26 d.
3y. 10m. 3d.
17 d.
1 y. 3 m.
ly. 4m. 25 d.
5y. 21d.
ly.9d.
10 m. 13 d.
5y. 7m. 2d.
2 m. 28 d.
2y. 8m. 19 d.
3 m. 16 d.
4y. 8m. 2d.
2y. 15d.
4y. 4d.
4 m. 14 d.
Id.
ly. 7m. 15d.
D E
Per cent.
5
8
3
2
7
4
11
1
10
9
12
3
2
9
5
11
10
1
4
8
7
12
50
100
7
DRILL EXERCISES.
253
628. Exercises upon the Table.
216. Find D per cent of A. t
217. Find E per cent of A.
218. Find D + E per cent of B.
219. A is D per cent of what sum ?
220. A is E per cent of what sum ?
221.*^ is what per cent of A ?
222. Find the commission for col-
lecting or investing A at
(D-E)%.
223. If A includes both the commis-
sion and sum to be invested,
what is the commission at
D%?
224. If A includes both the commis-
sion and sum to be invested,
what is the sum to be invest-
ed, the commission being D % ?
£25. Find the date, which is C years,
months, and days after Nov.
27, 1871.
226. Find the interest of $ 1 at 6 %
for the time in C.
227. Find the interest of $ 1 at 1 %
for the time in C.
228. Find the interest of $ 1 at D %
for the time in C.
229. Find the interest of $ 1 at E %
for the time in C.
230. Find the interest of $1 at
(D -I- E) % for the time in C.
231. Find the interest of A at D %
for the time in C.
232. Find the interest of A at
(D + E) % for the time in C.
233. Find the amount of A at 6 %
for the time in C.
234- Find the compound interest of
AatD% for2y. 9 mo. 18 d.
235. Find the compound interest of
A at D % for 1 y. and the
months and days in C, inter-
est payable semiannually.
236. Find the amount of A at com-
pound interest for 2 y. 6 mo.
15 d. at 6%.
237. Find the rate, A, B, C being
given.
(Let the fraction of the per cent
be changed to tenths, and the an<
swer be expressed thus : 8.3 ... %.)
238. Find the time, A, B, and (D + E)
being given.
239. Find the principal, B, C, D
being given.
240. Find the principal, A being the
amount, C the time, and 6 %
the rate.
'241. Find the present worth of A,
due in the time in C, at D % .
242. Find the discount on A, due in .
the time in C, at D % .
243. Find the discount on A, due
in the time in C, at 6 % .
244' Find the bank discount on a
note for A, payable in the
months and days in C, at D % .
245. Find the avails of a note for A,
payable in the months and
days in C, at D % .
246. Find the face of a note, which,
being discounted at a bank at
6 % for the months and days
in C, will yield A.
* See note after Exercise 237.
t See page 57, for Explanation of the Use of the Drill Tables.
254 RATIO AND PROPORTION,
SEOTIOI^ XYII.
RATIO AND PROPORTION.
SIMPLE RATIO.
629. Ten equals how many 2's. Ans. Five 2's.
In the above answer we express the relation of 10 to 2
by their quotient. The relation of two numbers expressed
by their quotient is ratio.
630. Oral Exercises.
a. What is the ratio of 8 to 2 ? of 2 to 8 ? of 9 to 3?
b. What is the ratio of 6 to 2 ? of f to f ? of f to f ?
c. What is the ratio of 5 to 2? of 0.5 to 0.2? of 21b. to 7 lb.?
631. The ratio of 10 to 2 is indicated thus, 10 : 2. The
expression is read, " The ratio of ten to two."
d. Indicate the ratio of 7 to 9 ; of 8 days to 15 days.
e. Eead the following expressions : 12 : 15 ; 1 4 : $ 18.
632. The numbers whose ratio is to be found are the
terms of the ratio. The two terms of a ratio form a
couplet. The first term of a couplet is the antecedent;
the second term is the consequent.
Note. The terms of a ratio must be numbers of the same denomination.
633. As the antecedent of a ratio is the dividend and
the consequent the divisor, it follows that
When the antecedent is multiplied or | ^^^ ^^^^^ -^ jn^itipiied.
the consequent is divided, )
When the antecedent is divided or the | ^j^^ ^.^^.^ -^ ^^^^^^^^
consequent is multiplied, )
When both terms of a ratio are multi- ) the value of the ratio is
plied or divided by the same number, S not changed.
COMPOUND RATIO. 255
634. Examples for the Slate.
Find the ratios of the following couplets :
. (1.) lQ:2m. (4.) 45:990. (7.) $9.00 : $12.50.
(2.) 8^ : 300. (5.) 28 : 910. (8.) $ 0.87^ : $ 0.12|.
(3.) 19:110^. (6.) 6^:75. (9.) 1001b. : 16| lb.
635. The ratio of two numbers is a simple ratio. A
simple ratio has one antecedent and one consequent.
COMPOUND RATIO.
636. Find the ratio of 2 to 5, and of 3 to 4 ; and then
find the product of these ratios. Ans. -| and | ; product ^.
The product of two or more simple ratios is a compound
ratio.
637. The compound ratio given above is indicated thus :
2:5) The expression is read,
3:4 J "The compound ratio of 2 to 5 and 3 to 4."
638. From Art. 636 it will be seen that when several
general numbers form a compound ratio, the value of the
ratio may he fouTid hy dividing the product of the ante-
cedents hy the product of the consequents.
639. Oral Exercises.
Find the value of the compound ratios indicated by each of
the following expressions :
a. 5:8) ^^ c. 3: 7| ^^
4:9) • 4:12) *
fe. 8 : 1 ) ^ d. 7 men : 5 men ) ^ 9
7:4) * $10.00 : $8.00 ) "*
Note. The ratio of numbers is the same whether the numbers are de-
nominate or general ; hence, in finding the value of the ratio in the last
example, the terms may be regarded as general numbers.
256 RATIO AND PROPORTION.
PROPORTION.
640. What is the ratio of 3 ft. to 6 ft. ? of $ 5 to $ 10 ?
These ratios are equal to each other.
An equality of ratios is a proportion.
641. The equality of the above-named ratios is ex-
pressed thus, 3 ft. : 6 ft. = $ 5 : $ 10.
This expression is read, " 3 ft. is to 6 ft. as $5 is to $10."
642. Exercises.
E-ead the following :
a. 5 : 7 = 15 : 21. c. 40 : 10 = 15 min. : 3| min.
b. 1:3 = 17:1105. d. 9:6 = 6:4.
643. The first and fourth terms of a proportion are the
extremes, and the second and third are the means.
Note I. In Example d above, 6 is the consequent of the first couplet
and the antecedent of the second ; and so 6 is a mean proportional be-
tween 9 and 4.
Note II. Four quantities are directly proportional when the first is
to the second as the third is to the fourth. Four quantities are inversely
proportional when the first is to the second as the fourth is to the third ;
or when one ratio is direct and the other inverse. Thus, the amount of
work done in any given time is directly proportional to the number of men
employed ; that is, the more men, the more work : but the time occupied
in doing a certain work is inversely proportional to the number of men
employed ; that is, the more men, the less time.
To supply a Missing Term of a Proportion.
644. Illustrative Example. Supply the missing term
denoted by x in the proportion, re : 5 = 4 : 10.
Explanation. — The ratios of the two coup-
lets are f and ^ ; these changed to fractions
having a common denominator are '^r^^o ^^^
4x5
10X5-
As these fractions are equal, and their de-
nominators the same, their numerators must
be equal. But one numerator is the product
of the means of the proportion, and the other
WRITTEN
WQ]
IK.
x:5 =
4:
10
OB _
^
aexio _
: JL_
X5
6x10
10X5
XX 10 =
.4x5
±X5_
10
-.2{
Missing
term.
2:5-
4:
10
ANALYSIS AND PROPORTION. 257
the product of the extremes. Therefore the missing extreme may be
found by dividing the product of the means (4 x 5) by the given extreme
(10). The missing term then is 2, and the proportion is 2 : 5=4 : 10.
645. From the preceding illustration may be derived
the following principles :
1. When four general numbers form a proportion, the
product of the means is equal to the product of the extremes.
2. A missing extreme may he found hy dividing the
product of the means hy the given extreme.
3. A missing mean may he found hy dividing the product
of the extremes hy the given mean.
646. Oral Exercises.
Supply the missing terms represented by x in the following
proportions :
a. 3 :4-9 : x. d. x : 7 = 8 : 9.
b. 8 : e = x : 3. e. 27 : S = x : 1.
c. 12:a; = 15:3. /. £c : 4 days = $ 5 : $15.
Note. To find x in Example f, disregard the denominations, and proceed
as if the terms were general numbers. ^-^. = 1 J. Then x equals 1^ days.
647. Examples for the Slate.
Supply the missing terms in the following :
(10.) 2 : 100 - 17 : x. (13.) 750 A. : 3 A. = cc : 13 tons.
(11.) 9 : 150 = 105 : x. (14.) x : 200 hats - $ 87.50 : $ 500.
(12.) 65:a; = $75:|850. (15.) 1 800 : $56 = 1390 : ».
ANALYSIS AND PROPORTION.
648. Illustrative Example. If 14 slates cost 98 cents,
what will 10 slates cost ?
By Analysis.
WRITTEN WORK. Explanation. — It 14 slates cost 98 cents,
7 1 slate will cost 1 fourteenth of 98 cents, and
98 x^ =70. 10 slates will cost 10 times 1 fourteenth of 98
^^ cents, which is 70 cents. An^. 70 cents.
Ans. 70 cents.
258 RATIO AND PROPORTION.
By Proportion.
WRITTEN WORK. Explanation. — The ratio of 14 slates to 10
14 : 10 = 98 : x. slates must be the same as the ratio of 98 cents,
J the cost of 14 slates, to the cost of 10 slates.
98 X 10 ^ W^ ^^y arrange the terms in any order which
Ta will express the equality of these ratios. Foi
rrn J. convenience, we make 98 cents the third term,
Ans. 70 cents. , ' /. i , r.
and X, the unknown cost oi 10 slates, the lourth
term. As the cost of 10 slates will be less than 98 cents, we make 10
the second term and 14 the first. Multiplying 98 by 10 and dividing
the product by 14, we have for the fourth or missing term, 70.
Ans. 70 cents.
649. Rule.
To solve examples by simple proportion :
1. Make the number that is of the same denomination
as the required answer the third term.
2. Determine from the statement of the example whether
the answer is to he greater or less than the third term.
3. Make the other two numbers in the example the first
and second terms of the proportion, taking the greater num-
ber for the second term if the answer is to be greater than
the third term, aTid the less number for the second term if
the answer is to be less than the third term.
4. Multiply the third term by the second term, and divide
the pvduct by the first term.
650. Examples for the Slate.
The following examples may be solved by analysis or by proportion, or by
both methods, at the option of the teacher.
16. If 4 yards of velvet cost $ 20, what will 14 yards cost ?
17. If 12 bushels of wheat cost $ 8, what will 30 bushels cost ?
18. What will 250 sheep cost if 24 sheep cost $ 72 ?
19. What will 75 pounds of cheese cost if 64 pounds cost
e 6.08 ?
SIMPLE PROPORTION. 259
20. How many feet of plank will be required for a bridge
528 feet long, if 17280 feet of plank are required for 288 feet ?
21. If 500 bushels of plaster were sufficient for the dressing
of 3^ acres of land, what would be required for 11^ acres of the
same kind of soil ?
22. If a building 13 ft. high casts a shadow of 4 ft., what
length of shadow will a church spire 346| ft. high cast at the
same time ?
23. If crackers can be sold at 10 cents a pound when flour
is worth $ 6.50 a barrel, for what can they be sold when flour is
worth $ 9.75 a barrel, the cost of making not being considered ?
24. If a hind wheel, which is 8f feet in circumference, turns
800 times in a journey, how many times will the fore wheel,
which is 6^ feet in circumference, turn in the same journey ?
25. If 400 bushels of potatoes were bought for $ 350.90, and
sold for % 425.50, what was gained on 25 bushels ?
26. If a 10-cent loaf weighs 1 lb. 2 oz. when flour is worth
% 7^ per bbl., what should it weigh when flour is $ 6 per bbl. ?
27. If my friend lends me $ 7000 for 15 days, for what time
should I lend him $ 4500 to requite the favor ?
28. If my friend lends me money for 4 months when inter-
est is 10 per cent, for what time should I lend him the same
sum to requite the favor when interest is 7 per cent ?
29. If 2 lbs. 5 oz. of wool make 1 yd. of cloth 32 inches wide,
how much will make a yard of the same quality 1| yards wide ?
30. How many yards of cambric 34 inches wide will be re-
quired to line 14^ yards of silk which is 22 inches wide ?
31. If 400 lbs. of coal are required to run an engine 12 hours,
whskt number of tons will be required to run three similar en-
gines for 30 days, day and night ?
32. A deer, 150 rods before a hound, runs 30 rods a minute ;
the hound follows at the rate of 42 rods a minute. In what
time will the deer be overtaken ?
260 RATIO AND PROPORTION,
COMPOUND PROPORTION.
651. A compound proportion is a proportion in which
one of the ratios is compound.
652. Illustrative Example. If it takes a man 5 days
of 9 hours each to earn $ 15, how many days of 8 hours
each will it take him to earn $ 20 ?
By Analysis.
WRITTEN WORK. Explanation. — If it takes a man 5 days to
g 2 earn $ 15, it will take him 1 fifteenth of 5 days
5 X 20 X 9 *° ^^^^ ^ ^' ^^^ ^^ times that to earn $ 20. If
— JbTZTq — ~ • 2* it takes him this number of days when the days
g 2 are 9 hours long, it will take him 9 times as
ji davs i^any days when they are I hour long, and
1 eighth of that number when they are 8 hours
long, which is 7^ days. Ans. 7^ days.
By Compound Proportion.
WRITTEN WORK. Explanation. — The number of days it
H -I will take depends, first, on the amount
g A of money to be earned, and, secondly, on
15 : 20 ) K T *he number of hoiirs a day the man
Q . J "" J ' ' works. We might get the answer by
2 3 using two simple proportions. In the
^ ^ ^ - 71 ^^* ^^ could find the number of days,
2 so far as it depends on the amount of
Ans. 7i- days, nioney to be earned ; and then, taking
this result as the third term of another
proportion, we could find the number of days so far as it depends on
the number of hours in a day's work. It will be more convenient,
however, to combine the two proportions, thus forming a compound
proportion.
To do this we make 5 days, which is a number of the same denomi-
nation as the required answer, the third term, and then consider the
statements of the example in order.
(1.) As $ 20 is a larger sum than $ 15, it will take a larger number
of days to earn it ; that is, the answer, so far as it depends on the
COMPOUND PROPORTION, 261
amount of money to be earned, will be larger than the third term ;
so we make 20 the second term and 15 the first term of the first ratio.
(2.) As it will take more days 8 hours long to earn this money than
days 9 hours long, the answer, so far as it depends on the length of
the days, will be larger than the third term ; so we make 9 the second
term and 8 the first term of the second ratio.
We now have the compound proportion,
15
8
* q r = 5 <iays : x days.
Multiplying 5 days by 20x9, and dividing the product by 15x8,
gives 7^ days. Ans. 7^ days.
The work may be shortened, as shown in the written work, by can-
celling.
653. Rule.
To solve examples by compound proportion :
1. Make the number that is of the same denomination
as the answer the third term.
2. Take the two numbers in each separate statement in
the example, and consider whether the answer, so far as it
depends on them alone, will be greater or less than the third
term. Arrange these two numbers accordingly as terms of
a ratio.
3. Multiply the third term by the product of the second
terms and divide this product by the product of the first
terms.
664. Examples for the Slate.
33. If $ 90 is paid for the work of 20 men 6 days, what
should be paid for the work of 5 men 8 days ?
34. If in 84 days 75 men can earn $ 68.75, in how many
days can 90 men earn $ 41.25 ?
35. If it costs $ 30 to paint the front of a building 140 ft.
long and 25 ft. high, what will it cost to paint the front of a
building 180 ft. long and 20 ft. high?
36. If 450 pounds of merchandise can be carried 26 miles
for 30/, how many miles can 3 tons be carried for $4 ?
262 RATIO AND PROPORTION.
37. If 85 tons of coal are required to run 6 engines 17 hours
a day for a certain number of days, how many tons will be
required to run 25 engines 12 hours a day for the same num-
ber of days ?
38. If 500 lbs. of wool worth 42 f a pound are given for 75
yds. of cloth If yds. wide, how much wool worth 36 / a pound
should be given for 27 yds. that is 1^ yds. wide ?
39. If it costs $ 135.00 to carry 855 pounds 64 miles, what
will it cost to carry 1288 pounds 15^ miles ?
40. If 200 rods of wall can be built by 25 men in 9J days
of 10 hours each, how many rods can be built by 12 men in 1
day of 12 hours ?
41. If it costs a certain family 1 700 a year to live in Brown-
ville, and the cost of living is twice as great in Chicago as in
Brownville, what will it cost to live 8 months in Chicago ?
42. How many men will be required, working 12 hours a
day for 250 days, to dig a ditch 750 ft. long, 4 ft. wide, and 3
ft. deep, if it requires 27 men, working 13 hours a day for (52
days, to dig a ditch 403 ft. long, 3 ft. wide, and 3 ft. deep ?
43. If 9 men working 12 hours a day for 7 days can make
7 cases of boots, in how many days of 11 hours each can 3 men
and 4 boys (one boy's work being equal to f of the work of a
man) make 33 cases of the same kind of boots ?
44. Wishing to find the number of bricks in a wall 6 rods
long, 4 feet high, and 13 inches thick, I found that a part of
the wall 6f feet long, 2 feet high, and 13 inches thick, con-
tained 330 bricks. How many bricks did the whole wall
contain ?
45. Wishing to find the weight of a block of marble 5 feet
long, 2 feet wide, and 1^ feet thick, I weighed a smaller block
6 inches long, 4 inches wide, and 2 inches thick, and found it
weighed 4 lb. 5 oz. What was the weight of the larger block ?
46. The weight of a cubical block of granite measuring
2 feet on each edge is 1352 pounds. What is the weight of a
cubical block measuring 4 feet on each edge ?
PARTNERSHIP. 263
PARTNERSHIP.
656. Illustrative Example. A and B associated them-
selves together in business for one year. A invested $ 500
and B % 700, agreeing to share their gains or losses in pro-
portion to their investments. They gained $ 2700. What
was each person's share ?
WKITTEN WORK. Explanation. — The whole invest-
$ 2700 X 5 ment was $ 1200, of which A put in
— =$1125. AS gain. ^^ and B 3!^. A should then have
i4 of $2700, or 1 11 25, and B should
^^'^^^^'^ = $ 1575. B's gain. have 3-^ of 1 2700, or $ 1575.
12 ^ws. A's,$ 1125; B's, $1575.
Each person's share of the gain can be found by proportion, thus :
$ 1200 : $ 500 = $ 2700 -.x. a; = $ 1125 A's gain.
1 1200 : $ 700 = $ 2700 -.x. x = % 1575 B's gain.
QbQ, The gains or losses of a partnership are shared
according to the agreement or contract of the partners.
667. In the following examples, when no agreement or
contract is mentioned, divide the gains or losses in propor-
tion to the capital invested by each partner and the time
it is employed.
Note. Some simple examples in partnership have already been given.
See page 113, Example m, and page 114, Example 196.
658. Ezamples for the Slate.
47. Two men, A and B, formed a partnership, A putting in
$5000, and B $3000. They gained $3000. What was the
share of each?
48. Blood and Searle shipped coal from Philadelphia to New
York. Blood had on board 450 tons, and Searle 900 tons. It
became necessary to throw overboard 250 tons. What was the
loss to each person ?
264 RATIO AND PROPORTION,
49. A bankrupt owed to M $ 900, to K $ 350, and to and
P 1 500 each ; his whole property was sold for $ 1584.80, of
which $158.48 was used to pay the expenses of the sale.
What was each person's share of the remainder ?
50. Of a store valued at 1 90000, A owned 1 fourth, B owned
1 third, and C the rest. The store was insured for | of its
value, and was entirely consumed by fire. What was the loss
to each owner ?
51. Divide $ 1500 among three persons so that their shares
shall be in the proportion of 3, 4, and 5.
52. Hinds, Bascom, and Ladd traded in company. Hinds
put in $2500 for 10 months, Bascom $2300 for 11 months,
and Ladd conducted the business, which was considered equal
to $ 2000 in trade for 12 months. They gained $ 1486. What
should each receive ?
53. X and Y received $ 857.50 for grading a road. X fur-
nished 5 hands for 20 days, and 6 others for 15 days ; Y fur-
nished 10 hands for 12 days, and 9 others for 20 days. What
was the share of each contractor ?
54. Band and Parker engaged in trade. Band had in trade
$ 1000 from January 1 till April 1, when he withdrew $ 550 ;
July 1 he added $ 700. Parker had in trade $ 3000 from Eeb.
1 to Oct. 1, when he added $ 300 ; Nov. 1 he withdrew $ 900.
The net gain during the year was $ 3500. What was the share
of each ?
669. Questions for Review.
What is a ratio ? Name the terms of a ratio. What is a simple
ratio ? a compound ratio ?
What is a proportion ? Which are the means of a proportion ?
the extremes ? What is a mean proportional between two numbers ?
When are four quantities inversely proportional ? When four general
quantities form a proportion, what two products are equal ? How do
you find a missing extreme ? a missing mean ? How do you solve
examples by simple proportion ? When is compound proportion
used?
How are the gains or losses of a partnership shared ?
INVOLUTION, 265
SEOTIOlSr XYIII.
POWERS AND ROOTS.
TNVOLUTIOK
660. Kame some products made by using 3's only as
factors. Ans. 9, which equals 3x3; 27, which equals
3x3x3; 81, which equals 3x3x3x3.
A product made by using only equal factors is a power.
661. A power made by using two equal factors is a
second power. A power made by using three equal fac-
tors is a third power. A power made by using four equal
factors is a fourth power; and so on.
What is the second power of 2 ? the third ? the fourth ? the fifth ?
662. The process of forming powers is involution.
Note. The process of forming any power of a number is sometimes
called raising the number to that power; the process of forming the second
power is called squaring the number ; the process of forming the third
power is called cubing the number.
663. The second power of 3 is indicated thus, 3^ ; the
expression is read, " The second power of three." The third
power of 3 is indicated thus, 3^; the expression is read,
" The third power of three " ; and so on.
Note. The names square and cube are often used for "second power"
and ** third power," because the contents of a square is found by raising to
the second power the number of units in one of its sides, and the contents
of a cube by raising to the third power the number of units in one of its
edges.
664. The small figure above and at the right, which
shows to what power a number is raised, is the index or
exponent of the power.
266 POWERS AND ROOTS.
QQb, Oral Exercises.
a. Kame the squares of the numbers from 1 to 10 inclusive.
b. What is the cube of 1 ? of 2 ? of 3 ? of 4? of5?
c. What is the fourth power of 2 ? of 3 ? the fifth power
of 2?
d. What is the square of ^ ? of ^ ? of § ? of 0.5 ?
e. What is the cube of ^ ? of i ? of | ? of 0.2 ?
QQQ. Examples for the Slate.
1. Find and commit to memory the cubes of the integers
from 1 to 10.
Find the powers indicated below.
(2.) 171 (6.) (H)^ (10.) 11^ (14.) (160^.
(3.) 281 (7.) 10.22. (11.) (1)6. (15.) 2.42.
(4.) {-hY' (8.) 15^ (12.) 0.5^ (16.) 0.242.
(5.) 0.172. (9.) 0.12^ (13.) 1.92. (17.) 0.16^.
EVOLUTION.
667. Name one of the two equal factors of 4 ; of 9 ; of
25; of 36; of 64; of 81.
Name one of the three equal factors of 8 ; of 27 ; of 125 ; of 216.
One of the equal factors which produce a number is the
root of that number.
668. One of the two equal factors of a number is its
second or square root. One of the three equal factors of
a number is its third or cube root. One of the four equal
factors of a number is its fourth root, and so on.
669. The process of finding the root of a number is
evolution.
Note. The process of finding the root of a number is sometimes called
extracting the root.
SQUARE ROOT, 267
670. The second or square root of 4 is indicated thus,
V/4; the third or cube root of 8 is indicated thus,.^; the
fourth, root of 16, thus, ^T6; and so on. These expres-
sions are read, " The square root of four "; " The cube root
of eight "; " The fourth root of sixteen."
The symbol V^ is called the radical sign. The small
figure at the left of the radical sign is called the index of
the root.
The root of a number is also indicated by a fractional
exponent. Thus, 4* means the same as V^4 ; 8* means the
same as ^.
SQUARE ROOT.
671. Oral Exercises.
a. What is one of the two equal factors of 25 ? of 49 ?
b. What is the square root of 100 ? of 400 ? of 10000 ? of 1 ?
c. What is the square root of J ? of ^ ? of ^ ? of ^^ ?
d. What is the square root of 0.01 ? of 0.09 ? of 0.81 ?
e. v/i21 = ? g. v/l44=? i. v/| = ? k. \ll2\ = ?
t. Vi^-? h. \llM = '^ j. v/i| = ? 1. v/30i = ?
To find the Number of Terms in the Square Root of a given
Number.
672. By squaring 1 and 9, 10 and 99, 100 and 999, etc.,
we obtain the followincr results :
1002= 10000.
etc.
These results show that when a number has one term its
square is expressed by one or two figures ; when a number
has two terms the square is expressed by three or four
figures ; when a number has three terms the square is ex-
pressed by five or six figures, and so on. Hence,
1/XA/i.XJ. UO-IV
J iV^XiVJ VV Xllii iOOlAlUO .
1.
102= 100.
1002= 10000.
81.
99^ = 9801.
9992 = 998001.
268 POWERS AND ROOTS.
673. If a numerical expression he se;parated into periods
of two figures eachy beginning with the units' figure, the
number of periods will be the same as the number of terms
in the square root.
To find the Parts of a Second Power.
674. To find what parts a second power is made up of,
we may take a number consisting of tens and units, 36 for
example, and raise it to the second power.
36- 30 + 6= 30 + 6
36= 30 + 6= 30 + 6
216 180 + 36 30x6 + 6^
108 900 + 180 302 + 30x6
1296= 900 + 360 + 36= 30^ + 2 x (30 x 6) + 6^
The written work above shows the partial products obtained by mul-
tiplying each term of the multiplicand by each term of the multiplier.
From this we see that the second power of a number consisting of
tens and units is made up of three parts, —
(1.) The square of the tens.
(2.) Twice the product of the tens by the units.
(3.) The square of the units.
These parts may be expressed by the formula,
Tens^ + 2 (tens x units) + units^.
676. Illustrative Example I. What is the square root
of 1296 ? *
WRITTEN WORK. Explanation. — We first find the
Formula, uumber of terms in the root by sepa-
Tens2 + 2(ten8 x units) + units2. rating the expression 1296 into periods
12'96 (36 of two figures each, beginning with the
(3 tens)2 = 9 units' figure. The square root of this
\~oo number will consist of tens and units.
6 tens X 2 = 6 tens) 39 ^^ ^^^ ^^^^ ^j^.^ number must have
6 tens X 6 = 36^ jj^ -^ ^^^ square of the tens of the root,
36 plus twice the product of the tens by the
6* = .36 units, plus the square of the units.
* For another method of finding square roots, see Appendix, page 312.
SQUARE ROOT, 269
As the first part of the power, the square of the tens^ is hundreds,
the 12 hundreds of the given number must have in it the square of the
tens of the root.
The greatest square in 12 (hundreds) is 9 (hundreds), the square root
of which is 3 (tens). This we write as the first term, or tens, of the root.
Taking the square of 3 (tens), or 9 (hundreds), out of 12 (hundreds),
there remain 3 (hundreds).
As the second part of the power, tivice the product of the tens by the
'xnits, is tens, we unite the 9 tens of the given number with the 3
hundreds remaining, and have 39 tens.
This 39 tens has in it a product of which twice the tens of the root
is one factor, and the units of the root the other factor. Dividing the
39 tens by twice 3 tens, 6 tens, we find 6 (units) to be the other fac-
tor, which we write as the second term or units of the root.
Multiplying the 6 (tens) by 6 (units), and taking the product 36
(tens) out of 39 (tens), we have 3 (tens) left.
As the third part of the power, the square of the units, is units, we unite
the 6 units of the given number with the 3 tens remaining, and have
36 units. This 36 units has in it the square of the units of the root.
Subtracting the square of 6 units, or 36, from 36, nothing remains.
Ans. 36.
676. Illustrative Example II. What is the square
root of 1159.4025?
Explanation. — To extract
the square root of the in-
tegral part of the number
1159.4025, we proceed as in
Illustrative Example I. Hav-
ing found this part of the
root, we consider it as tens in
reference to the next term,
double it for a new divisor,
form a new dividend, and
proceed as before.
So, whatever the number
of terms in the root, having
found a part of them, we
consider the part found to
WRITTEN WORK.
Formula, Tens^ -
f 2 (tens X units) + units^
11'59.40'25
(34.
05
9
3
X2:
= 6)25
6
x4^
4^
= 24
19
16
84
340
X 2-
x2--
^680)3402
680
x5--
3400
25
5'= 25
be the tens, double it for a new divisor, and proceed as before.
270 POWERS AND BOOTS.
677. Rule.
To extract the square root of a number :
1. Beginning with the units' figure, point off the expres-
sion into periods of two figures each.
2. Find the greatest square in the number expressed by
the left-hand period, and write its square root as the first
term of the root.
3. Subtract this square from the part of the number
used, and ivith the remainder unite the next term of the
given number for a dividend.
4. Double the part of the root already found for a di-
visor, and by this divide the dividend, writing the quotient
as the next term of the root.''^
5. Multiply the divisor by this term, and subtract the
product from the dividend. With the remainder unite the
next term of the given number and subtract from the num-
ber thus formed the square of the term of the root last found.
6. If there are more terms of the root to be found, unite
vnth the remainder the next term of the given number, take
for a divisor double the part of the root now found, and
proceed as before.
Note I. When, as in the work of Illustrative Example II., the divisor
is not contained in the dividend, place a zero as the next figure in tlie root,
place also a zero at the right of the divisor, and for a new dividend unite
the next two tertns of the given number ivith the previous dividend.
Note II. When there is a remainder after all the terms of the given
number have been used, annex zeros to the remainder, and continue the
work as far as desired.
Note III. The square root of a common fraction may be obtained by
extracting the roots of both numerator and denominator, when they are
perfect squares. When they are not perfect squares, first change the frac-
tion to a decimal, and then extract its square root.
* If this quotient shotQd prove to be too large, mqJce it less and repeat the work.
SQUARE BOOT, 271
678. Examples for the Slate.
Roots of numbers not perfect squares may be found to thousandths.
Find the square root
18. Of 18769. 21. Of 66.7489. 24. Of 54802.81.
19. Of 811801. 22. Of 55.2049. 2^. Of 5.004169.
20. Of 8779.69. 23. Of 0.1849. 2Q. Of 3656.6209
Find the square root-
27. Of ^5. 31. Off. 35. Of 37. 39. \J oJM = ?
28. Ofi§§. 32. Off^. 36. Of 5.39. 40. v/Mx40=.?
29. Of if. 33. Of7T\. 37. Of 0.78. 41. V272.25 = ?
30. Of^^. 34. Of8|-. 38. Of 2. 42. ^4x268'^-?
679. Applications.
43. What is the length of one side of a square which con-
tains 8836 square feet ?
44. A body of troops consisting of 2401 men has an equal
number in rank and file. How many are there in each ?
45. Find the side of a square that will contain as much
surface as a rectangle 280 feet long and 70 feet wide.
46. Find the mean proportional between 42 and 168.
Note. The mean proportional between two numbers is the square root
of their product. (See p. 256, Note I.)
47. Find the mean proportional between 56 and 224.
48. How many rods of fence will be required to enclose a
square lot of 3 acres ?
49. On one side of a roof there are laid 5000 slates, the
number in the length being twice the number in the breadth.
What is the number each way ?
50. A and B each own a 10-acre lot. A's lot is square, and
B's is twice as long as it is wide. How much greater length of
fence will B require to enclose his lot than A to enclose his ?
51. There is a rectangular court paved with 1728 paving-
stones each 15 inches square ; the length of the court is to thft
width as 4 to 3. What is the length and width of the court ?
52. A rectangular block of granite is 8 ft. high, square at
the base, and contains 162 cubic feet. What is the length of
one side of the base ?
272 POWERS AND BOOTS.
CUBE ROOT.
680. Oral Exercises.
a. What is one of the three equal factors of 8 ? of 27 ? of
64 ? of 125 ? of 216 ? of 1000 ? of 1728 ?
b. What is the cube root of 0.001 ? of 0.008 ? of 0.027 ? of
0.216? of 1.728?
^- ^i=? ^S=? M=? ^^=? f^ = ? ^3F=?
To find the Number of Terms in the Cube Root of a given
Number.
681. By cubing 1 and 9, 10 and 99, 100 and 999, etc.,
we obtain the following results :
13 = 1 103= jLOOO 100^= 1000000
9^ = 729 99^ = 970299 999^ = 997002999 -
These results show that when a number has one term, its
cube is expressed by one, two, or three figures ; when a
number has two terms, its cube is expressed by four, five,
or six figures ; when a number has three terms, its cube is
expressed by seven, eight, or nine figures, and so on. Hence,
682. If a numerical expression de separated into periods
of tliree figures each, heginning with the units' figure, tlic
n2imher of periods will be the same as the number of terms
in the cttbe mot.
To find the Parts of a Third Power.
683. To find what parts a third power is made up of,
we may take a number consisting of tens and units, 36, for
example, and raise it to the third power.
36^= 1296= 30^ + 2 X (30x6) + 62
36 = _ 30 + 6
7776 '(SO'xQ) + 2 X (30T 6')'+ 6'
3888 30^ + 2^ ( 30^ x 6 ) + ( 30 x 6^ )
36« = 16656 = 30^ + 3^ lsO^6)7s7(S0~x6')V6'
CUBE ROOT. 273
The foregoing written work shows the partial products obtained by
multiplying each term of the square of 36 by each term of 36.
From this we see that the third power of a number consisting of
tens and units is made up of four parts :
(1.) The cube of the tens.
(2.) Three times the product of the square of the tens by the units.
(3.) Three times the product of the tens by the square of the units.
(4.) The cube of the units.
These parts may be expressed by the formula,
Tens* + 3 (tens^ x units) + 3 (tens x units^) + units.*
684. Illustrative Example I. What is the cube root
of 262144 ? *
WRITTEN WORK. Explanation. — We first
Formula, find the uumber of terms
Te„»3+3(ie„s^x„„ii.)4-3(iensx„„lis^, + „„ii,3. .^ ^^^ ^^^^ ^^ separating
262'144 (64 the expression 262144 into
(6 tens)^ = 216 periods of three figures
(6 tens)2 X 3 = 108 luiud.) 461 ^^^h, beginning with the
lOShund. x4= 432 units' figure. The cube
294
6tensx42x3= 288
root of this number will
consist of tens and units.
We know that this number
64 must have in it the cube of
43 _ g4 the tens of the root, plus
— three times the product of
the square of the tens by the units, plus three times the product of the
tens by the square of the units, plus the cvhe of the units.
As the first part of the power, the cube of tens, is thousands, we find
the greatest cube contained in 262 (thousands), which is 216 (thou-
sands), and write its cube root 6 (tens) as the first term, or tens, of the
root.
Taking the cube of 6 (tens), 216 (thousands), out of 262 (thousands),
there remain 46 (thousands).
Now, as the second part of the power, three times the product of the
square of the tens by the units, is hundreds, we unite the 1 hundred of the
given number with the 46 thousands remaining, and have 461 hundreds.
* For another method of finding cube roots, see Appendix, page 313.
274
POWERS AND ROOTS.
This 461 hundreds has in it a product of which three times the square
of the tens of the root is one factor, and the units of the root is the other
factor. Dividing 461 (hundreds) by three times the square of the
tens, or 108 (hundreds), we find 4 (units) to be the other factor, which
we write as the second term, or units of the root.
Multiplying 108 (hundreds) by 4 (units) and taking the product 432
(hundreds) out of 461 (hundreds), we have 29 (hundreds) left.
Now, as the third part of the power, three times the product of the
tens by the square of the units, is tens, we unite 4 (tens) of the given
number with the 29 (hundreds) remaining, and have 294 (tens).
We take out of this number three times the product of the tens by
the square of the units, 288 (tens), and have 6 (tens) left.
Now, as the fourth part of the power, the cube of the units, is units,
we unite the 4 units of the given number with the 6 tens remaining,
and have 64 units.
We take the cube of the 4 units out of this number, and nothing
remains. Ans. 64.
What is the cube
Explanation. — To ex-
tract the cube root of
the integral part of the
number 126732.947167,
we proceed as in Il-
lustrative Example I.
Having found this part
of the root, we consider
it as tens in reference
to the next term, take
three times its square
for a new divisor, form
a new dividend, and
proceed as before.
So, whatever the num-
ber of terms in the root,
having found a part of
them, we consider that
part to be the tens, take
three times its square
for a new divisor, and
proceed as before.
685. Illustrative Example II.
root of 126732.947167 ?
WRITTEN WORK.
Formula,
Ten8*+3 (tens^x units) + 3 (tens x units^l + units^.
5' =
5a X 3 = 75
126'732.947a67 (50.23
125
60^x3=7500)17329
7500 X 2 =
50x2^x3 =
2« =
5022 ^ 3 =756012) 2269391
756012x3 = 2268036
15000
23294
600
226947
8
502x3^x3
13556
13554
3« =
27
27
CUBE ROOT. 275
686. Rule.
To extract the cube root of a number :
1. Beginning loith the units' figure, foint off the expres-
sion into periods of three figures each.
■ 2. Find the greatest cube in the number exjpressed by the
left-hand period, and write its cube root as the first term
of the root.
3. Subtract this cube from the part of the number used,
and with the remainder unite the next term of the given
number for a dividend.
4. Take three times the square of the part of the root
already found for a divisor, and by this divide the divi-
dend, writing the quotient as the next term of the root.*
5. Multiply the divisor by this term, and subtract tJie
product from the dividend. With the remainder unite the
next term of the given member.
6. Subtract from the number thus formed three times
the product of the first term of the root by the square of
the second. With this remainder unite the next term of the
given number.
7. Subtract from the number thu^ formed the cube of
the second term of the root.
8. If there are more terms of the root to be found, unite
with the remainder the next term, of the given number, take
for a divisor three times the square of the 'part of the root
now found, and proceed as before.
Note I. When, as in the work of Illustrative Example 11., the di-
visor is not contained in the dividend, place a zero as the next figure in
the root ; also place two zeros at the right of the divisor, and for a new
dividend unite the next three terms of the given number icith the previous
dividend.
* If this quotient sliould prove to be too large, make it less and repeat the yturk.
276 LOWERS AND ROOTS.
Note 11. The cube root of a common fraction whose numerator and de-
nominator are both perfect cubes may be found by taking the cube root of
die numerator and of the denominator.
But when the numerator or the denominator of the fraction is not a per-
fect cube, change the fraction to a decimal and then extract its cube root.
Note III. When there is a remainder after all the terms of the given
number have been used, annex zeros, and continue the operation as far as
desired.
687. Examples for the Slate.
Eoots of numbers not perfect cubes may be found to the fourth term.
What is the cube root
53. Of 12167? 57. Of 2924207? 61. Of 832 ?
54. Of 97336? 58. Of 14172.488 ? 62. Of 346?
55. Of 405.224? 59. Of 82881.856? 63. Of 0.353 ?
56. Of 941192? 60. Of 817.400375 ? 64. Of 0.80 ?
What is the cube root
m. Of tWf? 68. Of-i^^? 71. ^ 027 = ?
66. Of ^^% ? 69. Of 0.87 ? 72. ^0^1^ = ?
67. Of 42^ ? 70. Of 63^ ? 73. ^2~= ?
688. Applications.
74. What is the length of one edge of a cube which contain?
36594368 cubic inches ?
75. What is the length of a cubical pile of wood which con-
tains 2 cords ?
76. What is the depth of a cubical cistern which contains
300 gallons?
77. What is the depth of a cubical bin which will hold 65
bushels of grain ?
78. What will be the cost of lead, at 12 1 cents per lb.,
1^ lbs. to the square foot, to line a cubical box containing
15 f cubic feet ?
689. Exercises upon Drill Table No. 2 (page 60).
247. ''^'ind the sq. root of the numbers given in column B (25 examples).
248. Find the cu. root of the numbers given in column E (25 examples).
MENSURATION, 277
SEOTIOIsr XIX.
MENSURATION.
TRIANGLES.
690. A plane figure bounded by three straight lines is
a triangle. Fig. i.
691. A triangle having a right angle
in it is a right triangle.
Pig- 2. Note. All other triangles are oblique
triangles. Right Triangle.
692. A triangle having two sides equal is an
isosceles triangle. Kg. 3.
Isosc gIgs
Triangle. 693. A triangle having its three
sides equal is an equilateral triangle.
694. Any side of a triangle may be taken
to be the base. The length of a perpendicular Triangle.
Fig. 4. Fig. 5. drawn from the opposite
vertex to the base, or
the base prolonged, is
the height.
Base. Base.
QUADRILATERALS.
695. A plane figure bounded by four straight lines is a
quadrilateral. p. g Fig. r.
696. Two straight lines hav-
ing the same direction are
parallel lines.
697. A quadrilateral having ^p«"^- Trapezoid,
none of its sides parallel is a trapezium. (Fig. 6.)
278 MENSURATION.
698. A quadrilateral having two of its sides parallel is
a trapezoid. (Fig. 7.) ^. ^. ^
■^ \ o / Pig 8 Pig 9
699. A quadrilateral hav-
ing its opposite sides parallel
is a parallelogram. (Figs. 8,
9, 10, and 11.) Rectangle. Square.
Note. A rectangle is a parallelogram whose angles are all right angles.
A square is a rectangle whose
sides are all equal. Fig. 10. Fig. 11.
A rhomboid is a parallel©- r
gram whose angles are not /
right angles. A rhombus is a /
parallelogram whose angles are L
not right angles and whose Rhomboid. Rhombtw.
sides are all equal.
^'g- ^^- 700. Any side of a parallelogram may
A 7 be taken to be the base. The length of a
/ i / perpendicular drawn from the opposite side
/ to the base is the height.
Base
POLYGONS. Fig 13
701. A plane figure bounded by straight
lines is a polygon. A polygon having all
its sides equal and all its angles equal is a
regular polygon.
Regular Polygon.
702. Exercises.
a. Which of the following nained figures are regular poly-
gons : square, rhombus, equilateral triangle, trapezium ?
b. Draw a right triangle ; an equilateral triangle ; an
isosceles triangle; a right-angled isosceles triangle.
c. Draw a rhombus ; a rhomboid ; a rectangle ; a square ;
a. trapezoid ; a regular polygon of six sides.
AREAS OF POLYGONS. 279
AREAS OF POLYGONS.
703. The area of any parallelogram is equal to that of
a rectangle of the same base and height.
For, if we cut off a right triangle from
one end and put it on at the other end,
as in Fig. 14, we change the form to
a rectangle having the same base and
height as the parallelogram, though we do not change the
area. But the area of a rectangle is equal to the product
of its base and height. Hence the
Rule.
To find the area of any parallelogram : Multiply the base
hy the height. (See Art. 313.)
704. A TRIANGLE is half a parallelo- ^^«- ^^•
gram of the same base and height. For
two equal triangles may be placed, as
shown in Fig. 15, so as to make a
parallelogram. Hence the
Rule.
To find the area of a triangle : Multiply the base hy the
height, and divide the product hy 2.
To find the area of a triangle when the three sides are
given : Find half the sum of the three sides; from this sub-
tract each side separately; multiply together the four results
thus, obtained, and extract the square root of the product.
705. A TRAPEZOID may be divided into ^ig. le.
two triangles having the parallel sides of / y'''\
the trapezoid for bases and the distance be- / /' \ \
tween them for their height. Hence the [^l LA
280 MENSURATION.
Rule.
To find the aiea of a trapezoid : Multiply the sum of the
parallel sides hy the distance hetween them, and divide the
product hy 2.
706. Any polygon may be divided into triangles. Hence
for cases not met by the rules already given, we have the
Rule.
To find the area of a polygon : Divide it into triangles,
and find the sum of their areas.
707. Examples for the Slate.
The pupil will be aided by drawing figures to illustrate the examples
which follow.
1. How many square feet are there in a parallelogram
whose base is 3 feet 4 incheSj and height 1 foot 3 inches ?
2. What is the area of a triangle, the base being 20 feet,
and the height 10 feet 4 inches ?
3. What is the area of a right triangle, the sides, which in-
elude the right angle, being 20 feet and 18 feet long respec-
tively ?
4. How many square feet are there in a building lot having
two parallel sides respectively 140 feet and 116 feet long, and
83 feet apart ?
5. How many acres are there in a four-sided field having
two parallel sides, which are 60 rods apart and 40.05 rods and
64.08 rods long respectively ?
708. Exercises.
a. Draw a triangle having a base 8 centimeters long ; meas-
ure its height and find its area in square centimeters.
t. Draw a polygon of five sides, letting one of them be a
decimeter long ; divide the polygon into triangles, and find the
sum of their areas in square centimeters.
CIRCLES. 281
CIRCLES.
709. A line drawn from the centre of Jig. 17.
a circle to any point of the circumfer-
ence is a radius. A line drawn through
the centre of a circle terminated each
way by the circumference is a diameter.
A diameter is double the radius.
710. A circle may without sensible
error be said to consist of triangles whose bases form the
circumference and whose vertices
are at the centre, the height of
the triangles being equal to the
radius of the circle. Hence the
Rule.
To find the area of a circle :
Multiply the circumference hy the
radius, and divide the product hy 2.
711. If the circumference of a circle be divided by its
diameter, the quotient, expressed to the nearest ten-thou-
sandth, is 3.1416.* Hence the following
Formulas relating to the Circle.
1. Circumference = Diameter x 3.1416.
^ _. , Circumference
2. Diameter = » ^ . ^^^ •
3. Circumference = 2 Radius x 3.1416.
Circumference
4. Radius
2 X 3.1416
712. The rule in Art. 710 may be expressed thus :
. Circumference x Radius -n j.^- • xi i p ^t.
Area = Putting in the place of the
* The number 3| is accurate enough for most purposes.
282 MENSURATION.
circumference its value as given in Formula 3, we have :
A Radius x 2 x 3.1416 x Radius tt ^^ n i
Area = ^ — ^ Hence the formulas :
5. Area = Radius^ x 3.1416.
6. Radius = ' ^^^*
/ Area
y 3.1416
713. Examples for the Slate.
6. What is the distance round a circular pond which is 24
feet across the middle ?
7. If the radius of a circle is 4 feet, what is the circumfer-
ence ?
8. How. many feet is it across a circular grass-plot which
measures 100 feet around ?
9. How many square feet of glass in a circular window 4 ft.
10 in. across?
10. If a horse is tethered so that his head can reach 10^ feet
from the stake in any direction, over how many square feet can
he graze ?
11. How many rods of fencing will he required to enclose
a circular piece of ground which contains an acre ?
12. How many planks 2 inches thick can he sawed from a
log 10 feet in circumference, allowing \ of an inch for each saw^
cut, and 2 slahs, each at least 5 inches thick, to he cast aside ?
13. In the Yosemite Valley is the stump of a tree 32 feet
across ; what is the circumference ? Allowing 3 square feet as
standing-room for one person, how many persons can stand on
the top of the stump ?
RIGHT TRIANGLES.
714. In a right triangle, the side oppo-
site the right angle is the hypothenuse,
one of the remaining sides is the base,
and the other is the perpendicular.
Fig. 19.
RIGHT TRIANGLES.
283
Fig. 20.
~
—
vx;<^
w
X.
B
715. Suppose the figure A B c to be a riglit triangle,
whose sides are 3, 4, and 5 feet re-
spectively. A square formed upon
the hypothenuse, A c, will contain 25
square feet ; one formed upon the base,
B c, will contain 16 square feet; and
one formed upon the perpendicular,
A B, will contain 9 square feet. Thus
it appears that the square upon the
line A c is equal to the sum of the
two squares upon A B and B c.
716. It can be shown to be true in general that the
squo.re upon the hyjpothenuse of a right triangle is equal to
the sum of the squares upon the other two sides. Hence the
Rules.
I. To find the hypothenuse, having the base and perpen-
dicular given : Square the base and the perpendicular, add
the squares, and extract the square root of the sum.
II. To find the base or perpendicular, having the hy-
pothenuse and the other side given : Square the hypothenuse
and the given side, subtract the latter square from the former,
and extract the square root of the remainder.
717. Examples for the Slate.
14. The base of a right triangle being 18 feet, the perpen-
dicular 24 feet, what is the hypothenuse ?
15. The hypothenuse of a right triangle being 32.5 feet, the
base 26 feet, what is the perpendicular ?
16. If the foot of a ladder 20 feet long be put 8 feet from
the base of a wall, how high will it reach ?
17. Two boats started at the same point, and sailed one
north 5280 feet and the other east 3960 feet. How far apart
were they then ?
284 MENSURATION.
18. Near Lexington battle-ground was an elm partially
broken off and fallen across the street. If the height of the
fracture was 15 feet from the ground, and the distance from
the foot of the tree to the point where the top reached the
ground was Q>Q> feet, what was the entire height of the tree be-
fore it fell ?
19. B lives 40 rods east of A, 72 west of C, and 30 nort!
of the school-house. If A and C both call for B in going
to school, how much farther does each travel than if he goes
directly to school ?
20. Four boys, Dwight, Brown, Chase, and Doane, live in
different places, as follows : Dwight 40 rods north of the school-
house ; Brown 60 rods east ; Chase 57 rods south ; and Doane
36 rods west. What is the shortest distance one of the boys
must travel to visit all the rest, and reach his own home ?
21. What is the length of a diagonal, that is, the distance
from one corner to the opposite corner of the floor of a room
15 feet square ?
22. If the height of this room is 9 feet, what is the distance
from one corner of the floor to the opposite upper corner of the
ceiling ?
Note. The diagonal of the floor is the base of the right triangle of which
the required diagonal of the room is the hypothenuse. Hence the square of
the diagonal of a room, or of any rectangular solid, equals the sum of the
squares of its three dimensions.
23. What is the length of the longest rod that, without
bending, can be put into a box 1 yard long, 1 foot wide, and 9
mches high, measured on the inside ? Fig. 21.
24. Find the height and area of an equi- m
lateral triangle whose sides are 10 feet long ?
Note. The perpendicular m n divides the triangle /
into two equal right triangles. A n is then one half /
of AB. ^Z_ ^
25. Find the height and area of an isosceles
triangle whose base is 20 feet long, the other sides being each
26 feet long ?
Fig. 22.
SOLIDS.
SOLIDS.
Fig. 23.
^85
Kg. 24.
I
l|lil!.lll l ll ! l.|:{Jll|IIM I |||| ! |l||
Prism.
Cube. Eectangulax Solid.
718. A prism is a solid bounded by two equal and
parallel polygons and a number of parallelograms.
Note I. It will be seen that all rectangular solids are prisms.
Note II. The equal and parallel polygons are the hoses of the prism,
and the parallelograms taken together form its convex surface. When the
bases are regular polygons and the parallelograms are perpendicular to the
bases, the prism is a regular prism.
Fig. 26. Fig. 27.
Pyramid. Cylinder. Cone.
719. A pyramid is a solid bounded by one polygon,
which is the base, and a number of triangles which terminate
in a common point called the vertex.
Note. The triangles form the convex surface of the pyramid. "When
the base of a pyramid is a regular polygon, and a line drawn from the
vertex to the middle of the base is perpendicular to the base, the pyramid
is a regular pyramid.
720. A solid formed by turning a rectangle about one
of its sides as an axis is a cylinder.
Note. As the rectangle is turned, the side opposite the axis describes
the convex surface of the cylinder, and the other two sides describe parallel
circles, which are the bases of the cylinder.
286
MENStlRATIOi^.
Fig. 28.
Fig. 29.
721. A solid formed by turning a right triangle about
one of its shorter sides is a cone.
Note. As the triangle is turned, the hypothenuse describes the convex
surface of the cone, and the side perpendicular to the axis describes a
circle, which is the base of the cone.
722. If the upper part of
a pyramid or of a cone is cut
off by a plane parallel to the
base, the part that remains is a
frustum of the pyramid or of
the cone.
Frustum of
a Cone.
Fig. 30.
723. The height of any of the solids here defined is the
perpendicular distance from the highest point
above the base to the plane of the base. Thus,
in Fig. 30, the line A B indicates the height of
the solid.
Note. In a regular pyramid or in a cone, the shortest
distance from the vertex to the perimeter (boundary) of the ^^
base is the slant height.
In the frustum of a regular pyramid or of a cone, the shortest distance
between the perimeters of the two bases is the slant height. Thus, in Fig. 30,
the line o p indicates the slant height of the frustum. pj gj^
724. A globe, or sphere, is a solid
bounded by a curved surface, every point of
which is equally distant from a point within,
called the centre.
Note. A circle which divides a sphere into two
equal parts is a great circle of the sphere. A circum-
ference, a diameter, or a radius of a gi'eat circle of a
Sphere.
sphere is also a circumference, a diameter, or a radius of the sphere itself.
Volumes of Solids and Areas of their Convex Surfaces.
726. A PRISM or a cylinder 1 inch high contains as many cubic
inches as there are square inches in the base. If the height be in-
creased to 2, 3, or any n\imber of inches, the volume will be increased
in the same proportion. Hence the
SOLIDS. ^ 287
Rule.
To find the volume of a prism or cylinder : Multiply the
area of the base hy the height.
726. If a PRISM or a cylinder is 1 inch liigh, its convex surface
contains as many square inches as there are inches in the perimeter of
the base. If the height be increased to 2, 3, or any number of inches,
the convex surface will be increased in the same proportion. Hence
the
Rule.
To find the convex surface of an upright prism or cylin-
der : Multiply the perimeter of the base by the height.
727. It can be proved that a pyramid or a cone is equivalent to
^ of a prism or a cylinder of the same base and height. Hence the
Rule.
To find the volume of a pyramid or cone : Multiply the
area of the base by the height, and divide the product by 3.
728. The convex surface of a regular pyramid or cone may be
regarded as composed of triangles whose bases form the perimeter of
the base of the solid, and whose height is the slant height of the sohd.
Hence the
Rule.
To find the convex surface of a regular pyramid or cone :
Multiply the perimeter of the base by the slant height and
divide the product by 2.
120, It can be proved that the frustum of a pyramid or cone is
equivalent to the sum of three pyramids or cones, which have for their
common height the height of the frustum, and whose bases are the
lower base of the frustum, the upper base, and a mean proportional
between them. Hence the
Rule.
To find the volume of a frustum of a pyramid or cone :
Multiply the sum of the areas of the two bases, plus the
square root of their product, by the height, and divide the
product by 3.
288 MENSURATION.
730. The convex surface of the frustum of a regular pyramid or
cone may be regarded as made up of trapezoids whose parallel sides
form the perimeters of the bases, and whose height is the slant height
of the frustum. Hence the
Rule.
To find the convex surface of a frustum of a regular
pyramid or cone : Multiply the sum of the perimeters of
the two bases ly the slant height and divide the product
hy 2.
731. It can be proved that the surface of a sphere is equivalent
to that of four great circles of the sphere. Hence the
Rule.
To find the surface of a sphere : Find the area of a great
circle of the sphere, and multiply it hy 4-
732. A SPHERE may be regarded as composed of pyramids whose
bases taken together form the surface of the sphere and whose height
is the radius. Hence the
Rule.
To find the volume of a sphere : Multiply the convex
surface hy the radius and divide the product hy 3.
733. The last two rules may be expressed by the follow-
ing formulas :
Surface of a sphere = 4 Radius^ x 3.1416.
Volume of a sphere = f Radius^ x 3.1416.
Note. To find the capacity of a cask, see Appendix, page 314.
734. Examples for the Slate.
2Q. How many cubic inches are there in a prism whose base
is 8 inches square, and whose height is 7 inches ?
27. How many cubic feet in a prism 5 feet high and having
for its base a triangle, each side of which is 10 feet long ?
28. How many cubic feet in a pyramid 10 feet high and
having for the base 1 square rod ?
EXAMPLES. 289
29. How many cubic inches in the frustum of a pyramid
whose bases contain 12 and 108 square inches respectively, and
wliose height is 18 inches ?
30. How many bushels of corn can be put into a corn-crib
9 feet square at the bottom, 12 feet square at the top, and 8
feet high ? (See Art. 393, note.)
31. What is the convex surface of a prism, the perimeter of
whose base is 7 yards 2 feet, and whose height is 5 yards 1 foot ?
32. How many square feet in the surface of a four-sided
pyramidal roof, the slant height being 18 feet and the house
20 feet square ?
33. How many feet of boards will cover the sides of an
eight-sided tower, the length of each side of the base being 2
feet 9 inches, that of each side of the top 1 foot 10 inches,
and the slant height 12 feet ?
34. How many gallons will a pail contain that measures on
the inside 14 inches in depth and 11 inches across ?
35. How many square feet of sheet-iron are there in a piece
of stove-pipe 9 feet long and 6 inches in diameter, no allowance
being made for lapping at the joints ?
36. What is the height of a conical tent, if the diameter of
the base is 15 feet and the slant height is 1%^ feet, and how
many cubic feet will the tent contain ?
37. How many square yards of canvas will be required to
make this tent, no allowance being made for seams ?
38. How many gallons will a circular vat contain that meas-
ures across the bottom 12 feet, across the top 15 feet, the depth
being 6 feet ?
39. How many square feet in the surface of a foot-ball 1 foot
in diameter ?
40. At 38/ a square foot, what is the cost of painting a
globe 6 feet in diameter ?
41. How many cubic feet are there in this globe ?
42. If the diameter of the earth is 7900 miles, and 73| % of
the surface of the e^rth is water, how many square miles of the
surface is wav^ii a
290 MENSURA TION.
SIMILAR SURFACES.
735. Figures which have
the same shape are similar
figures.
Note. The corresponding sides
of similar figures are proportional.
Fig. 82.
736. We see from the
illustration above that a figure 1 inch square contains 1
square inch, a figure 2 inches square contains 4 square
inches, and a figure 3 inches square contains 9 square
inches, etc. In general,
The areas of similar figures are to each other as the
squares of their corresponding dimensions.
737. ILLUSTRA.TIVE EXAMPLE I. The area of a certain
triangle is 120 square feet and its base is 24 feet. What is
the area of a similar triangle whose base is 96 feet ?
WRITTEN WORK.
4 4
242 . 9g2 ^ 120 : aj ^^^^^f^^^^ = 1920. Ans. 1920 sq. ft.
738. Illustrative Example II. One side of a trianojle
is 40 feet long; wliat must be the length of a side of a
similar triangle containing twice the area ?
WRITTEN WORK.
1 : 2 - 402 ; a;2 y^ x 40 x 2 = 56.568... Am. 56.568... feet.
739. Examples for the Slate.
43. If a room 16 feet in length requires 22 yards of carpet-
ing to cover the floor, how many yards of carpeting will be
required for a room 20 feet long and of the same shape ?
SIMILAR SOLIDS.
291
44. There is a public park 1320 feet long, containing 25
acres. What is the length of a park of the same shape con-
taining 49 acres ?
45. If a circular lot of land which is 10 rods in diameter
contains 78.5398 square rods, what number of rods will a cir-
cular lot contain which is 5 rods in diameter ?
46. If a pipe 2 inches in diameter discharges 20 gallons of
water in a given time, how many gallons will a pipe 5 inches
in diameter discharge in the same time, no allowance being
made for friction ?
47. If it costs 17/ for tin to make a pail 6 inches high,
what will it cost for tin to make a similar pail 14 inches high ?
48. If it costs $ 72 for material to paint a spire 50 feet high,
what will it cost for material to paint a similar spire 75 feet
high?
49. If the cost of plating a pitcher 6 inches high is % 1.75,
what is the cost of plating a pitcher of the same shape 10
inches high ?
50. What is the height of a pitcher similar to that described
above, of which the cost of plating is $ 4.00 ?
SIMILAR SOLIDS.
, X ^<.
■v
\ N.
t
> ^ ^
s
N
740. Solids which have Fig. 33.
the same shape are similar
solids.
Note. The corresponding dimen-
sions of similar solids are propor-
tional.
741. We see from the illustration above that a cube
whose edoe is 1 inch contains 1 cubic inch, a cube whose
o
edge is 2 inches contains 8 cubic inches, a cube whose edge
is 3 inches contains 27 cubic inches, etc. In general,
The volumes of similar solids are to each other as the cubes
of their corresponding dimensions.
292 MENSURATION.
742. Illustrative Example I. If a cube of lead whose
edge is 3 inches weighs 12 pounds, what is the weight of a
cube of lead whose edge is 2 inches ?
WRITTEN WORK.
3» : 2» = 12 lb. : x %S^ = 3f . a.,. 3f lb.
743. Illustrative Example II. A pyramid 9 feet high
contains 48 cubic feet. What is the height of a similar
pyramid that contains 100 cubic feet ?
WRITTEN WORK.
48 : 100 = 9^ : cc» ^9x9x9x100 ^ 11.49. . . ^,,, 11,49. . . feet.
» 48
744. Examples for the Slate.
51. If an Q^^ 2\ inches in circumference weighs 1 ounce,
what would another of the same form and consistency weigh
whose circumference is 6 inches ?
62. An ox measuring 7 feet in girth weighs 1500 pounds ;
what is the weight of an ox measuring 9 feet in girth ?
53. If a bushel measure is 18^ inches in diameter and 8
inches deep, what must be the diameter and depth of a half-
bushel measure similar in form ?
54. Estimating the mean diameter of the earth at 7912
miles, and that of the moon at 2160 miles, how many bodies of
the size of the moon could be made from the bulk of the earth ?
55. If the bulk of Saturn be 1000 times as great as that of
the earth, what is the diameter of Saturn ?
56. At what distance from the top must a cone 12 inches
high be cut parallel with the base, that the cone may be divided
into two equivalent parts ?
57. Mr. Root has three stacks of hay of similar shape, the
diameters of their bases being respectively 10, 12, and 14 feet ;
if the smallest stack contains 2J tons, what will each of the
others contain?
GENERAL REVIEW. 293
745. General Review, No. 6.
58. Supply the 2d term in the proportion 3f : a; = 8 : 25.
59. What is the mean proportional between 0.8 and 0.72 ?
60. Divide 11900 between two men, in the proportion of 3 to 5.
61. Divide $ 45 among three boys, so that one shall have as
much as the other two, whose shares are as 2 to 7. ^
62. How many pounds can 5 horses draw, if 6 horses can
draw as much as 10 oxen, and 2 oxen can draw 2400 pounds ?
63. Smith and Lee formed a partnership. Smith put in
$1000 for 6 months and $800 for 2 months. Lee put in
$600 for 8 months and was allowed $800 for his services.
They gained $ 1435.50 ; what was each partner's share ?
64. What is the 5th power of 23 ? the cube of 96 ?
65. What is the largest number of men in a regiment of
1000 that can be arranged in a square ; and how many men
will remain ? How many men will there be on each side of
the square ?
66. How many feet of fencing are required to enclose a
square farm containing 15 acres ?
67. A ladder 27f feet long reaches a window 25f feet from
the ground. How far does the foot of the ladder stand from
the house ?
68. What is the diameter of a circle which contains 314^
square feet ?
69. How many rods of fencing on both sides of a road which
surrounds a circular park containing 15.708 acres, the road
. being 3 rods wide ?
70. What must be the depth of a pail that is 10 inches
across, to contain 5 gallons, the sides being upright ?
71. How many square feet of canvas are required to con-
struct a conical tent 14 feet across the bottom and 9^ feet from
the highest point to the ground ?
72. If a pipe 2^ inches in diameter will fill a cistern in two
hours, in what time will a pipe 5 inches in diameter fill the
same? *
294 QUESTIONS FOR REVIEW.
746. Questions 'for Review.
What is a power ? a second power ? a third ? a fourth ? What is
INVOLUTION ? What is squaring a number ? cubing a number ? Give
the squares of the numbers from 1 to 12 ; the cubes of the numbers
from 1 to 10.
What is the root of a number ? What is evolution ? Repeat the
formula used in extracting the square root ; the 'rule. What do you do
when a term of the root proves too large? when the trial divisor is not
contained in the dividend ? How do you extract the square root of a
fraction ? of a mixed number ? Repeat the formula used in extract-
ing the cube root ; the rule. How do you extract the cube root of a
fraction ? of a mixed number ?
What is MENSURATION ? Name and describe the different kinds of
triangles given ; the different kinds of quadrilaterals given. What is
the base of a triangle ? the height ?
How do you find the area of a square ? of a rectangle ? of any
parallelogram ? of a triangle ? of a trapezoid ? of any polygon ?
How do you find the circumference of a circle when the diameter is
given ? when the radius is given ? How do you find the diameter
when the circumference is given ? How do you find the area of a
circle when the radius is given ? the radius of a circle when the area
is given ?
What is a cube ? a prism ? a pyramid ? a cylinder ? a cone? What
is the frustum of a pyramid or a cone ? What is a sphere ? Draw or
name something in the form of each of these solids. What is the
height of any solid ? the slant height of a pyramid or of a cone ? the
slant height of a frustum of a pyramid or of a cone ?
How do you find the volume of a cube ? of any rectangular solid ?
of a prism or a cylinder ? of a pyramid or a cone ? of the frustum of
a pyramid or cone ? How do you find the convex surface of each of
these solids ?
When the diameter and height are given, how do you find the vol-
ume of a cylinder ? of a cone ? of a frustum of a cone ? How do you
find the volume of a sphere ? How do you find the convex surface of
a cylinder ? of a cone ? of the frustum of a cone ? of a sphere ?
When are plane figures similar ? What proportion is there between
the areas of similar figures ? When are solids similar ? What pro-
portion is there between the volumes of similar solids ?
MISCELLANEOUS EXAMPLES, 295
747. Miscellaneous Examples.
73. What is the weight of a bale of cloth contaiiling 13
pieces, 42 yards to the piece, every 3 yards weighing 1;^
pounds ?
74. The sum of three numbers is 55^ ; two of them are 14^
and 24|^ ; what is the third ?
75. ^ -I- ^ + ^ + ^ of a certain number increased by 3|^ equals
40. What is the number ?
76. A trader bought apples at $ 1.62^ per barrel, and imme-
diately sold them at $ 2.25, making % 234.37^. How many
barrels were bought ?
77. Suppose a dividend to be 241.3, and the quotient 0.127,
what was the divisor ?
78. The ridge-pole of a house is 46 feet from the ground, the
eaves 38 feet, the rafters on each side of the roof to the eaves
being 18 feet long. What is the width of the house ?
79. When the ice upon a pond is 10 inches thick, what will
be the value of the ice taken from one acre of the pond at \ of
a cent a pound, 1 cubic foot of ice containing 58^ pounds ?
80. The city tax of Lincoln being f fo, and the State and
county tax 0.15 % ; for what sum is James Otis taxed, who
pays $ 56.22, including 1 1.50 poll-tax ?
81. Of what number is f the | part ?
82. A city collector received 0.8 % for collecting taxes ; he
paid into the treasury $ 94625.64 after deducting his commis-
sion ; what was the whole sum collected ?
83. A coal-dealer purchased 500 tons of coal at $ 7.50 per
long ton, paid $1 per ton for freighting, and sold it for $11
by the short ton. What per cent did he gain ?
84. A can do a piece of work in 1^ hours, A and B in 48
min. ; in what time can B do it alone ?
85. Crane Brothers & Co. purchased oil stocks to the amount
of $ 5714.25, including their commission of :^ % ; the stock,
the par value of which was $ 50 per share, was purchased at
95 % . How many shares were purchased ?
296 MISCELLANEOUS EXAMPLES.
86. When butter is 25 cents a pound, and f of a pound will
pay for f of a dozen eggs, how many eggs will be required to
pay for 6 pounds of raisins, 7 pounds of which cost 98 cents ?
87. What is the difference between the true and bank dis-
count of % 700, due in 90 days, when the legal rate is 7% ?
88. How much would, you receive from a bank, June 12,
1878, for a note of $820, dated April 12, 1878, payable 6
months after date, discount being 6% ?
89. Write a note for 60 days, for which you should get
$300 at a bank, discount being 6% ?
90. If $ 1239 were paid for harvesting the wheat on a lot
of land 400 rods long, 350 rods wide, what should be paid for
harvesting the oats upon a lot 500 rods long, 450 rods wide, the
cost of harvesting oats being f as much as for harvesting wheat ?
91. Bates and Henricks traded in hides for one year.
Bates put in $ 2000 at first ; at the end of 3 months he with-
drew 1 700, and at the end of 7 months put in $ 1000. Hen-
ricks put in 1 1200 at first, and 1 500 more in 4 months. At
the end of 6 months he withdrew 1 200. The gain for the year
was % 2355.75, of which Henricks received 1 1000 for conduct-
ing the business. What was the share of each ?
92. The pyramid of Cheops, in Egypt, is said to contain
82111000 cubic feet of masonry, and to have been 480 feet
high. Allowing 7000000 cubic feet, which are required to
perfect its pyramidal form and to fill its chambers, what is the
length of one side of its base, which is a square ?
93. If you buy figs at the rate of 9 pounds for $ 1.50, and
sell them at the rate of 10 pounds for $ 2, what per cent do
you gain ?
94. What is the average time for paying for $ 200 worth
of goods purchased May 17, 1877, on 4 months' credit ; 1 500
worth, purchased June 18, 1877, on 60 days' credit^ and $300
worth, purchased June 19, 1877, on 90 days' credit ?
95. The captain of a ship at sea finds by his chronometer,
at 12 o'clock, noon, that it is 45 m. past 8 o'clock in the even-
ing at London. What is his longitude ?
MISCELLANEOUS EXAMPLES. 297
96. Pedrick & Closson sold at auction,
2 mattresses, at $ 16.00, which cost 1 13.50. -
8 chairs, at 4.62, " " 3.75.
1 rocker, at 17.50, " " 17.00.
1 set furniture, at 38.00, " " 42.00.
1 " " at 83.50, " " 62.00.
They also sold on commission, at 10%, 5 chairs, at $8 ; 12 chairs,
at % 1.70 ; 1 bureau, at $ 18 ; 1 table, at $ 8 ; 1 lounge, at $ 12 ;
1 stove, at $ 17. What were their net proceeds from the above
sales ?
97. How many bricks 8 in. by 4 in. by 2 in. in the walls of
a building 29 ft. long by 24 ft. wide and 20 ft. high, outside
measurement, having 10 windows 6 ft. by 4 ft. and 2 doors
7 ft. by 4^ ft., the thickness of the walls being 1 foot, and ^ of
the entire wall being mortar ?
Note. In estimating the number of bricks required, masons reckon by
outside measurements, and make no allowance for corners.
98. The circular outlet to a cistern being 4 inches in di-
ameter, what must be the width of a rectangular receiving-
pipe, whose depth is 2 inches, that its capacity may be the
same as that of the discharging-pipe ?
99. When it is 10 A. m. in X, which is 44° 15' 2" W. long.,
what is the time in Y, which is 8° 4' 40'' E. long. ?
100. A, B, and C shipped goods by the same vessel. The
value of A's goods was $50000; of B's, $40000; of C's,
$ 30000. During a storm half of A's goods and one iiftli of
B's were thrown overboard. What should be each man's share
of the loss, and how much should be paid to A by C and by B
to adjust the losses ?
101. I sold 6 sewing-machines at $72 each. On two of
them I gained 20%, on two others 33^%, and on the rest I
lost 25%. What was the balance of gain or loss?
102. A grocer imported 75 gallons of oil, which cost him $ 2
a gallon and a duty of 10%. Suppose 5 gallons to leak out, for
what must he sell the remainder per gallon to gain 10% on the
money spent ?
298 MISCELLANEOUS EXAMPLES.
103. At 1 cent per cubic foot, what will be tbe cost of digv
ging a ditch outside a square garden containing 12.75 square
rods, the ditch to be 7 feet wide and 5 feet deep ?
104. How many gallons in a cylindrical jar 2 feet across
and 4 feet high ?
105. I found, on going to Gile & Walcott's dry-goods store,
that they had that morning marked up their goods 15%. What
did I save by purchasing the day before the following goods :
18 yds. blk. silk, at $ 1.12 ; 13 yds. de laine, at $0.27 ; 9 yds.
cambric, at 1 0.15 ; 3 yds. silesia, at $ 0.25 ; 1 waterproof, at $ 8 ?
106. A grocer paid 21 cents a gallon for a cask containing
27 gallons of kerosene, 10 % of which leaked out. If the re-
mainder was sold 25% on the gallon higher than it cost, what
was the gain or loss on the money invested ?
107. Supposing a cubic foot of snow to weigh 21 lbs., what
will be the pressure of a body of snow 9 inches deep upon a flat
roof 100ft. by 25 ft.?
108. If an elephant's tusk 9^ feet long and 8 inches in
diameter at the base weighs 214 pounds, what would be the
dimensions of a similar tusk weighing 75 pounds ?
109. An engineer, having placed a mortar near the bank of a
river, wished to find its distance from a fort on the opposite
shore. To do this he
marked off a line from
the mortar towards the
fort ; went 8 rods up the
river, where he drove a
stake ; and 6 feet farther
on took his station. Then
he told his assistant to
start from the stake and mark off a line parallel with the first
line, till he came in range between him and the fort. This
line measured 480 feet. What was the distance sought ? (See
Art. 735, note.)
For other miscellaneous examples, see Appendix, page 315.
APPENDIX
Names of Numbers (Art 2).
1. The only compound names of numbers that do not show
plainly how they are made up are Eleven, Twelve, Twenty,
and the other names ending in -ty.
Eleven (in Old English endlif, in Gothic dinlif) is a compound of end
or en, meaning one, and Uf, meaning ten. So eleven means one and ten.
Twelve (in Old English twelf^ in Gothic twa-lif) is a compound of twa,
meaning two, and lif, meaning ten. So twelve means two and ten.
Twenty (in Old English twentig) is a compound of twen, meaning twain
or two, and tig, meaning ten. So twenty means two tens, thirty means
three tens, and so on.
Roman Numerals (Art. 12).
2. The Koman Numerals are so called because they were
used by the ancient Romans. They were in general use in
Europe as late as the twelfth and thirteenth centuries for
keeping accounts and other purposes of common life. They
were not used as the Arabic numerals are, to make computa-
tions with, but merely to record the results. The computations
were made mostly with counters.
By the Roman method of writing, seven letters are used to
denote numbers, as follows :
I.
V.
X.
L.
c.
D.
M.
One.
Five.
Ten.
Fifty.
One hundred.
Five hundred.
One thoxisand.
The method of using these letters to denote numbers is
shown in the following table: —
300
APPENDIX.
TABLE.
I stands for 1.
XI
II '
. .. 2.
XII
III •
' " 3.
XIII
IV '
< «« 4.
XIV
V
' " 5.
XV
VI
' " 6.
XVI
VII '
' " 7.
XVII
VIII '
' " 8.
XVIII
IX
' " 9.
XIX
X
' " 10.
XX
XI stands for 11
" 12,
" 13.
" 14,
" 15
" 16
" 17
" 18,
" 19,
.
XXI stands for 21.
CI Stan
XXX
' " 30.
CC
V
XXXI
' " 31.
CCC
L
XL
' " 40.
CD
)
L
' " 50.
D
>.
LX
' " 60.
DC
LXX
' " 70.
DCC "
5.
LXXX
' " 80.
DCCC "
).
XC
' " 90.
CM
)."
C
' " 100.
M
for 101.
" 200.
" 300.
'* 400.
" 500.
" 600.
" 700.
" 800.
" 900.
" 1000.
3. Names of Numbers higher than Trillions (Arts. 22, 23).
The names of the groups used to express numbers higher than
trillions are, in their order from trillions, quadrillions, quintillions,
sextillions, septillions, octillions, nonillions, decillions, undecillions, duo-
decillions, tredecillions, quatuordecillions, quindecillions, sexdecillions,
septendecillions, octodecillions, novemdecillions, vigintillions, etc.
4. To Read Decimals (Art. 35).
The following method of reading decimals is recommended
by its simplicity and its conformity to the method of reading
whole numbers.
Illustrative Example. Eead the number 0.279036205.
Begin at the decimal point, and point off three figures at the right of
the point for the thousandths^ group, three more for the millionths' group,
and so on; thus,
0.279,036,205.
Then read the number expressed in each group separately, pronouncing
the name of the group; thus, "Two hundred seventy-nine thousandths,
thirty-six millionths, two hundred five billionths."
The expression 0.36038 would be read, " Three hundred sixty thou-
sandths, thirty-eight hundred thousandths."
6. Contractions in Multiplication (Art. 83).
To multiply by 9, 99, 999, etc.
Since 9 times a number is the same as 10 times the number less
once the number, and 99 times a number is 100 times the number less
once the number, and so on,
To multiply by any number whose terms are all 9's : Annex as
Tnany zeros to the expression for the multiplicand as there are 9's in that
CONTRACTIONS IN MULTIPLICATION. 301
of the multiplier, and from the number thus expressed svhtract the mul-
tipUcand ; thus, 27 x 99 = 2700 - 27 = 2673.
Examples for the Slate.
(1.) 36x99 = ? (4.) 36841x9999 = ?
(2.) 264x999=? (5.) 7x9999999=-?
(3.) 68x9999=? (6.) 245x998 = ?
Note. In Example 6, multiply by 1000, and subti-act twice the mul-
tiplicand.
(7.) 356x9995=? (8.) 54932x999997=?
6. To multiply by a composite number.
Separate the multiplier into convenient factors, multiply the multi-
plicand by one of the factors, and that product hy another factor, and so
on, till all the factors have, been used ; the last product is the answer :
thus, 41 x 25 = 41 X 5 X 5.
Examples for the Slate.
9. Multiply 368 by 72 ; by 36.
10. Multiply 4079 by 81 ; by 48.
11. Multiply 2145 by 108 ; by 144.
12. Multiply 50411 by 55 ; by 150.
7. To multiply by aliquot parts of 10, 100, 1000, etc.
Multiply by 10, 100, 1000, etc., as the case may require, and then find
the required part ; thus,
To multiply by 25, multiply by 100 and divide by Jf.
By 125, multiply by 1000 and divide by 8.
By 33|-, multiply by 100 and divide by 3.
By 16f, multiply by 100 and divide by 6.
By 12^, multiply by 100 and divide by 8. (See Arts. 258 to 261.)
8. To multiply when the number of tens is the same in the
multiplicand and multiplier, and the sum of the units is ten.
27 Multiply the number of tens by the number of tens plus
23 one ; write the product as hundreds ; at the right express the
621 product of the units by the units.
302 ATPENDtJL.
Examples.
(13.) 23x27=? (16.) 45x45=? (19.) 55x55=?
(14.) 28 X 22 = ? (17.) 56 X 54 = ? (20.) 85 x 85 = ?
(15.) 31 X 39 = ? (18.) 87 X 83 = ? (21.) 105 x 105 = ?
9. To square a number consisting of an integer and \.
7^ Multiply the integer by the integer plus (me, and to the
7^ product add ^.
Examples.
(22.) 5^ X 5^ = ? (24.) 11^ x Hi = ? (26.) 99 J x 99i = ?
(23.) 8|x8i=? (25.) 13^x131 = ? (27.) 16^x16^ = ?
10. To square a number consisting of an integer and ^.
12J Square the integer ; to this square add 1 half the
12| integer plus ■^.
144 + 6 + A = 150Jg^
Examples.
(28.) 6i X 6J = ? (30.) 7i x 7 J = ? (32.) 50j x 50j = ?
(29.) 8i X 8i = ? (31.) 30J x 30^ = ? (33.) 61^ x 61^ = ?
Contractions in Division (Art. 111).
11. To divide by aliquot parts of 10, 100, 1000, etc.
To divide by 5, divide by 10 and multiply the quotient by 2.
By 25, divide by 100 and multiply by 4.
By 125, divide by 1000 and multiply by 8.
By 33|-, divide by 100 and multiply by 3.
By 16f, divide by 100 and multiply by 6.
By 166f, divide by 1000 and multiply by 6, etc. (See Arts. 258 to
261.)
12. Analysis of Oral Examples (Art. 116).
Analysis of Example a. If the car runs 69 miles in 3 hours, in
1 hour it will run 1 third of 69 miles, or 23 miles, and in 5 hours it
will run 5 times 23 miles, or 115 miles. Ans. 115 miles.
Example c. First find what $ 1 will buy.
Analysis of Example i. If a quantity of hay lasts 22 oxen 10
days, it will last 1 ox 22 times 10 days, or 220 days, and it will last
10 oxen (5 yoke) 1 tenth of 220 days, or 22 days. Ans. 22 days.
DIVISIBILITY OF NUMBERS. 303
Analysis of Example m. If the work can be done by 50 men
in 4 weeks, it will require 4 times 50 men, or 200 men, to do it in
one week. But 50 men are already employed. Then 200 men less
50 men, or 150 more men, must be employed to do it in a week.
In solving an example by analysis, we first take the number whose
denomination is the same as that of the required answer to work upon,
and then proceed according to the statements given in the example.
13. The Divisibility of Numbers (Art. 153).
1. Divisibility of numbers by 2, 4, 5, or 8.
Ten is divisible by 2, so any number of tens is divisible by 2.
Hence a number is divisible by 2 if the number of its units is divisible
by 2.
For a similar reason, a number is divisible by 5 if the number of its
units is divisible by 5.
One hundred is divisible by 4, so any number of hundreds is divisi-
ble by 4. Hence a number is divisible by J^ if its tens and units to-
gether are divisible by J^.
One thousand is divisible by 8, so any number of thousands is di-
visible by 8. Hence a number is divisible by 8 if its hundreds, ten^,
and units together are divisible by 8.
2. Divisibility of numbers by 9.
3486
3486= (333 + 44 + 8) X 9 + (3 + 4 + 8 + 6)
The number 3846 is now separated into two parts, one of which is
divisible by 9, and the other equals the sum of its digits. Any num-
ber can be so separated. Hence a number is divisible by 9 if the sum
of its digits is divisible by 9.
3. Divisibility of numbers by 3. Any number that is divisible by 9
is divisible by 3. Therefore, when a number has been separated into
two parts, as shown above, the first part is divisible by 3. The other
part is equal to the sum of its digits. Hence a number is divisible by
3 if the sum, of its digits is divisible by 3,
ILLUSTRATION.
3000 =
333 X 9
+ 3
+ 400 =
44 X 9
+ 4
+ 80 =
8x9
+ 8
+ 6 =
6
304 APPENDIX.
14. The Greatest Common Factor of Numbers (Art. 175).
The method of finding the g. c. f. of numbers, as given in
Article 175, depends upon the following principles.
I. Any factor common to two numbers is also a factor of
their sum and of their difference.
Thus 3, which is a common factor of 24 and 18, is a factor of 42 ( = 24 +
18) and of 6 ( = 24- 18). Since 24 is equal to a certain number of 3's, and
18 to a certain other number of 3's, their sum must be a number of 3's, and
their difference must be a number of 3's.
II. The greatest common factor of two numbers is equal to
the greatest common factor of the smaller of them, and the re-
mainder obtained by dividing one of them by the other.
52) 143 (2 Thus, the g. c. f. of 143 and 62 is equal to the g. c. f.
104 of 52 and 39.
^ Since 39 = 143-52-52, all factors common to 143
and 52 are also factors of 39, and hence common factors
of 52 and 39. Therefore the g. c. f. of 143 and 52 is a comvion factor of 52
and 39, and cannot be greater than the g. c. f. of 52 and 39.
Again, since 143 = 52 + 52 + 39, all factors common to 52 and 39 are also
factore of 143, and hence common factors of 143 and 52. Therefore the
g. c. f. of 52 and 39 is a common factor of 143 and 52, and cannot be greater
than the g. c. f. of 143 and 52.
The g. c. f. of 143 and 52 and the g. c. f. of 52 and 39 are, then, two num-
bers, neither of which can be greater than the other ; they are therefore equal.
16. Analysis of Illustrative Example (Art. 246).
WRITTEN WORK. Analysis. — If f of a dollar will buy 1 basket
1x3x5 rr of peaches, ^ of a dollar will buy -^ of a basket,
2 ^* and | of a dollar, or 1 dollar, will buy | of a
Ans. 7h baskets. basket. If 1 dollar will buy | of a basket, 5 dol-
lars will buy 6 times | or Jj^ of a basket, which
equals 7-| baskets.
16. To Divide an Integral Number or a Fraction by a
Fraction (Art. 248).
Note. The following methods of dividing by fractions are in common
use. To understand either method, the learner must bear in mind that the
smaller the divisor, the larger is the quotient, and the larger the divisor,
the smaller is the quotient.
FRACTIONS. 305
Illustrative Example. What is the quotient of f -?- f ?
WRITTEN WORK. Explanation \.i — The quotient of ^ divided
2 by 1 is I ; of I divided by 2 is J of |, or ^ ;
^ ^ ^ = f = 1^. ^^ i divided by f (which is ^ as large as 2) is
^ ^ ^ 3 times -J of |, or J^, which equals 1-^.
Ans. 1^. ^rw. \\.
Explanation 2. — The quotient of ^ divided by 1, is -J ; of -f divided
by \ (which is ^ as large as 1), is 3 times \ or ^ ; and of \ divided
by f, is ^ of 3 times | or 4^ which equals 1^. An^. \\.
17. To change a Circulating Decimal to a Common Fraction
(Art. 287).
Illustrative Example. Change 0.63 to a common fraction.
WRITTEN WORK. Explanation. — As the repetend consists
0.63 X 100 = 63.6363... of two figures, we multiply the given cir-
0.63 X 1 = 0.6363... culate by 100, and find that the decimal
0.63 X 99 = 63. part of the product is precisely the same
0.63 = M = tV ^'f^- ^ *^^ given circulate. Hence, if we sub-
tract the given circulate from this product
there will be no decimal fraction in the remainder. Thus we find that
99 times the given circulate equals 63 ; therefore once the given circu-
late is ff, or ^y. Hence the rule given on page 129.
18. To change a Mixed Circulate to a Common Fraction
(Art. 288).
Illustrative Example. Change 0.263 to a common frac-
tion.
WRITTEN WORK. Explanation. — As the repetend
0.263x100 = 26.3636... consists of two figures, we multiply
0.263 X 1 = 0.2636... the given mixed circulate by 100 and
0.263x99 =26.1 subtract from the product once the
0.263 = ^^ = ^fj- = ■^. mixed circulate, and have 99 times
Jifis^ Jia^, the mixed circulate, equal to 26.1.
Then once the mixed circulate will
equal ^^, or f|^. But 261 is the difference between 263, the mixed
circulate regarded as an integer, and the finite part regarded as an
integer. Hence the rule given on page 130.
306 APPENDIX.
19. Surveyors' and Mariners' Measures (Art. 302).
Surveyors, in measuring, use a chain called Gunter's chain
(ch.), which is 4 rods or &Q feet long. The chain is divided
into one hundred links (1.).
Surveyors' Long Measure.
7.92 in. = 1 link.
100 1. = 1 chain.
80 ch. = 1 mile.
Note. 25 links equal 1 rod.
Surveyors' Square Measure.
10 sq. ch. = 1 A.
640 A. = 1 sq. m.
1 sq. m. = 1 section.
36 sec. = 1 township.
Examples for the Slate.
34. A road was found to be 8 ch. 30 1. long, how many links in
length was it ?
35. Express 45 links as the decimal of a chain.
36. Express 11 ch, 561. in chains and decimals of a chain.
37. Change 18 ch. 5 1. to rods.
Operation : 18.05 x 4 = 72,20 rods.
38. Change 32 ch. 27 1, to rods and feet.
39. In a wall which measures 23 ch. 47 1., how many rods and how
many feet over?
How many acres are there in a rectangular piece of land
(40.) 50 ch, long and 30 ch, wide ?
(41.) 45 ch. long and 24 ch. wide?
(42.) 19.04 ch. long and 3.7 ch. wide ?
(43,) 84 ch, 8^ 1. long and 13 ch, 24 1. wide ?
20. Mariners, in measuring short distances at sea, use
cable-lengths and fathoms.
6 feet = 1 fathom (used in measuring depths at sea).
120 fathoms = 1 cable-length.
7^ cable-lengths = 1 common mile.
Longer distances at sea are estimated in nautical or geo-
graphical miles (Art. 335). 3 nautical miles = 1 marine league.
MISCELLANEOUS TABLES. 307
21. Apothecaries' "Weight (Art 329, note II.).
In mixing medicines, apothecaries use the Troy pound di-
vided into ounces (oz. or §), drams (dr. or 5), scruples (sc.
or 9), and grains. They also use the fluid ounce (f. | ), flui-
drachm (f. 5), minims or drops (iix\).
Weights.
20 grains = 1 scruple.
3 scruples = 1 dram.
8 drams = 1 ounce.
12 ounces = 1 pound.
Liquid Measures.
60 minims = 1 fluid drachm.
8 fluid drachms = 1 fluid ounce.
16 fluid ounces = 1 pint (O).
8 pints = 1 gallon (Cong).
22. Explanation of Leap Year (Art. 339).
The earth revolves around the sun in 365 days 5 hours 48 minutes and
50 seconds nearly, but we call 365 days a year. It will be seen that what
we call a year is nearly 6 hours less than the true year, and 4 such years
nearly one day less than 4 true years. To rectify this error, 366 days are
allowed to every fourth year. The year of 366 days is called a leap year.
The addition of a day in every fourth year is too much by a number of
minutes, which in one hundred years amounts to about three fourths of a
day. To balance this error, only 365 days are allowed to the final year of a
century in three centuries out of every four.
Hence any year is a leap year, when the number denoting the year is di-
visible by 4 ci'nd not by 100, and when it is divisible by 4OO.
23. Miscellaneous Tables.
Books.
A book formed of sheets folded
In 2 leaves is a folio. In 16 leaves is a 16mo.
In 4 leaves is a quarto. In 18 leaves is an 18mo.
In 8 leaves is an octavo. In 24 leaves is a 24mo.
In 12 leaves is a duodecimo, In 32 leaves is a 32mo.
or 12mo. In 64 leaves is a 64mo.
Iiengrth.
3 in. = 1 palm.
9 in. = 1 span.
4 in. = 1 hand (used in measuring the height of horses).
Surface.
100 sq. ft. = 1 square j ^^ "' tdS tcf 'l"St^'"*'*'«'
308 APPENDIX.
Capacity.
1 barrel of flour = 196 pounds.
1 barrel of beef, pork, or fish = 200 "
1 cental of grain or 1 quintal of fish = 100 "
1 cask of lime =240 "
Weights of Iron and Uead.
14 pounds = 1 stone.
21^ stones = 1 pig.
8 pigs = 1 fother.
24. To compute Interest by Aliquot Parts (page 211).
Illustrative Example. What is the interest of $ 720 for
2y. 7mo. 29d. at8%?
WRITTEN WORK.
To compute in-
incipal, $720
terest by aliquot
0.08
parts : Find the
$ 57.60 X
:2= $115.20 Int. for 2 y.
interest for one
^of 57.60
= 28.80 " " 6 mo.
period of time, as
J of 28.8C
4.80 " " Imo.
1 year or 1 month,
0.8 of 4.80
3.84 " " 24 d.
and then find the
J of 4.80
= 0.80 " " 5d.
interest for the
$153.44 ", " 2y.7m.29d.
balance of the time
by taking conven-
ient multiples or aliquot parts of this interest or of any of the results.
25. To compute Interest at 6 % by Aliquot Parts.
(First see Oral Exercises, Art. 541.)
At 6%, what part of the principal does the interest equal for
Months. Ans. Montlis. Ans. Months. Ans. Months. Ans.
a. 2? 0.01. d. 1? 0.00^. g-. 50? J. j. 66^ f
b. 20? 0.1. e. 10? 0.0^. h. 40? f k. 33^? ^.
C. 200? Prin. /. 100? ^. i. 25? ^. 1. 16f? ^.
At 6%, what part of the principal does the interest equal for
Days. Am. Days. Ans. Days. Ans. Days. Ans.
m. 60? 0.01. o. 30? 0.00^. q. 20? 0.00^. s. 10? 0.00^.
i2. 6? 0.001. p. 3? O.OOOJ. r. 2? 0.000^. t. 1? 0.000^.
ANNUAL INTEREST. 309
Illustrative Example. Find the interest of % 700 for
lOy. 5mo. 23d. at 6%.
WRITTEN WORK. Explanation. — 10 y. 5 mo.
Principal, $ 700 =125 mo. We first find the
^ of $ 700 =350 Int. for 100 mo. interest for 100 mo. by taking
^ of 350 = 87.50 " " 25 mo. \ of the principal, then for
i of 7 = 2.33 " " 20 d. 25 mo. by taking J of the in-
|of 0.70= 0.35 " " 3d. terestforlOOmo. We then
$440.18 find the interest for 20 days
by taking ^ of the interest for 60 days, or ^ of 0.01 of $700, and
the interest for 3 days by taking \ of the interest for 6 days, or ^ of
0.001 of $700. The sum of these items equals $ 440.18. Ans. $440.18.
To compute interest at 6 % by aliquot parts :
1. To find the interest for 200 monthsj take a sum equal to the prin-
cipal; for 20 months, equal to ^ of the principal; for 2 months, equal
to Y^ of the principal; and for 6 days, equal to ^^ of the principal.
2. For any other periods of time, take convenient multiples or aliquot
parts of the interest for the times expressed above ; and add the results.
• 26. Interest at any other % may be found hy first finding the inter-
tit at ^fo^a^ above, and then at any given fo, as in Art. 542.
27. Annual Interest (page 220).
Illustrative Example. What is the interest due on a
note for $1000, interest payable annually at 67o, if no pay-
ment be made till the expiration of 4 y. 6 mo. 12 d. ?
In some of the States the courts have sanctioned the taking of in-
terest upon interest in cases like the above, where interest is not paid
when it becomes due. Thus, interest is allowed on the above note
For the 4 y. 6 mo. 12 d. = $ 272.00
Also on each year's interest ($ 60) after it becomes due,
viz. :
On the Ist year's interest for 3 y. 6 mo. 12 d.
« « 2d " " " 2y. 6 mo. 12 d.
« « 3d " " " ly. 6 mo. 12 d.
« « 4th " " " 6 mo. 12 d.
Which equals the interest of $60 for 8 y. 1 mo. 18 d. » 29.28
Total interest . . . $301.28
310 APPENDIX.
Simple interest, taken upon the principal, and upon each year's
interest of the principal due and unpaid, is aiiJiua.1 interest.
Rule.
To compute annual interest : Compute simple interest on the princi-
pal for the time it is on interest. Also, on one yearns simple interest for
a period of time equal to the sum of the times for which the yearly inter-
ests severally remain unpaid. Add the results.
Examples for the Slate.
44. What is the annual interest of $ 334 for 3 y. 8 m. 10 d. ?
45. What is the annual interest of $ 118.50 for 5 y. 3 m. 18 d. ?
46. What is the amount at annual interest of $175 for 6y. 2 m.
25 d. ?
28. Vermont Rule for Partial Payments.
1. Compute annval interest upon the principal to the end of the first year
in which any payments are made ; also compute interest upon the payment or
payments from the time they are made to the end of the year.
2. Apply the amount of such payment or payments first to cancel any in-
terests that may have accrued upon the yearly interests, then to cancel the
yearly interests themselves, and then towards the payment of the principal.
3. Proceed in the same way with succeeding payments, computing, however,
no interest beyond the time of settlement.
29. The New Hampshire Rule
is the same as the foregoing, with the following provision :
If at the time of any payment no interest is due except what is accruing
during the year, and the payment or payments are less than the interest
due at the end of the year, deduct such payment or payments at the end
of the year, without interest added.
30. Connecticut Rule for Partial Payments.
1. When a year's interest or more has accrued at the time of a payment, and
always in case of the last payment, follow the United States Rule.
2. When less than a year's interest has accrued at the time of a payment,
except it be the last payment, find the difference between the amount of the
MONETARY UNITS OF FOREIGN COUNTRIES. 311
principal for an entire year, and the amount of the payment for the balance
of a y^ear after it is made ; this difference will form the new principal.
8. If the interest which has arisen at the time of a payment exceeds the
payment, compute interest upon the principal only.
31. Monetary Units of Foreign Countries (Art. 616).
The following table shows the par value in gold of the
monetary units of different countries, as published by the Sec-
retary of the Treasury of the United States, Jan. 1, 1878.
Countries.
Monetary Unit.
Standard.
Value in
U. S. Money.
Gold.
Austria
Florin
Silver . .
10.453
0.193
0.965
0.965
0.545
1.00
0.918
0.912
0.268
0.918
4.974
0.193
4. 866 J
0.193
0.238
0.436
0.193
0.997
1.00
0.998
0.385
0.268
0.918
1.08
0.734
1.00
0.193
0.268
0.193
0.829
0.118
0.043
0.918
Belgium
Bogota
Franc
Gold and silver...
Gold
Peso.
Bolivia
Dollar
Gold and silver. . .
Gold
Brazil
Milreis of 1000 reis
Dollar
British Possessions
in Nort;h America
Central America . . .
Chili
Denmark
Gold
Silver
Dollar
Peso ....
Gold
Crown
Gold ...
Ecuador
Dollar
Silver
EffVPt
Pound of 100 piasters...
Franc
Gold
France
Gold and silver...
Gold
Great Britain
Greece
Pound sterling
Drachma
Mark
Gold and silver...
Gold
German Empire....
India
Rupee of 16 annas
Ijira .
Silver .
Italy
Gold and silver...
Gold
Japan
Liberia
Yen
Dollar..
Gold
Mexico . . . ;
Dollar
Silver
Netherlands
Florin . .
Gold and silver...
Gold
Norway
Crown
Peru
Dollar .
Silver
Portugal
Milreis of 1000 reis
Rouble of 100 copecks...
Dollar..
Gold
Silver
Gold
Russia ... . .
Sandwich Islands..
Spain
Peseta of 100 centimes..
Crown . .
Gold and silver...
Gold
Sweden
Switzerland
Franc
Gold and silver...
Silver
Silver
TripoU
Tunis
Mahbub of 20 piasters...
Piastm-of 16caroubs....
Piaster .
Turkey ....
Gold
U. S. of Colombia .
Peso
Silver . .
312 APPENDIX.
32. Table of English Money.
4 farthings (qr.) = 1 penny (d.)
12 pence = 1 shilling (s.)
20 shillings = 1 pound (£).
Also, 2 shillings = 1 florin ; 10 florins = 1 pound.
33. French Money. 100 centimes = 1 franc (fr.)
34. German Money. 100 Pfenniges (pennies) = 1 Reichmark
(mark).
Note. The coin which represents the pqund value is gold, and called a
sovereign. The franc and the mark are both silver.
Square Root (Art. 675).
35. The following method of extracting square roots may
be substituted for that given in the body of the book, if pre-
ferred. Let it be required to extract the square root of 1296.
WRITTEN WORK. Explarmtion. — The formu-
Formuia, la Tcus^ + 2 (tcns X units)
Teiua + (2 X tens + unit.) X nnita. + ^^its^ may be changed to
12'96 (36 the form, Tens^ + (2 x tens
(3tens)2= ^ + units) X units. As the first
Trial divisor (3 tens) x 2 = 6 tens\ 396 part of the power, the square
rrr -t. . — I 396 of the tens, is hundreds, the
True divisor 66/ -,n -u j i e ^-u
^ 12 hundreds of the given
Or simply, number must have in it the
^ square of the tens of the root.
The greatest square con-
66) 396 tained in 12 (hundreds) is 9
396 (hundreds), the square root
of which is 3 (tens). This we
write as the first term, or tens, of the root.
Taking the square of 3 (tens) = 9 (hundreds) out of 12 (hundreds),
there remain 3 (hundreds), with which we unite the remaining part of the
number, 96, making 396, which must contain the product of two times
the tens plus the units multiplied by the units. If it contained only two
times the tens multiplied by the units, we should find the number of units
by dividing 396 by two times the tens. So we make two times the tens,
or 6 tens, the trial divisor, and find that it is contained in 39 tens 6 times.
Then 6 is probably the next term, or units, of the root. Adding 6 units to
CUBE ROOT. 313
the 6 tens (the trial divisor), we have now the true divisor, which, multi-
plied by 6, completes the square. So the given number is a perfect square,
and 36 is its square root.
36. Rule.
To extract the square root of a number :
1. Beginning with the unit^ figure, point off the expression into periods
of two figures each.
2. Find the greatest square in the number expressed by the left hand
period, and write its square root as the first term of the root.
3. Subtract this square from the part of the number used, and with the
remainder unite the next two term^ of the given number for a dividend.
4. Double the part of the root already found for a trial divisor ; and
by it divide the dividend (rejecting the lowest term of the dividend) and
write the quotient as the next term of the root. Also write it at the right
of the trial divisor to express the true divisor.
5. Multiply the true divisor by this term, and subtract the product
from the dividend.
6. If there are more terras of the root to be found, unite with the re-
mainder the next two terms of the given number, take for a trial divisor
double the part of the root now found, and proceed as before.
Cube Root (Art. 686).
37. The following method of extracting cube roots may be
substituted for that given in the body of the book, if preferred.
Let it be required to extract the cube root of 262144.
WRITTEN WORK. Explanation. — The
Formula, formula Tens'* + 3 (tens^
Tmus + (3 Xten82 + 3 X tern. X unite + nnitH3)x units. ^ units) + 3 (tens X
262'144(64 units2) + units^ may be
(6tens)8= 216 ^^^^^^^ to the form,
Trial divisor (6 tens)^ x 3 =108 hunds.) 46144 Tens^ + (3 x tens^ + 3 x
6tensx3x4= 72 tens. tens x units + units'-*) x
42 = 16 anits.
True divisor x 4 = 11536x4 = 46144 As the first part of
the power, the cube of
the tens, is thousands, we find the greatest cube contained in 262 (thou-
sands), which is 216 (thousands), and write its cube root 6 (tens) as the
first term, or tens, of the root.
314 APPENDIX.
Taking the cu"be of 6 (tens), 216 (thousands), out of 262 (thousands), ^hert
remain 46 (thousands), with which we unite the remaining part of the num-
ber, 144, making 46144, which must contain (3 x tens'-^ + 3 x tens x units
+ units^) X units.
If the number 46144 contained only 3 x tens'-^ x units, we should find the
number of units by dividing 46144 by 3 times the square of the tens. So
we make this number, 108 (hundreds), the trial divisor, and find that it is
contained in 461 (hundreds) 4 times. Then 4 is probably the next term, or
units, of the root.
Adding 3 times 6 tens x 4 units, and the square of 4 units to the trial
divisor, we have the true divisor, which multiplied by 4 completes the
cube. So the given number is a perfect cube, and 64 is its cube root.
38. Rule.
To extract the cube root of a number :
1 . Beginning with the unit^ figure, point off the expression into periods
of three figures each.
2. Find the greatest cube in the number expressed by the left-hand
period, and write its cube root as the first term of the root.
3. Subtract the cube from the part of the number used, and with the
remainder unite the next three terTns of the given number for a dividend.
4. Take three times the square of the part of the root already found
for a trial divisor, and by this divide the dividend (rejecting the lowest tivo
terms of the dividend) and write the quotient as the next term of the root.
5. To the trial divisor {which is hundreds) add three times the first
term of the root {tens)-multiplied by the last term, also the square of the
last term. .
6. Multiply this sum by the last term of the root, and subtract the
product from the dividend.
7. If there are more terms of the root to be found, unite with the re-
mainder the next three terms of the given number, take for a trial divisor
three times the square of the part of the root now found, and proceed as
before.
39. To find the Capacity of a Cask or Barrel in Gallons.
Add to the head diameter | of the difference between the head and bung
diameters {or, if the staves are but little curved, 0.6 of the difference).
This will give the mean diameter.
Multiply the square of the numher of inches in the mean diameter by
the number of inches in length, and this product by 0.0034.
MISCELLANEOUS EXAMPLES. 315
40. Miscellaneous Examples.
47. If I lose 10% by selling goods at 18 cents per yard, for what
should they have been sold to gain 20%?
48. If 30 men, working 11 hours a day, can do a piece of work in
a certain time, how many more men must be employed, when it is
half done, to finish it in the same number of days, working 10 hours
a day?
49. Two armies are in opposite directions from a certain point, one
being 300 miles east and the other 250 miles west of it, and marching
towards each other, the first at the rate of 15 and the other of 18 miles
in a day. In how many days will they meet, and where ?
50. If, by selling goods at 60 cents per lb., 20% is gained, what %
would have been gained by selling them at 75 cents per lb. ?
51. A broker purchases a lot of stocks at an average of 9% below
par, and sells them at an average of 7f % above par, and makes $ 300.
What was the par value of the stocks ?
52. How many bushels of com at 50 cents a bushel miist be mixed
with 30 busliels of grain at 80 cents a bushel, that the mixture may
be worth 75 cents a bushel ?
Note. Take such a quantity of corn as shall make the gain in selling it
at 75 cents a bushel equal the loss in selling the grain at 75 cents a bushel.
53. How much water must be mixed with a barrel of ink (31 gals.),
which cost $34.10, that it may be sold at $ 1.10 a gallon and 25% be
gained ?
54. 20% of a lot of barley, originally 5000 bushels, was destroyed
by fire, the cost having been %\\ per bushel. What per cent will be
gained on the lot by selling the remainder at $2 per bushel?
55. I sell ^ of a lot of goods for $9, and thereby lose 25%. For
what must I sell the remainder to make 8^ % on the whole ?
56. I sold 4 ploughs at % 24 each ; on 2 of them I made 20%, and
on 2 I lost 20%. What did I gain or lose on the whole ?
57. Divide 52 into two such parts that \ of one part shall equal f
of the other.
58. How many cubic yards of earth must be removed for a cellar
10 feet deep and measuring inside the walls 27 feet long and 15 feet
wide, the wall being 2 feet 6 inches thick ?
59. If 10% is lost by selling boards at $ 7.20 per M., what % would
be gained by selling them at 90 cents per C. ?
316 APPENDIX.
60. A person takes a note on 2 months for $ 110 in payment for a
watch. On getting the note discounted at a bank, he finds that he
has lost 40% on the first cost of the watch. What was the cost ?
61. What would be due May 1, 1878, on a note for $ 1000, dated
March 26, 1875, at 8% interest, on which $200 were paid at the end
of each year from the date of the note ?
62. If I buy coal at $4.12 per ton on 6 months' credit, for what
must I sell it immediately to gain 10%?
63. What will a pine log weigh whose length is 18 ft., measuring
3 ft. across the larger end, and 2jft. across the smaller, pine being 0.6
as heavy as water, which weighs 62^ lbs. to a cubic foot ?
64. Required the number of square feet in the surface of a ditch
surrounding a circular garden which is 25 yards across, the ditch being
2^ ft. wide.
65. An aeronaut ascends at the rate of ^ miles an hour for 40
minutes, after which he maintains the same elevation ; if his balloon
is driven east 7 miles during the first hour from the time of his start-
ing, and in an opposite direction at the rate of 10 miles an hour for
the remaining time, how far from his starting-point in a straight line
is he at the end of 5 hours ?
66. 13% is lost by selling a lot of land for $ 783. What would it
have brought if it had been sold at a loss of 8^%?
67. What will be the per cent of gain on the cost of a Gas Co.'s
stock, the par value of shares being $87.50, if it be bought at 15%
below par, and sold at 19|% above par?
68. What is the length of the edge of the largest cube that can be
sawed from a globe 9 inches in diameter?
69. Two boys tried their skill in running for pegs. Five pegs were
set up in a line 6 feet apart. The starting-point was in the same line
6 feet from the first peg. How far must each boy run to fetch all the
pegs one at a time to the starting-point ?
70. John Barnes bought, June 8, 1875, 10 bales of cotton cloth, 14
pieces in a bale, 43 yds. in a piece, at 8/- per yd., for which he gave
his note on interest at 6% On the 4th of Nov., 1877, he sold 1 bale
at 30/ a yd., and with the proceeds made part payment of his note.
On the 3d of May, 1878, he sold 1 bale at 40/, and paid on his note
the amount he received. On the I7th of Sept., 1878, he sold the re-
mainder at 60 /, and settled the note. What did he gain by his specu-
lation ?
INTEGRAL NUMBERS. 317
ADDITIONAL EXAMPLES TO BE SELECTED FROM
BY THE TEACHER.
A few of these examples are designed especially to test advanced pupils.
41. Integral Numbers.
71. Owing $2759, I gave in payment a house worth $1575, and
land worth $ 387. How much of the debt remained unpaid ?
72. In California, in 1870, there were 23724 farms, containing
11,427,105 acres. What was the average number of acres to a farm?
73. "What is the difference in the height of Lake Superior and the
Dead Sea, the former being 627 feet above and the latter 1317 feet
below the level of the ocean ?
74. A man walked 162 miles in 6 days, walking 8 hours each day.
What was the average rate per hour ?
75. Find the average of the following daily readings of a ther-
mometer : 43°, 47°, 50°, 39°, 38°, 41°, 45°, 48°, 51°, 53°.
76. At a concert there were 75 rows of seats containing 16 persons
each, and 50 rows containing 7 persons each ; the galleries contained
enough to make a total of 2339. How many were in the galleries ?
77. Of the above 2339 persons, those who bought tickets paid 25
cents apiece, the rest had complimentary tickets. If the amount
collected was $ 570, how many had complimentary tickets ?
78. The salmon caught in the Columbia River and canned in a
single season weighed 13,894,760 pounds before they were canned.
If the average weight of a salmon was 22 pounds, how many salmon
were caught ? If the loss in dressing was 3 pounds per salmon, how
many pounds were canned during the year ?
79. The metric system was legalized in France in 1795 ; its use
was enforced by law in 1845. If half as much time shall elapse be-
tween the time it was legalized in this country (1866) and the time it
shall be enforced, in what year will its use be enforced here ?
80. In 1868 the American whaling-fleet produced 1,485,000 gal-
lons of sperm-oil and 2,065,612 gallons of train-oil. If the average
yield for each whale was not less than 4500 gallons, what is the
greatest number of whales the fleet could have captured ?
81. At $1.15 a gallon for sperm-oil and $0.75 a gallon for train-oil,
what did the whole quantity given in Example 80 bring?
318 APPENDIX.
82. In a certain school building, in each of 4 rooms there are 58
desks, in each of 2 rooms there are 48 desks, in each of 3 rooms there
are 54 desks ; in the hall can be seated 396 pupils. How many more
seats must be placed in the hall that it may seat as many pupils as
there are desks in all the rooms ?
83. If in one year the production of carpetings in the United
States was as follows, ingrains 16,924,711 yards, tapestry Brussels
1,711,000 yards, Venetian 1,350,017 yards,' felt 586,000 yards, velvet
107,000 yards, and of Brussels enough to make 21,485,233 yards in
all, how many yards of Brussels were made ?
42. Examples with Decimals.
84. How many acres must be added to 26.79 acres that the sum
may equal 350 acres ?
85. If you have 5.23 meters, 8.72 meters, and 3.972 meters of satin,
how many more meters must you get to make 20.5 meters in all?
86. At 0.6 of a cent per pound, what is the cost of the following
lots of ice : 357 pounds, 900 pounds, and 465 pounds ?
87. At 45 cents per hundred for ice, what is the cost of 85 pounds ?
88. If 9 gallons of milk weigh 64.5 pounds, how many pounds
will 1 gallon weigh ?
89. If 0.72 of a pound of flour is obtained from a pound of wheat,
how many pounds of flour may be obtained from 100 pounds of
wheat? from 250 pounds ? from 4385 pounds?
90. During the year 1875, in Great Britain, 133,306,486 tons of
coal were mined by 525,843 men. What was the average number of
tons mined per man? [Ans. to hundredths.]
91. If in the coal-mines of Great Britain 15908 lives were lost
during the 14 years previous to 1875, and 1,608,576,193 tons of coal
were mined, what was the average loss of life per year, and how many
tons were mined to each life lost ? [Ans. to hundredths.]
92. If an engine can run a mile in 0.025 of an hour, how many
hours will it take to run 40 miles ?
93. How much are 17500 francs of Belgium worth, if 1 franc is
worth $0,193?
94. If a gas-jet consumes 5 cubic feet of gas an hour, and is
burned 4 hours every evening, what is the cost of gas for a week at
$ 0.003 a cubic foot ?
UNITED STATES MONEY. 319
43. United States Money.
95. If 4 bushels of beans cost $ 12.56, what will 9 bushels cost at
the same rate ?
96. If I can buy eggs at one place for 35 cents a dozen and at
another place for 28 cents a dozen, how much money do I save by
buying 300 eggs at the latter place ?
97. If $597.78 is the dividend and 18 the quotient, what is the
divisor ?
98. An oil-train containing 12 tanks of oil passed over the rail-
road this morning. If each tank contained 3745 gallons of oil worth
8 cents a gallon, what was the value of the whole quantity ?
99. Mr. Rice received $175 for his month's salary ; of this he paid
for rent |20, for help |12, meat bill $9.63, grocer's bill $13.41,
milk $2.79, fuel $8.25, clothing $43.18, and $11.14 for other ex-
penses. How much of his salary remained unexpended ?
100. Two hundred and forty people struck for higher wages, and
were out of work 3 weeks in consequence. If their average earnings
were $ 1.35 a day, what was their entire loss for the time?
101. A man bought a barrel of beef (200 pounds) for $16.10. In-
cluding 55 cents for freightage and 35 cents for cartage, how much
did the beef cost him a pound ?
102. Of a barrel of beef which cost 17 dollars, 136 pounds were
sold at 12 cents a pound and the remainder at 8 cents a pound. How
much was gained?
103. What is the average cost of milk per month for a family whose
bills for the year are as follows : $4.34, $3.92, $4.34, $4.20, $3.72,
$3.60, $3.72, $3.72, $3.60, $3.72, $4.20, $4.34?
104. Mrs. Brown's family expenses for the year 1879 were as fol-
lows : $114.48, $109.06,187.93, $138.07, $182.35, $97.16, $153.42,
$42.13, $ 103.42, $147.40, $ 132.58, $ 100.74. In 1878 they averaged
$ 1 13.71 a month. In which year were her family expenses the greater,
and how much greater were they ?
105. Mr. Sprague had $ 2145 on deposit in a bank. He drew out
$ 572 of his deposit at one time, $ 67 at another, and gave a check for
the balance towards the payment of a debt of $ 1700. What was the
amount of the check he gave, and how much of his debt remained
unpaid?
106. If I pay $29.45 for a load of wheat, when wheat is worth 95
cents a bushel, how much should I pay for a load twice as large when
wheat is worth 72 cents a bushel ?
320 APPENDIX.
107. Mr. Gane and Mr. Searle went on a journey, agreeing to stare
the expenses equally.
Mr. Gane paid : Mr. Searle paid :
Fares, % 25, $ 42, $ 36.19 ; Fares, 1 11.94, % 0.88, $ 1.72 ;
Hotel bills, ^ 5.25, $ 3.12, % 16 ; Carriage hire, $ 1.25, % 1.80 ;
Refreshments, 1 1 .50, $ 0.25, % 0.47. Porter, $ 0.50, $ 0.75.
Which is in debt to the other, and how much ?
108. If it costs 1 200 a year for tuition, $ 24 for books and station-
ery, $6 a week for board, and $ 1.62 a week for a room, how much
will it cost a person at college for a year of 38 weeks ?
109. Martha bought for a dress 6 yards of cambric @ 17 /, 6 yards
@ l^f, 5 yards of lace ®Z'^f, 2 spools of cotton @ bf, 1 dozen and
a half of buttons @ 18/^, and paid for work of the dressmaker $ 2.25.
What was the whole cost of her dress ?
110. A dealer bought 328 bushels of wheat at 87;^ a bushel, 745
bushels of oats at 56/' a bushel, and gave in part payment $425,
drawing a check for the balance. For how much was the check
drawn ?
111. Of 1073 bushels of wheat costing 85)^ a bushel, 142 bushels
were sold at 95/, and the remainder at $ 1.12 a bushel. How much
was gained ?
112. Of 745 bushels of oats which cost 34/ a bushel, 326 bushels,
being damaged, were sold at 25/ a bushel, and the remainder at 45/
a bushel. Was there a gain or loss by the sale of the oats, and how
much ?
113. I started out shopping with $30.72, and bought with the
money 2 collars @ 10 ^ apiece, 6 collars at the rate of 3 for 25/,
2 pairs of cuffs @ 20/, 4 yds. gingham @ 17/ and 2 yds. @ 33/,
14 yds. linen @ 33/, 1 and a half yds. lace @ 18/, 2 and a half yds.
edging @ 22/. How much money should I have left ?
114. When the introduction price for readers is 54 cents, and the
exchange price is 36 cents, what is the cost of a new set of readers in
a town requiring 675 new readers, and giving 250 old readers in
exchange ?
115. Mr. Lucas kept 42 bushels of cranberries through the winter,
but found in the spring that a third of them was decayed. He then
sold the remainder for $ 4.50 a bushel and received in payment the
price which he might have received for the 42 bushels in the fall.
What was the price per bushel in the fall ?
BILLS. 321
44. BiUs.
In the following examples supply dates, etc., when wanting.
116. John Hill sold to J. Sparks & Co. 14 barrels of kerosene at
13 cents a gallon, and 24 barrels at 17 cents a gallon, each barrel con-
taining 50 gallons. Make out Mr. Hill's bill and receipt it.
117. Two customers ordered the following at a restaurant : roast
beef 40)^, veal pie 35/*, 2 tumblers milk @ 5 /-, pudding 10/', pie 20/,
pickles 5 /, sauce 5 /. Make a statement of the above with amount
due ; also tell the proper bills and coins to return if $ 10 should be
given in payment.
118. Make out a bill for freightage of the carcasses of 5 hogs weigh-
ing, respectively, 442 pounds, 468 pounds, 524 pounds, 537 pounds,
and 419 pounds, freightage being 35/ a hundred pounds.
119. Make out a bill to Mrs. Drake for a turkey weighing 12
pounds at 21 /, 8 pounds of beef at 14/, 4 hours' work at 25/, and
for use of horse and cart 50 /.
120. Alvin Bliss cut 148 cords of hard wood which he sold at
$7.80 a cord, and 87 cords of pine wood which he sold at $5.35 a
cord. How much did he receive for the whole ?
Make out a bill of the above to the New York Central K. R.
Company and receipt it by Charles Brown for Mr. Bliss.
121. Make out a bill to yourself from the Boston and Albany
R. R. Company for the freight of 1 car-load household goods, 12000
lbs. ; 3 boxes household goods, 1450 lbs. ; 1 piano, 1200 lbs. ; freight-
age 24 cents a hundred.
122. Oliver Granger sold to Charles Hyde 57 bushels of corn at
88 / a bushel, and 29 bushels of turnips at 45 / a bushel, and received
1 28.38 in part payment. Find how much was still due Mr. Granger,
and make out his bill.
123. Dec. 20, 1879, Mrs. S. Coles bought a photograph scrap-book
for $ 1.25 and photographs as follows : 22 pictures illustrating ancient
art @ $1.50 a doz., 25 illustrating medieval art @ $1.50 a doz. ;
Dec. 24, she bought 2 of Landseers' @ 33/, 2 of Hunt's @ 35 /, and an
etching for 37/. If she gave a lO-doUar bill in payment, what
change should be returned? Make out and receipt the photog-
rapher's bill.
322 APPENDIX.
45. Common Fractions.
124. At 37^ f a dozen, what will 7 dozen and 6 peaches cost?
125. In the Lachine Canal there are 5 locks with a total rise of
44f feet ; what is the average rise to a lock ?
126. If 789 whales yield 5-| million dollars' worth of oil and whale-
bone, what is the average yield per whale ?
127. How many pairs of pillow-slips, each slip requiring 1-| yards
of cloth, can be made from 43J yards, and how much cloth will l)e
left ?
128. "When 286 pounds of cocoa are bought for $81.51, for what
must it be sold a pound to gain 7-| /' on a pound V
129. Find the cost of 10-| pounds of beef at 16| j^; a pound, and 24|
gallons of vinegar at 30 / a gallon, and add the results.
130. If by working -f of an hour a day a boy can hoe 18 rows of
corn in a week, how much can he hoe if he works f of an hour a day ?
131. If it costs 75 / a cord to saw wood, so as to make each stick
into 3 parts, what will it cost to saw it so as to make each stick into
4 parts?
132. At 8^ o'clock in the morning two persons started from Dun-
kirk and travelled in the same direction, A at the rate of 6| miles
an hour, and B at the rate of 8^ miles an hour. How far apart were
they at 12 o'clock ?
133. A young man spent $165 during the first term in college,
which was W of his year's allowance. How much had he left for the
rest of the year ?
134. I sold a horse for $ 255, which was i| of what he cost ; how
much did I lose ?
135. At 66| f a yard for cashmere, how many yards can be bought
for $ 7.60 ?
136. At 38 / a pound for butter, what is the cost of butter for one
year for a family of 7 persons, allowing half a pound a week for each
person, and 52 weeks and 1 day to equal a year ?
137. If 42 pounds of maple-sugar can be made from 150 gallons of
sap, how many pounds can be made in three weeks from the sap of
128 trees, the average daily yield per tree being 1\ gallons?
138. At 8 / per yard for calico, and 33 / per pound for cotton,
what is the cost of materials for a pair of calico bed-comforters, each
being in length 2| yards, and in width equal to 2J breadths of
calico, and requiring 3 pounds of cotton ?
COMMON FRACTIONS. 323
139. Two boys are to have $ 1.75 for hoeing a field of corn ; if one
hoes 5 rows while the other hoes 7 rows, till the work is dOne, how
much should each boy receive ?
140. A train leaving M at 10 o'clock, P. M., reaches N at 5^
o'clock the next morning. The distance from M to N by rail is 175 J
miles. What is the average rate of the train per hour?
141. At 7 o'clock, A. M., A started on a journey, and travelled at
the rate of 7 miles an hour ; at 9 o'clock B started at the same place,
and travelled in the same direction at the rate of 9^ miles an hour.
In how many hours did he overtake A ?
142. At 10 A, M. Brown starts at Springfield for Worcester, distant
54 miles, and travels at the rate of 8 miles an hour. At the same
time Smith starts at Worcester for Springfield, and travels at the
rate of Q\ miles an hour. How far apart will they be at the end of
an hour ? In how many hours will they meet ?
143. To what must you add the difference between 11^ and 5f,
that the sum may be 16^ ?
144. A man owned ^ of a hotel which cost $ 12000 ; he sold \ of
his share to a third party at cost. What part of the hotel did he
then own, and how much did he receive for what he sold ?
145. The coast of the continent of Europe is said to be 20700
miles long, and all but 9800 miles of this borders on the Atlantic
Ocean. What part of the entire coast borders on the Atlantic ?
146. If a person travels 8^ miles in | of an hour, how long will it
take him to travel 22 miles ?
147. What will | of | of a yard of broadcloth cost at $ 4.54 a yard ?
148. If it takes |^ of a day to do a certain piece of work, how long
will it take to do | of ^ of it ?
149. If ^^ of an acre of land is worth $400, how much will 2|
acres be worth at the same rate ?
150. If 9^ eggs weigh a pound, and a pound of eggs will go as far
for food as a pound of steak, at what price per dozen must eggs be
bought, that nothing may be lost or gained by using them in place of
steak at 22 )^ a pound ?
151. A flag-staff 58 feet long is fastened to a building in such a way
that y^- of what is above the roof equals ^ of what is below. How
much is above the roof ?
324 APPENDIX.
46. Decimal Fractions.
152. If $20 is paid for 45000 keg-hoops, what is the price of 100 ?
153. A liter is a measure which holds 0.092 less than a quart ; what
part of a quart is a liter ?
154. Find the average height of the mercury in a barometer from
the following record of daily observations : 29.73, 29.84, 29.90, 30.01,
30.14, 30.09, 30.11, 29.94, 30.15, 30.17, 29.89, 29.93.
155. January 1, a gas-meter registered 11800 cubic feet of gas con-
sumed ; April 1 it registered 14100 cubic feet. At $ 2.70 per thou-
sand cubic feet, what was the cost of gas used in the intervening
time?
156. Beckoning .£1 of English money worth $ 4.8665, what is the
worth of £ 10.25 ?
157. If the coin of Germany called a mark is valued at $0,238
how many marks will equal $ 1000?
158. The annual expense of dikes and water- works in Holland is
often 7,000,000 guilders. At $ 0.401 per guilder, what is the expense
in dollars ?
1 59. How many bushels of wheat, each 60 pounds, must be used to
make a barrel of flour, 0.28 of the wheat being lost in making the
flour?
160. A and B are walking the same way along a road. At noon A
was 3.6 miles ahead of B. When will B overtake A, if A walks 3.75
miles an hour, and B 4.125 miles an hour?
161. What number is that to which if you add 7.6849 the sum will
be 7.9?
162. What iTCimber is that to which if you add 0.35 of itself, the
sum will be 945.945 ?
163. If 7.75 ounces of gold be divided among 3 men and a boy,
the boy receiving half as much as a man, how much gold will each
person have ?
1 64. What part of $ 9.50 is $ 0. 1 25 ?
165. To 5.49 add 0.7 of 8.65 ; subtract the sum from 18 ; multiply
the remainder by f of 9.18, and divide the product by 0.007.
166. From what must you take the difference between 4.25 and
3y^^, that the remainder may be 7.63 ?
167. How many hours will it take a person to travel 40.5 miles at
the rate of 5| miles per hour ?
COMPOUND NUMBERS. 325
47. Compound Numbers.
168. At 2|- feet to a step, how many steps must you take to meas-
ure a mile ?
169. If 9-^ eggs weigh a pound, what do 6 dozen weigh ?
170. How many cups of coffee, each holding | of a gill, may be
contained in a coffee-pot holding 1 gallon ?
171. At a school examination 625 pupils used 2 sheets of paper
apiece. How much did all use ? [Answer in reams, quires, etc.]
172. When a gross of buttons can be bought for 9 cents, what part
of a cent will 1 button cost ? What decimal part of a'dollar?
173. When you can get 3 gills of water by melting a quart of
snow, dry measure, how much snow must you melt to fill a 10-gal-
lon tub full of water ? [Answer in bushels, pecks, and quarts.]
174. Find the cost of steak and potatoes for a family breakfast
when there are used 2-|- pounds of steak at 18/ a pound, and 2 quarts
of potatoes at 75 / a bushel ?
175. What is the value of a township of public land at $2.50 an
acre?
176. How many quarts of currants will a basket hold that is 8
inches square and 4^ inches deep ?
177. How many quarts of water may be held in a dish of the
same dimensions as the basket above ?
178. What will 2 bu. 3 pk. of plums cost at 12 / a quart ?
179. What will 2 bu. 3 pk. of plums cost at $ 3.00 a bushel ?
180. At $ 10 per ton, what is the cost of three loads of coal weigh-
ing, respectively, 1635 lbs., 1848 lbs., and 1715 lbs. ?
181. What profit do I make by buying 150 tons of coal at $3.80
per long ton, and selling it at $ 5.60 per short ton ?
182. Find the cost of f of an acre of land at 17 ^ a square foot.
183. When snow is 2 feet deep, how many cubic feet of snow must
be removed in digging a path 2 rods long and 3 feet wide ?
184. A barrel holds 2 bushels and 1 peck. If the price of apples
is $2.50 per barrel, what is that per bushel ?
185. A car goes 200 miles in 8 hours ; allowing 1 h. 40 min. for
stops, what is the average rate of speed per hour ?
186. The latitude of Washington is 38° 52' 20" N. How many
miles is it from the equator ? from the North Pole ? [See Art. 334.]
187. How jnany slats will be required for a fence 5 rods in length,
if the slats are 2J inches wide and set 3 inches apart ?
326 APPENDIX.
188. What is the value of a lot of land 4 rods wide and 90 feet
deep, at 18 cents per square foot ?
189. How many times did a common clock strike in the year
1879?
190. How many days from Jan. 10 to Oct. 3 of the year 1880?
How many months and days from July 5, 1879, to April 3, 1882 ?
191. If a barrel of sugar containing 232 pounds, and costing l\ f
per pound, lasts a family from May 1, 1879, till July 4, 1880, what is
the cost of the sugar per week ?
192. How much does it cost a family for milk for a common year,
if 2 quarts are used every day, and milk is 6 cents a quart from May 1
to Nov. 1, and 7 cents a quart the rest of the year ?
193. If, when coal is $5.75 a ton, it takes 13 tons to supply the
furnace from the first of October to the first of April in a common
year, what is the cost of coal per week ?
194. At 60 cents per hundred, what will be the cost of ice for a
family that takes 40 pounds 3 times a week, from Apr. 1 to Oct. 7 1
195. What would be the cost of the hay which would be required
to keep a horse from June 1 to Nov. 15, allowing him 15 lbs. a day,
the price of hay being 1 17 a ton ?
196. A boy picked from his pear-tree 2 bu. 3 pk. of pears, which
he sold at the rate of 3 for 10/. If each peck averaged 3^ dozen
pears, how much did the boy receive for the whole quantity ?
197. Allowing 6 lbs. clover seed, J bu. timothy, and 1 bu. red top
to the acre^ what is the cost of seed for 3|- acres when the price of
clover seed is 12/ per pound, timothy $ 2.65 per bushel, and red top
% 3.25 per bushel ?
198. If a person earns $2.87 a day, how much can he earn during
the working days of a common year which begins on Sunday ? How
much in a leap year which begins on Monday ?
When it is 6 p. M. in New York, what time is it
199. In Albany ? 201. In Mexico, 99° 7' 8" W. ?
200. In London ? 202. In Bombay, 72° 54' E. ?
203. A person going from Boston found at the end of two days
that his watch, which kept Boston time, was 1 h. 35 m. too slow. In
what longitude was he ?
204. A telegram sent at 9 o'clock in the morning from A, longi-
tude 72° W., reached B, at 8^ o'clock the same morning. The tele-
gram passing instantaneously, in what longitude was B ? *
COMPOUND NUMBERS, 327
205. What is the weight of a granite pedestal that is 6 feet long,
6 feet wide, and 4 feet high, the granite being 2.5 times as heavy as
water ?
206. How many bushels of rye can a bin contain that measures on
the inside 6 feet in length, 5 feet in width, and 4 feet in depth ?
207. How deep must a bin 6 feet by 4 feet be to contain 30 bushels
of rye ?
208. How many bricks 8 inches long, 4 inches wide, and 2 inches
thick, in a rectangular pile of bricks 10 feet long, 2 feet wide, and 4^
feet high, no allowance being made for waste of space?
209. The Washington elm in Cambridge has been estimated to
bear 7,000,000 leaves annually, averaging 4 square inches of surface
each. What is the aggregate surface of the leaves in feet ? in acres ?
210. How many square feet of glazing in a house containing 7 win-
dows of 4 panes, each 24 in. by 16 in., 5 windows of 12 panes, each 14
in. by 10 in., and 2 windows of 4 panes, each 30 in. by 16 in. ?
211. Find the cost of cement at 11 cents per sq. yd. for a cellar
floor 18 ft. by 24 ft.
212. How many bricks, each 8 in. by 2 in., will be required to
make a walk 2 rd. 7 ft. long and 6 ft. wide ?
213. How many Dutch tiles 3 inches square will it take to cover a
floor 18 feet long and 14 feet 6 inches wide ?
214. A speculator bought a field of 3 J acres ; from this he made 8
house-lots of 9650 square feet each, and then divided the rest equally
into 5 house-lots. How many square feet in each of the 5 house-
lots ?
215. A farmer divided a field into house-lots, making 18 lots of
16475 square feet each, and 15 lots of 15968 square feet each. How
many acres were there in the entire field ?
216. A school-room 32 feet long by 28 feet wide and 14 feet high
is occupied by 35 pupils. How long will it take to spoil the air in
the room, if each pupil spoils 4 cubic feet in a minute ?
217. From a field 20 rods long and 9 rods wide are cut three loads
of hay weighing 1875 lbs., 1950 lbs., and 2025 lbs., respectively. This
is a yield of how many tons to the acre ?
218. In the United States 20,000,000,000 of matches are manu-
factured yearly. If 50 matches are made from a cubic inch of wood,
how many cords of wood are required for these matches, no allowance
being made for waste ?
328 APPENDIX.
219. Some fire-wood cut 4 feet long is piled 6 feet high. How
long must the pile be to contain 17 cords?
220. How many yards of Holland will it take to curtain 5 windows
6 feet high, 8 windows 5^ feet high, and 3 windows 6^ feet high, 4
inches more than the height of each window being allowed to a
curtain ?
221. The force of waves against a sea-wall in a heavy storm is
sometimes '2\ tons to a square foot. At this rate, what is the force
exerted upon a sea-wall 20 feet in height and ^ of a mile long ?
222. Before Lake Haarlem was drained it was 15 miles in length
and covered 45000 acres. What was its average width ?
223. In digging a ditch A\ feet deep and 20 rods long, 220 cords of
muck were obtained. What was the width of the ditch ?
224. What is the cost of boards at $14 per thousand to make the
floor and sides of a bin 15 ft. long, 6 ft. wide, and 7 ft. deep, no allow-
ance being made for thickness of boards ?
225. What is the cost of boards at $20 per thousand to make 50
boxes, each 7 ft. 10 in. long, 3 ft. 8 in. wide, and 2 ft. 6 in. high, no
allowance being made for thickness of boards ?
226. Find the cost per pound of lead to line a tank 3 ft. square
and 4 ft. 3 in. deep, if the whole cost is $ 38.67, and the lead used
weighs 5 lbs. to the square foot ?
227. A water-tank 6 feet long, 2^ feet wide, and 3 feet high is
filled by 1242 strokes of a force-pump. How many gallons does the
tank contain ? How much water is raised by each stroke of the
pump ?
228. I have a rectangular field 80 rods long and 6 rods wide. If
this field be divided into 10 equal rectangular house-lots, having their
fronts on the longest side of the field, what will be the cost of fencing
to enclose and separate the lots at $ 6 a rod ?
229. A street 3 rods wide and 200 rods long is on an average 10
inches above grade ; how many cubic yards of earth must be removed
to bring the street to grade ?
230. What is the weight of the air in a room 25 ft. long, 20 ft.
wide, and 12 ft. high, water weighing 770 times as much as air, and
a cubic foot of water weighing 1000 ounces ?
231. In 100 parts by weight of air are 22 parts of oxygen. What
is the weight of the oxygen in a room 20 ft. long, 18 ft. wide, and
10ft. high?
METRIC MEASURES. 329
48. Metric Measures.
232. If a car- wheel is 1.25 ™ in circumference, how many times
must it turn in going 87.5 '^ ?
233. At 8 / per meter, what will be the cost of a wire fence on
both sides of a road 10.5 kilometers in length ?
234. At the rate of a meter a second, how many minutes will it
take a mountain torrent to run 10 kilometers ?
235. At the rate of a meter a second, how many kilometers will a
torrent run in an hour ?
236. How many liters of water may be contained in a reservoir 8 °*
long, 6 ™ wide, and 5 ^ high ? What weight in kilograms ?
237. In 1878 the artesian wells in Algeria yielded on an average
2200 dekaliters of water in a second. As a rule, each of these wells
can water 6 times as many palm-trees as it gives out liters of water in
a minute. How many palm-trees can these all water ?
238. Wishing to find the average length of my pace, I measured
off a distance of one dekameter on the ground. This I paced back
and forth, counting my paces as follows : 14, 13^, 14, 14, 14^, 14, 13^,
14, 14^, 14^, 13, 14^. What is the average length of my pace in
centimeters ?
239. What will a hektar of land cost at $ 0.75 per sq. meter ?
240. If 7 hektars of land be divided into 4 equal house-lots, how-
many square meters will each lot contain ?
241. How many grams does a dekaliter of water weigh ?
242. The specific gravity of lead being 11.445 (that is, lead being
11.445 times as heavy as water), what is the weight of a piece of lead
3 d™ long, 2 <i°^ wide, and 5 «i™ thick ?
243. The specific gravity of honey being 1 .456, how many kilos of
honey may be put into a box 2 ^^ square at the bottom and 1.5'*™ deep?
244. How many metric tons of Egyptian marble in a rectangular
block 0.75 ™ square at the base and 3 ™ high, the specific gravity of
the marble being 2.668 ?
245. If a bar of silver is 2 cm square at the end and 1 **'" long, and
has a specific gravity of 10.474, how many kilos does it weigh ?
246. How many sters in a pile of wood 15™ long, 3.5™ high,
and 1 ™ wide, and what will it cost at $ 3.50 a ster ?
247. How many pounds avoirdupois in 1000 letters, each weighing
18 grams ? [Use equivalent for gram.]
248. Philadelphia being about 160 kilometers from New York,
what is the distance in miles ?
330 APPENDIX,
49. Percentage.
249. The prime cost of an article being $ 230, at what price must
it be sold that a profit of 30 % may be made?
250. A baking powder, when analyzed, was found to contain
38.12 % of starch, 31.82 % of soda, and the remainder was burnt
alum. What per cent was burnt alum?
251. If the price of gloves has declined since 1864 from $ 2.25 per
pair to $ 1.25, what per cent has it declined?
252. If water weighs 62^ pounds to the cubic foot, and milk 64^
pounds, by what per cent is milk heavier than water ?
253. An estate sold for $ 45000, which was 37^ % below the ap-
praised value. What was the appraised value ?
254. How many pounds of bread can a baker make from a barrel
of flour, if the bread made is 32 % heavier than the flour used ?
255. How many pounds of flour must a baker use to make 200
pounds of bread, if the bread weighs 32 % more than the flour used ?
256. From a bill amounting to $2463, there is thrown off $63 ;
that is equivalent to a discount of what per cent ?
257. A owns \ of the copyright of a book, and B owns the remain-
der ; but for services on the book they agree that C shall have 5 % of
the whole copyright. What per cent will A then have ? What per
cent will B have ?
258. A person has a preparation worth $3.20 a pound (Av.).
What per cent does he gain by putting it up in powders of 10 grains
each, and selling them at the rate of 6 powders for 25 cents ?
259. If by buying collars at the rate of 3 for 25 cents, instead of
buying them singly, I save 16| %, what is the price per collar when
bought singly ?
260. What is the value of the gold in a ton of ore, if 3 % is gold
worth $ 16 an ounce ? [Reckon by the long ton.]
261. A seal weighing 56 pounds has been known to eat a quantity
of fish equal to 25 % of its own weight in a day. At this rate, how
many pounds would it eat in a common year ?
202. A savings-bank having suspended payment, its deposit books
are sold at 23 % discount. What could be realized on a book show-
ing a deposit of $ 952.17 ?
263. The entire cost of finishing a room was $115, If 5 % of
this was for laths at 25 /' a hundred, how many laths were used ?
PERCENTAGE. 331
264. What is the retail price of a book which is sold at 33|- % be-
low the retail price, and then brings 33^ cents ?
265. When chairs are sold at $ 4.89 per dozen, with a discount of
5 % for cash, what is the cash value of 200 chairs ?
266. A lamp is sold at 10 % below cost, and brings 54 f. What
would it bring if sold at 25 % above cost ?
267. If a jockey sells a horse for 90 % of his cost, he gets $45 less
than if he sells him for 120 % of his cost. What was his cost ?
268. A commission merchant gets 5 % commission on his sales.
His commissions during the year amount to 1 13685.50 ; what is the
whole amount returned to his consignors ?
269. At 51 %, what is the commission on the sale of 350 barrels of
apples at % 2.80 per barrel, and how much money should the con-
signor receive for them ?
270. Find the cost, including \ % commission, par value 1 100, of
17 Old Colony Railway @ 112J.
10 Atchison, Topeka & Santa F€ @ 137|.
23 Flint & Pere Marquette @ 24|.
5 Chicago, Burlington & Quincy @ 124 J.
12 Rutland preferred @ 29|.
271. If the wages paid for work on the backs of chairs is 10/ per
chair, and on seats is 8/, how much must be paid for one hundred of
each, if the wages are advanced 10 % ?
272. How large a bill of goods can be purchased for a remittance
of $ 1000, if 20 % discount is allowed for cash ?
273. I bought a lot of leather at 10 % below the asking price, and
sold it for $ 2400, and by so doing I gained 33^ % on the cost. What
was the asking price ?
274. I built a house costing $ 3000 upon a lot which cost $400.
The house being burned, the insurance company paid me 75 % of the
cost of the house. I then sold the land for % 1400 ; did I gain or
lose by the transactions, and what per cent ?
275. A lot of goods was marked for sale at 40 % above the cost ;
if the lot was sold at auction at 30 % below the marked price, was
there a gain or a I4ss on the cost, and of what jjer cent V
276. A house is insured for 1 4500 for 5 years, at a premium of
$ 45. What is the rate per cent of insurance per year ?
277. What is a building worth if the insurance premium of | % on
I of its value, including $ 1 for the policy, equals $ 43 ?
332 APPENDIX.
278. Wliat must I pay to insure some goods worth % 3500, at a
premium of 1^ % on | of their value ?
279. If these goods should be destroyed by fire, what would be the
loss to the owner ?
280. What would be the loss to the underwriters ?
281. What amount must be insured to cover property worth $1800
and a premium of -f % ?
282. A marine insurance company took a risk of |25000 at 3^ % ,
and reinsured -| of their risk in another company at 3^ % . Should
the property be destroyed at sea, how much would the first company
lose ?
283. The taxes in a school district are. assessed upon property
valued at % 87000 ; the whole tax is as follows : for building and
furnishing a school- house, % 2100 ; for fuel, care of house, and wages of
teacher, % 510. What is the tax on $ 1 ? What amount of tax is
paid by a person whose property is valued at $ 4500 ?
284. At 20 % ad valorem, what is the duty on 18 chests of tea, the
gross weight being 765 pounds, 6 pounds for tare being allowed on
each chest, the tea costing 35 f per pound ?
285. When the duty on a quantity of lace at 30 % ad valorem is
$ 115.80, what was the cost of the lace and duty in francs at $0,193
per franc ?
286. A merchant imported from Geneva 35 watches at % 65 each,
and 42 watches at $ 125 each. The duty being 35 % ad valorem,
what did the watches cost, and for how much apiece must they be
sold to make a profit of 15 % ?
60. Interest
287. Find the amount of a $ 300 note given the 25th of August,
and paid the 19th of the next December, interest 10 % per annum.
288. A trader borrowed $ 8000 at 5 % per annum. He used this
money in a way to yield him $ 1256.42 during the year. How much
did he clear by borrowing this money ?
289. What is the interest of 1 cent for 1 day at 1 % a year (365 d.) ?
290. Find the amount of each of the following sums at 7 % :
1820 from Feb. 10, 1880, to Dec. 4, 1880.
1 1500 from Dec. 15, 1880, to May 10, 1881.
291. A due- bill is given for | 68.50 with interest at 7^ %. What
will the bill amount to if it runs 72 days ?
INTEREST. 333
.ij2. Find the difference "between the accurate interest of $ 1500 at
6% and the interest, by the common method, the time being from
May 24 to Oct. 15, 1881.
293. What per cent on my investment do I save by buying, April 1,
% 30 worth of goods for which I should pay $ 48 the first of the fol-
lowing December, money being worth 6 % a year ?
294. A note given May 4, 1879, on demand at 7 % for % 475, was
indorsed July 2, 1879, $ 100 ; Oct. 1, 1879, $ 100 ; what sum re-
mained due April 1, 1880 ?
295. Jan. 1, 1879, I gave a note for $ 356 at 6 % . When will this
note amount to $ 400 ?
296. In what time will the interest on a sum of money be f of the
principal at 6 % per annum ?
297. At what rate per annum will the interest on a sum of money
be 0.0215 of the principal in 4 months ?
298. The use of money being worth 6 % a year, what is the pres-
ent value of $ 500 due four years hence ?
299. If a merchant sells flour at $ 9 a barrel, and has to wait 6
months for his pay, at what price could he aff'ord to sell for cash
down, the use of money being worth to him 2 % a month ?
300. Sold goods amounting to $4500, one half on 4 months*
credit, the rest on 6 months, and got the notes discounted at a bank
at 8 % . What was realized ?
301. A trader buys 900 pairs of gloves at % 0.75 a pair cash, and
immediately sells them for f 1090 on a note payable in 4 months with-
out interest. Suppose he gets the note discounted at a bank for the
4 months, what will he have made ?
302. Write a note on 3 months' time for such a sum that, when dis-
counted by a bank at 7 % , the proceeds may be $ 400.
303. What is the amount, at compound interest, of $ 100 from
April 1, 1876, to Jan. 1, 1879, at 6 % per annum, interest payable
semiannually ?
304. What is the equated time for paying the following bills ot
goods bought on 30 days' credit : June 2, f 420 ; June 8, $ 600 ; June
15, $560?
305. A person holding three promissory notes, one for $ 100 pay-
able in 2 months, another for $ 300 payable in 4 months, and another
for $ 200 payable in 6 months, exchanges them for a single note for
the sum total. When should this be payable?
334
APPENDIX.
306. A owes B $2000 clue Oct. 5. If he should pay 1 1200 of it
on the 8th of the previous September, when should the balance be
paid, that there may be neither gain nor loss of interest ?
307. F. Bates owes A. Smith $ 350, due May 29. If he should
pay $ 50 on the 29th of April previous, when should he pay the bal-
ance ?
308. Find the balance due Flint in the following account, Oct. 1,
1880, interest at 6 %, from the date of the items : —
Dr. SETH DAVIS in Acct. witli AMBEOSE FLINT. Cr.
1880,
Mar. 29
Apr. 22
To Mdse
" Cash
1880.
$Jt76
93
Apr. 24
869
82
May 15
By Mdse
" Mdse
379
51. Exchange.
309. The value of £ 1 being 1 4.866^, find the value of $ 1 in
shillings and pence.
310. The value of 1 franc being $0,193, what is the value of $ 1 in
francs and centimes V
311. The value of 1 franc being % 0.193, and the value of £1 being
$ 4.86&I-, find the value of <£ I in francs and centimes. Also find the
value of I franc in units of English money.
312. If <£I = 25.22 francs, find the value in English money of a
French coin worth 20 francs.
Note. — In solving the next four examples use the par value of money
as shown on page 311.
313. A railroad has been commenced, passing from Algiers to the
Niger, near Timbuctoo. It is estimated that the remaining 1700
miles will cost 400,000,000 francs. Find the estimated cost per mile
in francs ; in dollars.
314. Coal was 29 shillings a ton in London in 1879. What was
that in United States money ?
315. It is estimated that the actual cost of rearing a boy of one of
the poorest classes in England is £ 300. What would be the cost in
dollars of rearing a family of 5 boys?
316. If electric pens cost 18 francs each, what is the cost of a dozen
such pens in dollars and cents ?
317. I wish to remit £ 143 10 s. to London. Exchange is quoted
at $ 4.87^. What will my bill of exchange cost ?
BONDS. — PROPORTION. 335
62. Bonds.
318. If Mr. Wm. H. Vanderbilt owns $31,000,000 in U. S. 4 per
cent bonds, what is his income per day from these bonds ?
319. When the use of money is worth 5J% a year, what should
be the price of a thousand-dollar Government 4 per cent bond, no
account being made of interest upon the quarterly interests, nor of
the fact that when the bond becomes due, it will be paid at par.
320. A man has settled on his wife $ 1200 a year. What sum
must he invest in government 4 per cents at 107| to produce that
amount of income ?
321. By investing in the 6 per cent bonds of a city I get an annual
income of 4| % on my money. At what price were the bonds bought ?
322. From which investment would the larger returns be derived ;
from 7 per cent bonds at 115, or 6 per cent bonds at 98?
323. What per cent must I make on an investment of $ 1000 to
equal the income from three U. S. 4 per cents of f 500 each ?
63. Analysis and Proportion.
324. If 112 men can do a piece of work in 10^ days, how many
men can do it in 12 days ?
325. How many eggs at 24 per dozen will pay for 25 lbs. of but-
ter at 42 f per pound ?
326. My bill for gas being % 17.25 when gas is % 1.90 per 1000
cubic feet, what would it be if gas were $2.15 per 1000 cubic feet,
and I should use | as much?
327. If the cost of keeping 25 horses is f 57.50 a week, what is the
"-ost of keeping 13 horses from July 1 to October 31, both days in-
cluded?
328. At noon on a certain day the shadow of a man 5 ft. 11 in.
tall is found to measure 4 ft. 2 in. What is the height of a tree
whose shadow at the same time measures 47 feet 4 inches ?
329. The interest of % 500 at 6 % for 3 years is equivalent to the
interest of how much at 8 % for 2^ years ?
330. If 450 pounds of merchandise can be carried 26 miles for 30/,
how many miles can 3 tons be carried for $ 4 ?
331. If, when flour is $ 7.50 per barrel, a 3-cent loaf weighs 2 oz.,
what should a 12-cent loaf weigh when flour is $ 12 per barrel?
332. If a seamstress earns f 9 a week by working 10 hours a day,
how much can she earn in 3 weeks 2 days by working 8 hours a day ?
336 APPENDIX.
333. If 5 men can do a piece of work in 7 days of 11 hours each,
how long will it take to tinisli the work when it is half done, if 3
more men are set to work upon it, and all work 8 hours a day ?
334. If it takes 21 yards of cloth | of a yard wide to make 4 gar-
ments, how iwAw^ yards l^- yards wide will it take to make 2 dozen
garments of the same kind ?
335. It takes 8 men 15 days to build 28 rods of fence ; how long
will it take to build 80 rods if 4 boys help, each boy doing half as
much work as a man does ?
336. If the weight of a block of sandstone 3 ft. long, 2 ft. wide, and
2 in. thick, is 140 pounds, what is the weight of another block of
sandstone 4 ft. long, 3 ft. 6 in. wide, and 3 ft. thick?
337. If 3 silver spoons, each weighing 2 oz. 6 pwt. 16 gr., cost
$9.00, silver being worth % 1.25 per ounce, what should be paid for
6 dozen similar spoons, each weighing 3 oz. 3 pwt. 12 gr.,when silver
is worth f 1.40 per ounce, and the cost of making is the same ?
64. Partnership.
338. Four men hire a pasture in common, paying 1 34.50. A puts
in 3 cows 19 days, B 2 cows 15 days, C 3 cows 7 days, and D 1 cow
30 days. How much must each pay ?
339. Two contractors have finished work for which they have been
paid 1 12500. One of them employed 50 laborers for 125 days of 12
hours each ; the other 40 laborers for 90 days of 10 hours each.
Divide the money between them in proportion to the labor each
furnished.
340. A, B, and C invest $ 4860 in trade : A invests twice as much
as B, and C invests twice as much as A and B together ; they gain
40 per cent on their investment. What is each person's share ?
341. Divide % 9765 among 8 men, so that one half of them shall
have twice as much as the other half.
342. Mr. Bacon has failed for $ 14568 more than is covered by his
available resources. His creditors' claims amount to % 30548. How
much must M lose, if Bacon owes him $2500?
343. T. and V. formed a partnership to trade in coal. T. furnished
$ 3000 for the first 10 months, at the end of which time he added
$ 1000 more, and at the end of the second year, % 500 more. V. put in
$ 2500 for the first 18 months, at the end of which time he put in
% 3500 more. At the end of the third year they found their gain to
be % 5565 ; what should each receive ?
POWERS AND ROOTS, 337
65. Powers and Roots.
Find the powers indicated below :
(344.) 192. (347.) 2.5052. (350.) (2^)2. (353.) (| X li)8.
(345.) 1.92. (348.) 1.0043. (351.) (5^)2 (354.) (| + 1)2.
(346.) 0.192. (349.) 0.034. (352.) (161)2. (355.) 1.05*.
Find the square root
356. Of 6241. 359. Of 19600. 362. Of 44.009956.
357. Of 9409. 360. Of 1960. 363. Of 0.005648.
358. Of 1633. 361. Of 640900. 364. Of 369056.25.
365. Find the difference between the sum of the square roots of
6075 and 41616 and the square root of their sum.
366. V^l0S = ? 370. V^^=? 374. ^0.4 + 25=?
367. V9/^ = ? 371. V^l00|| = ? 375. V^0.9X0.16=?
368. 0711 = ? 372. ^J^xW = '> 376. V'^ of | of 1^ = ?
369. sj4^\ = l 373. V'|^f|| = ? 377. ^0.36 h-| X 6^ = ?
378. If a square contains 12J square feet, what is the length of one
of its sides ?
379. On a roof are laid 7200 slates, the number in the length being
twice the number in the width. What is the number each way ?
380. My orchard contains 7350 trees. The number of trees in
width is to the number of trees in length as 2 is to 3. What is the
number each way ?
Find the cube root
381. Of 39304. 384. Of 4104. 387. Of 0.27. 390. (^i55| = ?
382. Of 103.823. 385. Of 79.507. 388. Of 6.4. 391. v'^=?
383. Of 0.00043. 386. Of 2.24. 389. Of 0.8. 392. V^I| = ?
393. Maury estimates the annual amount of rainfall in the Missis-
sippi Valley to be 620 cubic miles of water, of which 107 cubic miles
is poured into the sea. What would be the depth of a cubical reser-
voir that would contain the remainder ?
394. It is estimated that in 1871 the world's consumption of petro-
leum was 6,000,000 barrels. If each barrel contained 50 gallons,
what would be the length of a cubical tank large enough to contain
it all?
338 APPENDIX.
&Q. Mensuration.
Kectilinear Figures.
395. Find the area of a rectangle whose base is 7 ™ and whose
height is 5.4 ™.
396. Miss Gove has a triangular flower-bed whose sides are 10 feet,
15 feet, and 9 feet long respectively. How many square feet are
there in the bed?
397. Of two triangular flower-beds, one having each of two sides
8 feet in length and the third side 12 feet, the other having a right
angle enclosed by two sides each 7 feet long, which is the larger,
and how much larger?
398. How many yards of lace must be used to trim the edge of a
"half-handkerchief," the length of the diagonal side being 28 inches,
and 4 inches being allowed for fulness around each corner ?
399. Wishing to find the distance between two points A and B, on
opposite sides of a swamp, I measured two lines A C and B C, from
A and B, making a right angle at C, and found them to be 60 feet
and 80 feet long respectively. How far was A from B ?
400. How many ars are there in a square field whose diagonal is
100 meters ?
401. Find the area in acres of a rectangular field whose length is
10 chains and breadth 8.49 chains. [See page 306, Art. 19.]
402. If the area of a parallelogram is 500 square yards and its
height is 50 feet, what is its base ?
403. Two parallel sides of a quadrilateral field are 45 chains and
36 chains respectively, the distance between them is 25.4 chains.
How many acres are there in the field ?
404. The sides of a quadrilateral field are 50 chains, 43.4 chains,
26.8 chains and 43.2 chains respectively ; the length of a diagonal
cutting off the first two sides is 53 chains. How many acres are
there in the field ?
Circles.
405. How far must I go to walk from one point in the circum-
ference of a circular pond to the opposite point, if it is 40 feet across ?
406. If it is 36 feet around the casing of a circular window, how
far is it across ?
407. How many square yards of turf will be required to cover a
circular plat of ground 27 feet across?
MENSURATION. 339
408. If the area of a circular park is 12 acres, how long must a
fence be to enclose it ? [Ans. in rods.]
409. If the area of a circle is 40 square feet, what is the area of
an inscribed square ?
410. If the side of a square inscribed in a circle is 40 chains, what
is the area of the circle ? **
Solids.
411. Find the entire surface of a cube each of whose edges is 2
feet long.
412. Find the entire surface of a rectangular solid whose dimen-
sions are 2 feet, 3 feet, and 4 feet respectively.
413. Find the convex surface of a regular pentagonal column
whose height is 10 ft. and the width of each of whose faces is 1 ft. 3 in.
414. Find the convex surface of an octagonal block whose slant
height is 6 inches, the length of each side of the upper base being
7 inches and that of the lower base being 9 inches.
415. Find the convex surface of a regular 4-sided pyramid 12 feet
high, the length of one side of the base being 10 feet.
416. Find the number of cubic feet in the pyramid named in
Example 415.
417. If the pyramid named in Example 415 be cut by a plane
parallel with the base and 2 feet from it, how many cubic feet will
be contained in the lower part?
418. How many cubic feet are there in a piece of timber 15 feet
long, 16 inches square at one end and 12 inches square at the other?
419. Find the convex surface of a pipe 2 meters long, the diameter
of which is 36 centimeters.
420. Find the convex surface of a cone a meter high, the radius of
the base being 3 decimeters.
421. How many square inches in the surface of the largest sphere
that can be put into a cubical box whose dimensions on the inside
are 14 inches each?
422. How many cubic feet of air may be contained within a conical
tent the diameter of the base being 25 feet and the slant height 20
feet 10 inches ?
423. How many square feet of canvas will cover a conical tent, the
diameter of the base being 28 feet and the height 13 feet?
424. How many cubic inches of water can be contained in a globe
whose diameter inside is 15 inches ?
340
APPENDIX.
e
1
d
L
■<*•
1
1
k;-H
-— J
18 feet.
16 feet.
eo5fl.
/
1
6
/
3 ft.
«
<o
15 feet.
8 ft.
/
a, study ; 6, hall ; c, sitting-room ; d, dining-room ; c, conservatory ; /, /, /, piazza.
425. Above is a plan of some rooms in a house. If you should
carpet the study with ingrain carpeting a yard wide, and lay it down
so as not to split any breadth nor turn any under, how many yards
would you use, nothing being allowed for matching the figures ?
426. How many yards would you use to carpet the study if you
should lay the carpeting down lengthwise of the room and should
not split any breadth, nothing being allowed for matching figures ?
What will be the cost of the carpeting at % 1.25 per yard ?
427= How many yards of Brussels f yd. wide will it take to carpet
the sitting-room, if the breadths are laid lengthwise of the room and
one breadth is split into two equal parts, no allowance being made
for matching figures ? What wdll be the cost of the carpeting at
$2 25 per yard ?
428. If the cost of carpet for carpeting the sitting-room with
Brussels carpet be % 87.75, what will be saved by carpeting it with
ingrain at $1.25 per yard, laying the breadths lengthwise of the
room, and splitting no breadth ?
429. How many square feet are there in the floor of the dining-
room if the bay window is 2 feet deep and 9 feet wide ?
430. How many square feet are there in the floor of the hall ?
Note. In solving Examples 430 and 432, no allowance is made for
thickness of walls.
MENSURATION. 341
431. How many square feet in the floor of the conservatory if it is
8 feet in front, 10 feet at the back, and 3 feet deep ?
432. How many square feet in the floor of the piazza if it is 5 feet
wide ?
433. Wishing to carpet the stairs as shown in the plan, I found
the height of the second floor from the hall floor to be 10 feet. Allow-
ing half a yard for the landing, etc., what would be the cost of carpet-
ing at $ 2.12^ per yard ?
434. The back stairs are 14 in number and carpeted with tapestry
at 87J f per yard, 5 % off for being damaged. Each stair is 8 inches
high, and 8 inches wide, and | of a yard is allowed for turnings, etc.
What is the cost of the carpeting ?
435. In the dining-room is an extension-table 4 feet wide. The
ends of the table are semicircular, and besides there are 6 leaves each
15 inches wide. How many square feet are there in the surface of
the table and leaves, and what is the greatest number of people that
can be seated aroimd it, allowing 2 feet for each person ?
436. When 3 leaves are in, how long must the table-cloth be to
cover the table and hang a quarter of a yard over at each end ?
437. What is the cost of a concrete walk 10 feet wide extending
diagonally across a square park containing 1 acre, the concrete costing
^f &. square foot ?
438. At $ 3.50 a yard, what is the cost of a piece of velvet, the
length of one side on the selvage being 2| yards, and of the other
being 2\ yards?
439. At % 2.60 a yard, what is the cost of a piece of satin 27 inches
wide cut off at one end at right angles with the selvage, and at the
other end being cut " on a bias " (making half a right angle with the
selvage), the length of the longer side of the satin being 2-|- yards?
440. On a piece of land 10 rods long, 7 rods wide at one end and
5 rods wide at the other, 1 bushel of oats was sown. This was at the
rate of how many bushels per acre ?
441. There are 3,000,000 cu. ft. of masonry in the Victoria Bridge
at Montreal. If this masonry was put into a pyramidal form whose
base was square and whose height equalled that of Bunker Hill Monu-
ment (221 feet), what would be the length of one side of the base ?
442. How many hectoliters of water may be contained in a reser-
voir 10 meters deep, 360 meters square at the bottom, and 400 meters
square at the top ?
342 APPENDIX.
443. The diameter of the large wheel of a bicycle is 46 inches.
How many times will this wheel turn in going a mile ? [Ans. to
tenths.]
444. The large wheel of John's bicycle is A\ feet in diameter, that
of Burt's is 46 inches. In going 2 miles and a half and returning,
how many more revolutions are made by Burt's bicycle than by
John's? [Ans. to tenths.]
445. If a point on the rim of a circular saw whose diameter is 40
inches goes at the rate of 1^ miles per minute, how many revolutions
does the saw make a minute ? [Ans. to tenths.]
446. How many revolutions would a circular saw whose diameter
is 30 inches have to make that the rim might go at the rate of 1^
miles a minute ?
447. What will it cost to enclose a hectar of land in a circular form
with a fence costing $ 1.25 per meter?
448. At $ 1.25 per meter for fencing, what will it cost to enclose a
square piece of land containing a hectar ?
449. Wishing to ascertain the number of cubic inches in an irregu-
lar piece of stone, I immersed it in water and found it displaced
enough to fill to the depth of 1-^ inches a cylindrical dish whose
diameter was 6 inches. How many cubic inches in the stone ?
450. How many barrels of water, 31|- gallons to a barrel, can be
contained in a cylindrical tank 8 feet deep and 12 feet in diameter ?
451. How many hectoliters in a circular reservoir 6 meters deep,
the diameter at the bottom being 60 meters and at the top 70 meters ?
452. Estimating the diameter of the earth to be 8000 miles, and
its specific gravity to be 5.6, what is its weight in tons ?
453. A pipe 4 inches in diameter will drain a certain reservoir in
96 hours. In how many hours would a pipe 16 inches in diameter
drain it ?
454. If a whale 95 feet long weighs 245 tons, what will a whale
similar in form 50 feet long weigh ?
455. What must be the height of a tree to contain 125 times as
much as a similar tree 30 feet high ?
456. Wishing to know the quantity of coal piled 3 meters high in
a corner of a coal-bin, I weighed a pile similar in form 4.85 deci-
meters high, and found it contained 85 pounds. What was the weight
of the larger pile ?
MISCELLANEOUS EXAMPLES. 343
57. Miscellaneous Examples.
457. What is the par value of stock of which 4 shares, at 72 %
discount, sell for $112 ?
458. When the government tax for matches was a cent for every
hundred, and $ 2,000,000 was received per year for the taxes on matches,
how many matches were manufactured during the year ?
459. From a bushel of fruit and 15 pounds of sugar 25 jars of pre-
serves were made. What was the cost of the preserves per jar, if the
cost of the fruit was $1.12^ and the sugar was l\^^ per pound ?
460. I sent to my agent a lot of nails which he sold for $ 963 ; on
this he was allowed a commission of 3| %. I directed him to invest
the balance in hops, after deducting a commission of 1 J % for invest-
ing. What sum remained to invest?
461. The first instalment of the fund for the Smithsonian Institute
was $515169; the residuary legacy was $26210.63. Congress has
since added $108620.37, and Mr. Hamilton bequeathed $1000.
What is the yearly interest on this fund at 6 % ?
462. The Smithsonian Institute also holds certificates and bonds
in Virginia worth at par $88125, present value $37000. What per
cent has this Virginia stock depreciated ?
463. In my History of England the third chapter begins at the top
of the 110th page and ends at the bottom of the 189th. How many
pages are there in the chapter?
464. Four per cent per annum paid quarterly is equal to what
per cent per annum paid annually, interest at 6 % being allowed on
each quarterly interest ?
465. A dealer in provisions loses 10 % by giving credit, and pays
2-^ % for having his bills collected. What per cent above cost must
he charge for goods, that he may clear 10 % ?
466. It is a principle in mechanics that the two arms of a lever
are inversely proportional to the pressures at their extremities.
What power 6 feet from the fulcrum will balance a weight of
80 pounds 5 feet from the fulcrum? 2 feet from the fulcrum? 15 feet
6 inches ?
467. How far from the support of a tilt must a boy weighing
120 pounds be placed to balance another boy weighing 96 pounds
and seated 10 feet from the support ?
344 APPENDIX.
468. What is the compound interest at 6 % per annnm, payable
semiannually, on a note for $1000, dated September 14, 1880, and
paid December 2, 1881 ?
469. A water-tank 3 feet in depth and 4 feet square at the base,
is supplied by rain from a flat roof 30 feet long and 20 feet wide.
What depth of rain must fall to fill the tank ?
470. Find the cost of materials and labor for 100 square yards of
lath and plaster-work, 3 coats, hard finish, as follows, and make out a
bill for the same. Common lime, 4 casks @ $ 1.00 ; lump lime, | cask
@ 1 1.35 ; plaster of Paris, ^ cask @ $ 1.50 ; laths, 2000 ©20/ per
hundred ; common sand, 7 loads @ 30 / ; white sand, 2-^ bu. @ 8 / ;
nails, 13 lbs. @ 5/ ; hair, 4 bu, @ 15/' ; mason's labor, 3J days @
$2.50 ; laborer, 3 days @ $ 1.25 ; cartage, $2.00.
471. A speculator had 10 rectangular lots of land, situated side by
side and having their fronts in the same straight line. Each lot was
66 ft. wide in front and 132 ft. deep. What was the cost of fencing
these lots at $ 20 a rod for the front fences, and $ 10 a rod for the
remainder ?
472. Suppose coal to be worth $ 6 a ton, and the net cost of manu-
facturing coal-gas to be 15 % of the price of the coal. If a ton of coal
yields 12000 cubic feet of gas, what is the cost of gas per thousand
feet?
473. North Eastland, near Spitzbergen, is said to be covered with
a glacier from 2000 to 3000 feet deep. At the average depth of 2500
feet, what pressure is exerted by the iceberg upon a square foot of
earth beneath, if the ice is 0.9 as heavy as water ?
474. Wishing to get $ 400 from a bank on 3 months, the cashier
added the discount for 3 mo. 3 d. to the $ 400 and directed me to draw
my note for the amount. When this note was discounted, how much
did the proceeds fall short of $400?
475. A collector received $5.87 as his commission at 5 % on the
amount of his collections for 1 day. If the debtors were allowed 10 %
discount from the face of their bills, what was the total due on the
bills collected ?
476. If Mr. Cook gets $25 a month for rent of a house worth
$6000, and on which he pays $70 a year for taxes, how much does
he lose a year, money being worth 6 % V
At what price can Mr. Cook afford to sell the house ?
MISCELLANEOUS EXAMPLES. 345
477. A party of emigrants bought a township of government land
at $ 1.25 an acre. Reserving 150 acres for public purposes and setting
aside 200 acres which were worthless, they divided half of the re-
mainder into farms which sold at f 4 an acre, and the other half into
farms which sold at $5 an acre. After selling all the farms, and
paying the first cost of the township, how much money was left ?
478. A person buys coffee at $ 29 per hundred pounds, and chicory
at $ 11.75, and mixes them in the proportion of 2 of chicory to 5 of
coffee. He sells the mixture at 38 J^^ a pound. What is his gain per
cent?
479. In Lombardy 60,000,000 cubic yards of water are daily distrib-
uted over 1,375,000 acres of land. If this water were equally distrib-
uted over the surface, what would be its depth in inches ?
480. If a body moving at the rate of 40 miles per hour could go
from the sun to Jupiter, whose mean dfstance is 475,692,000 miles,
how many years of 365^ days each would it require to make the
passage ?
481. How many years would it take the body mentioned above to
go from the sun to Neptune 27,270,308,000 miles further from the
sun than Jupiter is ?
482. On a note for $400 having three years to run find the differ-
ence between bankers' discount and true discount, at 6 % ?
483. The rate of discount at the bank being 5-| % , what would be
the total proceeds on the 14th of June, 1880, of the following notes :
$500, given for 6 mo., dated May 17, 1880.
$750, given for 3 mo., dated May 23, 1880.
$254, given for 4 mo., dated May 31, 1880.
$435, given for 2 mo., dated June 6, 1880.
484. The largest known diamond in the world in its original state
weighed 900 carats of S\ Troy grains each ; what was its weight in
Troy units ?
485. The Kohinoor, now owned by Queen Victoria, once weighed
186yig^ carats, but lost in cutting 82^^^ carats. What is its present
weight in Troy units ? What per cent was lost in cutting ?
486. Among the crown jewels of Russia is a diamond that weighs
194 carats, which was bought for $450000 and an annuity of $20000.
What is the yearly expense of the jewel to Russia if money is worth
4 % per annum ?
487. In the temple at Baalbec is a stone 66 feet long 12 feet wide
and 12 feet thick. If its specific gravity is 2.6, what is its weight '?
346 APPENDIX.
488. What is gained by selling a hundred powder-kegs at 16f ;*
apiece, the cost per hundred being $1.87 for making and $1.75 for
hooping, eight hoops at $ 0.045 per hundred being required for each
keg, and the value of the other stock being 8 f per keg 1
489. The rate of a clock is 0.000375 fast. How much time does
it gain in one week ?
490. What is the average date for making the following payments :
$ 100, due May 28, 1879 ; $250, due July 3, 1879 ; $ 150, due August
31, 1879 ; $200, due September 26, 1879 ; $400, due November 30,
1879.
491. Which is the more valuable, and how much more, $1200 paid
to-day or $ 1500 paid four years hence, the use of money being worth
6 % a year ?
492. A man agreed to pay $800 with interest at 6 % annually.
Suppose the payment of interest was deferred at the end of the first
and second years, and that simple interest was allowed upon the de-
ferred payments, what would be due at the end of the third year ?
493. Mr. Lamb offered to sell his house for $4500, but, finding no
customer, he was obliged to keep it five years ; he then sold it for
$ 8000. If he received $ 200 per year for rent more than he paid for
taxes and repairs on the house, money being worth 6 % a year, how
much did he gain by keeping .the property ? What per cent on the
$4500?
494. Pure iron weighs 7,79 times as much as an equal bulk of
water. A cubic foot of water weighs 1000 oz. Av. How many cubic
inches in a cube of iron weighing 1 lb, Av. ?
495. Find the diameter of a cast-iron ball weighing 9 lbs., supposing
that the iron weighs 7.2 times as much as an equal bulk of water.
496. Intensity of light from a given object varies inversely as the
squares of the distances. If your book is 27 inches from the light
and mine is 6 feet 9 inches, how many times as great is the light on
your book as on mine ?
497. There is a cypress-tree in Lombardy said to have been quite
a large tree 42 years b. c. Suppose it to have been planted 87 years
B. c, what is its age in 1880 a. d. ?
498. This tree is 121 feet high. What must be the length of a
cord that will reach from the top to the ground 50 feet in a horizontal
line from the base ?
MISCELLANEOUS EXAMPLES. 347
499. This tree is 23 feet in circumference at the "base. If the trunk
were a perfect cone to the top of the tree (121 feet), hpw many cords
of timber would it contain ?
500. August 8, 1866, I insured my life, paying a premium of
$ 121.30 a year. After paying my insurance for 11 years, the com-
pany went into bankruptcy, and on February 14, 1880, the sum of
i 162.12 was sent me in full for all demands on the company. Al-
lowing the use of money to be worth 6 %, how much money did I
lose by the transaction ?
501. K. Fales & Co. imported from France 220 meters of cloth
costing 15,75 francs per meter, paying \b f per yard for duties. If
the cloth be sold at .|4.50 per yard, what is gained ?
502. How many square yards of canvas in a circular tent of 200
feet diameter, the vertical wall standing 16 feet high, and the roof
extending from the top of the wall to a height at the centre of 50 feet
from the ground ?
503. A hot-air register in a school-room is 2 ft. long by 1 ft. 6 in.
wide, and half the area is taken up by the grating. How much air
per minute must pass through each square foot of the opening of this
register into the room to supply each of 42 scholars with 4 cubic feet
of fresh air every minute ?
504. A schooner beating against the wind sails S. E. 6 miles, then
S. W. 12 miles, then S. E. 12 miles, then S. W. 12 miles, and finally
S. E. 6 miles. How many miles is it in a straight course from the
point she left to the point she arrives at ?
505. Exchange on Paris is quoted here to-day at 5.17 and on Lon-
don at 4.85. At London, exchange on Paris is quoted 25.32^ francs
to ^ 1. Which will be the cheaper, and how much, to remit to Paris
directly or to remit to my correspondent in London and let him
remit to Paris ?
506. If I sell 22 rakes for as much money as I paid for 36, what
per cent is gained ?
507. Find the first cost of an article of which 100 can be made
with raw materials costiiig $350, labor $ 150, and other fixed charges
S| 200 ; and find the price to gain 25 % , How much would this price
be aff'ected by raising the wages of all the laborers 15 % ?
508. Find the average age of the pupils in the graduating class of
a certain school, the ages being as follows : 14 y. 3 mo., 15 y. 2 mo.,
15 y. 1 mo., 14 y. 4 mo., 14 y. 5 mo., 14 y. 1 mo., 13 y. 11 mo., 13 y.
9 mo., 14 y. 2 mo., 15 y. 6 mo., 14 y. 8 mo., 14 y, 7 mo., 15 y. 2 mo.
348 APPENDIX.
509. In the Centigrade thermometer the freezing-point of water is
marked 0° and t^e boiling-point 100°. In the Fahrenheit thermometer
the freezing-point is marked 32° and the boiling-point 212°. When
the Centigrade thermometer stands at 37° at what decree will the
Fahrenheit thermometer stand? What degree of the Centigrade
thermometer corresponds to 92° Fahrenheit ?
510. A builder hired money at b\ % per annum, and built with it
a house which, with the land, cost | 8572. At the end of 18 months
he sold the house for % 10500. How much did he gain ?
511. In an election three candidates were voted for, and the total
number of votes cast was 4214. The winning candidate. A, had a
plurality of 613 over B, and 1125 over C. How many votes did each
candidate receive ?
512. To meet the appropriations made in a town meeting $158400
must be raised by taxation. What must be the amount of the tax
levy if a margin of 10 % is allowed for uncollected taxes and a com-
mission of 7-^ % on taxes collected ?
513. The cost of making a certain book is 38/ per copy. What
must be the retail price of the book that one third may be taken off
for wholesale buyers, a further discount of 5 % from the face of their
bills allowed, and yet a profit of 50 % remain for the publisher ?
514. A travelling salesman is allowed 12 % on his sales. His
employer makes a profit of 20 % on the goods sold. What is the
first cost of goods which are sold by the salesman at •! 7.65?
515. How far from the wall of a house 24 feet high must a ladder
23 feet long be placed that a person may ascend to within 5 feet of
the top of the wall ?
516. If from a point between two houses a ladder 32 feet 6 inches
\ong will reach to a window 26 feet high in one house and 30 feet
high in the other, what is the distance between the houses ?
517. What can he paid on the dollar by a bankrupt having assets
worth $ 375240 and liabilities amounting to J| 681426? How much
more could he pay on the dollar if $ 150000 of these liabilities should
fail to be proved by the supposed creditors? \^Ans. to mills.]
518. A contractor has 20 days in which to do a piece of work. He
hires 30 men, who after working 8 days strike for higher wages, re-
maining idle 5 days and then returning to their work. But for the
strike the work would have been done at the end of the 18th day.
How many more men must the contractor hire that he may finish his
work within the time set?
Copyright^ 1878,
ANSWEES.
Page 5.
Ex.
Ans.
9.
4,000.
10.
4,400.
11.
4,040.
12.
4,004.
Page 6.
13.
8,422.
14.
356,890.
15.
60,065.
16.
1878.
Page 7.
17.
46 thous. 520 units.
18.
406 tho. 502 units.
19.
1 million, 1 thous.
110 units.
20.
85,503,007.
21.
906,218,028.
22.
3,037,900,200.
23.
18,000,000,004.
24.
40,700,000.
25.
37,099,009,000,000
Page U.
53.
0.7.
54.
0.07.
55.
0.007.
56.
0.25.
57.
0.039.
58.
7.005.
59.
25.0049.
60.
0.00306.
61.
0.05047.
Page 15.
1.
12,880.
2.
757.
3.
11,347.
4.
78,239.
5.
664.
6.
1,135.
7.
135.237.
Ex. Ans.
8. 4.019.
9. 20.429.
Page 18.
10. 2,316 bushels.
11. 6,776 pounds.
12. $2,785.
13. 3,925 yards.
14. 3,023 pounds.
, K ) 1,191 barrels.
• J $8259.
16. 1604 miles.
1 Y ) 7,329 acres.
^' J 621,472 trees.
Page 19.
18. 1,038 miles.
19. $77,277.
20. 12.297.
21. $1814.155.
22. 5.17 miles.
23. $33.70.
24. 39.023 acres.
25. 35.625 miles.
26. $160.13.
27. $1,123.66.
28. $1,034.33.
29. $886.84.
30. $289.87.
31. $1,932.19.
Page 22.
1. 141 oranges.
2. 321 acres.
Page 23.
3. 571.
4. 102.
5. 8,272.
6. 654.
7. $357.
8. 78 years.
9. 972 years.
Page 24.
Ex. Ans,
10. 2,771 oxen.
11. 473 birds.
12. 178.5.
13. $11.72.
14. 999.999.
Page 26.
15. 750 bushels.
16.4297.
17. -2,776 acres.
18. 3,948 feet.
19. $1,881.87.
20. 84 years.
21. $6,''l75.
22. 215,174 bushels.
23. 1,522 miles.
24. 35,884.
25. 27,333.
26. 78,883.
27. 110,121.
28. 202,837.
29. 298,122.
30. 128,623,
31. 881,803.
32. 3 times; 440,111
over.
33. 139,886 feet.
Page 27.
34. 194.85 acres.
35. 0.315.
36. 0.515.
37. 76.87.
38. 7,551.5544.
39. 906.975.
40. 584.995.
41. 11.086.
42. 883.83.
43. $89.
44. 715 persons.
45. $ 2,058. -
360
ANSWERS.
Ex. Ans.
46. $20,060.
47. 162 miles.
48. 338 miles.
49. 712.
50. $1,340.
Page 30.
1. 980, pounds.
2. 1,032 pounds.
3. $2,191.
4. $4,275.
5. 2,560 miles.
6. 407 days.
7. 63,360 feet.
Page 31.
8. 5,124/^.
9. $2,916.
10. $2,856.
11. 5,156/.
12. $2,352.
13. 3,913 cents.
14. 2,403.
15. 11,304.
16. 57,817.
17. 1,890,441.
18. 1,325,700.
19. 1,692,501.
20. 650 cents.
21. $10,000.
22. $8,250.
Page 32.
23. 5,930,720.
24. 7,599,240.
25. 40,907,020.
26. 150,000.
27. 112,000.
( 168,300.
28. 1 1,683,000.
( 16,830,000.
29. 21,440,000.
30. 25,628.
31. 66,912.
32. 18,763.
33. 32,215.
34. 2,573,284.
35. 2,740,356.
Page 33.
Ex. Ans.
36. 850.
37. 4,802.
38. 79,576.
39. 368,400.
40. 4,378,125.
41. 488,151.
42. 46,215.
43. 1,030,945.
44. 3,834,432.
45. 233,473,500,000.
46. 381,840,480,000.
Page 34.
47. 43.24; 9,082.5.
48. 42.5454; 0.009.
49. 24.
Page 36.
50. $540; $2,700.
51. $5,100.
( 41,400 stitches;
^n ) 496,800 stitches;
^^•) 2,980,800 stitches;
( 155,001,600 "
53. 1,950,000 inhab.
r, { 1,170,000 yards.
I 60,840,000 yards.
55. 180,000,000 lbs.
56. 86,400 seconds.
57. 91,289,796 miles.
58. 16.25 hours.
59. 36.225 acres.
60. $446.50.
61. $7,962.50.
62. 336.8336.
Page 37.
63. 10,980 pounds.
64. 3,000 feet.
65. 3,586 feet.
66. $352.
67. $783.
68. 479,232.
69. $518.
70. 39,296 strands.
71. $27,000.
72. $121.
73. $148.
74. 33.
Page 41.
Ex. Ans.
1. 634 cords.
2. 268 hours.
3. 405 hours.
4. 983 packages.
Page 42.
5. 104 weeks; 2 rem.
6. 323 cans; 3 rem.
7. 16 years; 8 rem.
8. 134 rows.
9. 542 yds.; 10/ rem.
10. 3,268 or.; 3/ rem.
Page 43.
11. 225/.
12. 195| miles.
13. $315|.
14. 410| thousand.
15. $7,544f.
16. 40,598|.
17. 1,1111
18. 3,07 l^V
19. 37,341 f.
20. l,443,002f
21. 19,661,314f
22. 123,432.
23. 20,551.
24. 41,089.
25. 10,917.
26. 17.2811.
27. 11,759.6.
28. $26.06.
29. $129.62.
Page 45.
30. 151^\ or 151.096...
31. 224.
32. 108ff or 108.740...
33*. 108|| or 108.838...
34. 528lf or 528.565...
35. 345-f|or345.805...
36. 438-|f or 438.952...
37. 396|f or 396.395...
Page 46.
39.
l,2364f or
1,236.506...
ANSWERS.
351
Er. Ans.
40. 4,158.
7,753|^ or
7,753.920...
191i|| or
191.783...
133^ or
133.175.
3,249^ or
3,249.059...
,107|H or
107.843...
Page 47.
46. $123.67.
47. $148.63.
48. $56,849.
49. 2,775.
50. 42,303.3248.
41.
42.
43.
44.
-II;
51. 6|^ days.
52. 28 barrels.
53. 403^ days.
54. 103^2^ days.
55. 5f|f| miles.
56. 197ff acres.
57. 133 boxes.
58. 228 eggs.
59. $85.00.
f.r. { $336,768;
^^•jl, 754 lbs.
61. 295.875 acres.
62. $1.12.
63. $0.03.
64. 0.142... acres.
65. 0.312... tons.
66. 8.566... miles.
67. 0.385...
68. 0.015...
69. 0.008...
Page 50.
1. 5,095,768.
2. 4,643.
3. 53,291.
4. 15,672 by 698.
5. 13,702.
6. 931,055.
7. 76,350.
Ex. Ans.
8. 405,095,040.
9. 17,240,192.
10. 4.
11. 876,232.
12. 190.
13. 2,488,496.
14. 5,800.
15. $1,678.00.
16. 480 acres.
17. $1,592.00.
18. 5,718 gallons.
19. 23,831 men.
20. $390.00.
21. $35.25.
22. 588 miles.
23. 4,474 sq. m.
24. 5 States; 1,270sq.m.
25. 5,538 sq. m.
26. 3; 4; 5,270 sq. m.
27. 30,000 lbs.
Page 53.
28. 660 pupils.
29. 2,122 feet.
30. 182 girls.
31. $1,960.00.
32. $136.00.
33. $1,436.40.
34. 3,276 bush.
35. $71.04.
36. $331.00.
37. $52.00.
38. Gained $39.
39. 90 days.
Page 54.
40. 14 posts ; 65 rails.
41. $12,200.
42. $2,028.
43. 28 days.
44. 15 months.
45. $8.00.
46. 134i| acres.
47. $118.00.
48. 4,997.
49. 225 miles.
50. Gained $ 161.
51. 2,271; 2,053.
Page 56.
Ex. Ans.
52. $52,458. '
53. 2,323,200,000 eu. ft.
54. 883; $56.15 left.
55. 23f| hours.
56. $85.20.
57. 50 days.
58. 40 pieces.
59. 3,711 years.
60. 170 b. c.
61. 1438 a. D.
Page 56.
62. lOfff years.
63. 499i\\\7^seconds.
Pages 59, 61, and 63.
Answers to the drill ex-
ercises on tables 1, 2,
and 3 will be found in
the Teachers' Key.
Page 65.
1. $118.98.
2. $62.91.
Page 66.
3. $19.16; $42.50.
4. $ 16.60.
5. $61.31.
6. $ 36.88.
7. $21.75.
8. $952.56.
9. 24.
10. 288.
11. 60 boxes; 20/ rem.
12. 32 veils; 56/ rem.
,« (35 dinners;
12|^ rem.
Page 69.
14. $1,085.18.
15. $92.75.
16. $167.50.
Page 70.
17. $ 74.69.
18. $5.16.
Page 71.
19. $306.28.
20. $248.30.
352
ANSWERS.
Ex. Ans.
21. $65.88.
22. $1,035.84.
23. $2.42.
24. $9.13.
25^
26. $14.50.
Page 73.
For answers to drill ex-
ercises on Table No. 4,
see Teachers' Key.
Page 74.
27. $6.09.
28. Lost 50;^.
29. $4.80.
30. $36,649.62.
31. $292,318.04.
32. $9.92.
.33. $1.25.
34. 16 boys.
35. $137.05.
36. Whole, $9.66; ap-
ple, 25/; peach,
28^; ijear, 55/.
37. $570.
Page 78.
1. 2, 2, 3, 3, 5.
2. 2, 2, 2, 2, 2, 2, 3.
3. 2, 2, 2, 2, 11.
4. 2, 2, 2, 2, 13.
5. 2, 2, 5, 13.
6. 13. 13.
7. 7, 47.
8. 2, 13, 13.
9. 3, 7, 17.
10. 2, 2, 7, 23.
11. 2,2,3,3, 19.
12. 2, 2, 5, 5, 5, 5.
13. 11, 31.
14. 2, 2, 2, 43.
15. 2, 181.
16. Prime.
17. 2, 2, 2, 3, 17.
18. 2, 3, 3, 5, 5.
19. 2, 5, 59.
20. 2, 2, 2, 2, 5, 7.
21. 2, 2, 149.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
a.
b.
c.
d.
e.
/.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
Ans
13, 53.
2, 2, 2, 2, 2, 2, 11.
3, 3, 79.
2, 3, 127.
3, 3, 3, 5, 7.
2, 2, 3, 3, 3, 3, 3.
2,2,2,2,2,2,2,3,7,
2, 2, 3, 97.
2, 2, 2, 2, 7, 29.
Page 80.
70. g. 15.
46. h. 18.
10. i. 2.
2. j. 11.
11. k. 8.
5. ;. 75.
Page 81.
4if.
3.
$ 2.50.
338 bushels.
175 feet.
306 dozen.
$12.,
70 men's work.
10 weeks.
24 days.
38-| yards.
4 months.
1,260 overcoats.
Page 83.
4. 53. 1.
6. 54. 6.
9. 55. 3 feet.
6. 56. 6 feet.
14. 57. 3 feet.
13. 58. 21 feet.
Page 84.
59.
17.
61.
1.
60.
21.
62.
17.
63.
64.
Page 86.
2,520.
180.
Ex. Ans.
65. 360.
66. 840.
67. 7,644.
68. 3,744.
69. 5,040.
70. 2,520.
Page 87.
71. 75,712.
72. 35,880.
73. 616.
74. 19,824.
75. 940,500.
76. 554,664.
77. 9,680.
78. 1,007,930.
Page 9L
1. i 7. i.
2. |. 8. ,%■
3. |. 9. fif
4. |. 10. 1^.
5. ^l 11. ^V
6. A- 12. m-
Page 92.
13. 611
14. 8^V
15. lOlf
16. 47ii
17. 280f
18. 22-,^.
19. 49| days.
20. 18^ years.
Page 93.
21. ^^.
22. ii^8-a.
23. ^^^.
24. -5-^.
25. ^'^•
26. ^.
27. ^^.
28. -^fp.
29. 4F-
ANSWERS.
353
Page 96.
Ex. Arts. Ex. Ans.
30. 2^. 37. liff.
31. 2/3. 38. Hi-
32. l^V 39. l^-ii,.
33. H. . 40. 9i|.
34. ifi. 41. 168^VV
35. 1. 42. 172^V
36. IH. 43. V37H-
Page 98.
44. ^. 54. 141^.
45. H- 55. 33V
46. ^j. 56. If.
47. 35|. 57. i||.
48. 16^. 58. iVj.
49. 59f. 59. 4||.
60. 11|. 60. 15^5^.
51. 4|. 61. 12i|f.
52. 24|. 62. 6l|.
63. 5-,V 63. 4^01.
64. 7f| miles.
65. 4,650j7^ feet.
Page 99.
66. 9^ barrels.
67. $51.70.
68. Ill
69. 13f.
Page 100.
70. 15|A.
71. 42| yd.
72. 103^ rd.
73. I92I miles.
74. 660 ft.; 5,280 ft.
75. 187^ pounds.
76. 2,150f in.
77. 502^ days.
Page 101.
78. 160^1 days.
79. 132f
80. IM^V
81. 262|.
Ex. Ans.
82. 14,244|.
83. 4,623|.'
84. 7,906|.
Page 102.
85.
$403.13.
86.
124§ bu.
87.
217-^ oz.
88.
262f qt.
89.
$80.63.
90.
177| lb.
91.
$34.29.
92.
204.
93.
473^.
94.
1071.
95.
15,746i.
96.
20,143|.
97.
25,7S8f
Page 104.
98.
,\-
99.
A-
100.
*•
101.
^■
102.
M-
103.
m-
104.
202^.
105.
72M-
106.
n m.
107.
$267,55, or$267.81
108.
$470T^,or$470.31
109.
ix-
110.
$9t^,or$9.14.
111.
420.
112.
A-
113.
1-
Page 105.
114.
A-
115.
mk-
116.
for If
117.
ih-
118.
m-
119.
7f.
Page 106.
Ex. Ans.
120. H in.
Page 107.
121. 15iVm.
122. 5|yd.
123. 5|rd.
124. 22^ift.
125. ^.
126. ^1^.
127. ^^.
128. ToW
129. 10^.
130. 4if.
131. m
132. 13|f.
133. lii.
134. 13f|.
Page 109.
135. ^V
136. ^\^.
137. i,.
138. 14,661.
139. 1,152.
140. 9,720.
141. 1321b.
142. 205 ft.
143. 21f.
144. 70.
145. 140.
146. 96.
147. l^p
148. Iff
149. 3.
150. ^f
151. 24 bu.; 16|bu.
152. 250.
Page 110.
153. 12,307y\yd.
154. 6^rd.
155. 28| breadths.
156. ISf^yd.
157. 240^1 lengths.
354
ANSWERS.
158.
159.
160.
161.
162.
163.
164
165.
166.
167.
168.
169.
170.
171.
172.
173.
174.
175.
176.
177.
178.
179.
180.
181.
182.
183.
184.
185.
186.
187.
188.
189.
190.
191.
192.
Ans.
2f|| sq. rods.
8 francs.
I-
TtW-
A-
14-
28|.
2 8 1 (5 •
51-
iL268
3893-
Page 112.
18.
38^.
131.
135
5t.
HI-
$363.80.
|- acre.
$ 660,000.
$576.
$2,724.15.
$4,225.92.
( Infantry, 4,200;
\ Cavalry, 600.
1st y. 1,840 bu.
2d y. 2,300 bu.
165.
330 fowls.
Page 114.
H.
h
196. $
197.
198.
199.
200.
201.
206.
207.
208.
209.
210.
211.
212.
213.
214.
215.
216.
217.
218.
219.
220.
221.
222.
223.
224.
225.
226.
227.
228.
229.
230.
231.
232.
233.
234.
235.
236.
Ans.
21; $15;
$12.
202.
$24;
203. |.
r^-
i I
Page 115
204. l|i
205. \^.
h I h I
35,'750.
39^.
45,600.
18,262^.
Page 117.
2x2x3x5x7.
3§.
7.
252.
137|; 6.
-0-
9f ; 78^.
iif; A-
29f; 37; S^V
459
Too'
If
39^ miles.
$6.25.
$31.50.
Amt. $ 85.78.
Page 119.
$8.75.
$8.17.
$ 8,523.54.
$ 14.63.
Ex.
237.
238.
239.
240.
241.
242.
243.
244.
245.
246.
247.
248.
249.
250.
251.
252.
253.
254.
255.
256.
257.
258.
259.
260.
261.
262.
263.
264.
265.
266.
267.
268.
269.
270.
271.
272.
273.
274.
Ans.
$7.29.
25/'.
$11,623.13.
$14,262.50.
$21.66.
$55.91.
5^Q mo.
$ 52.80.
114ff.
$ 37.34.
$ 2.60.
Page 120.
25f f days.
17^/.
$ 2.16.
$18,000; $27,000.
$ 36.50.
$3,600; $450.
144| feet.
793|i days.
15f rods.
781
$ 143.75.
95i|| sq. yds.
$ 33.89.
Page 121.
($ 119.931;
} 623^ fr.
648 lb.
25f| lb.
$5,833.33.
$8.75.
2 If miles.
27^ hours.
$ 825.42.
16f|.
$42,666.67.
ANSWERS.
355
Ex. Ans.
{ 800 in all.
275. < 160G.,200F,80Sc.,
/ 100E.,120Sw.
276. $600.00.
277. 3^^ days.
278. 4f days.
279. 5^ days.
280. 6^Vdays.
Page 123.
For answers to drill ex-
ercises on Table No. 5,
see Teachers' Key.
Page 126.
1. 0.070.
( 0.400 ; 0.750 ;
I 2.500; 1.060.
i 0.0030; 1.7500;
I 0.0060.
3.000; 3.0000.
7.0 7.00 7.0000
40.0 40.00 40.0000
L37.0 37.00 37.0000
I-
6.
7.
8.
9.
10.
11.
12. §.
13. t.
14. f
15. I.
7
3
5-
h
16. |.
18. I.
20.
21.
22.
23.
24.
25.
A-
1
Page 127.
26. 0.625.
27. 0.35.
28. 0.024.
29. 3.0625.
30. 5.125.
31. 20.03125.
32. 1.078125.
33. 8.75.
34. 17.1171875.
35. 1.0612.
Ex. Ans.
36. 0.0475.
37. 0.03125.
38. 2.225.
39. 2.975.
40. 1.6075.
41. 16.9875.
42. $54.59.
43. 111.875.
44. $4.70.
45. 16.485 acres.
46. 1.9307734...
Page 128.
47. 2.1071.
48. 58.6881.
49. 0.3724.
50. 11.5953 hours.
51. 139.8837 rods.
Page 129.
52. 0.3.
53. 0.83.
54. 0.42857 i.
55. 0.5.
56. 0.916.
57. 0.571428.
58. 0.38.
59. 0.045.
60. 0.63.
61. 0.0416.
62. 1.153846.
63. 3.i35.
Page 130.
64. f 74. f .
65. f. 75. f
66. M- 76. If.
67. M- 77. \\.
68.
TT-
69. 14. 79. ^f^.
70. ^. 80. ^.
71. If. 81. ^^.
72. ^. 82. flffl
73. JoV 83..i^f
Page 131.
Ex. Ans.
84. 0.432. ■
85. 0.918.
86. 1.5.
87. 4.704.
88. 42.
89. 4.704.
90. 10.23.
91. 0.01648.
92. 0.37418.
93. 0.648.
94. 128.94.
95. 1.516.
96. 220.
97. 93.16f.
98. 1.0125.
99. 153.5625.
100. 462.825.
101. 380.
102. 7,501.
103. 7,500.0000001.
104. 5,290.5687.
105. l,136.296f.
106. $2,471.10.
Page 133.
107. 48 books.
108. 19f yards.
109. 6 rods.
(75qt.dry.
( 87i\ qt. liquid.
111. 0.5 ton.
112. 200 yards.
113. 2.13.
114. 5.5.
115. 4.
116. 0.36.
117. 111.
118. 7,720.
119. 15.49.
120. 0.3.
121. 0.094.
122. 0.1003...
123. 1538.4615...
124. 6,275.
125. 950.
126. 10,050.
127. 0.0071.
356
ANSWERS.
Ex. Ans.
128. 0.0497—
129. 0.0066—
130. 42.08.
131. 649.084.
132. 43.08.
133. 0.1234.
134. 25.1045...
135. 0.2342—
136. 74,0762—
137. 11,728.8.
138. 2,877.617.
Page 135.
For answers to drill ex-
ercises on Table No.
6, see Teachers' Key.
Page 147.
1. 133,608 oz.
2. $640.00.
3. 62|ft.
4. $191.40.
5. 43,560 sq. ft.
6. 1,955 acres.
7. 46,656 oil. in.
8. $7.21.
9. $16.56.
Page 148.
10. 42,048,000 times.
11. 14,608 hours.
12. 52,594,560 min.
,^ ( 1,625.26 common
^'^' I miles.
Page 149.
14. Im. 274 rcl. 1yd.
15. 284 rd. 1ft.
16. 60 A. 87sq. rd.
,w UA. 28 sq. rd. 5
I sq. yd.
ift i 3 sq. yd. 6 sq. ft.
^^' I 39 sq. in.
19. 14° 54' 44".
20. 4T. 1,3281b.
21. 1,2211b. 11 oz.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
l53.
.4ns.
38 lb. 10 oz. 1 pwt.
4 oz. 7 pwt. 1 gr.
158 bu. 3pk. 7qt.
2y.53d. 4h.33m.
1 lb. 5 oz. 3 pwt.
18 gr.
lib. 5oz. 10 pwt.
7,899^ miles.
( 30^1 bn., or
( 30 bu. 1 pk. 5 qt.
Page 150.
3 yd. 2 ft.
203 rd. 3 yd. 1 ft.
6 in.
203 A. 101 sq. rd.
24sq.yd. 6 sq.ft.
108 sq. in.
j 42sq.rd.20sq.yd.
( 1 sq. ft. 72 sq. in.
1,750 lb.
10 oz. 10 pwt.
10cu.ft.864cu.in.
1 pt. 1 gi.
2 pk. 3 qt. pt.
Ifgi-
33' 20".
81d. 2h. 40 m.
274 d. 12 h.
$ 6,534.00.
64 doses.
6,930 planks.
Page 151.
^gal.
|rd.
i cu. yd.
0.2738— miles.
Page 152.
0. 8449 A. + or 0.8450 A. -
0.55 y.; 0.73jy.; 0.49jy,
Page 153.
Ex. Ans.
54. 135 gal. 2 qt.
^- j 50 A. 130 sq. rd.
^^' I 49 sq. ft.
56. 78 m. 207 rd.
57. 13° 39' 8".
58. 16 d. 13 h.
59. 2 pk. 3^ qt.
60. 205f rd.
61. 5541 lb.
Page 154.
62. 3 bu. 1 pk. 3 qt.
63. 2oz. 14 pwt. 19 gr.
64. 15° 52' 40".
65. 84 rd. U yd.
66. 5 ft. 6 in.
67. 511 A. 154isq.rd.
68. 7oz. 16^ pwt.
69. 8h. 19 m. 46.2 s.
70. C.Horn, 21° 36' 4".
71. 13° 50' 7".
72. 11° 3' 21".
73. 86° 27' 17".
74. 149° 14' 43".
75. 74°.
Page 157.
76. 6qt. Igi.
77. 40 bu. Ipk. 7qt.
78. 258 m. 166 rd.
79. 1 pk. 3 qt. 1 pt.
80. 1 A. 50| sq. rd.
81. 8 m. 14|s.
82. 49 men.
83. 154 bins.
84. 6h. 14 m. 10 s.
Page 158.
85. 64° 15'.
86. 32° 15'.
87. 93° 2' 30".
88. 16° 21' 15".
ANSWERS.
357
93.
Ex. Arts.
89. 45° W.
90. 1° 15' E.
91. 18° E.
92. 62° W.
Page 159.
16° 15' 13";
90° 15' 16" W.
94. i67° 48' 45" E.
95. 10 m. 45^^?.
96. 4 h. 55 m. 37f s.
97. 1 h. 6 m. 48 s.
98. 49 m. 40 s.
Page 160.
99. 7 m. 31-^ s. p.m.
100. llo'cl. 7m. 43^ s. A.M,
101. llo'cl. 17m.51is.A.M
102. 8o'cl.58m. 26i-s. A.M.
103. 5 o'cl 58m. -^ s. p.m.
104. 6 o'cl. Im 46^ig 8. p.M
105. 5 o'cl. 17m. 321 s. p.m
106. Uo'cl. lm.27is. p.m
Page 161.
107. 28^^ yd.
108. 37^ yd.
109. $41.25.
110. 5,476 sq.ft.
111. $5,197.50.
112. 121 ft.
113. $2,178.00.
114. 164sq. id.
115. 2 sq. rd.
116. 20^117 A.
117. $34,456.64.
Page 162.
118. $400.00.
119. 1,696 cu. in.
120. l,546§lb.
121. 6f ft.
122. n^sq.yd.
123. 22f ft.
Page 163.
124. S^T ft.
125. $54.66.
126. 9|ft.
127. 42/^.
Ex. Ans.
128. 99 ft.
129. 45 ft.
130. 2,400 ft.
131. 120 ft.
132. 81 ft.
Page 164.
133. $12.21.
134. $0.95.
135. 51f qt.
136. 168^ gal.
,„^ ( 1.024 cu. ft.
^'^'' I 7.66... gal.
138. 45^8^^^ bii.
139. 191 bu.
140. 4^^^Tbii.
141. 7.644 cu. in.
142. 40 bu.; 32 bu.
Page 165.
143. 27,101 ft.
144. lOoz. 2pwt. 17gr.
145. 16cu.ft.l,512cu.in.
146. $48.00.
147. $0.75.
148. 0.695°.
149. 0.1045... rd.
150. I A. or 0.83|- A.
151. 185 d. 19h. 28 m.
sq. rd. 3 sq. yd.
sq. ft. 36 sq in.
152.
,-o ^ 21b. 5oz. 3pwt.
1^3. I 8gr.
154. 2,079 sq. ft.
155. 1,920 cu. ft.
156. 4 ft. 7 in.
157. lib. 11^ oz.
158. 900 bricks; $8.10.
159. 1.
160. ^V
161. 26 yd.
162. 28 m. 32^ s.
163. 46° 1' 30'.
164. 423 d.
Page 166.
165. 25/.
166. $2.91.
Ex. Ans.
167. 60 furrows.
168. lloz.Opwt. 15gr.
169. 31 sq. ft.
170. $379.50.
171. 768.
172. 2° 13' 47f".
173. 40 rd. 2 yd. 8 in.
174. 17 d. Ih. 30 m.
175. $68.75.
( 321 jWb^i ^i^i-. or
176. •\320 bu.. rough esti-
( mate.
177. $16.50.
m( 78 A. 13.5 sq. rd. sons'.
• ( 87 A. 115 sq. rd. dau's.
179. 48 yd.
Page 167.
180. 1,860,000 yd.
181. 18 lb. lO^V oz.
182. 72 sq. ft.
183. $0.27^V ^
184. $5.50.
185. 11|| cd.
186. $7.45.
187. $37.75.
188. 2864f gal.
189. $51.75.
190. $44.55.
191. 92^% d.
192. If yd.
193. $40.08.
194. 66,825 stones.
Page 168.
,QK flh. 53m.35^
^•-'^" (. after midnight.
196. 7gr.gr.3gr. 6doz.
mp h. SgVm.
• Ish. 543 m.
Page 171.
For answers to drill ex-
ercises on Table No. 7,
see Teacher's Key.
Page 174.
1. 0.1™-
2. 1.3"-
3. 21.4™-
4. 0.01™
5. 0.38™-
dbb
ANSWERS.
Ex. Ans.
Page 182.
Ex. Ans.
6. 5.29™-
Ex. Ans.
26. 60/'.
7. 0.001 ™-
47. $3.09.
27. $500.00.
8. 0.048™-
48. $2.31.
28. $ 1,041.67.
9. 3.675™-
49. 135 grams.
29. $1,250.00.
Page 177
50. 5.4 T.
Page 191.
10. 4,250™-
Page 184.
30. 40 years.
11. 3,500™-
51. 157.4 ft.
31. 600.
12. 23,500™-
13. 9,460™-
14. 9,240™-
15. 397™-
52. 31.0685 m.
53. 44.478 A.
54. 2.04724 in.
32. $4.00.
33. $11,250,000.
55. 188.713761b.
34. 25%;50%;17|%.
16. 720™-
56. 6,340.08 gal.
35. 4%; 125%; 1%.
17. 7.3™
57. 22.046 T.
36. 166|%.
18. 50™-
19. 67.678™-
58. 40.2325 ^m.
59. 80.94"^-
37 {931%; 94|^^.
■\ 89|%; 90f%.
20. 649.115™-
60. 2,265.36'-
21. 415,990™-
22. 5.26749 Km.
61. 15.087™-
62. 17.506541-
Page 192.
38. 73J^%.
23. 35,000™-
63. 618.9^-...
39. 6^%'%.
24. 86.62™-
64. 88.9056 Kg-
Page 193.
25. 15,480™-
Page 189.
40. $7,936.72.
26. 71 1 ™-
1. $112.80.
41. $1,000.00.
27. 73.50672 Km.
2. $62.65.
42. $1.05.
28. 22.2075™-
3. 1,240 men.
43. $5,970.00.
29. 300 times.
4. $0,108.
44. $2,400.00.
5. 90 days.
45. $7,000.00.
Page 179.
6. 14.28 lbs.
46. $330.00.
30. 23.85^1-™-
7. 438,120.
47. 24|%.
31. 20™-
8. 1,500 ft.
48. $577.78.
32. $ 8.40.
9. 10,500.
Page 194.
33. 217.5^?-; 2.175^-
10. 9,396.
49. 25%.
50. 371%.
61. $1.02;$1.44;$1.80.
34. 4,585.76^-
11. $1.27.
35. $6,556.57.
12." 131.4 ft.
Page 180.
36. 12.043044049^"-™-
13. 45 in.
14. $8.32.
62. 6%.
63. 128f%.
37. 153.41^"™-
38. $373,721
39. 369.6 loads.
Page 190.
15. $1,600; $480.
54. 108^\.
55. 331%.
16. l,816|d.; 5,450 d.
Page 195.
40. $29.25.
17. 627; 209.
56. $3.56.
41. 12™-; $20.25.
18. 4,000; 6,000.
57. $91,575.
19. 210; 294.
58. $6.50; $74.79.
Page 181.
20. 31; 160.
59. $8.50; $161.50.
42. 20/.
21. 6,375.
60. $81.60.
43. $1.23.
22. $5,208; $15,624.
61. $6,151.56.
44. 75 '•
23. $200.00.
62. $5,729.75.
45. 39«'-
24. $1,488.00.
63. 3%.
46. 4.5°^
25. $214.29.
64. $1;200.00.
ANSWERS.
35<
Page 196.
Ex. Ans.
Ex. Am.
Ex. Ans.
103. $117.99.
145. $2.81..
65. 500 barrels.
104. $102.37.
146. 40%.
66. 20 acres.
105. $194.27.
147. $3.33.
67. 2%.
Page 204.
148. $16,000.
68. 30%.
69. $637.80.
Page 198.
70. $1,101.50.
106. $714.00.
107. $235.20.
Page 205.
108. $80.33.
149. 86 shares.
150. $ 100.
151. 4%.
Page 208.
71. $1,765.00.
109. $5.77.
152. $ 1,440.72.
72. $1,505.25.
110. $36.38.
153. 20%.
73. $3,281.25.
111. $3,830.40.
112. $361.64.
154. 15%; 5%; 10%
74. $3,615.00.
155. $8.75; 5/.
75. $8,903.50.
113. $238.99.
156. $65,909.09.
Page 199.
114. 7^%%.
115. 0.0 W^.
116. \l.
157. 30 shares.
76. $1,468.25-
77. $2,100.
158. $ 12.00 loss.
159. $ 1,008.09.
78. $2,500.
117. $0.0807.
160. $700.
79. 50%.
118. $500.00.
Page 211.
80. $205.
119. $18,000.
I. $207.36.
81. $258.75.
120. $500.00.
2. $41.56.
82. $30.07.
121. $6,000.00.
3. $ 156.20.
83. 97j^jfo.
122. $96.00.
4. $26.99.
Page 200.
123. 10^-^%.
5. $23.41.
84. $10.00.
85. $66.00.
124. $608.30.
125. $9,657.86 worth.
Page 213.
6. $0,076.
86. $145.80.
Page 206.
7. $0.242f.
87. $81.25.
126. $2,750.
8. $0.020|.
9. $0.009j.
88. li%.
127. 8 shares; $133 rem.
89. $13.50.
128. $313.50.
10. $0.066f.
Page 201.
90. $24.31.
91. $2,297.10.
129. $11,200.
130. $22.50.
131. $372.60.
132. $5,495.85.
11. $0.10.
12. $1.00.
13. $0,473.
14. $43.50.
Page 202.
133. $400.00.
15. $6.55.
92. l^f; $188.50.
134. $2.00.
16. $325.08.
Page 203.
135. $282.00.
17. $21.82.
93. 21 m.; $8.75.
94. $4,902.00.
95. $1,500,000.
136. $427.50.
137. $250.00.
138. $1,161.56.
18. $161.64.
19. $230.33.
20. $1.66.
96. 18 mills.
Page 207.
21. $59.79.
97. $76.50.
139. $373.50.
Page 214.
98. $54.00.
140. $15,126.50.
22. $ 185.61.
99. $50.40.
141. $180.00.
23. $3.75.
100. $141.30.
142. $302.17.
24. $403.28.
101. $28.17.
143. 1%.
25. $50.15.
102. $ 163.22.
144. $40.00.
26. $0.51.
360
ANSWERS.
Ex. An$.
27. $3.23.
28. $653.40.
29. $624.46.
30. $1,184.13.
31. $143.96.
32. $1,047.82.
33. $504.65.
34. $939.04.
35. $ 165.30.
36. $408.95.
37. $937.54.
Page 215.
38. $4.65.
39. $4.62.
40. $9.17.
41. $11.85.
42. $4.27.
43. $4.92.
44. $28.09.
45. $113.00.
46. $44.38.
47. $9.06.
48. $360.55.
49. $18.90.
Page 218.
50. $381.38.
51. $397.80.
52. $742.00.
53. $191.99.
54. $430.01.
55. $322.43.
56. $830.28.
Page 219.
57. $445.16.
Page 220.
58. $756.56.
Page 221.
59. 10 mo.
60. 3 years.
61. ly. 6 mo.
62. ly. 4 mo. 12 d.
63. 5 y. 6 mo. 20 d.
64. 5y. 4 mo. 3d.
65. ly. 6 mo.
66. 6 mo.
70.
71.
72.
73.
74.
75.
Ex. Ans.
^ ( 100 y. ; 50 y. ; 33 y.
67. { 4 mo. ; 16 y. 8 mo. ;
( 10 y.
68. 8%.
69. ^%.
Page 222.
4%.
^%.
10/^.
Page 223.
76. $125.00.
77. $100.00.
78. $1,020.00.
79. $68.20.
80. $72.00.
81. $600.00.
82. $600.00.
83. $2,063.65.
84. $400.
85. $83.86.
86. $300.00.
87. $200.00.
Page 225.
88. $25; $2.50.
89. $97.08; $3.88.
90. $197.04; $2.96.
91. $167.69; $8.11.
92. $650; $11.37^.
93. $1,540; $69.30.
94. $590.16.
95. $123.64.
96. $1,533.85.
97. $207.69.
98. $83.87.
Page 227.
99. $4.13; $745.87.
100. $18.08; $981.92.
101. $8.20;" $291.80.
102. $14.00; $686.00.
103. $9.90; $490.10.
Page 228.
104. $4.57; $285.43.
105. $5.25; $494.75.
Ex.
Ans.
106.
$4.11; $252.73.
107.
$9.33; $1,190.67
108.
$336.99.
109.
J $ 102.75;
\ $ 4,100.58.
Page 229
110.
$ 80.44.
111.
$ 500.00.
112.
$ 1,500.00.
113.
$ 950.00.
114.
$2.70.
115.
1 7.83.
116.
$492.61.
117.
$497.17.
Page 230.
118.
$ 506.00.
119.
$4.95.
120.
$5.33.
121.
$ 117.80.
122.
$421.38.
123.
$ 310.05.
124.
$ 33.33.
125.
$ 1.20.
126.
$ 65.00.
127.
$1.85.
Page 232.
128.
$ 238.20.
129
$ 426.03.
130
$ 2,513.04.
131
$ 2,217.44.
132
$2.86.
133
$ 60.00.
Page 233.
134
$234.32.
135
. $915.71.
136
. $85.16.
137
. $45.06.
138
. $2,396.56.
Page 237.
139
. April 23.
140
. Dec. 16.
141
. Oct. 14.
142
. 6 mo. 29 d.
143
. 6 mo. 13 d.
144
. 15 mo-
ANSWERS.
3e
Bx. Ans.
Ex. Ans.
Page 255.
145. June 30, 1876.
30. $600.00.
Ex. Ans. , Ex. Ans
146. May 17, 1876.
Page 240.
31. $179.25.
32. $49.27.
33. $400.
1. ^^. 6. ^.
2. ^. 7. i|.
147. May 22, 1877.
34. $20.46.
3. ^,. 8. 7.
148. Aug. 18, 1876.
4. ^. 9. 6.
149. Dec. 6, 1877.
Page 248.
S- -is-
150. Jan. 2, 1877.
35. $2.06.
D
151. Nov. 25, 1876.
36. $99.50.
Page 25J.
Page 243.
1. $808.00.
2. $2,505.00.
37. $1,308.74.
38. 3 mo.
39. $1,251.13.
40. 6 T. 772 lb.
10. 850.
11. 1,750.
12. 736f.
13. 3,250 tons.
3. $703.50.
4. $1,960.00.
5. $ 696.50.
6. $506.76.
7. $273.63.
41. $39.36.
42. $437.93.
43. $2,687.50.
44. 1 Length, 2^\ yds.;
( breadth, |-^ yds.
14. 35 hats.
15. $27.30.
Page 258.
16. $70.
17. $20.
Page 244
.p. ( Com. $ 13.27.
^^' \ Rem. $ 530.73.
18. $750.
8. $ 2,430.00.
19. $7.13.
Page 245.
46. $43.75.
47. 3^i#>.
48. b% gain.
Page 259.
9. $4,840.00.
20. 31,680 feet.
10. $106.80.
21. 2,500 bushels.
11. $109.60.
Page 249.
22. lOe^^ieet.
23. 15 cents.
24. l,076f| times.
12. $47.50.
13. $366.55.
49. $4,365.00.
50. $32.19.
14. i; 2,000.
51. 31^%.
25. $4.66.
15.^400 105.
52. 50 years.
26. 1 lb. 6^ oz.
Page 246.
53. $124.00.
27. 23Hays.
16. $4,340.00.
17. $905.25.
18. $2,500.00.
19. $3,387.50.
20. $3,277.50.
54. $5>.50.
55. $1.57.
56. 8 mills; $16.
57. $502.77.
58. $2,247.05.
59. $3,754.00.
28. 5f mo.
29. 31b. 4^ oz.
30. 9^1 yds.
31. 36 tons.
32. 12^min.
Page 261.
Page 247.
60. $400.00.
21. $369.90.
22. ^j^%.
23 i Mortgage; |i %
\ more.
Page 250.
61. $147.92.
62. $146.55.
33. $30.
34. 42 days.
35. $30.86.
36. 26 miles.
24. $21,050."
25. 16f %.
63. $154.31.
64. $196.08.
Page 262.
37. 250 tons.
26. $94.20.
Page 252.
38. 193l|-lb.
27. $ 373.49.
For answers to drill ex-
39. $49.25.
28. 3%.
ercises on Table No. 8,
40. $ 12H rods.
29. 11 mo.
see Teacher's Key.
41. $933.33.
362
ANSWERS.
Ex. Ans.
42. 18 men.
43. 54 days.
44. 9,680 bricks.
45. 1 T. 328| lb.
46. 5T. 8161b.
Page 263.
4^ (A's, $1,875;
^''\ B's, $1,125.
B's,83rr.;
^^•|S's, 166fT.
Page 264.
( M., $570.53;
4q )K, $221.87;
^^•)0., $316.96;
(P., $316.96.
( A's, $ 5,625 ;
50. <^ B's, $ 7,500 ;
( C's, $ 9,375.
51. $375; $500; $625.
.„ (H.$500;B.$506;
^"^'l L. $480.
.^ i X's, $ 332.50;
^•^•| Y's, $525.
54 (R's, $908.30;
^^* (Fs, $2,591.70.
Page 266.
( 1, 8, 27, 64, 125,
l.<^ 216, 343, 512,
( 7^, 1,000.
2. 289.
3. 784.
5. 0.0289.
6. 3§.
7. 104.04.
8. 3,375.
9. 0.001728.
10. 14,641.
11.-^.
12. 0.03125.
13. 3.61.
14. 272J.
15. 5.76.
16. 0.0576.
17. 0.004096.
Page 271.
Ex. Ans.
18. 137.
19. 901.
20. 93.7.
21. 8.17.
22. 7.43.
23. 0.43.
24. 234.1.
25. 2.237.
26. 60.47.
27. ^V
28. II
29. I, or 0.666...
30. 0.529...
31. 0.866...
32. 1.590...
33. 2.723...
34. 2.924...
35. 6.082...
36. 2.321...
37. 0.883...
38. 1.414... •
39. 0.3794...
40. 5.059...
41. 16.50.
42. 536.
43. 94 ft.
44. 49 men.
45. 140 feet.
46. 84.
47. 112.
48. 87.63... rods.
.Q ( Length, 100;
^^' I breadth, 50.
50. 9.704... rods.
^, ( Length, 60 ft.
^^- (width, 45 ft.
52. 4.5 feet.
Page 276.
53. 23.
54. 46.
55. 7.4.
56. 98.
57. 143.
58. 24.2.
59. 43.6.
Ex. Ans.
60. 9.35.
61. 9.405...
62. 7.020...
63. 0.7067...
64. 0.928...
65. ^V
66. A = i.
67. 3.5.
68. 0.9654...
69. 0.9546...
70. 3.986...
71. 0.6463...
72. 0.9718...
73. 1.259...
74. 332 in.
75. 6.349... ft.
76. 41.07... in.
77. 51.89... in.
170 ( Without cover, $5.86;
'^'t with cover, §7.03.
For answers to drill ex-
ercises 247 and 248, see
Teacher's Key.
Page 280.
1 (4Jsq. ft.= 4sq.ft.248q.
^- \ in.
2. 103| sq.ft.
3. 180 sq. ft.
4. 10,624 sq.ft.
5. 19.52iVA.
6. 75.3984 ft.
7. 25.1328 ft.
8. 31.8309... ft.
9. 18.3478... sq
10. 346.3614... sq.
11. 44.8369... rods.
12. 12 planks.
o i 100.5312 ft.
'^' I 268 persons.
ft.
ft.
14.
15.
16.
17.
Page 283.
30 ft.
19.5 ft.
18.330... ft.
6,600 ft.
ANSWERS.
363
Ex.
18.
19.
20.
21.
22.
23.
24.
25.
27.
28.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
61.
52.
ft.
Page 284.
Ans.
82.683... ft.
A,20rd.;C,24rd.
276.099... rods, or
276.1..., if roots are
found to ten-thou-
sandths.
21.213... ft.
23.043... ft.
39 inches.
(8.660... ft.;
j 43.3... sq.ft.
24 ft.; 240 sq,
Page 288.
448 cu. in.
216.5 cu. ft.
907.5 cu. ft.
Page 289.
936 cu. in.
j 710.4 bu.unsheird.
\ 568.32 bu. shelled.
40 sq. yd. 8 sq. ft.
720 sq. ft.
220 ft.
5.7596 gal.
14.1372 sq. ft.
18ft.; l,060.29cu.ft.
51.051 sq. yd.
6,450.9696 gal.
3.1416 sq.ft.
$ 42.98.
113.0976 cu. ft.
(144,109,433.16... sq
Page 290.
34§ yards.
Page 291.
1,848 ft.
19.63495 rods.
125 gallons.
$ 0.93.
$ 162.00.
$ 4.86.
9.071 in.
Page 292.
18^ oz.
3,188.046... lb.
Ex.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
m.
67.
68.
69.
70.
71.
72.
Ans.
14.68... in.;
6.349... in.
49.14... bodies.
79,120 miles.
9.524... in.
4.32tons; 6.86 tons
Page 293.
11§.
0.7589...
712^; l,187f
22^; 5; 17^.
10,000 lb.
( S's, $ 389.50;
\ h% $ 1,046.00.
( 6,436,343;
\ 884,736.
! 961 ; 39 rem.
f 31 on each side.
3,233.32... ft.
lOf ft.
20 ft.
374.280.
14-lf in.
256.56..
\ hour.
Page 295.
2271 lb.
16^.
30A.
375 Darrels.
1,900.
32.249... ft.
$ 5,274.84.
$ 6,080.00.
rds.
ft.
95,388.75.
44^f%.
2h. 24 m.
120 shares.
Page 296.
4 dozen.
$ 0.62.
$803.19.
Face of note^
$ 1,593.00,
Ex. Ans.
Q, ( Bates, $771.32;
^^- \ Henr. $1,584.43.
92. 746.286 ft.
93. 20%.
94. Sept. 2, 1877.
95. 131° 20' 38" W.
Page 297.
96. $41.50.
97. 40,882^ bricks.
98. 6.2832 in.
99. lh.29m.l8|s.P.M.
f A'sloss,$13,750;
J B'sloss,$ll,000;
1^0- 1 C's loss, $8,250.
Cpays A, $8,250.
V. B pays A, $3,000.
101. $ 12 gain.
102. $2.59f.
Page 298.
103. $92.28.
104. 94.0032 gal.
105. $5.07.
106. Gained $0.71.
107. 19 T. 1,3751b.
,^« jL'gth, 6.697... ft.
^^°*(Diam. 5.640... in.
109. 2 miles 480 feet.
Page 301.
1. 3,564.
2. 263,736.
3. 579,942.
4. 368,373,159.
5. 69,999,993.
6. 244,510.
7. 3,558,220.
8. 54,931,835,204.
9. 26,496; 13,248.
10. 330,399; 195,792.
11. 231,660; 308,880.
,2 (2,772,605;
^^- \ 7,561,650.
Page 302.
13. 621.
14. 616.
364
ANSWERS.
Ex. Ans.
Ex. Ans.
Ex. Ans.
15. 1,209.
55. $30.
92. 1 hour.
16. 2,025.
56. Loss, $ 4.00.
93. $3,377.50.
17. 3,024.
57. 12 and 40.
94. $0.42.
18. 7,221.
58. 2372Vcii.yd.
19. 3,025.
59. 12^ %.
Page 319.
20. 7,225.
95. $28.26.
21. 11,025.
Page 316.
96. $1.75.
22. 30|.
23. T'2h
60. $181.40.
97. $33.21.
61. $615.18.
98. $3,595.20.
24. 132^.
25. 182|.
62. $4.40.
99. $54.60.
63. 2T. 20.261b.
100. $5,832.
26. 9,900^.
64. 608.685 feet.
101. $0.08^ or $0,085.
27. 272^.
65. 33.136... miles.
102. $4.44.
28. 39^^.
66. $825.00.
,„„ 5$3.95-,2^or
^^2- j$3.95l .
29. 68^.
67. 40-1^%.
30. 52,V
68. 5.196... in.
104. 1,879; $44.22.
31. 915^V
69. 180 feet.
,_ 5 Check, $1,506;
^^^- I unpaid, $ 194.
32. 2,525^V-
70. $2,745.61.
33. 3,75 l^V
Page 317.
106. $44.64.
Page 306.
71. $797.
Page 320.
34. 830 links.
35. 0.45 ch.
72 5 481i|ffl A. or
^'^' 1 481.668... A.
^ 5 Mr. Searle ;
^^^' 1$ 55.47.
36. 11.56 ch.
73. 1,944 feet.
108. $513.56.
37. 72.2 rd.
74. 3| m. or 3.375 m.
75. 45 5 o or 45.5°.
109. $6.37.
38. 129 rd. 1.32 ft.
110. $277.56.
39. 93 rd. 14.52 ft.
1 V^. J.T^ .J^ Q
76. 789 persons.
111. $265.57.
40. 150 acres.
77. 59 persons.
112. Gain, $16.75w
41. 108 acres.
„^ ( 631,580 salmon.
'^- I 12,000,020 lbs.
113. $22.84.
42. 70.0448 acres.
114. $319.50.
43. 111.32854 acres.
79. 1,891.
80. 789 whales.
11.5. $3.
Page 310.
81. $3,256,959.
Page 321.
44. $80.15.
116. $295.
45. $42.59.
Page 318.
117. $1.25. •
46. $250.82.
82. 94 seats.
118. $8.37.
83. 806.505 yards.
119. $5.14.
Page 315.
84. 323.21 acres.
120. $1,619.85.
47. 24/.
85. 2.578 meters.
121. $35.16.
18. 3 more.
86. $10.3.3.
122. $34.83.
49 5 16fdays;50miles
• ( east of the point.
87. 38.25 j^.
88. 7.166... pounds.
,g^ ($1.14; amtof
1^^- \ bill, $8.86.
60. 50 %.
Q^ 5 72 lbs.; 180 lbs.;
^^- 13,157.2 lbs.
51. $1,800.00.
Page 322.
52. 6 bushels.
90. 253.51... tons.
124. $2.81.
53. 7f gal.
Q, j 1,136.28... lives.
^^' 1 101,117.43... T.
125. 8^1 feet.
54. 28 %.
126. $ 6,970.85 worth.
ANSWERS.
365
Ex. Ans.
,07 n 5 pairs;
^"^'^ j 21 yds. left.
128. 36<*.
129. 19.18.
130. 20 rows.
131. ^1.12f
132. 6^\ miles.
133. $210.
134. $17.
135. 1 If yards.
136. $69.35.
137. 5,456^1 gallons.
138. $3.98.
Page 323.
139. T^f;% 1.02.
140. 23| miles.
141. 5-| hours.
142. 36f m. ; 2-^^ h.
143. 10,%V
144. ^^; $2,333.33...
145. m-
146. 1 If hours.
147. $2.04^V
148. ^\ of a day.
149. $4,266.67.
150. $0.27i|perdoz.
151. 40 ft.
Page 324.
152. $0.04f.
153. 0.908 quart.
154. 29.991
155. $6.21.
156. $49.88.
4,201^V9i^'^i'ks
^^'- \ or 4,201.68... m.
158. $2,807,000.
159 ]4|fbu.(,r
^^^- I 4.537... bu.
160
5 In 91
• I 9 o'clc
lock 20 m,
161. 0.2151.
162. 700.7.
,«« ( Boy, l.lOf- oz. ;
^^'^' I man, 2.21f oz.
164. ^ or 0.013^V
Ex.
165
Ans.
,643.5f or
643.514...
166. 8.5675.
167. 7.2 hours.
M
Page 325.
168. 2,262f steps.
16Q 571^ lbs. or
170. 48 cups.
,^, (2 reams 12
' ( quires 2 sheets.
172. Jg/-; $0.000625.
173. 3bu. Ipk. 2|qt.
174. 45/.
175. $57,600.
176.
177.
178.
179.
4| qts. or
j 4.2857... qts.
5 4^f qts. or
I 4.987... qts.
$ 10.56.
$8.25.
180. $25 99.
181. $370.80.
182. $4,628.25.
183. 198 cu. ft.
184. $1,111
185. 31iimile.s.
,Q^ ("2,688.403— m.
^^^- 1 3,535.997... m,
187. 180 slats.
Page 326.
188. $1,069.20.
189. 56,940 times.
190. 267d.;32m.29d.
191 If^H^or
^^^- ($0.415...
192. $47.42,
193. $2.87^.
194. $19.44.
195. $21.29.
196. $ 15.40.
197. $17.37.
$895.44;
$901.18.
198. I
199
200.
201.
202.
203.
204.
5 6 o'c:
• |lm.
Atis.
'clock
3. p. M.
10 o'clock 55 m.
\ 37f s. p. M.
j 4 o'clock 19 m.
I 31f s. P. M.
( 3 o'clock 47 m.
I 361 s. A. IT. of
( the next day.
47° 18' 30" W.
79° 30' W.
Page 327.
205. 11 T. 5001b.
'^^^- \ 96.427.
207 Plti^fJ-
^^'- (1.555... ft
208. 2,430 bricks.
209.
bu.
or
214,
215,
216
218.
194,444| sq.ft.;
mn A. or
4.4638... A.
210. 159-1 sq.ft.
211. $5.28.
212. 2,160 bricks.
213. 4,176 tiles.
15,052 sq.ft.
( 12^\Y^ A. or
\ 12.306... A.
1 h. 29| m.
217. 2|tons.
1,808^^^^ cd. or
1,808.449... cd.
Page 328.
219. 90f feet.
220. 3211 yards.
221. 66,000 tons.
999 S 4H niiles, or
^^^' ■J4m. 220rd.
223. 18|f ft.
224. $5.38.
225. $114.94.
226. 12,V^^.
227 5 336Hgal.;
228. $1,356.
dm
ANSWERS.
Ex. Ans.
229. 5,041f cu. yds.
230. 487yiylbs.
231. 64f lbs.
232.
233.
234.
235.
236.
237.
238.
239.
240.
241.
242.
243.
244.
245.
246.
247.
248.
Page 329.
70,000 times.
$1,680.
166§ minutes.
3.6 ^"'•
j 240,0001-;
\ 240,000 ^^
7,920,000 trees.
(71.428... '^•^•
17,500.
17,500 ^1- •"■
10,000 ^^•
3.4335 ^•
8.736 K-
4.502^ metric T.
0.41896 K-
52.5 ^t- ; 1 183.75.
39.67875 lb.
99.4192 miles.
Page 330.
249. $299.
250. 30.06 % .
251. 44|%.
252. 3^ %.
253. $72,000.
254. 258.72 lbs.
255. 151^1 lbs.
256. 2||f %.
257. A, 76%; B,19%.
258. 8114^ %.
259. 10 f.
260. $17,203.20.
261. 5,110 lbs.
262. $733.17.
263. 2,300 laths.
Page 331.
264. 50;^.
265. $76.
266. Ibf.
267. $150.
Ex.
269.
268. $260,024.50.
j $ 53.90 com. ;
I $926.10 net.
270. $4,829.75
271. $19.80.
272. $1,250.
273. $2,000.
274. 1-^ %.
275. Loss, 2 %.
276. \%.
211. $9,333.33...
Page 332.
278. $32.81.
279. $907.81.
280. $2,592.19.
281. $1,811.32.
282. $5,984.38.
283. 3/ ; $135.
284. $45.99.
285. 2,600 fr.
( Cost$10158.75;
286. <^ $100.91^
($194.0(
287. $309.50.
288. $856.42.
289. $0.0000002ff
900 i ^ 866.88 ;
^'^^' j$ 1,542.29.
291. $69.53.
Page 333.
292. $0.26.
293. 56 %.
294. $296.98.
295. Jan. 22, 1881.
296. 13 y. 4 m.
297. 6^9^%.
298. $403.23.
299. $8.04.
300. $4,347.00.
301. $392.65.
302. $407.37.
303. $117.67.
304. July 9.
305. 4 m. 10 d.
Page 334.
Ex. Ans.
306. Nov. 15.
307. June 3.
308. $596.12.
309. £0 4s. 1.317— d.
(5fr. 18.135—
( centimes.
I 25 fr. 21.503—
\ centimes ;
) 9.518... d. or
f 9d. 2... far.
( 15 s. 10.325... d.
I or 15 s. 10 d.
(l...far.
( 235,294^\ fr. or
I 235294. 1 1 8- fr.
($15411.76.
314. $7.06.
315. $7,299.75.
316. $41.69.
317. $699.20...
Page 335.
318. $3,397.26.
319. $727.27...
320. $32,137.50.
$126^<Vor .
$126.31H-
^7^syield6^%-
310.
311.
312.
313.
321.
6'syield6^%
323. 6 %.
324. 98 men.
325. 43f doz.
326. $13.01.
327. $525.39.
328. 67 ft. ^\ in.
329. $450.
330. 26 miles.
331. 5 oz.
332. $24.
Page 336.
334. 75|yd8.
335. 34|days.
ANSWERS.
367
Ex.
336.
337.
338.
339.
340.
341.
342.
343.
344.
345.
346.
347.
348.
349.
350.
351.
352.
353.
354.
355.
356.
357.
358.
359.
360.
361.
362.
363.
364.
365.
366.
367.
368.
369.
Ans.
\ 5,880 lbs. or
] 2 T. 1,880 lbs.
$329.1^8.
\ A, $ 14.25 ;
) B, $ 7.50 ;
)C, $5.25;
(D, $7.50.
] % 8,445.95 ;
1^4,054.05.
A, $432 ;
< B, $216;
( C, $ 1,296.
( 1st half
)$l,627.50each;
) 2d half
($813.75 each.
$1,192.22.
j T. $ 2,659.04 ;
( V. $ 2,905.96.
Page 337.
361.
3.61.
0.0361.
6.275025.
1.012048064.
0.00000081.
30|."
272^.
JJJL.
T096"
1.21550625.
79.
97.
40.410...
140.
44.2718...
800.562...
6.634.
0.075...
607.5.
63.560...
3.023...
4i
6i
Ex.
370.
371.
Ans.
f
10.025...
372. i|.
373. ^V
374. 5.039...
375. 0.379...
376. |.
377. 2\.
378. 3|-feet.
i 120 slates in
379 ) length;
'^'^- ) 60 slates ill
( width.
I 105 trees in
380. r^nf^'-
j / trees m
( width.
381. 34.
382. 4.7.
383. 0.07547...
384. 16.01...
385. 4.3.
386. 1.308...
387. 0.6463...
388. 1.856...
389. 0.9283...
390. ^.
391. 0.4261...
392. 1.040...
393. 8.005... miles.
394. 342 ft. 3... in.
Page 338.
395. 37.8 sq-™-
396. 43.634... sq. ft
397 J The first by
(Tt
17.^
249... sq.ft.
398. 2.211... vards.
399. 100 feet.'
400. 50 ars.
401. 8.49 acres.
402. 90 feet.
403. 102.87 acres.
404. 158.7226... A.
405. 62.832 feet.
406. 11.459... feet.
407. 63.617... sq. yd.
Page 339.
Ex. Ans,
408. 155.330... rods.
409. 25.4647... sq.ft.
..f. 5 2,513.28 sq.ch.
^^''' tor 251.328 A.
411. 24 sq. ft.
412. 62 sq. ft.
413. 62| sq. ft.
414. 2 sq. ft. 96 sq. in.
415. 260 sq. ft.
416. 400 cii. ft.
417. 168.519— cu. ft.
418. 20.555— cu. ft.
419. 2.2619... ^q""
420. 0.9839... s^"*
421. 615.7536 sq. in.
422. 2727.08^ cu. ft.
423. 840.279... sq.ft.
424. 1,767.15 cu. in.
Page 340.
425. 23|yds.
426. 25 yds.; $31.25.
427. 39 yds; $87.75.
428. $50.25.
429. 228 sq.ft. '
430. 118 sq.ft.
Page 341.
431. 27 sq.ft.
432. 335 sq.ft.
433. $ 15.94.
434. $5.61.
42.566... sq.ft.
13 people.
436. 3 yds. 3 in.
437. $232.13.
438. $10.94.
439. $4.55.
440. 2|bu.
441. 201.801... ft.
442. 14,453,333^ "••
Page 342.
443. 438.4 times.
444. 324.8 — more rev.
445. 756.2 rev.
435.
368
ANSWERS,
446.
447.
448.
449.
450.
451.
452.
453.
454.
455.
456.
457.
458.
459.
460.
461.
462.
463.
464.
465.
466.
467.
463.
469.
470.
471.
Ans.
1,008.4... rev.
1443.11.
$500.
42.41 .. . cu. in.
214.86... bbls.
199,491.6 HI-
6,905,727,150,
981,120,000,
000 tons.
6 hours.
35.719... tons.
150 ft.
\ 10.058. . . T. or
\ lOT. 116.7... lb.
Page 343.
% 100.
5 20,000,000,000
( matches.
$0,114.
$915.56.
$ 39,060.
58,1,%.
80 pages.
25f|f %
' 66f lbs
lbs.
8 feet
26f
206| lbs.
Page 344.
$74.69.
0.96 in.
$27.70.
$2,080.
Ex.
472.
473.
474.
475.
476.
477.
478.
479.
480.
481.
482.
483.
484.
485.
486.
487.
488.
489.
490.
491.
492.
493.
494.
Ans,
$0.57f
140,625 lbs. or
70 T. 625 lbs.
$0.10.
$130.44...
5 Loss, 130 ;
\ $ 3,833.33.
Page 345.
$ 73,305 left.
0.324... in.
5 l,356,Wiyr. or
1 1,356.639... vr.
C 791291111 y.V
( 79129.591... yr.
$11.18.
$ 1,910.46.
Page 346.
5oz. 18pwt.l8gr.
5l3pvvt.l6^|gr.;
$ 38,000.
j 1,544,400 lb. or
\ 772 T. 400 lb.
$4.69.
3.78 min. or
3 min. 46.8 sec.
Sept. 16, 1879.
Latter, $12 more.
$952.64.
$3,150; 70%.
3.55 — cu. in.
Ex.
495.
496.
497.
498.
499.
500.
501.
502.
503.
504.
505.
506.
507.
508.
609.
510.
511.
512.
513.
514.
515.
516.
517.
518.
An$.
4.04... in.
9 times.
1,967 years.
130.923... feet.
Page 347.
13.264... cords.
$1,854.01.
$377.82.
4,803.89... sq.vd
112cu. ft.
33.941... miles.
i Through Lon-
l don, by 0.05lf
( fr. on a dollar.
C Gain, ,'^y, or
163,1
h %■
$7; $8.75;
wonld be raised
to$9.03|.
14 y. 6,"^^ mo.
98.6° F.; 33^° C.
$1,220.81.
A1984;B1371;
C859.
$ 190,270.27.
$0.90.
$5.61.
12.961... ft.
32 ft.
r 1st 55,^114,/
)or $0.5506...
%n$l. 2d Ans.
( $0.155... more.
13 more men.
YB 35840
54 I ^ryS
UNIVERSITY OF CALIFORNIA LIBRARY
feft-
mmt
.SViV.S?
mm'