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ARITHMETIC,
IN TWO PARTS
PART FIRST,
ADVANCED LESSONS IN MENTAL ARITHMETIC.
PART SECOND,
RULES AND EXAMPLES FOR PRACTICE IN
WRITTEN ARITHMETIC.
FOR COMMON AND HIOH SCHOOLS.
BY FREDERIC A. ADAMS,
PRINCIPAL OF DUMMER ACADEMY.
LOWELL:
PUBLISHED BY D. BIX-BY & CO.
Boston : B. B. Mussoy & Co. ; W. J. Reynolds & Co. New Yobk : D. Appleton & Co
PmLADELPHiA : Thomas, Cowperthwait & Co. Baltimore : *Cushing & Brother.
Richmond: Nash" & Woodhouse. Charleston : McCarter & Allen. Mo-
bile : J. Dobler. New Orleans : J.' B. Steele. St. Louis : S. B. Meech.
Louisville : Morton & Griswold. Cincinnati : Derby, Bradley & Co.
Detroit : C. Morse. Chicago : A. H. & C. Burley. Provi-
DENCB : C. Burnet, Jr. Portland : Hyde, Lord ^ Duren.
- 1848.
■^■.K
/,&-69f
U__l
Entered according to Act of Congress, in the year 1848,
By DANIEL BEXBY,
In the Clerk*8 Office ocT the District Court of the District of Massachusettt.
aTBHBOTTPED AND PEINTKB BY DIOKINHOK & CO., 52 WASUIKGTON ST., BOaTOBT.
PREFACE.
The book here offered to Schools and Academies, had ite origin
in the urgent want the author has found, in the case of his own
pupils, of a higher work on Mental Arithmetic. Such a work,
he has thought, should be constructed with reference to several
important objects.
It should habituate the pupil to perform, with ease and readiness,
mental operations upon somewhat large numbers.
It should present these operations in their natural form, freed
from the inverted and mechanical methods which belong of neces-
sity to operations in written Arithmetic.
It should train the student to such a power in apprehending
the relations of numbers, as shall give him an insight into the
grounds of the rules of Arithmetic ; and, consequently, shall re-
lease him from dependence on those rules; and it should free
him from the liability to those wide mistakes often made in written
Arithmetic, which appear so absurd, and are yet too frequent to
excite the teacher's surprise.
A higher training in Mental Arithmetic would also, it is be-
lieved, prepare the members of our schools, when they should leave
their studies and engage in the active pursuits of life, to solve
mentally, and with ease and delight, a large share of those ques-
tions,* of business or curiosity, for which a process of ciphering is
^ wrdi^arily thought indispensable.
iv PREFACE.
The study of Arithmetic in the schools of this country received
its best impulse, unquestionably, in the publication of " Colburn's
First Lessons." So completely has this little book performed the
work within its prescribed sphere, that there is little reason to
desire a change in that particular, or to expect that the work will, •
for the present, be superseded. ^Vhoever would now write a book
of First Lessons in Arithmetic, must, it is believed, if he would
write a good one, walk the most of his way in the steps of one, at
least, who has gone before him.
The " Advanced Lessons " are designed to continue and extend
the course of discipline in numbers, which is begun in the elemen-
tary book above named. Consequently it requires, for its success-
ful study, an acquaintance with the elements, as taught in that
work, or in some other occupying essentially the same ground.
In all the mental calculations in large sums, it will be found a
uniform characteristic of this work to begin with the highest order
of numbers in the sum, — ■ hundreds before tens, tens before units.
In this way, the numbers are presented in the same order in which
they are presented in the common usage of our language. In
most of the operations of written Arithmetic, however, the smallest
number is taken first ; and thus a method is pursued, the reverse
of what the genius of our language would naturally suggest.
Another advantage of taking the highest numbers first, in Mental
Arithmetic is, that we thus obtain a large approximation to the
final answer, at the first step. When the first step, however,
as in written addition, or multiplication, furnishes only the
units of the answer, leavingj the hundreds or thousands still
unknown, only a minute fraction of the answer is at first ob-
tained. It is too plain to require proof,^that that method will be
most interesting and gratifying to the mind, which secures the
largest portion of the answer at the first step. Another advantage
PREFACE. V
of the method here used, is found in the fact, that we naturally
make the higher order the standard, and the lower order takes its
value in the mind from a comparison with the^higher, as a certain
part of it. Thus 150 is apprehended by the mind, as one hundred
and half a hundred. This is not, indeed, the method of acquir-
ing the idea of large numbers, but the method of combining them
after the idea has been acquired ; consequently, it is the legitimate
method of instruction, just as soon as the pupil is qualified to enter
on the study of such combinations. If, now, we obtain the num-
ber of the highest order first, we have a standard, under which all
the succeeding orders naturally fall, and from a comparison with
which they successively take their value. If we begin with units,
however, and work upward through the higher orders, we obtain
no standard ; we must hold the successive numbers in suspense,
until the last term shall furnish the nucleus for the group, — the
standard under which all the lower orders shall take their rank.
It is on the basis of these facts, which are only indications of the
laws of the mind, that, throughout the Mental part of this Arith-
metic, the author has in all operations, taken the highest order of
numbers first. The increased interest which the persevering use
\ of this method will awaken in the minds of pupils, will be, to
teachers, a better commendation of it»- correctness, than any more
•-extended mental analysis. *--
There are other features of the Advanced Lessons which are,
perhaps, sufficiently distinctive to justify their mention here ; but
as the truest test of a school book is its use in the school room,
the work is referred to that ordeal.
The Second Part contains examples in "V^tten Arithmetic on
all the most important rules. They are designed to be sufficiently
numerous to lead the student to ready and accurate practice in
ciphering. In this Part the author has aimed to interest, the
1*
Bcholar by furnishing liim with natural and reasonable questions,
and to aid both teacher and scholar by arranging them progres-
sively.
The rules and explanations will, probably, be found sufficient,
after a thorough mastery of the First Part. It is not necessary
that the pupil complete the First Part before beginning the Second.
He may carry on both Parts at the same time ; but, under each
particular head, the mental part should be thoroughly mastered
before the written examples are begun.
The answers to the questions in the Second Part are given in
a separate work. This course has seemed to the author, on the
whole, the best, notwithstanding some incidental disadvantages
that may arise from it. It will enable the teacher to oversee a
much larger amount of work in Arithmetic, than he could other-
wise attend to.
The Key will be bound up with the Arithmetic, for the use of
teachers ; and such copies will be lettered Teacher's Copy.
The present contains a considerable number of examples more
than the Third Edition, but no change In the numbering of the
sections or of the examples, to occasion inconvenience to the teacher.
To aid in awakening a higher interest and zeal in this branch
of study, the author will offer a few suggestions.
Let the key be used as little as the teacher's necessities will
permit.
Let original questions be proposed by the teacher in connexion
with every Section.
Each member of the class should be encouraged to propose
original questions to be solved by the class.
It will often be useful, especially in a review, to alter some one
figure in the conditions of each question. This often produces a
happy excitement, and gives quite a new zest to the study.
DuMMER Academy, April 18, 1846.
CONTENTS.
PAUT FIRST.
Section. Page.
Preface, 3
^ Explanations, 11
I. Multiplication of Tens and UniA, 13
n. Multiplication of Tens and Units. — Complement, • • .16
in. Practical Questions, 18
IV*. Division, 20
V. Time. — Linear Measure, 25
VI. Federal Money. — Sterling Money. — Dry Measure.
— Avoirdupois Weight. — Troy Weight. — Apothe-
caries' Weight. — Cloth Measure.-^Wine Measure.
— Beer Measure. — Measure of the Circle, 33
Vn. Prime numbers, 42
VILl. Multiplication and Division of Fractions. — .To Find
the Divisors of Numbers, 48
IX. Multiplication of Fractions by Fractions. — Division
of Fractions by Fractions. — Addition of Frac-
tions. — To find a Common Denominator, 53
X. The least Common Multiple, • 60
XI. Practical Questions, • • 62
Xn. Decimal Fractions. — Addition and Subtraction of
Decimals. — Multiplication of Decimals. — Division
of Decimals, • 65
Xm. Keduction of Vulgar Fractions to Decimals, 71
XIV. Interest. — Banking. — Discount. — Loss and Gain.
— Per Centage, 75
vm , CONTENTS.
XV. Square Measure, 81
XVI. Construction of the Square. — Practical Questions,- • -85
XVn. Practical Questions in Square Measure, 91
XVIII. Analysis of Problems, 95
XIX. Solid Measure. — Construction of the Cube, 98
XX. Ratio. — Proportion. — Comparison of Similar Sur-
faces.— Comparison of Similar Solids, 103
Notes to Part First, "• 114
PART SECOND.
4
Numeration of whole Numbers. — Numeration
of Decimals, * 117
I. Addition, 1 20
II. Subtraction, 122
in. Multiplication, 124
IV. Division, / 126
V. Reduction, 129
VI. Reduction, • 131
VII. Compound Addition, 132
Vin. Compound Subtraction, 134
IX Compound Multiplication, ' 135
X Compound Division, 136
XI. Miscellaneous Examples, 137
Xn. Divisibility of Numbers, 138
Xni. Reduction of Fractions, 139
XIV. Change of Numbers and Fractions to Higher
Terms, 141
XV. Multiplication and Division of Fractions, 142
XVI. Multiplication and Division of Fractions, 143
XVn. Addition and Subtraction of Fractions, 144
XVIII. Reduction of Denominate Fractions, 145
XIX. Change of Denominate Integers to Fractions, • • • 146
XX.
XXL
xxn.
xxni.
XXIV.
XXV.
XXVI.
xxvn.
xxvm.
XXIX.
XXX.
XXXI.
xxxn.
xxxm.
XXXIV.
XXXV.
XXXVI.
xxxvn.
xxxvin.
XXXIX.
XL.
XLI.
XLH.
XLm.
XLIV.
CONTENTS. IX
Practical Examples, 147
Decimal Fractions. — Addition and Subtraction.
— Multiplication of Decimals. — Division of
Decimals, 14.8
Reduction of Vulgar Fractions to Decimals. —
Repeating and Circulating Decimals, • • • 1^9
Reduction of Denominate Integers to Decimals, -151
To find the Integral Value of Denominate Deci-
mals, 151
Practical Examples, 152
Practical Questions in Vulgar and "Decimal Frac-
tions, • 154
Reduction of Currencies. — English Currency.
— Federal Money to Sterling. — Canada Cur-
rency. — New England Currency. — New
York Currency. — Pennsylvania Currency.* • -155
Interest, 157
Partial Payments. — Annual Interest, • • 160
Discount, • 163
Banking, 1 64
Loss and Gain. — Per Centage, 165
Alligation, 168
Equation of Payments, 171
Square Measure, 172
Duodecimals, 1 74
Extraction of the Square Root, 1 75
Extraction of the Cube Root, t^^.'^:i^'l79
Proportion. — Practical Question^. — Partnership, 181
Arithmetical Progression, 188
Geometrical Progression, • 191
Mensuration of Surfaces, 192
Mensuration of Solids, 193
Lllscellaneous Theorems and Questions. — Spe- ^.^^
clfic Gravlt}^;^ MecTianlcnl Po wcrsb- — Tho
Lever. — The Wheel and Axle. — The Screw.
— Strength of Beams to resist Fracture. —
Stiffness of Beams to resist Flexure, 195
X CONTENTS.
XLV. Business Forms and Instruments. — Promissory Notes
— On Demand, witli Interest ; on Time, with In-
terest ; on Time, without Interest ; Payable by In-
stalments, with Periodical Interest. — Remarks on
Promissory Notes. — Receipts — A general Form ;
for Money paid by another * Person ; for Money
received for Another; in Part of a Bond; for
Interest due on a Bond ; on Account ; of Papers.
— Order at Sight. — Order on Time. — Award
by Referees. — Letter of Credit. — Power of At-
torney, 204
XL VI. On the Standard of Weights and Measures. — The
English System ; Adopted by the Government of
the United States. — French Decimal System. —
French Long Measure. — French Square Measure.
— French Decimal Weight, • • • 208
XL VII. Appendix, 213
EXPLANATIONS
1. The sign = indicates equality; as 7 times 3=21.
2. The sign -j- indicates addition; as 15-j-7=22.
3. The sign — placed between two numbers, indicates that
the latter number is to be taken from the former ; as 9 — 4
t=5.
The larger number is called the minuend ; the smaller, the
subtrahend.
4. The sign X indicates multiplication; as 6X7=42.
The two numbers are called factors ; the number multiplied
is called the multiplicand ; the number by which it is multi-
plied, the multiplier.
5. The sign H- indicates that the number placed before it,
is to be divided by the number after it ; as 15-r-5=3.
The number to be divided is called the dividend ; the num-
ber by which it is divided is called the divisor.
6. When a number is multiplied by itself, the product is
called the second power of that number, or the square of it ;
as 2 X 2=4, which is the second power, or the square of 2 ;
so 9 is the square of 3 ; 25 the square of 6.
7. When a number is multiplied hy itself, so as to be taken
3 times as a factor, the product is called the 3d power, or the
cube of the number ; thus 8 is the cube of 2, for it is formed
by multiplying 2X2X2; 27, or 3X3X3, is the cube or third
power of 3 ; 125, or 5x5X5, is the third power of 5. The
number thus used as a factor, is called the root of the power ;
thus 3 is the square root of 9, and the cube root of 27 ; 5 is
the square root of^S.
The number of the power may be expressed by a small
figure, thus 2^ is the 3d power of 2; 3 2 is the 2d power of 3 ;
53 is the 3d power of 5.
12 EXPLANATIONS.
Ah angle 'is formed when two lines meet, run-
ning in different directions.
A triangle is a figure bounded by three straight
lines. It is called a triangle,.because it has three
angles. An equilateral triangle has all its sides
equal.
A right angle is formed when one line meets
_, another, making the angles on both sides equal.
0A square is a four-sided figure, the sides of which
are all equal, and the angles of which are right angles.
The diagonal divides it into two equal parts.
A rectangle is a four-sided figure, the oppo-
site sides of which are equal, and the angles of
which are right angles. The diagonal divides
it into two equal parts.
A parallelogram is a four-sided figure the
opposite sides of which are equal and parallel.
The diagonal divides it into two equal parts.
A circle is a figure bounded by a curved line,
called the circumference, every part of which is
equally distant from the centre.
A straight line from the centre to the circumference is
called the radius.
The diameter is a line drawn from side to side of the cir-
cle, through the centre. It follows that the diameter is equal
to twice the radius.
Any portion of the circumference considered by itself is
called an arc.
A sector of a circle is a portion of it bounded by two ra-
dii and the arc between them.
A sphere is a solid bounded by a curved surface every part
of which is equally distant from the centre of the solid.
O
MENTAL ARITHMETIC
PART FIRST
SECTION I.
MULTIPLICATION OF TENS AND UNITS.
1. A man drove six oxen to market, and sold three of them
for 50 dollars apiece. What did they come to ?
Three times 50 are 150. Ans. 150 dollars.
He sold the remaining three for 52 dollars apiece. What
did they come to ?
Three times 50 are 150, and three times 2 are 6, which
added to 150 makes 156. Ans. 156 dollars.
What did they all come to ?
Twice 100 is 200, and twice 50 is 100, which added to 200
makes 300, and 6 added to 300 makes 306. Ans. 306 dollars.
2. A merchant bought 45 barrels of flour for 6 dollars a
barrel. What did it come to ?
6 times 40 are 240, and 6 times 5 are 30 ; 30 added to 240
makes 270. Ans. 270 dollars.
He bought 75 barrels more at 5 dollars a barrel. What
did it come to ?
5 times 70 are 350 ; 5 times 5 are 25, which added to 350
makes 375. Ans. 375 dollars.
What did all the flour come to ?
300 and 200 are 500, 70 and 70 are 140, which added to
500 makes 640, and 5 are 645. Ans. 645 dollars.
3. What will 87 barrels of flour come to at 6 dollars a
barrel ?
6 times 80 are 480, and 6 times 7 are 42, which added to
480 makes 522. Ans. 522 dollars.
2
14
MENTAL ARITHMETIC.
4. What are 7 times 68 ? What are 8 times 72 ?
What are 9 times 84 ? What are 4 times 96 ?
8 tunes 64? 5 times 72 ? 7 times 83 ? 5 times 79 ?
4 times 98 ? 3 times 81 ? 6 times 73 ? 6 times 86 ?
The preceding examples will show the importance of being
able readily to multiply tens by units. This becomes easy,
after acquiring the Multiplication Table. It may be con-
nected with a review of the Multiplication Table in the follow-
ing manner.
Twice 1 are how many ? Twice 10 are how many ?
Twice 2 are how many ? Twice 20 are how many ?
Twice 3? Twice 30? Twice 4? Twice 40?
Twice 5? Twice 50? Twice 6? Twice 60?
Twice 7 ? Twice 70 ? Twice 8 ? Twice 80 ?
Twice 9 ? Twice 90 ? Twice 10 ? Twice 100 ?
3 times 1 ?
3 times 3 ?
3 times 5 ?
3 times 7 ?
3 times 9 ?
4 times 1 ?
4 times 3 ?
4 times 5 ?
4 times 7 ?
4 times 9 ?
5 times 1 ?
5 times 3 ?
5 times 5 ?
5 times 7 ?
5 times 9 ?
6 times 1 ?
6 times 3 ?
6 times 5 ?
6 times 7 ?
6 times 9 ?
7 times 1 ?
7 times 3 ?
7 times 5 ?
7 times 7 ?
7 times 9 ?
3 times 10 ?
3 times 30 ?
3 times 50 ?
3 times 70 ?
3 times 90 ?
4 times 10 ?
4 times 30 ?
4 times 50 ?
4 times 70 ?
4 times 90 ?
5 times 10 ?
5 times 30 ?
5 times 50 ?
5 times 70 ?
5 times 90 ?
6 times 10 ?
6 times 30 ?
6 times 50 ?
6 times 70 ?
6 times 90 ?
7 times 10 ?
7 times 30 ?
7 times 50 ?
7 times 70 ?
7 times 90 ?
3 times 2 ?
3 times 4 ?
3 times 6 ?
3 times 8 ?
3 times 10 ?
4 times 2 ?
4 times 4 ?
4 times 6 ?
4 times 8 ?
4 times 10 ?
5 times 2 ?
5 times 4 ?
5 times 6 ?
5 times 8 ?
5 times 10 ?
6 times 2?
6 times 4 ?
6 times 6 ?
6 times 8 ?
6 times 10 ?
7 times 2 ?
7 times 4 ?
7 times 6 ?
7 times 8 ?
7 times 10 ?
3 tunes 20 ?
3 times 40 ?
3 times 60 ?
3 times 80 ?
3 times 100 ?
4 times 20?
4 times 40 ?
4 times 60 ?
4 times 80?
4 times 100 ?
5 times 20 ?
5 times 40 ?
5 times 60 ?
5 times 80 ?
5 times 100 ?
6 times 20?
6 times 40 ?
6 times 60?
6 tunes 80?
6 times 100 ?
7 times 20 ?
7 times 40 ?
7 times 60 ?
7 times 80?
7 times 100 ?
MULTIPLICATION OF TESS AND UNITS.
15
8 times 1 ? 8 times 10 ? 8 times 2 ? 8 times 20 ?
8 times 3 ? 8 times 30 ? 8 times 4 ? 8 times 40 ?
8 times 5 ? 8 times 50 ? 8 times 6 ? 8 times 60 ?
8 times 7 ? 8 times -70 ? 8 times 8 ? 8 times 80 ?
8 times 9 ? 8 times 90 ? 8 times 10 ? 8 times 100 ?
9 times
9 times
9 times
9 times
9 times
10 times
10 times
10 times
10 times
10 times
11 times
11 times
11 times
11 times
11 times
1?
3?
5?
7?
9?
1?
3?
5?
7?
9?
1?
3?
5?
7?
9?
9 times
9 times
9 times
9 times
9 times
10 times
10 times
10 times
] 0 times
10 times
11 times
11 times
11 times
11 times
11 times
10?
30?
50?
70?
90?
10?
30?
50?
70?
90?
10?
30?
50?
70?
90?
11 times 11? 11 times 110?
12 times
12 times
12 times
12 times
12 times
1?
3?
5?
7?
9?
12 times
12 times
12 times
12 times
12 times
10?
30?
50?
70?
90?
12 times 11 ? 12 times 110 ?
9 times 2 ?
9 times 4 ?
9 times 6 ?
9 times 8 ?
9 times 10 ?
10 times 2 ?
10 times 4 ?
10 times 6 ?
10 times 8 ?
10 times 10 ?
11 times 2 ?
11 times 4 ?
11 times 6 ?
11 times 8 ?
11 times 10 ?
11 times 12?
12 times 2 ?
12 times 4 ?
12 times 6 ?
12 times 8?
12 times 10 ?
12 times 12 ?
9 times 20 ?
9 times 40 ?
9 times 60 ?
9 times 80 ?
9 times 100 ?
10 times 20 ?
10 times 40 ?
10 times 60 ?
10 times 80?
10 times 100 ?
11 times 20 ?
11 times 40?
11 times 60 ?
11 times 80?
11 times 100?
12 times 120 ?
12 times 20 ?
12 times 40 ?
12 times 60 ?
12 times 80?
12 times 100?
12 times 120 ?
A number which contains another number a certain number
of times, is a multiple of that number.
Thus 6 is a multiple of 2 ; 15 of 3 ; 28 of 7.*
Name all the multiples of 2, from 2 to 60.
Name the multiples of 20, from 20 to 600.
What are the multiples of 3 up to 75 ? of 30 up to 750 ?
What are the multiples of 4 up to 80 ? of 40 up to 800 ?
What are the multiples of 5 up to 100 ? of 50 up to 1000 ?
of 6 to 72 ? of 60 to 720 ? of 7 to 84 ? of 70 to 840 ? of 8
to 96 ? of 80 to 960 ? of 9 to 108 ? of 90 to 1080 ? of 10 to
120 ? of 100 to 1200 ?
* See Note 1, at the end of Part First-
16
MENTAL ARITHMETIC.
SECTION II.
MULTIPLICATION OF TENS AND UNITS. — COMPLEMENT.
1. What will 17 tons of liay come to at 8 dollars a ton ?
Aus. 8 times 10 are 80, and 8 times 7 are 56 ; 56 added
to 80 makes 136. 136 dollars.
2. What will 37 pounds of sugar come to at 9 cents a pound ?
3. A man drove 87 sheep to market, and sold them for 6
dollars apiece. What did they come to ?
4. A man travelled on foot 8 days ; he travelled 29 miles
each day. How many miles did he travel in all ?
In each of the above examples the second j^roduct when
added to the first makes a sum e^^eeding the next even hun-
dred: thus, in the 1st ex. — 80+56 ; in the 2d, 270+63 ; in
the 3d, 480+42 ; in the 4th, 160+72.
In order to perform such examples with ease, quickness,
and without mistake, each step in the process should be made
the subject of distinct practice. To illustrate these steps by
the first example, 80+56, the first thing to be done is *to
think of the number which must be added to 80 to make 100,
namely, 20 ; the next is to take this 20 from the 56, aad what
remains, — 36, — will belong to the next hundred.
The number which in such cases must be added to a given
number to make up an even hundred may be called the Com-
plement of that number. Thus the complement of 80 is 20 ;
of 60, 40 ; of 90, 10 ; of 56, 44. What is the complement of
10 ? 30 ? 50 ? 70 ?
* What is the complement of
10?
20?
30?
40?
50?
60?
70?
80 ?~ 90?
11?
21?
31?
41?
51?
61?
71?
81? 91?
12?
22?
32?
42?
52?
62?
72?
82? 92?
13?
23?
33?
43?
53?
63?
73?
83? 93?
14?
24?
34?
44?
54?
64?
74?
84? 94?
15?
25?
35?
45?
55?
65?
75?
85? 95?
16?
26?
36?
46?
56?
66?
76?
86? 96?
17?
27?
37?
47?
57?
67?
77?
87? 97?
18?
28?
38?
48?
58?
68?
78?
88? 98?
19?
29?
39?
49?
59?
69?
79?
89? 99?
How many are
25+83? 36+71?
37+84? 45+76?
40+76? 80+34?
45+82? 56+73?
88+37? 94+17?
70+91
43+82
76+87
? 90+17?
t? 95+36?
' ?
* See Note 2.
MULTIPLICATION OF TENS AND UNITS.
17
* How many are
How many are
12X2,3,4,5,6,7,8,9,10?
46X2,3,4,5,6,7,8,9,10?
13X2,3,4,5,6,7,8,9,10?
47X2,3,4,5,6,7,8,9,10?
14X2,3, 4,5, 6, 7, 8/9, 10?
4^X2,3,4,5,6,7,8,9,10?
15X2,3,4,5,6,7,8,9,10?
49X2,3,4,5,6,7,8,9,10?
16X2,3,4,5,6,7,8,9,10?
50X2,3,4,5,6,7,8,9,10?
17X2,3,4,5,6,7,8,9,10?
51X2,3,4,5,6,7,8,9,10?
18X2,3,4,5,6,7,8,9,10?
52X2,3,4,5,6,7,8,9,10?
19X2,3,4,5, 6,7,8,9,10?
53X2,3,4,5,6,7,8,9,10?
20X2,3,4,5,6,7,8,9,10?
54X2,3,4,5,6,7,8,9,10?
21X2,3,4,5,6,7,8,9,10?
55X2,3,4,5,6,7,8,9,10?
22X2,3,4,5,6,7,8,9,10?
56X2,3,4,5,6,7,8,9,10?
23X2,3,4,5, 6,7,8,9,10?
57X2,3,4,5,6,7,8,9,10?
24X2,3,4,5,6,7,8,9,10?
58X2,3,4,5,6,7,8,9,10?
25X2,3,4,5,6,7,8,9,10?
59X2,3,4,5,6,7,8,9,10?
26X2,^3,4,5,6,7,8,9,10?
60X2,3,4,5,6,7,8,9,10?
27X2,3, 4, 5, 6,7,8,9, 10?
61X2,3,4,5,6,7,8,9,10?
28X2,3,4,5, 6,7,8,9,10?
62X2,3,4,5,6,7,8,9,10?
29X2,3,4,5,6,7,8,9,10?
63X2,3,4,5, 6,7,8,9, 10?
30X2,3,4,5,6,7,8,9,10?
64X2,3,4,5,6,7,8,9,10?
31X2,3,4,5,6,7,8,9,10?
65X2,3,4,5, 6,7,8,9,10?
32X2,3,4,5,6,7,8,9,10?
66X2,3,4,5,6,7,8,9,10?
33X2,^,4,5, 6,7,8,9,10?
67X2, 3, 4,5, 6,7,8, 9, 10?
34X2,3,4,5,6,7,8,9,10?
68X2,3,4,5,6,7,8,9,10?
35X2,3,4,5,6,7,8,9,10?
69X2,3,4,5, 6,7,8,9,10?
36X2,3,4,5,6,7,8,9,10?
70X2,3,4,5,6,7,8,9,10?
37X2,3,4,5,6,7,8,9,10?
71X2,3,4,5,6,7,8,9,10?
38X2,3,4,5, 6,7,8,9, 10?
72X2,3,4,5,6,7,8,9,10?
39X2,3,4,5,6,7,8,9,10?
73X2,3,4,5,6,7,8,9,10?
40X2,3,4,5,5,7,8,9,10?
74X2,3,4,5,6,7,8,9,10?
41X2, 3,it, 5, 6, 7,- 8, 9, 10?
75X2,3, 4,5, 6, 7,8, 9, 10?
42X2,3,4,5,6,7,8,9,10?
76X2,3,4,5,6,7,8,9,10?
43X2,3,4,5, 6,7,8,9,10?
77X2,3,4,5,6,7,8,9,10?
44X2,3,4,5, 6,7,8,9, 10?
78X2,3,4,5,6,7,8,9,10?
45X2,3,4,5,6,7,8,9,10?
79X2,3,4,5,6,7,8,9,10?
To multiply any number less than 10 by 11, repeat the fig-
ure expressing the number : as 3 times 11 is 33, 4X11=44.
To multiply by 11 any number of two figures. Think of
the first figure, then of the sum of the two figures, then of the
last figure. These three figures will express the answer.
2*
^ Note 3.
18 MENTAL ARITHMETIC.
Thus 11 X 23 ; the first, 2 ; the sum of the two, 5 ; the last, 3.
Ans. 253. 11X24=264, 11X32=352, 11X43=473.
Remember, if the sum of the two is as much as 10, you
must increase the first figure by one.
How many are 11X26? 11X28? 11X29? 11X41?
11X43? 11X45?. 11X61? 11X62? 11X64? 11X71?
11X73? 11X81? 11X94? 11X75? 11X86? 11X89?
11X82? 11X84?
SECTIO^N III.
- PRACTICAL QUESTIONS.
1. If a rail-road car travels 23 miles in one hour, how far
will it travel in 9 hours ?
2. If a horse travels 38 miles in one day, how far will he
travel in 6 days ?
3. If a man earns 14 dollars a month, how much will he
earn in 7 months ?
4. If a man spends 6 cents a day for ardent spirit, how
much will that amount to in 10 days ? How much in 30
days ? How much in 300 days ? How much in 60 days ?
How much in 5 days ? How much in 365 days ?
5. If a man earns 10 cents in an hour, and works 12 hours
in a day, how much will he earn in a week, there being 6
working days in a week ? How much in 10 weeks ? How
much in 50 weeks ?
6. If a scholar in school is idle 18 minutes in the forenoon,
and 18 minutes in the afternoon, how much time will he lose
in a week, if there are 6 forenoons, and 4 afternoons of school
time in a week ?
7. If a town is 6 miles long, and 5 miles broad, how many
square miles does it contain ? If there are 40 inhabitants on
every square mile, how many inhabitants does the town con-
tain ? 40 times 30. 4 times 30 are 120. 40 times 30 are 10
times as many. If one in 12 of the inhabitants were able-
bodied men, how many able-bodied men would there be ? If
one in 6 are able-bodied men, how many such are there ?
8. What will 146 yards of broadcloth come to at 5 dollars
a yard ?
PRACTICAL QUESTIONS. ^ 19
9. What will 86 yards of broadcloth come to at 6 dollars
and a half a yard ?
10. What will 740 barrels of flour come to at 6 dollars
a barrel ? at 5 dollars a barrel ? at 5 dollars and a half a
barrel ?
11. What will 33 gallons of molasses come to at 31 cents a
gallon ? at 34 cents a gallon ? at 40 cents a gallon ?
12. What "s\dll 38 pounds of coffee come to at 14 cents a
pound ? at 16 cents a pound ?
13. If a room is 14 feet long and 9 feet high, how many
square feet are there in one of the side walls ? How many
in both the side walls ? If the same room is 13 feet wide,
how many square feet are there in one of its end walls ?
How many in both its end walls. How many square feet in
the ceiling ? •
14. A man wishes to know how many shingles he must buy
in order to shingle his house. His touse is 40 feet long, and
it is 18 feet from the eaves to the ridgepole. How many
square feet are there in one half of the roof? How many in
the whole roof?
One thousand shingles will cover 10 feet square, how many
thousand shingles will cover the roof? If shingles cost 4 dol-
lars a thousand, how much must be paid for shingles enough
to cover the roof? If the labor, the boards, and the nails,
added together, cost as much as the shingles, what will be the
whole expense of boarding and shingling the roof?
15. What are 8 J tons of hay worth, at 13 dollars a ton ?
16. If one acre of ground produce 65 bushels of corn, how
much would grow on 9 acres ?
17. If an acre of ground produce 228 bushels of potatoes,
how many bushels would grow on 5 acres ?
18. If standing wood is worth 2 dollars a cord, what is the
value of the wood on 7 acres, each of which furnishes 18 cords ?
19. If there are 200 families in a town, and each family
consumes 12 cords of wood annually, how many cords are
used in the town each year ?
What is the whole value of the wood at 3 J^ dollars a cord ?
How much money will be saved in the town if each family
bums 2 cords less than before ?
20 MENTAL ARITHMETIC-
SECTION IV.
DIVISION.
1. What is one half of 20? of 40? of 60 ? of 80? of 100?
of 120? of 140? of 160?
2. What is one half 22? of 42? of 62 ? of 82? of 102?
of 112? of 122? of 142? of 162? of 182?
3. What is one half of 44? 64? 86? 48? 66? 28? 84?
68? 46? 24? 26? 62?
4. What is one half of 70 ? divide it into 60 and 10.
What is one half of 90? divide it into 80 and 10.
What is one half of 50 ? of 30 ? of 1 10 ? of 130 ? of 150 ?
5. What is one half of 32 ? of 54 ? divide it into 50 and 4.
. What is one half of 76? of 74? of 78? of 96? of 98? of
92? of 94? of 72? of 7S? of 58? of 56?
6. What is one half -of 43? One half of 40 is 20. One
half of 3 is 1^, this added to 20 makes 21^.
What is one half of 47 ? of 49 ? of 63 ? of 65 ? of 67 ? of
69? of83? of85? of87? of89?
7. What is one half of 33 ? divide into 30 and 3.
What is one half of 35? 37? 39? 51? 53? 55? 57? 59?
of 71? of 73? of 75? of 77? of 79? of 91? of 93? of 95?
of 97? of 99?
8. What is one half of 367 ? divide into 300, 60 and 7.
What is one half of 674? of 895? of 724? of 632? of
945? of 424? of 688? of 546? of 392?
We can now find a very quick way of multiplying any
number by 5. Take one half the number : multiply that by
10. We will take the numbers in question 2, and multiply
them by 5 in this way.
Multiply 22 by 5 : half of 22 is 11, and ten times 11 is 110.
Multiply 42 by 5 : half of 42 is 21 : ten times that is 210.
Multiply 62 by 5 : half of 62 is 31 ; 310.
Multiply 82 by 5 : half is 41 : 410.
Multiply 102 by 5 : half is 51 : 510.
Multiply 112 by 5 : half is 56 : 560.
Multiply 122 by 5 : half is 61 : 010.
Multiply 142 by 5 : half is 71 : 710.
Multiply 162 by 5 : half is '81 : 810.
Multiply 182 by 5 : half is 91 : 910.
DIVISION. 21
9. Multiply by 5 in this way the numbers in question 3.
44. 64. 86. 48. QQ. 28. 84. 68. 46. 24. 26. 62.
Tou can, if you wish, perform these examples by both
methods, and thus prove the work correct.
Multiply 862 by 5 : half is 431 : 4310. ' Multiply 672 by
5: half is 336: 3360.
10. Multiply 686 by 5. 748 by 5. 932 by 5. 896 by 5.
1262 by 5.
If the number to be multiplied is an odd number, so that
half of it will show tbe fraction ^, this, when you multiply by
10, will become 5 : for ten halves are 5.
Multiply 781 by 5 : half is 390 J: 3905 Ans.
11. Multiply 963 by 5 : half is 481^ : 4815. Multiply 845
by 5. 381 by 5. 953 by 5. 845 by 5. 637 by 5. 429 by 5.
12. What is one fourth of 40 ? one fourth of 80 ? one fourth
of 120 ? One fourth of 12 is 3, a fourth of 120 is 10 times as
much, 30. What is one fourth of 160 ? one fourth of 200?
of 240 ? of 280 ? of 320 ? of 360 ? of 400 ?
13. What is one fourth of 60 ? Take half of it ; then half
of that half; half of 60 is 30, half of 30 is 15. What is one
fourth of 100? one fourth of 140? one fourth of 180? one
fourth of 220 ? one fourth of 260 ? one fourth of 300 ?
14. Another way of finding one fourth of the numbers in
the last example, is as follows :
What is one fourth of 60 ? Divide 60 into 40 and 20 ;
one fourth of 40 is 10 ; one fourth of 20 is 5. 15.
What is one fourth of 100 ? divide into 80-f20.
What is one fourth of 140? divide into 120+20.
What is one fourth of 180? divide into 160-|-20, &c.
15. What is one fourth of 30? of 50? of 70?
Find the best way of answering these, for yourself.
What is one fourth of 90? of 110? of 130? of 150? of
170? of 190? of 210? of 230? of 250?
16. What is one fourth of 76 ? divide the number into 40
and 36. What is one fourth of 96? divide the number into
80 and 16?
What is one fourth of 52 ? of 64 ? of 84 ?
-17. What is one fourth of 368? There are several ways
of dividing this number. First, into 200+100+60+8; a
second way would be, into 200+1 60+8 ; another way, into
320+48. This is shorter than either of the former. A bet-
ter division still is into 360+8.
What is one fourth of 496 ? Into what different sets of
22 MENTAL ARITHMETIC.
numbers, each divisible by 4, can you divide this ? What is
one fourth of 964? of 336? of 836? 596? 472? 1324?
1728? 2236?
18. What is one tenth of 10? of 20? of 30? of 40? of 50?
of 60? 70? 80? 90? 100? 110? 120? 130? 140? 150?
19. What is one tenth of 5 ? Ans. 5 tenths of one, or j%
equal to J.
What then is one tenth of 15 ? of 25 ? 35 ? 45 ? 55 ? 65 ?
75? 85? 95? 14? 17? 36? 47? 52? 91? 43? 28? 65?
86? 47?
20. What is one fifth of 25? 40? 45? 50? 55? 60? 65?
70 ? divide 70 into 50 and 20. Of 75 ? divide into 50 and
25. Of 80 ? divide into 50 and 30. Of 85 ? of 90 ? of 95 ?
of 100?
21. What is one fifth of 64? of 82? of 91? of 67? of 73?
of 59? of 63? of 72? of 78? of 83? of 87?
22. What is one fifth of 140? of 385? of 260? of 480?
of 390 ? of 580 ? of 470 ? of 865 ? of 395 ?
23. The following is a short way of dividing a number by
5: Take one tenth of the number and double it. That of
course gives 2 tenths, which is equal to one fifth. Take the
numbers in the last example, and divide by 5 in this way.
One fifth of 140 ; one tenth is 14, double that is 28. One
fifth of 385 ; one tenth is 38 and 5 tenths, twice that is 77.
One fifth of 260 ; one tenth is 26, twice 26 is 52. One fifth
of 480 ; one tenth is 48, twice that is 96. What is one fifth
of 390? of 580? 470? 865? 395?
24. The following is a short way of multiplying a number
by 25. Take one fourth of the number ; multiply that by
100. This will give 100 fourths, which are equal to 25 whole
ones.
Multiply 40 by 25 ; one fourth is 10, one hundred times
that are 1000. Multiply 60 by 25 ; one fourth is 15, 1500.
Multiply 80 by 25 ; one fourth is 20, 2000.
Multiply 120 by 25 ; a fourth is 30, 3000.
Multiply 112 by 25 ; a fourth is 28, 2800.
Multiply 116 by 25 ; a fourth is 29, 2900.
25. Multiply 22 by 25 ; one fourth is 5 and a half; 100
times this are 5 hundred and half a hundred, 550.
Multiply 26 by 25 ; one fourth is 6^, 650.
Multiply 28 by 25 ; one fourth is 7, 700,
Multiply 30 by 25 ; 32 by 25 ; 34 by 25 j 36 by 25 ; 40
by 25.
DIVISION. 23
Multiply 42 hj 25 ; 44 by 25 ; 46 by 25 ; 48 by 25 ; 50
by 25.
26. Multiply 13 by 25 ; one fourth is 3 and one fourth ;
one hundred times this i3 300, and one fourth of a hundred, or
25, 325.
Multiply 15 by 25; one fourth is 3. and three fourths, one
hundred times this are 3 hundred and 3 fourths of a hundred,
or 75, 375.
Multiply 17 by 25 ; 19 by 25 ■; 21 by 25 ; 23 by 25 ; 27
by 25 ; 29 by 25 ; 31 by 25 ; 33 by 25 ; 35 by 25.
27. Multiply 116 by 25 ; one fourth is 29, 2900.
Multiply 117 by 25 ; one fourth is 29J, 2925.
Multiply 121 by 25 ; 87 by 25 ; 156 by 25 ; 960 by 25.
28.* What is one third of 60 ? of 90? of 120? of 15? of
150 ? of 45 ? of 450 ?
What is one third of 18? of 180? of 21? 210? of 36? of
360?of30?of390?
What is one third of 72 ? divide into 60 and 12. "
Wliat is one third of 54 ? of 85 ? of 98 ?
What is one sixth of 60 ? of 80 ? divide into 60 and 20. Of
74? of84?of 96?oflOO?
What is one sixth of 12 ? of 120 ? of 130 ? of 140 ? of 144 ?
What is one sixth of 18 ? of 180 ? of 200 ? of 210 ? of 220 ?
What is one sixth of 384 f of 492 ? divide into 480 and 12.
Of 555 ? divide into 540 and 15. Of 620 ? of 726 ? of 947 ?
29. What are the two factors of 18 ? of 180 ?
What are the two factors of 27 ? of 270 ? of 22 ? of 220 ?
of 35 ? of 350 ? of 54 ? of 540 ? of 45 ? of 450 ? of 21 ? of 210 ?
of 28 ? of 280 ? of 42 ? of 420 ?
30. What two numbers multiplied together will produce
24?
What other two factors wiU produce 24? What other two ?
What two factors will produce 240 ? What other two ?
What others ?
What two factors will produce 30 ? What others ?
What two factors will produce 300 ? What others ?
What two factors will produce 18 ? Wliat others ?
What two will produce 180 ? What others ?
Name all the pairs of factors that will produce 36 ? 860 ?
48?480? 60? 600? 64? 640 ? 72 ? 720 ?
*Note 4. ' .
24 MENTAL ARITHMETIC.
31. What is one ninth of 27 ? A man divided 270 dollars
equally among 9 persons ; how much did he give to each ?
32. What is one fourth of 48 ? If 480 dollars are divided
into 4 equal shares, what will each share be ?
What is one eighth of 480 ? one sixth of 480 ? one twelfth
of 480?
33. What is one seventh of 63 ? If a ship sails at a uni-
form rate, 630 miles in a week, how many miles does she sail
in a day ?
What is one ninth of 630 ? What is one sixth of 630 ?
What is one third of 630 ?
34. What is one fifth of 25 ? If 250 trees are placed in 5
equal rows, how many will there be in each row ?
If placed in 50 equal rows, how many would there be in
each row ?
35. What is one fourth of 36? In a circle there are 360
degrees ; how many are there in one. fourth of a circle ? How
many in one eighth of a circle ? How many in one sixteenth
of a circle ?
36. What is one eighth of 56 ? If 560 trees were planted
in 8 equal rows, how many would there be in each row ? If
planted in 1 6 rows, how many would there be in each row ?
37. What is one eleventh of 55 ? If you place 550 trees in
11 equal rows, how many will there be in a row? If you
place them in fifty rows, how many will there be in each
row ? If you place them in 25 rows, tow many will there be
in each row?
38. What is one twelfth of 96 ? If a man spends 960 dol-
lars in a year, how much will be his average expense for
each month?
39. What is one tenth of 40 ? of 400 ? of 4000 ?
What is one fourth of 40 ? 400 ? of 4000 ? of 80 ? of 800 ?
of 8000 ? One fourth of 12 ? of 120 ? of 1200 ? of 12,000.
40. What is one fifteenth of 60 ? of 600 ? of 6000 ?
What is one thirtieth of 60 ? of 600 ? of 6000 ? of 1200 ?
of 12,000?
41. What is one fifth of 92? What is one third of 51 ?
One fourth of 65 ? One fifth of 78? One sixth of 96 ? One
seventh of 100? Divide into 70 and 30. What is one
ninth of 117 ? Ajis. One ninth of 90 is 10 ; one ninth of 27
is 3 ; 10 and 3 are 13.
42. What is one third of 49 ? one sixth of 84 ? one fifth
of 79 ? one eighth of 100 ? one seventh of 91 ? one sixth of
DIVISION. S5
79? one fourth of 76 ? of 92? of 57? of 60? of 52? of 65?
of 70?
43. What is one fourth of 480 ? What is one fifth of 155 ?
Divide into 150 and 5. What is one fourth of 920 ? What
is one fifth of 15,76*5. This number may be divided into
15,000, 750 and 15, or 15,000, 500, 250 and 15.
44* What is one sixth of 4836? One eighth of 336?
Divide into 320 and 16. What is one seventh of 574? one
third of 684 ? one sixth of 43,248 ? one ninth of 72,108 ? of
64,827 ? one fifth of 5275 ? one fourth of 92,648 ? ,
r 45. What is one third of 6156 ? of 8436 ?
46. What is one fourth of 6428 ? of 9648 ?
47. What is one fifth of 7655 ? of 12,535 ?
48. What is one sixth of 13,218 ? of 1944^.
49. What is one seventh of 10,542 ? of 14,280 ?
50. What is one eighth of 1632 ? of 2560 ? fy-C^
SECTION V.
TABLE OF TIME.
60 Seconds, [sec] make 1 Minute, marked m.
60 Minutes 1 Hour, h.
24 Hours 1 Day, d.
7 Days 1 Week, w.
4 Weeks 1 Month, mo.
52 Weeks, 1 day, 6 hours 1 Year, y.
365 Days, 6 Hours 1 Year, - y.
12 Calendar Months 1 Year, y.
In common reckoning, 4 weeks are called a month, but this
is merely for convenience in doing business. The number of
days in a calendar month is 30 or 31 ; except February,
which has 28 days, and in leap year 29. The 6 hours over
and above the 365 days in a year, will in 4 years amount to
a whole day ; it is then added to February, making 29 days,
and that year is called leap year. The number of days in
* Note 5.
26 MENTAL ARITHMETIC.
the Other months may be seen in the line below. The months
connected by a tie drawn over the words have 31 days ; those
connected by a tie underneath have 30. V
Jan. Feb. March. April. May. June. July. August. Sept. Oct. Nov. Dec.
28. 29. ^ ' > '
You observe that beginning with January, every alternate
month has 31 days, till you come to July and August. Here
there are two months together that have 31, and then the al-
ternation goes on as before to the end of the year. *
The leap year may be easily known from the fact that the
number of the year is exactly divisible by 4. Thus 1844
was leap year ; the number can be divided by 4.
What years in the present century have been leap years ?
What years will be leap years from now to the close of the
century*?
• 1. ia onp mmut§ t6e|;e'^re 60 seconds ; in one hour there
are 60 minutes. iTo^ niar^y s^c^ds af^^ there in one hour ?
How many in 10 hours ? How man/ in 20 h»urs ? How
many in one day ? How many in seven days, or one week ?
How many in ten days ? In 100 days ? In 300 days ? In
350 days? In 365 days?
2. If you save 30 minutes from idleness each day, how
many hours will you save in a week ? How many in 5
wfeeks ? How many in 50 weeks ? How many in 52 weeks ?
3. If you read 40 pages each day, how many pages will you
read in one week ? How many in 10 weeks ? How many
in 52 weeks ?
4. If a printer sets 4 pages of type in a day, in how many
days will he set the type for a book of 500 pages ? What
will his wages come to at Sl,50 a day ?
5. If there are 300 members in the Legislature of Massa-
chusetts, and each member receives 2 dollars a day during
the session, what does the pay of all the members come to for
one day ? What does the pay of the Legislature amount to
for one week ? For 10 weeks ?
6. The number of members in Congress is about 275. At
8 dollars a day, what is the amount of their pay each day ?
Wliat would be the amount of their pay for 10 days ? For
100 days ?
7. How many days are there in the 3 months of spring ?
TIME. 27
How many days in the 3 months of summer ? How many
days in autuimi ?
8. How many days in the winter of leap year ? How
many days were there in the winter of 1844 ? How many
days in the winter of 1845 ?
9. If January comes in on Monday, on what day of the
week will February come in ?
If March comes in on Wednesday, on what day of the
week will April come in ?
If August comes in on Saturday, on what day of the week
will September come in ?
10. If April comes in on Sunday, on what day of the week
will it go out ?
If June comes in on Tuesday, on what day will it go out ?
If September comes in on Saturday, on what day will it go
out?
11. If January comes in on Friday, how many Sundays
will there be in that month ?
If it comes in on Thursday, how many Sundays will there
be in that month ?
If June comes in on Friday, how many Sundays will there
be in that month ? If it comes in on Saturday, how many ?
If February comes in on Saturday, and that year is leap
year, how many Saturdays will there be in the month ? If
it is not leap year, how many ?
In 1845 February came in on Saturday, how many Sat-
urdays were there in that month ?
In 1844 February came in on Thursday, how many Thurs-
days were there in that month ?
12. If January comes in on Monday, on what day of the
week will March come in, if it is leap year ? On what day,
if it is not leap year ?
13. K June comes in on Wednesday, what day of the weei
will the 1st of August be ? The 9th ? the 12th ? the 15th ?
28 MENTAL ARITHMETIC.
TABLE OF LINEAR MEASURE.
12 inches, 1 foot, ft.
3 feet, 1 yard, • yd.
5J yards, 16J feet, 1 rod, rd.
40 rods, 1 furlong, • • fur.
8 furlongs=;320 rods, 1 mile, m.
3 miles, 1 league, 1.
69 J miles, 1 degree of latitude, • • • • deg.
For lengths less than an inch, the inch is divided into
fourths, eighths, tenths, or twelfths.
1. How many inches in 2 feet? In 4 feet? In 5 feet?
In 7 feet ? In 10 feet ? In 12 feet ? How many inches in
4 yards ? In 1 rod ? In 3 rods ? How many feet in 1 fur-
long ? In 2 furlongs ? In 4 furlongs ? In 1 mile ?
2. How many miles in 46 leagues ? In 132 leagues ?
How many miles in 2 degrees of latitude ? In 3 degrees ?
In 4^ degrees ? In 6 degrees ?
In estimating the miles in any number of degrees of latitude,
it is most convenient to call a degree 70 miles, and then, if
we wish to be accurate, we may subtract from the answer half
as many miles as there are degrees. In this way the distance
of places from each other may be determined on a map : the
degrees of latitude on the margin may be used as a scale of
miles. If the distance of two places from each other is equal
to 6| degrees of latitude, how many miles are they apart ?
3. How many yards in 10 rods ? In 20 rods ? In 30 rods ?
In 1 furlong ? In 8 furlongs, or 1 mile ?
4. In measuring land or a road with a chain 4 rods long,
how many times must the chain be applied to the ground in
measuring one mile ? How many times in measuring the road
from Boston to Salem, 15 miles ? How many in measuring
from Boston to Providence, 40 miles ?
5. If a man walks three miles in an hour, how many min-
utes will he be in walking 1 mile ? How many minutes in
walking 1 fourth of a mile ? How many rods will he walk
in 1 minute? Ans. 16.
How many seconds will he be, then, in walking 1 rod ?
16 will go into 60, 3 times and 12 over. He will be, then, a
little less than 4 seconds in walking 1 rod.
Let us now suppose he is precisely 4 seconds in walking
REDUCTION OF LINEAK MEASUHE. 29
1 rod ; how many rods would he walk in a minute ? How
many in 10 minutes? How many in 60 minutes? How
many miles ?
6. If a man in walking takes 6 steps to a rod, how many
steps will he take in walking a mile ? How many in walking
10 miles ? How many in walking 40 miles ?
7. If a man in walking takes 6 steps to a rod, and takes 2
steps in a second, how many seconds will he be in walking
one rod ? How many seconds in walking 10 rods ? 20 rods ?
If a man walks 20 rods in one minute, how many minutes
will it take him to walk a mile ? 20 are contained in 320 just
as many times as 2 are contained in 32.
8. If a man walks 20 rods in one minute, how long will it
take him to walk 4 miles ?
9. If a rail-road train goes 30 miles in an hour, how faf
does it go in one minute ? How many rods in one second ?
Ans. 30 miles in 60 minutes is 1 mile in 2 minutes ; half a
mile in one minute ; quarter of a mile in half a minute ; that
is 80 rods in 30 seconds ; that is 8 rods in 3 seconds ; and in 1
second, one third of 8 rods or 2 rods and two thirds.
10. How many rods in 14 miles ?
In one rod there are 16^^ feet. In one mile there are 320
rods ; how many feet are there in a mile ? There are various
ways of finding the answer to'this question ; some of them will
be suggested, and the pupil left to take his choice.
First, how many feet are there in 300 rods ?
This is not difficult, for in 3 rods there are three times 16 J
feet, and in 300 rods there 100 times as many. 3 times 15
feet are 45 feet ; 3 times 1^ feet are 4^, which added to 45
make 49 J feet in 3 rods. Now 100 times 49 are 4900, and
100 halves are 50 ; 4950 feet in 300 rods. In 20 rods, there
are ten times as many feet as in 2 rods ; in 2 rods there are
twice 16^ or 33 ; in 20 rods therefore there are 330 feet; 300
added to 4900 make 5200, and 30 added to 50 make 80 ;
there are then 5280 feet in a mile.
Another method would be to multiply 320 first by 8, and
that product by 2, for 8 and 2 are the factors of 16 ; then as
there was ^ a foot in each rod left out, there must be added
half as many feet as there are rods, or half of 320.
Another method would be to mtiltiply 320 by 10, then by
six and add the products, and lastly by J and add that to the
other products.
3*
30 MENTAL ARITHMETIC.
The pupil can tiy each of these ways, and see if he obtains
the same answer.
Let us now see if our answer is correct. If there are 5280
feet in a mile, how many are there in half a mile ? One half
of 5200 is 2600, one half of 80 is 40 ; there are then 2640
feet in half a mile. How many in 1 fourth of a mile ? One
half of 2640 feet, which is 1320. Now 1 fourth of a mile is
80 rods. If then there are 1320 feet in 80 rods, how many
will there be in 8 rods ? One tenth as many. One tenth of
1320 is 132. Now how many are there in one rod ? One
eighth of 132 : dividing 132 into 80 and 52 ; one eighth of
80 is 10, and one eighth of 52 is 6| or J, which added to 10
make 16|. We have now come down from 5280, and arrived
by successive divisions to 161^, the number from which we
started at first. The answer is thus proved to be correct.
11. How many feet are there in 2 rods ? In 20 rods ? In
200 rods ?
12. How many feet are there in 3 rods ? In 30 rods ? In
300 rods ? In 8 rods ? In 80 rods, or quarter of a mile ?
13. How many rods are there in 2 miles ? In 4 miles ?
In 8 miles ? In 20 miles ? In 30 miles ? In 50 miles ?
14. How many rods are there in half a mile ? In three
fourths of a mile ? In one mile and a half? In one mile and
three furlongs ? In two miles and five furlongs ? In 4 miles
and 7 furlongs ?
15. How many yards are there in 2 rods ? In 20 rods ?
In 3 rods ? In 30 rods ? In 300 rods ?
How many yards are there in 3 rods and 4 feet ? How
many yards are there in 17 rods 11 feet?
16. How many inches are there in 7 feet ? In 9 feet ? In
6 feet ? In 8 feet and 6 inches ? In 11 feet 9 inches ?
17. How many inches are there in 1 rod, or 16 J feet?
How many inches in 2 rods ? In 3 rods ? In 4 rods ?
18. A house is 46 feet and 5 inches in length, how many
inches long is it ?
A creeping vine grows on an average 3 inches a day ; how
many days will it take to grow from the ground to the top of
a house that is 25 feet high ?
19. A stage-horse travels 13 miles and "20 rods each day;
how far will he travel in 6f) days ? How far in 120 ?
20. How far will the horse travel in a year, if he rests 5
days in the year ?
VELOCITY OF SOUND. 81
21. What is the weight of iron used in one mile of rail-
road, allowing 55 pounds for a yard of rail ?
One yard of heavy rail weighs 55 pounds. Twice this, or
110 pounds, would be the weight of both rails for one yard
of a single track. Five and a half-times this would be the
weight for one rod. From this may be obtained the weight
for 10 rods ; for 100 rods ; for 300 rods ; for 320 rods.
22. What would be the cost of the iron for a single track
of one mile of rail-road at 4 cents a pound? How much
would be saved in the expense of the iron for one mile of rail-
road, if the price of iron should be reduced one cent a pound ?
23. If the cost of the iron for a single track of rail-road is
6000 dollars a mile, and the cost of the land and the labor of
construction equals that of the iron, what would be the cost
of 15 miles of rail-road ? of 24 miles ?
24. What would be the cost of constructing one mile of
common road at $2,25 a rod ?
25. What would be the cost of building 80 rods of common
wall at 54 cents a rod ?
26. If a horse travels 10 miles in an hour, how long is he
in traveling 1 mile ? How long in traveling J of a mile, or
80 rods ? How many seconds is he in traveling 8 rods ?
How long in traveling 1 rod ?
27. A body falling -through the air, falls in the first second
16 J feet, and each succeeding second it falls twice 16 J feet
further than in the preceding second. How far would a stone
fall in 2 seconds ?
28. How far would it fall in the third second ? How far
would it fall in 3 seconds ?
29. How far would it fall in the fourth second ? How far
would it fall in 4 seconds ?
30. -Sound moves through the air at the rate of 1090* feet
in a second. How many feet will it move in 3 seconds ? How
many feet in 4 seconds ? How many feet in 5 seconds ?
As sound is found thus to pass 5450 feet in 5 seconds, and
as there are 5280 feet in a mile, we see that in 5 seconds
sound moves 170 feet more than a mile. Now as 165 feet is
just 10 rods, we say, without much error, that sound moves
1 mile and 10 rodsAn 5 seconds. This is accurate enough for
all common purposes, and you will do well to fix it in your
memory, and make your calculations from it.
* Professor Pierce on Soands.
32 MENTAL ARITHMETIC.
81. How many rods will sound move in 1 second? One
fifth of 320+10 rods, = 66 rods.
32. How many rods in 2 seconds ? How many rods in 4
seconds ?
Thus, if you watch the stroke of an axe used by some one
at a distance, and observe that the sound comes to you one
second later than you see the stroke, you may know that the
distance is 66 rods. If the sound of a bell comes to you two
seconds after the stroke is given, you must be distant from the
bell 132 rods. In these cases no allowance is made for the
transmission of light. You are supposed to see the motion as
soon as it occurs. This is not strictly the fact ; but the time
is so exceedingly small that it need not be taken into the
account.
33. In a still night a church bell is sometimes heard at the
distance of 12 miles ; how many seconds, or nearly how many,
after the stroke would the sound be heard at that distance ?
34. If the report accompanying a flash of lightning is heard
4 seconds after the flash is seen, how far from the hearer was
the discharge ? How far, if the time between the flash and
the report is 6 seconds ? How far, if the time is 8 seconds ?
How far, if the time is 10 seconds ? How far, if the time is
15 seconds ?
35. The report of a cannon has, in some instances, been
heard at the distance of 100 miles : in how many seconds, or
nearly how many, after the discharge, would the report be
heard at that distance ? In how many minutes ?
36. By means of a magnetic telegraph it is possible to com-
municate intelligence instantly from New Orleans to Boston, a
distance of 1500 miles. If this intelligence could be commu-
nicated by sound passing through the air, how long would it
be traveling that distance, allowing 5 seconds to a mile ?
A ball discharged from a gun moves at first with a
greater speed than sound, but it moves slower and slower,
and before it is spent the report overtakes it, and passes by
it : for sound moves always at the same rate.
37. If a cannon ball moves a mile in 8 seconds, how long
would it be in moving 3 miles ? How long in moving one
fourth of a mile ? How long in moving one eighth of a mile ?
How long in moving If miles F
REDUCTION, MONEY. 83
SECTION VI.
TABLE OF FEDERAL MONEY.
10 IVIills make • • • • 1 Cent, marked • • . • ct.
10 Cents, 1 Dime, d.
10 Dimes, 1 Dollar, D.
10 DoUars, 1 Eagle, E.
This is established by law as the currency of the United
States.
The general mark for Federal Money is $, as $5.14, five
dollars fourteen cents. A period must always be placed be-
tween dollars and cents.
1. How many mills in 2 cents ? In 10 cents ? In 12
cents? In 5J cents? In 12J cents? In 36 cents? In 1
dollar?
2. How many cents in 5 dimes? In 11 dimes? In 16
dimes ? In 4^ dollars ? In 17^ dollars ? In 12| dollars ?
3. How many dimes in 7 dollars ? In 13 J dollars ? In 3
eagles ? In 56 dollars ? In 100 dollars ?
4. How many cents in 35 mills ? In 180 mills ? In 600
mills ? How many dimes in 80 cents ? In 210 cents ? In
740 cents ?
5. How many dollars in 350 cents ? In 325 cents ? In
700 cents ? In 850 cents ? In 1400 cents ? In 1675 cents ?
In 925 cents ?
TABLE OF STERLING MONEY.
4 farthings [qr.] make* • •! penny, marked* 'd.
12 pence, • 1 shilling, s.
20 shillings, 1 pound, £.
This is the currency of Great Britain.
1. How many farthings are there in 3 pence ? In 7 pence ?
In 8 pence ? In 10 pence ? In 11 pence ?
2. How many pence in 2 shillings? In 12 shillings? In
15 shillings ? In 18 shillings ? In 16 shillings ?
3. How many shillings in 4 pounds ? In 7 pounds ? In
18 pounds ? In 36 pounds ? In 84 pounds ?
34 MENTAL ARITHMETIC.
4. How many farthings in 1 shilling and 6 pence ? In 2
shillings and 6 pence ? In 15 shillings and 4 pence ?
How many pence in 10 shillings ? In 20 shillings ? In 2
pounds ? In 4 pounds ? In 12 pounds ?
5. How many farthings in 1 pound ? In 5 pounds ? In
8 pounds ? In 1 pound 2 shillings ?
6. How many pence in 45 farthings ? In 128 farthings ?
In 464 farthings ? In 1296 farthings ? In 648 farthings ?
7. How many shillings in 80 pence ? In 67 pence ? In
372 pence ? In 649 pence ? In 840 pence ?
8. How many pounds in 267 shillings ? In 845 shillings?
In 432 shiUings ? In 640 shillings ? In 4000 shillings ?
9. How many pounds in 890 pence ? In 16,000 farthings ?
In 720 pence ? In 1200 pence ? In 456 pence ?
10. How many pence in 5 pounds 4 shillings ? In 7
pounds 8 shillings ? In 12 pounds 3 shillings ?
How many farthings in 4 shillings^pence ? In 9 shilhngs ?
How many farthings in 6 pounds 3 shillings 8 pence ?
11. A man set out on a journey with £4 8s 6d in his
pocket : before spending any thing, he received in payment of
a debt £2 3s 8d. How much had he then ? When he arrived
home he had spent £1 4s 6d. How much had he then ?
These denominations, you must bear in mind, have not the
same value in English currency, that they have in the United
States.
In our country they have different values in the different
States, but in none of them so high a value as in England.
In the New England States a shilling is equal to 1 6 cents and
two thirds, -and 6 shillings make a dollar. In New York 12
and a half cents are a shilling, and 8 shillings a dollar. In
other States the values are still different ; but these denomi-
nations are gradually giving way to those of the Federal
currency. They are now used only in naming prices. Ac-
counts are not kept in them, and all that is important in them
may be learned by practice without further notice here.
In the Sterling currency, used in England, a pound is equal
to 4 dollars, 44 cents and 4 mills ; 10 shillings, therefore, or
half a pound, are 2 dollars, 22 cents, 2 mills ; and 1 shilling is
one tenth part of that, or 22 cents, 2 mills. An English six-
pence is, therefore, 11 cents 1 mill. The following table will
be useful in exchanging English money to our own.
REDUCTION. 35
1 pound, £, is $4.44 4
10 shillings, or half a pound, 2.22 2
1 shilUng, 22 2
6d, or half a shilling, Ill
4 shillings, 6 pence, 1.00 0
1 guinea, 21 shillings, 4.66 6
The actual value of the English money is a little higher
than is here stated, but this is sufficiently accurate for a gen-
eral table.
1. What is the value in dollars and cents of 2£ ? 3£ ? 4£ ?
5£? l£6s? 2£8s? 3s 6d? 5s 9d?
TABLE OF DRY MEASURE.
2 pints, [pt.] make • 1 quart, marked qt.
4 quaa-ts, • • ^ 1 gallon, gal.
8 quarts, 1 peck, pk.
4 pecks, 1 bushel, bu.
8 bushels, 1 quarter, qr.
36 bushels, 1 chaldron, ch.
These denominations are used for measuring grain, fruit,
and coal. The pint, quart, and gallon are larger than the
same denominations in wine measure, and less than those of
beer measure.
1. How many pints in 1 peck? In 3 pecks? In 1 bushel?
In 3 bushels ? In 4 bushels ?
2. How many quarts are there in 1 bushel ? In 4 bushels ?
How many pecks in 7 quarters ? In 2 chaldrons ?
3. If a horse eat 4 quarts of oats each day, how many
bushels will he eat in 10 weeks ? How many bushels in 50
weeks ? In 52 weeks ?
What will they cost at 50 cents a bushel ?
4. In 80 quarts how many pecks ? How many bushels ?
In 644 quarts how many pecks ? How many bushels ?
In 7840 quarts how many pecks ? How many bushels ?
5. In 100 pints how many pecks ? How many bushels ?
In 620 pints how many pecks ? How many bushels ?
36 MENTAL ARITHMETIC.
TABLE OF AVOIRDUPOIS WEIGHT.
16 drams, [dr.] make 1 ounce, marked oz.
16 ounces, • • 1 pound, lb.
25 pounds, 1 quarter, (net wt.) qr.
28 pounds, 1 quarter, • • • (gross wt.) qr.
4 quarters, 1 hundred weight, cwt.
20 hundred weight, 1 Ton, T.
, These denominations are used in weighing hay, grain, meat,
flour, and all the most common articles bought and sold by
weight. On account of the waste in handling such articles,
their shrinking in drying, and worthless admixtures sometimes
found in them, 112 pounds are sometimes allowed for one
hundred weight ; this makes 28 pounds one quarter, and is
called gross w^eight. In all the following questions of Avoir-
•dupois wt., understand gross wt., unless net wt. is expressed.
1. How many drams in 3 oz. ? In 5 oz. ? In 8 oz.? In
11 oz. .? How many oz. in 12 lbs.? In 15 lbs.? In 20 lbs.?
In 32 lbs.? In 45 lbs.?
2. How many lbs. in 4 cwt. net weight ? In 4 cwt. gross ?
In 6 cwt. net weight ? In 6 cwt. gross ? In 5 cwt. 2 qrs.
net ? In 5 cwt. 2 qrs. gross ? In 7 cwt. 3 qrs. net weight ?
In 7 cwt. 3 qrs. gross ?
37 How many lbs. in a ton, net weight ? In a ton, gross ?
How many lbs. in 5 tons, 3 cwt, net ? In 5 tons, 3 cwt. gross ?
4. There are 2 loads of hay whose net weight is as follows ;
the first, 25 cwt. 3 qrs. 17 lbs. ; the second, 17 cwt. 2 qrs. 21
lbs. "What is the weight of both ?
5. A man set out for market with a load of hay weighing
36 cwt. 2 qrs. 15 lbs., net weight ; he lost a part of it ; the re-
mainder weighed 25 cwt. 1 qr. 8 lbs. How much did he lose ?
6. If there are 196 lbs. in a barrel of flour, how many
pounds net weight are there in 10 barrels ?
196 lbs. are 7 quarters gross ; how many cwt. gross are there
in 10 barrels of flour ?
7. How many pounds are there in 100 oz.? In 650 oz.?
8. A barrel of flour weighs 7 quarters gvoss ; how many
tons gross, are there in 100 barrels of flour ?
9. What will be the expense of transporting by rail-road
100 bai*rels flour, 100 miles, at the rate of 3 dollars a ton ?
What will be the expense of transporting a single barrel ?
REDUCTIO¥. 37
100 barrels are 700 qrs. gross weight, 400 qrs.=100 cwt=
5 tons : 300 qrs.=75 cwt.=3 tons, 15 cwt. ; this added to 5
tons, makes 8 tons, 15 cwt.
10. The freight of goods by wagon is about 20 dollars a
ton' gross for 100 miles ; at this rate what will be the cost of
carrying a barrel of flour 100 miles ?
TABLE OF TROY WEIGHT.
" 24 grains [gr.] make 1 pennyweight, dwt.
20 pennyweights, 1 ounce, oz.
12 ounces, 1 pound, lb.
This is used for weighing gold and silver. The pound
Troy is nearly one fifth less than the pound Avoirdupois.
1. How many grains in 6 pennyweights ? In 8 penny-
weights? In 12 pennyweights? Inloz. ? In2oz. ? In
4 oz. ? In 6 oz. ?
2. How many pennyweights in 8 oz. ? In 11 oz.? In 1
lb.? In 3 lbs.? In 8 lbs.? In 5 lbs. ? Inllb.3oz.? In
Slbs.Soz.?
3. How many oz. in 120 dwt.? In 480 dwt.? In 960
grs. ? How many lbs. in 100 oz. ? In 860 dwt. ? In 1200
dwt.?
TABLE OP APOTHECARIES' WEIGHT.
20 grs. make • • • • 1 scruple, marked 9
3 scruples, 1 dram, 5
8 drams, 1 ounce, S
12 ounces, 1 pound, ib
This table is used only by apothecaries in mixing medi-
cines. The pound and ounce are the same as in Troy weight.
TABLE OF CLOTH MEASURE.
2^ inches [in.] make • • • • 1 nail, marked na.
4 nails, 1 quarter, qr.
4 quarters, 1 yard, yd.
3 quarters, 1 ell Flemish, Fl. e.
5 quarters, 1 ell English, E. e.
6 quarters, 1 ell French, Fr. e.
38 MENTAL ARITHMETIC.
1. How many inches in 1 qr. ? In 1 yd.? In 3 yds. ? In
1 EU. Eng. ? In 1 Ell. Fr. ? In 1 Ell. Fl. ?
2. How many inches in 4 yds. ? In 7 yds. ? In 12 yds. ?
In 10 yds. ? In 20 yds. ? In 6 yds. 3 qrs. ? In 4 yds. 1 qr. ?
TABLE OF WINE MEASURE.
4 gUls [gi.] make* • 1 pint, marked pt.
2 pints, 1 quart, qt.
4 quarts, 1 gallon, gal.
^1^ gallons, 1 barrel, bl.
63 gallons, 1 hogshead, hhd.
2 hogsheads, 1 pipe, • • p.
2 pipes, 1 tun, T.
This table is used for measuring wine, spirits, cider, and
wftter.
) . How many gills in 1 quart ? In 1 gal. ? In 4
In 6 gals. ? In 10 gals.? In 13 gals.? In 15 gals. ?
2. How many pints in 1 gal. ? In 4 gals. ? In 20 gals. ?
How many qts. in 1 barrel ? In one hogshead ?
3. How many gallons in 5 barrels?. In 8 barrels? How
many gals, in half a barrel? In one fourth of a barrel ?
4. In 100 gals, how many barrels? In 300 gals, how
many bis. ?
5. At 14 cents a gallon, what is 1 qt. of vinegar worth ? 3
qts.? 6 qts.? 10 qts.? 15 qts.? 21 qts.? 30 qts.?
6. What is one barrel of vinegar worth at 15 cts. a gallon ?
How much, if the price is 20 cts. a gal. ?
A TABLE OF. ALE OR BEER MEASURE.
(Used in measuring malt liquors, and milk.)
2 pints [pt.] make 1 quart, qt.
4 quarts, 1 gallon, gal.
36 gallons, • • • -1 barrel, bl.
The beer gallon is a little more than one fifth larger than
the wine gallon. There are other measures of beer besides
those in the tables ; as the firkin of 9 gallons; the kilderkin.
REDUCTION. 39
18 ; the hogshead, 54 ; but these are not much used in this
country. A barrel of wine contains not quite three fourths
as much as a barrel of beer.
1. In 1 bl. how many pints ? How many pints in 3 bis. ?
How many gallons in 5 bis. ? In 12 bis. ? In 15 bis. ? In
21 bis.?
2. In 100 gallons, how many bis. ? How many bis. in 400
gals. ? First consider how many bis. there are in 360 gals. ?
MEASURE OF THE CIRCLE.
Every circle is supposed to have its circumference divided
into 360 equal parts, called degrees ; and each degree into 60
parts, called minutes ; and each minute into 60 parts, called
seconds. Whether the circle is great or small, it is still
divided into 360 degrees; a degree therefore is always the
same fixed part of the circumference of a circle, although its
actual length is longer or shorter, according as the circle is
great or small. The line passing from the centre to the cir-
cumference is called the radius of the circle. To give you
some idea of the length of a degree in circles of different
magnitudes, I will state that, on comparing a degree in any
circle with its radius, it has been found to be about one fifty-
eighth part of it. In other words, 58 degrees on the circum-
ference of a circle are about equal to the radius. If a degree
is 1 inch, the radius of that circle is 58 inches. If the radius
of a carriage wheel is 29 inches, a degree on the rim of the
same wheel wiU be half an inch.
If we take for illustration one of the largest sized water
wheels, 29 feet in diameter, a degree on its rim would meas-
ure only 3 inches.
You may enlarge the circle in your mind, tiU you suppose
it extending over a plain, with a radius of 58 rods ; a degree
on such a circle will measure 1 rod. If the radius is 58
miles, a degree will measure 1 mile. Now the circle round
the earth is so great in extent that a degree measures 69 J
miles. This may aid you in forming a conception of the vast
magnitude of the earth.
Each of these degrees is divided into 60 minutes, or geo-
graphical miles ; a geographical mile therefore is about one
sixth greater than a common mile. The table of circular
measure is as follows :
40 MENTAL ARITHMETIC.
60 seconds [''] make 1 minute, '.
GO minutes (or geog. miles) 1 degree, *'.
360 degrees a circle.
The term miles instead of minutes, can be used only in
reference to the great circle of the earth.
As the earth turns round on its axis once in 24 hours,
every place upon it passes in that time through the 360 de-
grees of its circle ; and on the equator, which is the great
circle, each of these degrees, we have seen, is 69 ^^ miles.
How swiftly then does a body lying on the equator move
in consequence of the daily .revolution of the earth ?
In 24 hours it passes through 360 degrees ; in one hour
then it will pass through one twenty-fourth part as many,
which is 15 degrees. If it pass through 15 degrees in one
hour, how many minutes will it be in passing through 1 de-
gree ? One fifteenth of 60 minutes is 4 minutes. If it pass
through a degree in 4 minutes, what part of a degree will it
pass through in 1 minute ? -One fourth of a degree, or 15
geographical miles. If it pa^s through 15 geographical miles
in 1 minute, in how many seconds will it pass through 1 geo-
graphical mile ? In 4 seconds ; and in 1 second it will pass
through one fourth of a geographical mile.
Now a geographical mile on the equator is, as we have seen,
longer than a common mile. We will here suppose it no
longer, but of the same length, and it appears that an object
on the equator moves, as the vast earth whirls round on its
axis, one quarter of a mile every second of time. Reflect
now, that, while the surface of the earth moves with such
amazing speed, so vast is its size, that it occupies an entire
day and night in turning once round.
If, as above stated, the earth turns from west to east at the
rate of fifteen degrees in an hour, we can, by knowing the
time of day in any place, ascertain what time it is at a place
any particular number of degrees east or west of it. It is
noon at any place when the meridian of that place passes
'under the sun.
1. When it is noon at Boston, what time is it at a place
15 degrees west of Boston ? At a place 15 degrees east of
Boston ?
2. When it is 12 o'clock at Boston, what time is it at a
place 1 degree west of Boston ? At a place 1 degree east of
LONGITUDE AND TIME. 41
Boston ? At a place 2 degrees west of Boston ? At a place
2 degrees east of Boston ? 3 degrees east ? 3 degrees west ?
4 degrees east? 4 degrees west? 5 degrees east? 5 degrees
west?
3. Indianapolis is 15 degrees west of Boston; when it is
noon at Boston, what time is it at Indianapolis ? When it is
sunset at Boston, where will the sun be at Indianapolis ?
4. Niagara Falls is 8 degrees west of Boston ; when it is
noon at Boston, what time is it at Niagara Falls ? When it
is 4 o'clock at Niagara Falls, what time is it at Boston ?
5. Washington city is 6 degrees west of Boston ; if you set
your watch with the sun at Boston, and then carry it to
Washington, your watch keeping accurate time all the while,
when you arrive at Washington, will it be too fast or too
slow ? and how much ?
6. Two travelers met at a public house ; when one of them
said to the other, " Friend, are you traveling east or west ? "
" I am direct from home," said the other, " where my watch
agrees exactly with the sun, but here I find it is 10 minutes
too fast : now if you can tell which way I am traveling you are
welcome to know."
Had he traveled east, or west ? and how far ?
7. Boston is 71 degrees west of London; when it is noon
at Boston, what time is it in London ?
8. The English convicts are transported to Botany Bay,
150 degrees east of London ; when it is noon at London, what
time is it in Botany Bay ?
9. Enghsh traders are settled on Columbia river, 120 de-
grees west from London ; what time is it there when it is
noon in London ?
10. If a man is on the equator, which way must he travel,
and how many geographical miles, to have the day 4 minutes
longer than 24 hours? How far to have the day 2 minutes
longer ? How far to have it one minute longer ? How far
must he travel to have the day one minute and a half longer ?
Which way must he travel and how far, to have the day one
minute shorter ? 2 minutes shorter ? 5 minutes shorter ?
11. Suppose two birds start from the same place on the
equator, and fly, one east and the other west, at the rate of
60 geographical miles an hour, and at the end of the hour it
is just sunset to the bird flying east; how high is the sun
then at the place where the other bird is ?
4*
42 MENTAL ARITHMETIC.
How high was the sun at the place of their starting, when
they set out ?
12. A shipmaster sails from New York for Europe, and
for three days it is so cloudy that he cannot see the sun ; on
the fourth day he takes an observation of the sun at noon ;
and by his chronometer, which gives the New York time, it
is half past eleven ; how many degrees east from New York
has he sailed ?
In what longitude is he then, if New York is 74° 1' west
from Greenwich?
SECTION VII.
PRIME NUMBERS.
Numbers may be divided into two great classes. The first
class comprises such numbers as cannot be formed by the mul-
tiplication of any two or more numbers together, as 1, 2, 3, 5,
11, 17. These are called Prime numbers. The other class
may be formed by multiplying two or more numbers together,
as 4, which is formed by multiplying 2 by 2 ; 6, which is
equal to 2X3 ; 10, which is equal to 2X5, &c. These are
called Composite numbers. These may always be formed by
multiplying two or more prime numbers together. Thus all
numbers are either Prime, or are formed by the multiplica-
tion of Prime numbers together.
In separating numbers into their factors care should be
taken that the factors be all prime. Thus in resolving 30
into its factors we may say it is formed by multiplying 5 by
6, but this is not sufficient, for 6 is not prime ; it is formed of
the factors 2 and 3. The prime factors of 30 therefore are
2, 3 and 5. We may say that 30 is formed of the factors 3
and 10, but here again the analysis is not complete, for 10
is not prime ; it is composed of the factors 2 and 5. Thus
we are brought to the same three factors as before, namely,
2, 3 and 5.
The following table of numbers from 1 to 100 will show
what of them are prime, and what are the prime factors of
those which are composite. This table should be carefully
PRIME AND COMPOSITE NUMBERS.
43
studied and made perfectly familiar. The analysis of compo-
site numbers into their prime factors lies at the foundation of
so^e of the most important operations in numbers, and affords
ai» insight into some of the most intricate rules of Arithmetic.
1 prime.
2 prime.
3 prime.
4=2X2.
5 prime.
6=3X2.
7 prime.
8=2X2X2.
9=3X3.
10=2X5.
11 prime.
12=2X2X3.
13 prime.
14=2X7.
15=3X5.
16=2X2X2X2.
17 prime.
18=2X3X3.
19 prime.
20=2X2X5.
21=3X7.
22=2X11.
23 prime.
24=2X2X2X3.
25=5X5.
26=2X13.
27=3X3X3.
28=2X2X7.
29 prime.
30=2X3X5.
31 prime.
32=2X2X2X2X2.
33=3X11.
34=2X17.
35=5X7.
36=2X2X3X3.
37 prime.
38=2X19.
39=3X13.
40=2X2X2X5.
41 prime.
42=2X3X7.
43 prime.
44=2X2X11.
45=5X3X3.
46=2X23.
47 prime.
48=2X2X2X2X3.
49=7X7.
50=2X5X5.
5l=r3Xl7.
52=2X2X13.
53 prime.
54=2X3X3X3.
55=5X11.
56=2X2X2X7.
57=3X19.
58=2X29.
59 prime.
60=2X2X3X5.
61 prime.
62=2X31.
63=3X3X7.
64=2X2X2X2X2X2.
65=5X13.
66=2X3X11.
67 prime.
68=2X2X17.
69=3X23.
70=2X5X7.
71 ^rime.
72=2X2X2X3X3.
73 prime.
74=2X37.
44
- MENTAL ARITHMETIC.
75=3x5X5.
76=.2X2X19.
77=7X11.
78=2X3X13.
79 prime.
80=2X2X2X2X5.
81z:r3X3X3X3.
82=2X41.
83 prime.
84=2X2X3X7.
85=5X17.
86=2X43.
€7=3X29.
88=2X2X2X11.
89 prime.
90=2X3X3X5.
91=7X13.
92:1=2X2X23.
93=3X31.
94=2X47.
95=5X19.
96=2X2X2X2X2X3.
97 prime.
98=2X7X7.
99=3X3X11.
100=2X2X5X5.
On examining this table several things may be observed.
1. All the even numbers are composite ; for they are all
divisible by 2. So it appears in the table, with the exception
of the number 2, which is regarded as prime because it is
divisible only by itself.
2. Several of the numbers given above are powers of their
prime factors. Thus 4 is the 2d power of 2, 8 the 3d power
of 2, 16 the 4th power of 2, 32 the 5th, 64 the 6th. 9, 27
and 81, are the 2d, 3d and 4th powers of 3. 25 is the 2d
power of 5, 49 the 2d power of 7.
3. If you double the number of times a factor is taken,
you obtain the square of the number they at first made.
Thus 4 is obtained by taking 2 twice as a factor. If you take
it twice as many times, that is, 4 times as a factor, you obtain
16, which is the square of 4.
9 is obtained by taking 3 twice as a factor. If you double
the number of times it is taken, thus, 3X3X3X3, you ob-
tain the square of 9.
8 is obtained by taking 2 three times as a factor. If you
take it 6 times you obtain 64, the square of 8.
So universally, if you double the number of times a factor
is taken to produce a certain number, you obtain, not twice
that number, but the square of it.
I will make a single remark here about the prime numbers,
and then call your attention to the composite numbers.
Since the 'prime numbers are not formed by multiplying
any two or more numbers together, they cannot be divided by
any number. You will observe, however, that any number
PROPERTIES OF PRIME AND COMPOSITE NUMBERS. 45
whatever may be divided by itself, and may also be divided
by 1 ; but 1 is a unit and not a number; and by dividing a
number by itself, or by 1, you obtain no new number. Di-
viding the number by itself you obtain 1, and dividing by 1,
you obtain the number itself. Such an operation, therefore,
brings out nothing new. It is only another way of express-
ing what was just as plain before. In the same way we may
sometimes regard a number as produced by multiplying itself
into 1 ; thus, 7=7 X 1 ; but this is not multiplication, but
only an expression in the form of multiplication. It pro-
duces no new number, and is employed only for convenience
in order to make the reasoning more plain.
Composite numbers can be divided by their factors. Thus
you can divide 10 either by 2 or by 5, and by no other num-
ber. If you divide by 2, you obtain 5 for the answer, or
quotient; if you divide by 5 you obtain 2 for the answer.
Dividing by a number then, is the same as erasing that num-
ber as a factor, and will always give for the answer the other
factor, or factors. Thus dividing 10 by 2 you may represent
thus, ^X5, leaving the factor 5 for the answer: dividing 10
by 5, thus, 2X^j leaving 2 for the answer. Divide 21 by 3,
thus, ^X7. Divide 1 2 by 3 thus, 4X3, or^X^X3.
It is plain, therefore, that if you express any number by its
factors you can at once see what numbers you can divide it
by. You can divide it by each of its prime factors, or by any
combination of them, and by no other number. Thus 6=
2X3, you can divide by 2 or by 3 : 8=2X2X2, you can
divide by 2, and that quotient by 2, and that by 2 again ;
30=2X3X5, you can divide by 2, or 3, or 5, or by any two
of them combined.
Any composite numbers may be divided by any of its prime
factors, or by any combination of them.
By what numbers can you divide 15? 18? 20? 21? 26?
27? 36? 42? 46? 48? 49? 50?
Sometimes we have two numbers, and we wish to know if
there is any number that will divide them both. This we
can ascertain if we express each number by means of its
prime factors, and then see if the same factor is found in both :
if vSO, they are both divisible by that number. Thus, if we
wish to know whether any number will divide both 9 and 15,
we express them thus, 3X3 and 3X5. Now 3 appears as a
factor in both ; they can both therefore be divided by 3.
46 MENTAL ARITHMETIC.
This number 3 is called the common divisor, because it is a
divisor common to several numbers. If we wish to know
whether . any number will divide both 15 and 8, we express
15 bj its factors, 5x3 ; and 8 by its factors, 2X2X2. Now
there is no factor common to both ; no number therefore will
divide them both ; in other words they have no common
divisor. Numbers which have no common divisor are said
to be prime to each other. They may be composite consid-'
ered by themselves, as is the case with 8 and 15, but if they
have no common divisor they are said to be prime to each
other. Numbers which have a common divisor are said to be
composite to each other. If there are more than two numbers
they must be treated in the same way. Each must be writ-
ten in the form of its prime factors, and then, if any one
number appears as a factor in them all, they are divisible by
that.
Is there any common divisor to 9, 14 and 27? Written
in the form of their factors they stand thus, 3X3. 2X7.
3X3X3. They have therefore no common divisor ; for,
though 3 or 9 will divide both the first and the third number,
it will not divide the second ; and neither 2 nor 7, which are
the factors of the second number, appear in the first or third.
9, 14 and 27 are, therefore, prime to each other.
What is the common divisor of 15 and 27 ? of 14 and 22 ?
of 21 and 49 ? of 35 and 28 ? of 6 and 21 ?
Let us now take the following question: What is the
common divisor of 18 and 30 ? By inspecting their factors
2X3X3, and 2X3X5, we find that 2X3 or 6, is common to
both ; 6 is therefore the greatest common divisor.
What is the greatest common divisor of 18 and 27? of 4,
8 and 36? of 15 and 45? of 27 and 45? of 40, 64, and 16?
of 44 and 24 ? of 75 and 15 ? of 80 and 100 ? of 60 and 24 ?
of 35, 21 and 49 ? of 15 and 50?
We have seen that a composite number can be divided
only by its factors ; and that prime numbers cannot be divided
at all. It is frequently necessary, however, to attempt the
division of prime numbers ; and to divide composite numbers
by some number different from their factors. For example,
we may wish to divide 9 by 4, or to obtain one fourth of 9.
Now 4 is not a factor of 9, and the actual division of 9 by 4
is, strictly speaking, impossible. We proceed in this way.
We divide 8 by 4, and obtain 2 for the answer, and we have
INCOMPLETE DIVISION. TRACTIONS. 47
a remainder of 1 which we have not divided. To show that
w^e design this to be divided bj 4 we write the 4 "under it
with a Une between, thus, ^. ~ In this way we indicate plainly
enough what the answer is, although we have no one figure
that will express it.
This operation introduces us to a new class of quantities
called Fractions, fractions are expressions for quantities less
than a unit. The word Fractions here means the same as
broken numbers. In this class of expressions each unit is
regarded as broken up, or divided into a number of parts.
The figure below the line shows into how many parts the
unit is divided ; the figure above the line shows how many
of those parts are taken ; (or, what is just equivalent, the
number above the line is regarded as divided by the number
below it.) The fraction y indicates that each of the 3 units is
regarded as divided into 7 equal parts ; and that one of these
parts is taken from each of them. The number below the
line is called the Denominator ; that above the line, the Nu-
merator. If the Numerator is just equal to the Denomina-
tor, as I, J, 7, the value of the fraction is just equal to 1. If
the Numerator is smaller than the Denominator, the value of -
the fraction is less than 1, and is called a proper fraction ; if
the Numerator is greater than the Denominator, the value of
the fraction is greater than 1, and is called an improper
fraction. This, however, may always be changed to a whole
number, or a whole number and a proper fraction. Hence
the propriety of the definition, that fractions are expressions
for quantities less than unity.
Questions*
What is meant by a Common Divisor ?
What is meant by the greatest Common Divisor ?
When are numbers prime to each other ?
When are numbers composite to each other ?
What is the process of dividing 13 by 4 ?
In dividing 16 by 5 ? In dividing 25 by 6 ?
What are Fractions'^
Explain what is signified by each of the numbers in the
fraction |. In f In A- In y\. In |. In ||.
48 MENTAL ARITHMETIC.
A man bought a barrel of flour, and gave away two fifths
of it ; what fraction will express what he gave away ? What
fraction will express what he kept ?
A man bought a load of hay, and sold two elevenths of it ;
what fraction will express what he sold ? What fraction wiU
express what he kept ?
What is a proper fraction ? Give an example.
What is an improper fraction ? Give an example.
When is the value of a fraction just equal to 1 ?
SECTION VIII.
MULTIPLICATION AND DIVISION OF FRACTIONS.
We have seen that a fraction is not a simple expression, but
composed of two numbers ; and its value cannot be determined
by one of these numbers alone, but by both taken in connec-
tion. By looking at the numerator, you cannot tell the value
of the fraction unless you know what the denominator is.
By looking at the denominator you cannot tell the value of
the fraction, unless you know what the numerator is.
Let us now observe the effect of altering one of the terms
of the fraction without altering the other. We will take the
fraction f . If we increase the numerator by 1, making it f ,
we increase the value of the fraction, for we take one fifth
more than we had before. So, if we multiply the numerator
by 2, making it J, we double the value of the fraction ; and so
of any other numbers, if we multiply the numerator, we mul-
tiply the value of the fraction. And, by the same reasoning,
if we divide the numerator by 2, we divide the fraction by 2,
for J is plainly one half as great as J. So of all other num-
bers, by dividing the numerator we divide the fraction.
Let us now observe the effect of altering the denominator.
If we increase the denominator of the fraction f by 1, making
it f , we have not increased the fraction, but diminished it, for
one sixth is less than one fifth, and any number of sixths are
less than the same number of fifths. We will multiply the
denominator ,of the fraction § by 2, making f*^. Wliat effect
has been produced on the value of the fraction ? One tenth
FRACTIONS. 49
is half as great as one fifth ; and two tenths are half as great as
two fifths. The fraction is therefore half as great as it was
before ; that is, it has been divided bj 2. Multiplying the
denominator, therefore, divides the value of the fraction.
We will now divide the denominator. Take the fraction | ;
dividing the denominator by 2, we have |. Now as this is
twice as great as |, we have multiplied the fraction, by di-
viding the denominator.
There are, then, two ways of multiplying a fraction.
We may multiply the numerator ; or, if the multiplier is a
factor of the denominator, we may divide the denominator.
Thus, to multiply | by 2, we may multiply the numerator,
which gives f , or divide the denominator, which gives |, equal
to|.
To divide a fraction, we may either divide the numerator, if
the divisor is a factor of it ; or we may multiply the denomi-
nator. Thus, to divide f by 3, we may divide the numerator,
giving f , or we may multiply the denominator, which gives
^®y, which is equal to f .
We will now multiply both terms of the fraction by the
same number. Multiplying both terms of the fraction f by 3,
we have |. Here the denominator, expressing the number
of parts into which the unit is divided, is three times as great
as it was before, consequently each of the parts is only one
third as great ; but the numerator has also been multiplied by
three, so that three times as many parts are taken, and this
makes the value of the fraction just equal to what it was
before. So we may multiply by any number whatever, both
terms of the fraction §, and the value will still be the same as
before ; for example, |, f , y\, yf , yf, each of which is equal
to §. We may then at any time multiply both terms of a
fraction by the same number, without altering the value of
the fraction. By the same reasoning we may divide both
terms of a fraction by the same number without altering its
value. Taking the examples above, we may divide the terms
of f by 2, and we obtain f ; dividing the terms of | by 3 gives
us §, and so of the others ; f is the same fraction as |, f , y^,
&c., but it is expressed in lower terms, and therefore is more
convenient. It is easier to write J than it is to write Jf ,
though both have the same value.
To reduce a fraction to its lowest terms, we divide both the
numerator and denominator by their greatest common divisor.
5
50 MENTAL ARITHMETIC.
To find the greatest common divisor, separate each term into
its prime factors, and erase those which are common to both.
The remaining factors will express the value of the fraction
in its lowest terms.
Treating the above fractions in this way thej appear thus,
4_^X2 6_2X^ 8_^X^X2 10_2X^ 12_^X2Xg(
6~^X3' 9"~3X3' 12~^X^X3' l5~3X?' 18""^X3X$'
leaving in each case |-.
In how many ways can you obtain the answer to the follow-
ing questions ? JX2? 1X3? 1X4? t\X2?
In how many ways can you obtain the answer to the follow-
ing? fX3? 1x2? 1x4? t\X6? AX5? t\X3?
In how many ways can you obtain the answer to the follow-
ing? f^3? f-r-4? /^-~.3? 1*^2? ifH-4?
In how many ways can you obtain an answer to the foUow-
Reduce to their lowest terms each of the following' fractions,
e 14 3 20 40 16 18 24 26 6 9 16 8 30 70
H> ITTJ S^J ^^? ¥0J ST^) TSi 57> SS^ TF> T3") ^^i '2'^y ?F5 ^Z-
/ TO FIND THE DIVISORS OF NUMBERS.
Reduce the fraction -^J J to its lowest terms ?
You will not see immediately that these two numbers have
any common divisor. To assist you to reduce fractions of this
kind, something will here be said about the way of finding the
divisors of numbers. Let us first inquire what numbers can
be divided by 2.
We have seen that all even numbers, and only those, can
be divided by 2.
What numbers can be divided by 4 ?
If you examine you will find that all even tens are divisible
by 4, as 20, 40, 60, &c. K, therefore, the tens are even, and
the units are divisible by 4, then the whole is divisible by 4.
But the only unit numbers divisible by 4 are 4 and 8 ; there-
fore if the tens are even, and the unit number is 4 or 8, the
whole is divisible by 4 ; as 84, 88 ; 124, 128 ; 148, 364, &c.
Again, as 10 when divided by 4 leaves a remainder of 2,
any odd number of tens will do the same, as 30, 50, 70, 90 ;
for every odd number of tens is an even number of tens-(-10.
If, then, the number of tens is odd, the units must be two less
FRACTIONS. 51
than 4 or 8, in order to be divisible bj 4. That is, if the
tens are odd, and the units 2 or 6, the whole is divisible by 4 ;
as 72, 96, 52, &c.
Are the following even numbers divisible by 4 or only by 2 ;
and why ? 126, 82, 94, 92, 138, 156, 346, 548, 76, 58, 392.
What numbers can be divided by 8 ?
As 100 divided by 8 leaves a remainder of 4 (8X12=96,)
it follows that 200 will be exactly divisible by 8, for the two
remainders of 4 will make 8. If 200 is divisible by 8, it
follows that all even hundreds are divisible by 8 ; as 400,
600, 1400, &c.
If, therefore, the hundreds are even and the tens and units
are divisible by 8, the whole number will be 'divisible by 8 ;
ii< 248, 672, 1456, &c.
Again, if the hundreds are odd and the tens and units are
4 less than- some multiple of 8, the whole number will be di-
visible by 8 ; for the odd hundred, divided by 8, leaves a re-
maiiuler of 4; and this, added to the tens and units, will make
an exact multiple of 8.
Are the following numbers divisible by 8, or by 4 ; and
why? 444, 944, 136, 1328, 712, 532, 816, 516, 384, 128,
1236.
What numbers are divisible by 5 ? All tens are divisible
by 5 ; consequently if the unit figure is 5 or 0, the whole
number is divisible by 5.
What numbers are divisible by 3 ? By examining the
multiples of 3 we shall find this singular fact, that the sum
of -the figures which express any multiple of 3 is itself a mul-
tiple of 3. Take the multiples of three from 12 to 24 ; 12,
15, 18, 21, 24 ; by adding the figures which express any one
of these multiples we find that the sum is a multiple of 3.
The figures of 12 added are 1+2=3, of 15 are 1+5=6, of
18 are 1+8=9, of 21 are 2+1=3, of 24 are 2+4=6. The
same is true of all multiples of 3.
It will also be found that if you add the figures of any
number and the sum is a multiple of three, the whole number
is a multiple of three. To know, then, if a number is a
multiple of 3, add together the figures that express the num-
ber, and if the sum is a multiple of 3, the whole number is a
multiple of 3.
Are the following numbers divisible by 3 ? 471, 59, 115, 642,
624, 138, 234, 742, 894.
52 MENTAL ARITHMETIC.
It follows from what has been said, that if any number is
divisible by 3, any other number expressed by the same fig-
ures differently arranged will also be divisible by 3 ; for the
sum made by adding the figures will be the same in whatever
order they are taken.
Thus, if 936 is divisible by 3 ; 369, 396, 963, 639, 693 are
each divisible by 3.
We will next inquire what numbers are divisible by 6.
As 6=2X3, any number that is divisible by 2 and by 3 is
divisible by 6. You have learned what numbers are divisible
by 3, and what by 2. If a number combines both these con-
ditions, it is divisible by 6 ; that is, all numbers are divisible
by 6, the sum of whose figures is a multiple of 3, and whose
last figure is an even number.
What combinations of the figures 1, 2, 3, will give numbers
divisible by 6 ; and what by 3 only ?
Next let us inquire what numbers are divisible by 9.
If the figures which express any multiple of 9, as 18, 27,
36, 45, 54, be added together, the sum will be a multiple of 9.
Also, if the figures of any number be added together, and
the sum is a multiple of 9, the whole number is divisible by 9.
Are the following numbers divisible by 9 ? and why ? 936,
972, 396, 423,387, 527, 441, 416, 315, 756.
Any number divisible by 9 and by 2 is divisible by 9X2,
or 18 ; M'hich of the above numbers are divisible by 18 ?
Any number divisible by 9 and by 4 is divisible by 9X4,
or 36 ; which of the above numbers are divisible by 36 ?
Any number divisible by 9 and by 8 is divisible by 9X8,
or 72 ; is either of the above numbers divisible by 72 ?
Any number divisible by 9 and by 5 is divisible by 9X5,
or 45 ; which of the above numbers is divisi])ie by 45 ?
Whatare divisors of 124? of 176 ? of 252 ? of 384? of
153 ? of 186 ? of 207 ? of 702 ? of 4041 ?
We will now return to the fraction that was first given.
Reduce yJJ to its lowest terms.
Reduce to lowest terms, ||^ ; f || ; |§f .
Reduce to lowest terms, f||; %\%', f||.
FRACTIONS. 53
SECTION IX.
MULTIPLICATJON OF FRACTIONS BY FE ACTIONS.
We have seen how we may multiply or divide a fraction
by a whole number. We will now inquire how we can multi-
ply or divide one fraction by another. Let us multiply f by f .
First multiply f by 2, which gives y for the answer. But
here we have multiplied by 2, instead of the real multiplier, §.
Now 2 is 5 times greater than § ; the product ^ then is 5 times
greater than it should be. It must therefore be divided by 5.
We divide f by 5 by multiplying the denominator by 5, giving
^3" for the answer.
In the same way multiply f by yy. yXJ- f Xj.
DIVISION OF FRACTIONS BY FRACTIONS.
Let us now divide f by f. First divide f by 3. This we
do by multiplying the denominator by 3, giving for the an-
swer j*3-. Here, however, we have divided by 3, instead of the
true divisor, f . We have used a divisor seven times too large.
The quotient, therefore, will be seven times too small ; y%- must
therefore be multiplied by 7, making the answer ff . In the
same way perform the following : f-i-f • f -i-|. f -t-|. j ', 7.
The above analysis shows the grounds of the rules usually
given in Arithmetics for the multiplication and division of
fractions.
For Multiplication, multiply the numerators together for a
new numerator, and the denominators for a new denominator.
For Division, invert the divisor and proceed as in multipli-
cation.
Sometimes we wish to find the value of a compound frac-
tion, as § of f ; in such cases we may understand the sign of
multiplication, X ? to stand in the place of the word of, and
treat it as a case of multiplication. For in the above example
it is plain that one third of | is j\, and two thirds is twice as
much, that is, y^.
Wliat is I of I of f ? Multiplying as we have done above,
we have for the answer y^. But {his operation may be
shortened. We see that 4 appears as a factor both in the
5*
54 MENTAL ARITHMETIC.
numerator and the denominator. We may then cancel them
both, which will have the same effect as dividing both terms
of the answer by 4. Again, 3 appears in both the numerator
and the denominator, for in the denominator it is a factor of 9.
We may therefore cancel 3 in both terms.
The question will then appear thus, qX-vX^, substituting
3 in place of the 9. Multiplying together the terms that now
remain, we have f for tl;e answer. This is the same fraction
as yW. If you separate the terms of y/^ into their prime
factors, and cancel what are. common to both, the remaining
factors will give the fraction |.
Multiply the fractions IXIXtIXt) writing the terms
that are composite in the form of their prime factors, and
canceling factors that are common in both, it will stand
^X^X2><?^X3X?'<r which gives ,V
Multiply IXIXf. liXllXf.
Multiply |?XtV liX-,V J|X|X|.
TO MULTIPLY OE DIVIDE WHOLE NUMBERS BY FEACTIONS.
The above examples will show how to multiply or divide a
whole number by a fraction.
Multiply 7 by |. Multiplying 7 by 4 gives 28, which is 5
times too great, because 4 is five times greater than J. We
must therefore divide the answer by 5, thus \®. As this is
more than 1, we can reduce it to a whole number and a frac-
tion. As |- is equal to 1, ^/ will be equal to 5 ; ^-^ therefore
is equal to 5|.
In this way multiply 6 by J. 9 by |. 8 by y.
This operation is in fact the same as multiplying a fraction
by a whole number, which has been treated of already.
Let us next divide 7 by |. Dividing 7 by 3 we have J ;
here, however, we have divided by a number 4 times too great,
for 3 is four times greater than |, If the divisor is 4 times
too great, the quotient will be 4 times too small ; J, therefore,
must be multiplied by 4, giving ^-g for the answer.
Divide 8 by 4. 9-^|. 11-^.|. 10~fV
To reduce an improper fraction, as ^:j , to a whole number
and a proper fraction, we have only to consider how many
FRACTIONS. 55
whole ones the fraction is equal to, and how much remains.
Thus J^ is equal to 3 ; J^a therefore, is equal to 3^.
Reduce f , Y-> V;, ¥, ^, ^, ih H, ^'
d[n like manner, if we have a whole number and a fraction,
we may always reduce it to an improper fraction.
ADDITION AND SUBTRACTION OF FRACTIONS.
Suppose we wish to add together 3^^ so that its value shall
be expressed in a single expression ; we must change 3 to
halves, which will be | ; adding ^ to this we have J for the
answer.
In order to unite separate numbers into one expression,
they must be of the same kind. We cannot unite 2 bushels
and 3 pecks in one expression. It is still 2 bushels and 3
pecks, and we can make nothing else of it ; but if we change
the bushels to pecks, making 8 pecks, we can then add the
3 pecks, and bring it all into one expression, 11 pecks. So
to unite 5f we must change the 5 to thirds, making ^/, and
add the |, making y . This is called reducing a mixed, num-
ber to an improper fraction.
Reduce to an improper fraction 7J, 8|, 4y, 5J, 6J, 9J, 3|,
5?, 15^, 16f, 13|, 20|, 2U.
Supposing we wish to add ^ to i, we must change the ^ to
fourths, making | ; adding these, we have | for the answer.
Add J to yV- h=T%^ A-h:V=Aj ans.
Add jto^V 1+A. ^+#iT..f+#^.. l+il
Let us now add f and |. This question you perceive has
a difficulty which the former ones had not ; for § is no number
of fifths, and therefore we cannot bring the fraction into fifths
by any multiplication. We want a number for the denomina-
tor which can be divided both by 3 and by 5. Now if you
examine, you will find no such number until you come to 15.
This, is of course, divisible by 3 and by 5, for these are its
factors. We will then take 15 for the denominator. This
we call the common denominator. Taking now the fractions
§ and I, and changing the denominator 3 to 15, we see that
we have made it 5 times as large as it was before ; that is, we
have multiplied it by 5. We must therefore multiply the
numerator by 5, to preserve the value of the fraction. The
fraction § then becomes yj- without altering its value. Pass-
56 MENTAL ARITHMETIC.
ing now to the second fraction, |, we see that, in changing the
denominator to 15, we have multiplied it by 3 ; we must there-
fore multiply its numerator by 3. This will make the fraction
-}-|. The two fractions will stand, then, if +t|-, which added
together are y|==lTV-
TO FIND A COMMON DENOMINATOR.
We can always obtain a common denominator, by multi-
plying all the denominators together ; then, for the numera-
tors, consider, in the case of each fraction, what its denomina-
tor has been multiplied by, in order to change it to the com-
mon denominator, and multiply the numerator by the same
number. Thus each fraction will have had its numerator and
its denominator multiplied by the same number, and so its
value will not be changed.
What is the value of J+f? of f+f? of f+J? of i+f ?
ofJ+f?off+§?off+|?of|+|?
Supposing we wish to add the fractions f and §. We can
proceed as above, and with the common denominator, 24, the
fractions will be ^|-t-||. But we need not employ so large a
denominator as 24. We seek the smallest denominator that
shall contain both 4 and 6 as a factor. If now we separate
4 and 6 into their prime factors, we shall find the factor 2
belonging both to 4 and to 6 ; thus, 2X2, 2X3. Now one
of these may be cancelled, and we shall still have 2X2 for the
number 4, and 2X3 for the number 6. Multiplying the fac-
tors which remain, 2X2X3, we have 12 for the smallest
common denominator.
From this we see, that, when both the denominators con-
tain the same factor, we may reject it from one of them, and
multiply together the factors that remain.
Add I to y^-- Here 2 X 2 is common to both denominators,
rejecting it in one, and multiplying, we obtain 24 for the least
common denominator.
Add i\ to 3jy. Here 3X3 is common to both denomina-
tors, rejecting it in one, and multiplying what remains, we have
54 for the least common denominator.
Add^Jto^V Add I to ^V Add^to^jV
When more fraction? than two are to be added it is often
most convenient to add two together first, and then add a third
to the sum of these, and so on.
FRACTIONS. • 67
Add l+i+f • -f'irst add f and ^, which equal f . Next,
ffl ; ^=\h and f=f^; ig+^:H=i|=lT^^, Ans.
Add i+f+i^d- First add ^ and ^jj ; then to the sum of
these add f .
AddJ+,V+f- Addf+I+T^. Addi+f+f.
Addf;j+^-. AddA+f. AddT\+^+^.
From ^1^ subtract /^. From | sub. -^^. From -j^^ sub. /^j.
From If sub. |. From f § sub. y^^. From f ^ sub. ^^.
Miscellaneous Examples.
1. A man spends y of a dollar in a day; what part of a
dollar will he spend in 5 days ? How much -will he spend in
9 days? How much in 11 days?
2. A man earns | of a dollar in a day ; how much will he
earn in half a day ? How much in |: of a day ? How much
in "I of a day ?
Here consider whether you can divide the numerator.
3. A man earns J of a dollar in a day ; how much can he
earn in half a day ? How much in ^ of a day ? How much
in J of a day ?
Consider whether you can divide the numerator ; and if
you cannot, what you must do.
4. A vessel filled with water leaks so that f of its contents
will leak out in a week ; at this rate, what part will leak out
in a day ?
What is I of I?
5. If a team ploughs y of an acre in 6 hours, how much will
it plough in one hour? How much in 3 hours?
What is J of I What is ^ of ^ ?
6. If a horse runs | of a mile in one minute, how far will
he run in | of a minute ?
How far will he run in ^ of a minute ?
What is I of J ? What is f of J ?
7. A man has J of a dollar, which he wishes to distribute
equally among several persons, giving -^^ of a dollar to each ;
how many can receive this sum, and what will be the re-
mainder?
How many times is xV contained in7? -^in?? -T^inf?
How many times is -^ contained in 4 ? -^^ in 4? ^f^ in ^?
How many times is ^ contained in 6? ^in6? ^inf?
58 MENTAL ARITHMETIC.
8. A man gave y of a bushel of oats to some horses, giv-
ing to each |^ of a bushel; to how many did he give it? and
what was the remainder ? ' ^
How many times will y^^ go in 5 ? In ^ ? How many
limes will y\ go in f ?
9. A man has J of a dollar ; he gives ^ of a. dollar to one
person, and f of a dollar to a second, what part of a dollar
has he left ?
How many cents had he at first ? How many cents did he
give away ? How many cents had he Jeft ?
10. If 13 pounds of figs cost | of a dollar, what is that a
pound ?
11. If 5|- lbs. of figs cost y\ of a dollar, what is that a
pound ? Find first what one half pound will cost.
12. If f of a cwt. of iron cost 4J dollars, what will a hun-
dred weight cost ?
13. If 34^ lbs. of tea cost 11| dollars, what will 1 pound
C06t?
Here you find ^^ pounds cost \^ of a dollar : therefore 69
pounds must cost \^ of a dollar.
14. If § of a barrel of flour cost 3| dollars, what is that a
barrel ?
15. If wood is 5 1- dollars a cord, what will y^^ of a cord
cost ? What will 4J cords cost ?
16. If 33 J gals, of molasses cost 11| dollars, what is that
a gallon?
17. If 31 J gals, of vinegar cost 4| dollars, what is that a
gallon ?
18. If a bottle of wine containing IJ pints cost f of a dol-
lar, what would a bari'el of wine come to at that rate ?
19. In a pile of wood there a 13|^ cords ; how many loads
of f of a cord each are there in the pile ?
20. How many times will 2^ go in 7| ? In 9J ? In 11 ?
21. How many loaves, of 8^ oz. of flour each, can be made
from 7 pounds of flour ?
22. If a family consume 3 J pounds of flour a day, how long
will a barrel of flour, that is 196 pounds, last them?
How long will it last if they consume 2| lbs. a day ?
23. If a barrel of flour last a family 40 days, how long will
14 pounds last them ?
24. A garrison of 100 men is allowed 12 oz. of flour a
day to each man ; how long will 10 barrels last them ?
FRACTIONS. 59
25. Two men hire a horse, a week, for 5 dollars ; one trav-
els with him 30 miles, the ether 45 miles ; what oflght each to
pay?
26. Two men hire a pasture in common for $4,80 ; one
pastures his horse in it 7^ weeks ; the other pastured his horse
9 weeks ; what ought each to pay ?
27. A boy "bought 3 doz. of oranges for 372- cents, and sold
them for 1^ cents apiece ; what did he gain ?
28. A man bought 7 yds. of cloth for 16 dollars, and sold
it for 3 dollars a yard ; what did he gain on each yard ?
29. A man worth 1690 dollars, left f of his property to his
wife ; how much did she "receive ? The remainder he divided
equally among 3 sons ; what did each one receive ?
30. A man bequeathed his estate of 14,000 dollars, one
third to his wife, and the remainder to be divided equally
among four sons ; what did the wife and what did each son
receive.
31. In an orchard one third of the trees bear apples, two
fifths as many bear plums, and the rest bear cherries ; what
portion of the trees bear plums ? What portion Jbear cher-
ries ? The number of cherry trees is 40 ; what is the whole
number of trees in the orchard ?
32. What is f of 549 ? What is | of 374 ?
33. What is i of 175^ ? What is | of 198 ?
34. What is ^ of I of 1640 ? What is § of 972 ?
35. If 2 barrels of flour cost 111 dollars, what will 17 bar-
rels cost ? What vAll 22^ barrels cost ?
36. If 2^ cords of wood cost 15 dollars, what will 68f cords
cost ? What will 200 cords ?
37. If a horse eat 2^ tons of hay in 30 weeks, what part
of a ton will he eat in 1 week ?
38. What is the cost of 23 J yds. of cloth at J of a dollar a
yard ?
39. What is the cost of 31 J gallons of molasses at y^ of a
dollar a gallon ?
40. A grocer drew from a cask containing 31J^ gallons, J
of its contents. Now how much did he draw out? How
much remained ?
60 MENTAL ARITHMETIC.
SECTION X.
THE LEAST COMMON MULTIPLE.
The method stated in the foregoing section for finding the
smallest common denominator, serves to introduce a topic
which requires some more extended and careful study.
It often becomes desirable to ascertain, with respect to sev-
eral numbers, what number the^e is which contains them all
in itself as factors. A number which contains another num-
ber as a factor of itself is a multiple of that number. Thus
6 is a multiple of 2, and also of 3. A number which contains
several numbers as factors of itself, is a common multiple of
those numbers. Thus 12 is a common multiple of 2 and 3.
The smallest number which contains several numbers as
factors of itself, is the least common multiple of those numbers.
Thus, though 12 is a common multiple of 2 and 3, it is not
the least common multiple ; for 6 contains them both as its
factors ; 6 is therefore a smaller common multiple of 2 and 3
than 12 is ; and as no number smaller than 6 does contain 2
and 3 as its factors, 6 is the smallest common multiple of 2
and 3.
Suppose now we wish to find the smallest common multiple
of 3 and 5. The number, it is clear, must he a certain num-
ber of 3s, and also a certain number of 5s. Now bj multiply-
ing 3 and 5 together we evidently obtain such a number ; for
it will be 3 times 5, and it will be 5 times 3. Multiplying
the two numbers together then, will always give their com-
mon multiple. The next question is, will this product of the
two numbers be their least common multiple ? This will de-
pend on the character of the numbers. If the numbers are
prime to each other their product will be their least common
multiple. For example, in the numbers 3 and 5, if we take
any number of 5s less than 3, as 2X5, the factor 3 has disap-
peared, and the number is no longer a multiple of 3. If we
take any number of 3's less than 5, as 4X3, the factor 5 has
disappeared, and the number is no longer multiple of 5. The
product, therefore, of numbers prime to each other, is their
least common multiple. In the above example, the numbers
of 3 and 5 were prime in themselves, and not merely prime
to each other. To make the principle more clear, we will
FRACTIONS. 61
take two numbers that are not prime in themselves, but are
only prime to each other.
What is the least common multiple of 8 and 9 ? Multiply-
ing them together we Jiave 72. 72 is, then, a common multi-
ple of 8 and 9. The question is, is it their smallest common
multiple ? Writing the numbers with their factors they are
2X2X2 and 3X3. Now if we erase one of the 2's we have
no longer the factors of 8.. and the product of the factors will
not be divisible by 8. In the- same way, if we erase one of
the 3's the product will not be divisible by 9.
If, then, the numbers are either prime, or prime to each
other, the product is their least common multiple.
Next let us inquire, what is the least common multiple of
4 and 6 ? Their product is 24, but this is evidently not their
least common multiple, for 12 contains both 4 and 6 as fac-
tors. To show why it is, that in this case, something less
than the product of the numbers is their least common _ multi-
ple, we will express each by its factors, thus, 2X2, 2X3.
Now it is clear that any number of times which you take
2X2 as a factor will be a multiple of 2X2. If then we
^hrow out the 2 in the 2X3, and multiply by the remaining
3, the product will be a multiple of 2X2, or 4. Looking
now at the 2X3, or 6, it is evident that any number of times
which you may take that as a factor will be a multiple of
2X3. But the 2 we may take from the 2X2, throwing
away that in the 2X3; this leaves us to multiply the
2X3 by 2 ; as we before multiplied the 2X2 by 3, making
12 as the least common multiple. The rule, therefore, is :
Retain of each prime factor the highest power which appears
in any of the given numbers ; erase the rest, and multiply to-
gether what then remain.
Find the least common multiple of 8, 24 and 36. Ex-
pressed by the factors they are 2X2X2. 2X2X2X3.
2X2X3X3. Now 2X2X2 is common to 8 and 24; it may
be thrown out of the latter, leaving only 3. Examining again
you observe that 2X2 is common to 8 and 36; we throw
this out of 36, leaving 3X3. Finally 3, we find, is common to
24 and 36 ; throwing this out of 24, we find the numbers
appear as follows : 2X2X2. ^X^X^X0. ^X^X3X3.
These multiplied together give for the least common multiple,
72. This conforms to the rule; for 2X2X2 is the highest
power of the factor 2, and 3X3 of the 'factor 3. What is the
6
62 MENTAL ARITHMETIC.
least common multiple of 24, 60 and 100 ? These factors are
2X2X2X3; 2X2X3X5 ; 2X2X5x5. We see that
2 X 2 is common to them all ; expunge it in the second and
third number. Next, 3 is common to the 1st and 2d ; expunge
it in the 2d. Lastly, 5 is common to the 2d and 3d ; expunge
it in the 2d, and the numbers will stand, 2X2X2X3.
^X^X'^X$' ^X^XoX^- These multiplied together,
give 600.
To multiply these most easily, first take 2X2X5X5=100 ;
then the remaining factors, 2x3, multiplied by 100, give 600.
What is the least common multiple of 24, 40 and 72 ?
What is the least common multiple of 18, 54, 81 ?
What is the least common multiple of 15, 4, 7 ? of 15, 40,
27? of 16, 14, 6? of 60, 12, 18?
From the foregoing reasoning and examples you will per-
ceive that the least common multiple of several numbers is
the product of all their prime factors, each taken in the high-
est power in which it appears in any of the numbers.
SECTION XI.
PRACTICAL QUESTIONS.
1. What part of a shilling is 1 penny ? 2 pence ? 3 pence ?
4 pence ? 5 pence ? 6 pence ? 7 pence ?
2. What part of a penny are 2 farthings ? 3 farthings ?
4 farthings ? 5 farthings ? 6 farthings ? 8 farthings ?
3. What part of a shilling is 1 farthing ? 2 farthings ? 3
farthings ?
What part of a shilling is 1 penny and 1 farthing ? 1 pen-
ny, 2 farthings ? 3d 3 qrs. ? 4d 2 qrs. ? 6d 1 qr. ? 9d 2 qrs. ?
4. What part of a pound is 1 shilling ? 2 s.? 3 s.? 5 s.?
Is. Id. ? 2s. Id. ? 4s. 3d. ? 5s. 6d.? 7s. 9d. ? 3s. 8d. ?
5. What part of a pound is 1 farthing? 2 qrs.? 3 qrs. ?
2d. 3 qrs. ? 5d. 2 qrs. ? Is. Id. 1 qr. ? 6s. 7d. 3 qrs. ?
6. What part of a pound avoirdupois is 2 oz. ? 3 oz. ?
4 oz. ? 5 oz. ? 6 oz. ? 7 oz. ? 8 oz. ? 9 oz. ? 10 oz. ?
F][lACTIONS. ' 63
7. What part of one ounce is one dram ? What part of 1
pound is one dram ? 2 drs. ? 3 drs. ? 1 oz. 1 dr. ? 1 oz.
2 drs. ? 2 oz. 4 drs. ? 3 oz. 6 drs. ? 8 oz. 3 drs. ? 9 oz. 1 1 drs. ?
8. Yv^'hat part of a pound is -^^ of an oz. ? -^-^ of an oz. ?
What part of a* pound is ^ an oz. ? 2^ oz. ? of 3^ oz. ?
4^oz.?
9. What part of a pound Troy is 1 dwt. ? 5 dwt. ? 6 dwt. ?
9 dwt. ? 11 dwt. ? 10 dwt. ? 1 oz. 1 dwt. ? 3 oz. 4 dwt. ?
What part of oz. Troy is 1 dwt. ? 3 dwt. 1 gr. ? 4 dwt. 6 gr. ?
7 dwt. 3 grs. ? 8 dwt. 9 grs. ? 10 dwt. ? 12 dwt. ? 16 dwt. ?
10. What part of an ell English is 1 qr. of a yard ? 2
qrs. ? 3 qrs. ? What part of a qr. is 1 nail ? 3 nails ?
11. What part of a yd. is 1 qr. 1 nail ? 2 qrs. 3 n. ? 3 qrs.
2 n.? What part of an ell English is 3 nails ? 1 qr. 3 n. ? 4
qrs. In.?
12. What part of a yd. is 1 inch? 4 inches? 7 inches?
9 inches ? What part of a yard is 1 qr. 2 in. ? 2 qrs. 3 in. ?
3 qrs. 1 in. ?
13. From a vessel containing 3 gallons of wine, 3 gills
leaked out ; what part of a gallon leaked out ? What part
of a gallon remained ?
14. From a barrel full of wine 7 quarts were drawn ; how
many quarts remained ? What part of the barrel had been
drawn out ? What part of the barrel had remained ?
15. If f of a barrel of beer be divided into 4 equal parts,
what part of a barrel will each of the parts be ? How many
gallons will each part be ?
16. If one quart be taken from a barrel full of beer, what
part of a barrel will remain ? If 3 pints be taken out, what
part will remain ? If 7^ gallons be taken out, what part of
a barrel is taken out ? What part of a barrel remains ?
17. A man distributed 7^ gallons of milk among 5 persons ;
what part of a gallon did he give to each ?
18. If you have 3^ gallons of milk, and distribute it to
some poor persons, giving f of a gallon to each, how many
persons will you give it to ? How much will remain ?
19. What part of 1 foot is 1^ in.? 2iin.? 5^ in.? e^in.?
8|in.? 9iin.? lOfin.? 11^ in.?
20. What part of a yard is 2 inches ? 3^ inches ? 14 in. ?
5iin.? 6iin.? 17^ in.? 24^ in.?
21. What part of a rod is ^ a foot? 1^ feet? 2^ feet?
4 ft, 8 in,? 6 ft 7 in.? 10 ft. 5 in.?
64 MENTAL ARITHMETIC).
22. What part of 3 rods is | a foot? 1 foot? 3^ feet?
What part of a furlong are 2^ rods ? 5^ rods ?
23. What fraction of a foot is ^ of a yard ? ^ of a yd. ?
What fraction of a foot is -^ of a rod ? y^/of a rod ? ^ of
a rod ?
24. A man measured the length of his barn with a stick
half a yard long, arid found the barn 31^ times the length of
his stick ; how long was it ?
25. A carpenter is cutting up a board 17^ feet in length,
into pieces 2^ feet long ; how many pieces will there be, and
how long will be the piece that remains ?
26. A man measures a piece of fence with a pole 9^ feet
long; the fence is 15 1 times the length of the pole; how
many rods is it in length ?
27. What part of a peck is -^jj of a bushel ?.
What part of a gallon are -5^ of a peck ? f of a peck ?
What part of a quart is ^^^ of a peck ? /^^ of a peck ?
What part of a quart are ^-^ of a bushel ? ^^ of a bushel ?
28. What part of a peck is -^ of a bush. ? | of a bush. ?
f of a bush. ? ^ of a bush. ? |- of a bush. ?
29. Two men bought a lot of standing wood in company,
for 11 dollars; one cut off 2 cords, the other 1 cord; what
ought each to pay ?
30. Two boys bought the chesnuts on a tree for 50 cents ;
one had 11 quarts, the other 6 quarts and 1 pint; what ought
each to pay ?
31. Three men bought a piece of cloth for 24 dollars?
the first took 2|- yds., the second the same quantity, and on
measuring the remainder it was found to be 3 yards ; what
ought each to pay ?
32. Two men hire a horse for a month for 12 dollars ; one
travels 200 miles with the horse, the other 150 ; how much
should each pay ?
JH
FRACTIONS. 65
SECTION XII.
DECIMAL FRACTIONS.
[3ee Numeration, Part IT. J
In tlie calculations in common fractions, a great inconve-
nience arises from their irregularity. There is no law regu-
lating the magnitude of either of the terms. The denomina-
tor may be in any ratio whatever to the numerator. From
seeing one you can make no inference at all respecting the
magnitude of the other. In calculations of addition, it is
often more than half the work to bring the fractions into a
common denomination.
Now it is evident that if fractions could be written in the
same manner as whole numbers, that is, increasing in a ten-
fold rate as you advance to the left, and decreasing in a
ten fold rate as you advance to the right, an immense gain
would be made in the convenience of calculating them. Op-
erations in fractions would then be just as ea^ as operations
in whole numbers. Now this advantage is gained in decimal
fractions. They are brought under the same law as whole
numbers. Let us observe the manner in which whole num-
bers are written. Take the number 222 ; the right hand
figure signifies two units, the next two tens, the next two hun-
dreds; just as if it were written in this manner, 2Xl00-f-
2X10+2: two multiplied by 100 plus two multiplied by 10,
plus two; making two hundred and twenty-two. But thi
cumbersome method of writing is unnecessary, because the
law of notation determines what number the figures in each
place shall be multiplied by. It must not be forgotten that
the figure 2 in the above example in no case signifies of itself
more than"^ two. It is the place it occupies that gives it the
higher value of tens or hundreds.
Now it would evidently be a great convenience if we could
reduce fractions to the same law, so that they would, like
whole numbers, decrease in a decimal ratio, in advancing
from the left to the right. To show this regularity to the
eyf, we will write the following numbers : two multiplied by
1000, two multiplied by 100, two multiplied by 10, two units,
two divided by 10, two divided by 100, and two divided by
1000. Written in full they would stand thus: 2X1000+
2 X 100+2 X 10+2+A+Tf tt+t/^^.
6*
66 ■ MENTAL ARITHMETIC.
But we have seen that we may write the whole numbers
without the multipliers, thus, 2222, because we know from the
place each, figure occupies what its multiplier must be. Just
so we can write fractions without the denominators, provided
we know, from the place of the numerator, what the denomi-
nator must be. Thus the whole of the above series may be
written as follows ; 2222.222. A decimal, therefore, is the
numerator of a fraction, whose denominator is never written,
but is always understood to be 1, with as many ciphers as
there are places in the decimal.
You observe that, in writing the series given above, there
is a period placed at the right hand of the whole numbers,
separating the unit figure from that of tenths. The period
must never be omitted when there are fractions, for it enables
you to determine the value of each figure in the sum. In-
stead of reading .22 two tenths and 2 hundredths, we may
call it 22 hundredths, which is more convenient and amounts
to the same ; for 2 tenths is equal to 20 hundredths ; so
.222 is two hundred and twenty-two thousandths. So, in all
cases, read the decifaal numbers as whole numbers, and for
their denominator take 1 with as many ciphers as there are
places in the written decimals.
In all your study of decimals, be careful not to confound
the words which express frictions with the similar words
which express whole numbers ; as tenths with tens, hundredths
with hundreds. The following questions will aid you in fixing
this distinction clearly in mind.
1. How many tenths are equal to ten whole ones ?
2. How many tenths are equal to two and a half whole
ones?
3. How many hundredths are equal to three and- a quarter
whole ones ?
4. How many hundredths are equal to one hundred whole
ones ?
5. How many thousands are equal to ten whole ones ?
6. In fifteen whole ones how many tenths ? How many
hundredths ?
7. In seventy-five hundredths how many tenths ?
8. In three tenths how many hundredths ?
9. In six tenths how many thousandths ?
Thus, you observe, fractions have been brought under the
same law that regulates the writing of whole numbers. .They
ADDITION AND SUBTRACTION OF DECIMALS. 67
may now be added, subtracted, multiplied, and divided, like
whole numbers. But in doing this it is important to
determine the place of the period that separates the whole
numbers from the. fractional part of the sum. Where must
the period be placed in the answer ?
ADDITION AND SUBTKACTION OF DECIMALS.
Let us first observe how important it is that the rule in this
case be entirely correct. If I have this number, 32.5, to write,
and by any mistake I should write it 3.25, it would denote a
quantity only one tenth as great as it should be ; or, if I should
write 325. it would denote a quantity ten times greater than it
should be. Moving the period one place to the right, makes
the number ten times as great as it was before, for tens be-
come hundreds, and hundreds, thousands ; and each figure ten
times as great as before. So, by moving the period one place
to the left, the number becomes just one tenth what it was
before. Removing the period two places from its true place,
makes the number 100 times larger or smaller than it should
be, according as you remove it to the right or the left. Hence
you may see that in order to multiply a number that has
decimals, by 10, you have only to remove the period one place
to the right; to multiply by 100, remove it two places, and so
on. To divide by 10, remdve the period one place to the
left ; to divide by 100, remove it two places, and so on. From
the above you may see the importance of being perfectly ac-
curate in fixing the place of the decimal in the answer to any
question.
We will begin with addition. Add 4.46 to 3.21. Here you
observe the two whole numbers make 7, and 46 hundredths
added to 21 hundredths make 67 hundredths : the answer,
then, must be 7. 67, having two decimal places. Add 6. 8 to
5. 23. The 3 hundredths must evidently stand alone, since
there is nothing like it to add to it ; 2 tenths added to 8 tenths
make 10 tenths, or one whole one ; this we carry to the 5,
which gives us for the answer, 12. 03. This will serve to
suggest the rule for placing the period in the answer to ques-
tions in addition. The number of decimal places in the
answer must be as great as can be found in any one of the
numbers to be added.
68 MENTAL ARITHMETIC.
The same rule holds in subtraction. Take for illustration
the numbers given in the second example of addition. From
6. 8 subtract 5. 23. Now as in the minuend there are no
hundredths, we must borrow 10 in this place, and we shall
have a remainder of 7 hundredths ; adding 1 tenth to the sub-
trahend, to compensate for the 10 hundredths added to the
minuend, we have in the place of tenths a remainder of 5 ;
finally, in the place of units we subtract 5 from 6 : the answer
is 1. 57. In performing this operation, you may, if you please,
call the 8 tenths 80 hundredths ; then 23 hundredths from 80
hundredths leaves 57 hundredths. By performing slowly
and with care examples of your own selection, you will see
the verification of the rule given above, both for addition and
subtraction.
Add 2.4 to 3.8. Add .6 to 1.3. Add .4 to .3. Add .37
to .25. Add 3.7 to .24. Add 1.08 to .05.
From 4.6 subtract 2.4. From 7.1 subtract 6.4. From .18
subtract .13. From 4.5 subtract .6.
In these examples each step should be explained by the
pupil as he performs it.
MULTIPLICATION OF DECIMALS.
The rule in multiplication we shall find to be different from
the above.
1. First, we will multiply 2.4 by 3. If we regard the
multiplicand as a whole number, the answer Avill be 72. But
by regarding'' the multiplicand as a whole number, — as 24
instead of 2 and 4 tenths, — we regarded it ten times greater
than it really is ; the answer, therefore, is ten times too great.
Instead of 72 it must be 7.2.
2. Multiply 6.2 by 3.4. By regarding both as whole
numbers we obtain the answer 2108. Now in calling the
multiplicand 62 instead of 6.2 we treated it as 10 times
greater than it is. The answer must therefore be 10 times
too great, even if the multiplier were a whole number. Wp
must therefore divide it by 10, or write 210.8. But the mul-
tipHer also is 10 times too great; the answer must therefore
be divided again by 10, in order to bring it right. Thus the
answer will, stand 21.08.
3. Again ; multiply .62 by 3.4. Here we obtain the same
i
DIVISION OF DECIMALS. 69
figures as before, 2108 ; but by treating the multiplicand as a
whole number, we regarded it as 100 times too great ; the
answer therefore must be divided by 100, or written 21.08.
But the multiplier, calUng it a whole number, was taken 10
times greater than it is ; the answer must be again divided by
10, and thus it will stand 2.108.
4. Once more ; multiply .62 by .34. The figures of the
answer are as before, 2108, but by regarding both the factors
as whole numbers, we take each 100 times greater than it is ;
we must therefore divide by 100 to correct the error in the
multiplier, and again by 100 to correct the error in the multi-
plicand. This will remove the point four places to the left,
and the true answer will be .2108. By examining these
examples you will see that the pointing in each case conforms
to the following rule.
Point off as many figures for decimals in the ^answer as
there are decimal places in both the factors taken together.
5. Multiply 2.7 by .3. 6. Multiply .6 by .7. 7. Multiply
6. by .7. 8. Multiply .02 by .3. 9. Multiply .02 by .03.
10. Multiply .01 by .01.
DIVISION OF DECIMALS.
1. Divide 48 by 12. Ans. 4.
2. Divide 4.8 by 12. The figure expressing the answer
is 4, as in the first case ; but, observe, the dividend is only
one tenth as large as before ; the quotient, therefore, is only
one tenth as large. Instead of 4. it is .4.
3. Divide .48 by 12. The figure of the quotient is still 4,
but as the dividend is only one hundredth part as large as in
the first example, the quotient will be only one hundredth part
of 4, or 4 hundredths, written thus, .04.
4. Again ; divide 48 by 1.2. The quotient is still 4, but we
must investigate the question to see where this 4 must stand.
You observe that the divisor is now only one tenth of 12.
Now if the divisor is only one tenth as great as it was before,
you must consider how that will affect the quotient. You will
perceive on reflection that as you diminish the divisor you
increase the quotient. If you make the divisor half as great,
the quotient will be twice as great, and so proportionally of
other numbers. Now'as, in this instance, thr^ divisor is one
70 MENTAL ARITHMETIC.
tenth as great as before, the quotient must be ten times greater.
The figure 4, then, which is the quotient figure, instead of
standing in the place of units, as before, must stand in tho
place of tens ; that is, it must be 40, the cipher merely show-
ing that the 4 stands in the place of tens.
5. Once more : divide 48 by .12. Here again you have 4
for the quotient figure, for you can have no other ; but on
comparing this example with the first, you perceive the divisor
is only one hundredth part as great ; the quotient must there-
fore be one hundred times greater, that is, it is 400, the
ciphers merely removing the 4 into the place of hundreds,
On examining these examples carefully, you will see that
each answer is unquestionably correct. " But by what rule,"
you ask, "are these examples wrought?" They are not
wrought by rule, but by reasoning on the numbers themselves ;
and the more you habituate yourself to reason in arithmetic,
the less need you will have to depend on rules.
With this suggestion I will now state a rule, which you may
at any time follow, when you have not time to look into the,
reason of the operation.
There must be as many decimals in the quotient as the
decimals in the dividend exceed those in the divisor : when
there are fewer decimals in the dividend than there are in
the divisor, ciphers must be added so as to make the number
equal.
We will now review the foregoing examples, and observe
their conformity with the above rule. Example 1 has no
decimals in the divisor or the dividend, therefore none in the
quotient. Ex. 2, the dividend has one decimal, the ' divisor
none ; the quotient has therefore one. Ex. 3, the dividend
has two decimals, the divisor none ; the quotient has two.
Ex. 4, the dividend has none, the divisor one ; there must
then be a cipher added to the dividend, and then the quotient
will be in whole numbers. Ex. 5, the dividend has none, the
divisor two ; there must then be two ciphers added, and then
the quotient will be in whole numbers.
6. Divide 45 by 15. Divide 4.5 by 15. Divide .45 by
15. Divide 45 by 1.5. Divide 45 by .15.
7. Divide 66 by 11. 6.6 by 11. .66 by 11. 66 by 1.1.
66 by .11.
In calculations of Federal money, cents and mills are re-
garded as decimals ; the point therefore separating the whole
ANALYSIS OF DECIMALS. 71
numbers from the fractions must be placed between th#dol-
lars and the cents. Thus 24.00 is 24 dolls. ; 2.40 is 2 dolls.
40 cents ; 0.24 is 24 cents.
8. A man divided $24.00 among 3 men ; how much did
each receive ?
9. A man divided $2.40 among 3 men ; how much did each
receive ? Divide 2.4 by 3.
10. A man divided $0.24 among 3 men ; how much did
each receive ? Divide $0.24 by 3.
11. A man divided 36 dollars among 4 persons; how much
did each receive? Divide 36 by 4.
12. A man divided $3.60 among 4 persons ; how much did
each receive ? What is one fourth of $3.60?
13. A man divided $0.36 among 4 men; how much did
each receive ? What is one fourth of .36 ?
SECTION XIII.
REDUCTION OF VULGAR FRACTIONS TO DECIMALS.
We have now seen that Decimal Fractions have this great
advantage over Vulgar Fractions, — that they conform to the
same law of notation as whole numbers, and may be added,
subtracted, multiplied and divided in the same manner, and
with the same ease as whole numbers^ It is desirable, there-
fore, to introduce them in a great many cases instead of
Vulgar Fractions. The next question that arises, therefore,
is, can a Vulgar Fraction be changed to a decimal, ha\-ing
the same value ; and how can it be done ? Take the fraction
^ ; we wish to reduce it to tenths ; or in other words to ex-
press it in tenths. Now we can change any number to tenths
by multiplying it by 10. Thus 3 is 30 tenths, 4 is 40 tenths.
We will now take ^ and change the numerator 1 to tenths,
and it will stand .10 : but the fraction was not one, but one
half of one ; 10 therefore is twice as great' as it should be ;
we must divide it, therefore, by 2 ; that is, by the denomina-
tor, and it will be .5. To reduce a vulgar fraction, then, to a
decimal : add a cipher to the numerator, and divide by the
72 MENTAL ARITHMETIC.
denominator. If one cipher is not enough to render the
division complete, add more.
Reduce to a decimal -^ ; change the numerator to tenths ; it
will be .10, but the quantity to be reduced to tenths was not
one, but one fifth of one ; 10, therefore, is 5 times greater than
it should be ; dividing by 5, the answer is .2.
Reduce to a decin^al the fraction f , explaining each step in
the operation.
Reduce to a decimal the fraction ^.
Reduce to a decimal the fraction |.
Reduce to a decimal the fraction ^.
Reduce to a decimal the fraction | .
I will here direct your attention to a fact that it is interest- ,
ing to notice. If the denominator of the vulgar fraction is
one of the factors of 10, that is, if it is either 2 or 5, the
decimal figure will be as many times the other factor as
there are units in the numerator of the vulgar fraction.
This will appear self-evident when we express the numbers
by their factors. Thus in obtaining the decimal for ^ we
divide 10 by 2; but 10 is 2X5, therefore in dividing by 2*
we simply expunge the factor we divide by, and leave the
other : 2) ^X5. So in the fraction ^, we obtain the decimal
by dividing 10 by 5, which expunges the factor 5, 5)^X2;
in reducing f we divide 2 X 1 0 by 5, thus : 5) 2 X 2 X ^, leaving
twice the factor 2 ; in f , 5) 3 X 2 X ^, leaving 3 times the fac-
tor 2; in I, 5) 2X2X2X^, leaving 4 times the factor 2.
2. We will now take the fraction ^ ; proceeding as before
we wish to divide 10 by .'4, thus, 2X2) 2X5; here we seethe
division cannot be complete, for the divisor contains the fac-
tor 2 twice, while the dividend has it only once. If, however,
we had multiplied the original numerator 1 by 100, instead
of 10, we should have had 10 twice as a factor in the dividend,
and of course each factor of 10 twice; 100 is 10X10, and 10
is 2X5. It would have stood then thus, 2X2)2X5X2X5;
the division is now complete, for the dividend contains the
factor 2 as many times as the divisor has it. Expunging
these we have remaining the factor 5 taken twice, or .25.
This process you may observe conforms to the rule, to
add as many ciphers as may be necessary to render the
division complete.
3. Reduce the vulgar fraction f to a decimal. 30 is com-
posed of the prime factors 3X2X5; it contains 2 only once,
ANALYSIS OF DECIMALS. 73
and therefore it is not divisible by 2X2; 30 must therefore
be multiplied bj 10. This will introduce another 2, and
it will stand thus, 2X2)3X2X5X:'X5. By expunging the
two 2's and multiplying together the other factors, we have
.75 for the answer.
4. Reduce the fraction ^^ to a decimal. 10 expressed by its
factors is 2X5, and 8 is 2X2X2. We. must therefore mul-
tiply 2X5 by 10 till it shall contain the factor 2 as many
times as 8 contains the same factor. That is, the numerator
1 must be multiplied by a thousand. It will then stand,
2X2X2) 2X5X2X5X2X5. Expunging the three twos
there remains for the answer .125.
By examining the above examples you may observS this
fact, that if the denominator of the vulgar fraction contains
one of the factors of 10, that is, 2 or 5, one or more times as
a factor, the decimal will contain the other factor, just as
many times. Thus, ^=.5. ^ or ■i^^^=.25, or .5 X.5 ; ^ or
7x^xif^-^^^i ^^ '^X'5X'5. In the same way i=-2 ; ^V ^^
^^^=.04, or .2X.2; ^^ or ^x|^^=.008, or .2X.2X.2.
In this way you may determine that y^^ ^^^" reduced to a
decimal, will contain 5 four times as a factor, because 16 con-
tains 2 four times as a factor. So ^V will contain 5 five times
as a factor.
This is conveniently expressed by saying, whatever power
of one of the factors of 10 the denominator of the vulgar
fraction contains, the same power of the other factor will
appear in the decimal.
5. Reduce ^ to ^ decimal fraction. Preparing the num-
bers as before, it will stand 3)2X5. You observe that 3 is
different from either of the factors of 10. Now as 10 has
only the factors 2 and 5, it is not divisible by 3 without a
remainder.
If you add to the numerator ever so many ciphers, you will
only increase the number of times that 2 and 5 appear in it
as its factors, and the number can never become divisible by
3 without a remainder. The answer becomes .333-f- and
this indefinitely, as far as you may please to carry on the
operation. On the same principle we shall find that it is not
possible to express accurately in decimals any vulgar fraction
whose denominator contains as a factor anything different
74 MENTAL ARITHMETIC.
from the factors of 10 ; for this denominator becomes, in the
reduction, a divisor of 10 or some power of 10, and if it has
anything in it as a factf r which is prime to the factors of 10,
the complete division is impossible. Thus ^ cannot be ex-
actly expressed in decimals ; because, though one of its fac-
tors, 2, is a divisor of 10, the other, 3, is prime to 10. On
this principle the following questions .may be examined.
Can ^ be accurately expressed in decimals ? Why ?
Can ^ be accurately expressed in decimals ? Why ?
Can I be accurately expressed in decimals ? Why ?
Can -^^ be accurately expressed in decimals ? Why ?
Can^? tV? a? a? iiV? rV? iV? ^? T>s? A?-Ar?
A? A? iV? , ^
6. Name all the denominators, from 2 up to 20, of such
fractions as can be accurately expressed in decimals ?
From 20 to 40 ? From 40 to 60 ? From 60 to 80 ?
7. Name all the denominators, from 2 to 20, of such frac-
tions as cannot be expressed accurately in decimals? From
20 to 40 ? From 40 to 60 ? From 60 to 80 ?
8. What is the value of 4 shillings expressed in the deci-
mal of a £ ? As 1 shilling is ^ of a £, 4 s. is -^. We can
change 4 to tenths by adding a cipher ; it will then be 40 ;
4, however, was not the number we wished to reduce to tenths,
but /^; the answer, 40, is therefore 20 times too great ; dividing
by 20 it stands .2. 4 shillings, then, is 2 tenths of a £.
9. Now reverse the operation ; what is the value in shil-
lings of .2 of a £ ? Now shillings are twentieths ; we can
change any number to twentieths by multiplying it by 20, as
1 is 20 twentieths, 2 is 40 twentieths, &c. Multiplying the
.2 by 20 we have 40 ; but observe the two was not two wholes,
but two tenths ; the answer, 40, therefore, is ten times too
great ; dividing by 10 the answer is 4 shillings.
10. Reduce to the decimal of a £, 2 shillings. 5 shillings.
11. What is the value in shillings of .1 of a £ ? of .25 of
a£?
12. Reduce to a decimal of a shilling, 3 pence. 3 pence
are y\ of a shilling ; reducing to hundredths to render the
division complete, the ans. is .25.
13. What is the value in pence of .25 of a shilling ?
14. Reduce 9 pence to the decimal of a shilling.
15. Reduce 1 peck to the decimal of a bushel.
16. Reduce 3 pecks to the decimal of a bushel.
17. Reduce .5 of a bushel to pecks. .75 of a bu. to pecks.
INTEREST. 75
18. Reduce 15 minutes to the decimal of an hour.
1 9. Reduce 45 minutes to the decimal of an hour.
20. Reduce to minutes .5 of an hour. .25 of an hour. .75
of un hour.
21. Reduce 6 in. to the dec. of a foot. 9 in. to dec. of a
ft. 3 in. to the dec. of a ft.
SECTION XIV.
INTEREST.
Interest is. the sum piiid by the borrower to the lender for
the use of money. The rate of interest is established by law,
and varies in different countries. In England it is 5 per
cent., that is, 5 for the use of 100 for 1 year ; in the New
England States it is 6 per cent. ; in New York it is 7 per
cent. When no particular rate is mentioned in this book, 6
per cent, will be understood.
If I borrow 100 dollars fof 1 year, at the end of the year
I owe the sum I borrowed, 100 dollars, and 6 dollars for the
use of it, making 106 dollars. The sum borrowed is the prin-
cipal ; the sum paid for the use of it is the interest ; the prin-
cipal and interest added together make the amount.
1. What is the interest of 100 dolls, for 2 years? 3 years ?
4 years ? 5 years ? 6 years ? 7 years ?
2. What is the interest of 200 dolls, for 2 years ? 3 years ?
4 years ? 5 years ? 6 years ?
3. What is the interest of 300 dolls, for 2 years ? for 4
years ? of 400 dolls, for 3 years ?
4. What is the interest of 50 dolls, for 1 year ? for 3 years ?
of 25 dolls, for 1 year ? 2 years ?
5. What is the interest of 100 dolls, for 1 year ?
What is the interest of 100 cents for 1 year?
What is the interest of 2 dolls, for 1 year ? of 3 dolls. ? of
4 dolls.? 5 dolls.? 6 dolls.? 7 dolls.? 8 dolls.? 9 dolls.?
6. What is the interest of 36 dolls, for 1 year ? of 47 dolls. ?
of 57 dolls. ? of 34 dolls. ? of 62 dolls. ? of 89 dolls. ? of 125
dolls. ? of 136 dolls. ? of 207 dolls. ? of 561 doUs. ?
7. What is the interest of 50 cents for 1 year ? of 25 cents ?
76 MENTAL ARITHMETIC.
of 10 cents ? of 20 cents ? of 30 cents ? of 40 cents ? of 50
cents ? of 70 cents ? 6f 80 cents ? of 90 cents ?
8. What is the interest of 50 doll. 60 cents for 1 year ? of
84.30? of 96.40? of 112.25? of 230.75?
9. What is the interest of 100 dolls, for 6 months ? for 3
months ? for 2 months ? for 1 month ? for 4 months ? for 5
months ? for 7 months ? for 8 months ? for 9 months ? for 10
months ? for 11 months ?
10. What is the interest of 10 dolls, for 6 mo. ? 3 mo. ? 2
mo. ? 1 mo. ? 4 mo. ? 5 mo. ? 7 mo. ? 8 mo. ? 9 mo. ? 10 mo. ?
'11 mo.?
11. What is the interest of 1 doll, for 6 mo. ? 1 mo. ?
The interest of 1 dollar for 1 month is half a cent, and for
any number of months, it is half as many cents.
12. What is the interest of 1 dollar for 5 mo.? 7 mo.? 8
mo.? 9mo. ? llmo. ? 12 mo. ? 15mo. ? 16mo. ? 17mo. ?
18 mo. ?
The interest of any number of dollars for 1 month is half
as many cents.
13. What is the interest of 12 dollars for 1 mo.? of 15
dolls. ? 25 doUs.? 37 dolls.? 42 dolls.? 67 dolls.? 93 dolls.?
104 dolls.?
14. What is the interest of 12 dolls, for 3 months?
What is the interest of 25 dolls, for 6 months %
In computing interest a month is reckoned 30 days. As
the interest on a dollar for 30 days is half a cent, that is 5
mills, the interest on a dollar for 1 fifth of 30 days will be 1
mill. One fifth of 30 is 6 ; the interest therefore on 1 dollar
for 6 days is 1 mill, and the interest on any number of dollars
for 6 days will be as many mills as there are dollars.
15. What is the interest of 15 dollars for 6 days ? of 25
dolls. ? of 40 dolls. ? of 65 dolls. ? of 75 dolls. ? of 100 dolls. ?
of 500 dolls. ? of 360 dolls. ? of 840 dolls. ? of 1000 dolls. ?
As the interest of 1 doll, for 6 days is 1 mill, for 12 days
it will be 2 mills, for 18 days 3 mills, &c.
16. What is the interest of 1 doll, for 24 days ? of 2 dolls,
for 6 days ? of 2 dolls, for 12 days? of 2 dolls, for 18 days?
of 5 dolls, for 6 days? for 12 days? for 24 days? of 36 dolls.
for 18 days?
17. What is the interest of 125 dolls, for 1 year and 6 mo. ?
IKTEREST. 77
18. What is the int. of 268 dolls, for 3 years 4 mo. ?
19. What is the int. of 45 dolls, for 4 years 7 mo. ?
20. What is the int. of 60 dolls, for 1 year 3 mo. 18 days ?
,21. What is the int. of 100 dolls, for 2 years 1 mo. 12 days?
22. What is the int. of 165 dolls, for 3 years 2 mo. 6 days?
23. What is the int. of 50.45 for 1 year 7 mo. 12 days?
24. What is the int. of 94 dolls, for 8 mo. 24 days ?
25. What is the int. of 132.25 for 6 mo. 3 days?
26. What is the int. of 81.20 for 4 months 15 days ?
27. What is the int. of 64.50 for 10 months 16 days?
28. What is the int. of 86 dolls, for 9 days ?
29. What is the int. of 340 dolls, for 15 days ?
30. What is the int. of 875 dolls, for 22 days ?
When interest is more or less than 6 per cent., first find
the interest at 6 per cent, and then make a proportional addi-
tion or subtraction for the required per cent. If it is 7 per
cent, add one sixth ;. if 5 per cent, subtract one sixth.
31. What is the int. of 140 dolls, for 1 year, at 7 percent.?
32. What is the int. of 200 dolls, for 1 year and 6 mo. at
5 per cent.
33. What is the int. of 460 dolls, for 1 year at 4^ per cent.
Remark. — 4|^ is three fourths of 6.
34. What is the int. of 500 dolls, for 1 mo. at 9 per cent. ?
BANKING.
When money is obtained at a Bank, the note which is given
for it promises to pay it at a certain time, as 60, 90, or 120
days. The interest on this note, instead of being paid at the
end of the time, when the note is taken up, is paid before
hand ; that is, it is subtracted from the sum named in the
note ; so that, when you take up the note, you have only to
pay the face of it, as the interest has been paid already.
If you give a note to a Bank for 100 dolls, to be paid in
90 days, they subtract from the sum named in the note the
interest of the sum for 90 days, and three days besides, called
days of grace ; the balance is the sum you receive. The in-
terest of 100 dolls, for 90 days is $1.50; for 3 days it is
7*
78 MENTAL ARITHMETIC.
5 cents ; $1.55 subtracted from $100.00 leaves a balance of
$98.45, which is the sum you will receive.
If the note is given for 60 days, the interest is cast for 63
days, and subtracted from the sum named.
The interest thus subtracted is called the bank discount ;
and the bank, when it lends money on such a note, is said to
discount the note.
35. What is the bank discount on a note of 100 dollars
payable in 30 days ? and how much will be received on such
a note ?
The interest on 100 dollars for 30 days is 50 cents ; for 3
days it is 5 cents ; the discount, 55 cents, subtracted from 100
dollars, leaves $99.45, the sum received.
36. What is the bank discount on a note for 200 dollars
for 60 days ? and what is the cash value of the note ?
37. What is the bank discount, and what is the cash value
of a note for 150 dollars payable in 30 days ?
38. What is the bank discount, and what is the cash value
of a note for 200 dollars payable in 90 days ?
39. What is the bank discount, and what is the cash value
of a note for 300 dollars payable in 90 days ?
DISCOUNT.
When money is paid by the debtor before it becomes due,
an allowance is made, which is called discount. If I owe 100
dollars, to be paid in three months from this time, and I pay
it now, I ought not to pay the full hundred dollars, for I am
entitled to the use of the money three months longer. The
sum which should be paid now, to cancel a debt due at some
future time, is called the present worth of the debt.
To find the present worthy of a debt due at some future
time, first find the interest on the debt from the time of pay-
ment to the time when the debt is .due ; subtract this interest
from the debt, and the remainder will be the present worth.
Thus, if I pay a debt of 100 dollars three months before it
is due, I subtract the interest of 100 dollars for three months,
= $1.50, — from 100 dollars, leaving $98.50 for the sum
which I must pay.
This rule is not strictly equitable, because % 98.50, with
three months' interest added, will not amount to $100. The
above method, therefore, gives the present worth a little too
LOSS AND GAIN. 79
small ; but it is the method uniformly adopted in business,
and the error is on the right side, for it encourages the debtor
to be prompt in his payments.
40. What is the present worth of 200 dollars payable in
1 year ?
41. What is the present worth of 150 dollars payable in
2 years ?
42. What is the present worth of 60 dollars payable in
6 months ? _
43. What is the present worth of 530 dollars payable in
1 year?
What is the present worth of 400 dollars payable in 1 year
and 6 months ?
LOSS AND GAIN. — PER CENTAGE.
44. A boy bought a penknife for 25 cents, and sold it for
28 cents ; how many cents did he gain on quarter of a dollar ?
45. Suppose he had bought 4 knives at the same price
each, and sold them at the same profit, he would then have
traded with a dollar ; how much would he have gained on a
dollar ?
This is called so much per cent, which only means so much
on a hundred.
46. A boy bought a bushel of apples for 50 cents, and sold
them for 59 cents ; how much did he gain per cent. ?
47. A bookseller bought a book for 75 cents, and sold it
for 84 cents ; how much did he make per cent.?
As 75 is |- of 100, what he gained on the book will be |-
of what he would gain on a hundred ; or what he would gain
per cent.
48. A boy bought some melons for 40 cents, and sold them
for 60 cents ; wliat did he make per cent. ?
Ans. His gain was equal to half his outlay.
49. A grocer bought a lot of flour for 5 dollars a barrel,
but finding it damaged, he sold it for 4 dollars a barrel ; what
did he lose per cent. ?
50. A man bought a share in a bank for 80 dollars, and
Bold it for 82 dollars ; what did he gain per cent. ?
51. A man bought a lot of apples for $1.50 a barrel; what
must he sell them for to gain 10 per cent. ?
80 MENTAL ARITHMETIC.
52. A hatter bought some hats for $3.50 each ; he is willing
to sell them at a profit of 4 per cent. ; at what price will he
sell them ?
53. A manufacturing company declare a dividend of 7^
per cent. ; what ought a stockholder to receive who owns 350
dollars in that factory ?
54. A has a note against B for 140 dollars, which he sells
for cash at 4 per cent, discount ; what does he receive for the
note ?
55. A merchant buys 100 barrels of flour for 5 dollars a
barrel, and sells it so as to lose 5 per cent. ; what does he sell
it for a barrel ?
He afterwards buys 250 casks of lime at 1 dollar a cask ; he
wishes to sell it so as to make good his loss on the flour ; at
what per cent, profit must he sell it, and for how much a cask ?
^You observe that the money invested in lime is only one
half as much as was invested in flour.
56. A lends B 10 dollars for 2 months without interest ;
afterwards B lends / 5 dollars ; how long can A keep it to
balance the favor he iid to B ?
57. C lends D ICO dollars without interest for 4 months ;
afterwards D lends C 25 dollars ; how long can C keep it to
balance the favor ?
In these cases you will see that the money multiplied by
the time it was kept must, in the two cases, be equal. If 10
dollars is lent me by A without interest for 6 months, I can
balance the favor by lending A 5 dollars for 12 months, or 4
dollars for 15 months, or 15 dollars for 4 months, or 30 dollars
for 2 months, or 2 dollars for 30 months, or 20 dollars for 3
months.
58. A lends B 60 dollars for three months, without requir-
ing interest ; afterwards B lends A 90 dollars ; how long may
A keep the money to balance the favor ?
59. A lends B 40 dollars for three months ; afterwards B
lends to A, for two months, a certain sum, the use of wliich
should balance the favor ; how large must the sum be ?
60. A lends B 150 dollars for 4 months; B afterwards
lends A 100 dollars ; how long can A keep it, to balance the
favor ?
SQUARE MEASURE. 81,
SECTION XV.
.SQUARE MEASURE,
Linear measure is measure in a straight line, having length
onlj. Square measure is the measure of surface, having
length and breadth.
Thus a linear inch, ■ — —
A square inch,
*A line of 2 inches, when square, will therefore make 4
square inches, thus,
1. A line of three inches, when square, will make how
many square inches ?
2. The square of 4 inches is how many square inches ?
The square of 0 inches ? of 6? of 7 ? of 8 ? of 9 ? of 10?
of 11? of 12?
3. How many square inches are there in a square foot ?
* Note 6.
82
MENTAL ARITHMETIC.
4. How many linear feet are there in a linear yard ? How
many square'feet in a square yard ?
5. How many square inches are there in a piece of board
"12 inches long and 3 inches wide ?
6. How many square inches are there in a piece of board 8
inches long and 6 inches wide ? in a board 5 inches long and
3 inches wide ? in a board 9 inches long and 5 inches wide ?
7. How many square feet are there in the floor of a room
12 feet long and 10 feet wide ?
8. How many square feet are there in the floor of an entry
15 feet long and 4 feet wide ?
9. How many square yards of carpeting will cover a room
6 yards long and 5 yards wide ?
How many square rods are there in a piece of land 14 rods
long and 8 rods wide ?
10. If a road 4 rods wide passes through my land for the
distance of 60 rods, how many square rods of my land does it
occupy .''
We will now return to the measure of an inch,
of squaring a linear inch, we take
only half an inch and square it, we
shall hav©^ but one fourth of a square
inch, thus,
If, instead
So if we square one third of an inch,
it will give us ^ of a square inch. If
we square one fourth of an inch, it
willgive us y*^ of a square inch.
11. What part of a square inch will -J- of an inch when
squared be ? What part of a square inch will ^ of an inch
be when squared ? ^? ^? ^? -j^^? ■^? ■^^?
12. What part of a square inch is a piece of paper 1 inch
long and half an inch wide ? One inch long and three fourths
of an inch wide ?
13. What part of a square foot is a board 1 foot long and
half a foot wide ? One foot long and 9 inches wide ?
14. How many square inches will there be in the square
of a line 1^ inches long ?
[This, and the following questions may be answered by
drawing the figure on a slate or on a board.]
How many square inches will there be in the square of a
line 2^ inches long? 8^? 4^? 5^? 6^? 7^? 8^? 9^? 10|?
SQUARE MEASURE. 88
15. How many square feet in ^ sl yard squared ?
16. How many square incbes are there in 1 square foot?
17. How many square feet in 1 square yard ?
18. How many square yards in 1 square rod ?
19. How many square feet in 1 square rod?
40 square, rods make 1 rood ; 4 roods make 1 acre.
20. How many rods make' 1 acre ?
21. If a piece of board is 6 inches wide, how long must it
be to contain a square foot ?
22. If a piece of board is 3 inches wide, how long must it
be to contain a square foot ?
23. How long must it be to contain a square foot, if it is 2
inches wide ? If 1 inch wide ? If 4 inches wide ?
■■ 24. If cloth is 1^ a yard wide, how much in length will
make a square yard ?
25. How much lining f of a yard wide will line one yard
of cloth one yard wide ?
26. If cloth is two thirds of a yard wide, how much in
length will it take for a square yard ?
■ 27. How much cloth ^ a yard wide will it take to line 7
yards of cloth | of a yard wide ?
How much f wide will line l^- yards f wide ?
28. How long must a strip of land one rod wide be to con-
tain an acre ? How long, if 2 rods wide ? If 3 rods wide ?
If 4 rods wide ? If 8 rods wide ? If 10 rods wide ?
29. How long must a piece of land be to contain | of an
acre, if it is 4 rods wide ?
30. If a piece of land is 10 rods in length how wide must it
be to contain ^ an acre ?
31. A man has an acre of land 16 rods in length ; how wide
is it?
32. How many steps must the owner take to walk round it
if he take 5 steps to a rod ?
33. A man has an acre of land 8 rods wide ; how long is it ?
How many rods of fence will it take to fence it ?
34. If a road 4 rods wide is laid out through ray land, how
much of the road will it take in length to make an acre ?
How many acres will there be -in one mile of the road ?
35. If it passes through my land for half a mile, and I am
paid at the rate of 30 dollars an acre for the land occupied by
the road, what will be the amount of damages- due me ?
84 MENTAL ARITHMETIC.
36. If land in the city- is worth 45 cents a square foot, what
will be the cost of a building-lot 30 feet front and 60 feet from
front to rear ?
37. There are two pieces of land ; one of them 12 rods
square, the other 13. Which is nearest an acre ?
38. There is a piece of land 12^ rods square ; how much
does it fall short of an acre ?
39. A painter tells me it will cost 20 cents a square yard
to paint the floor of a room in my house ; supposing the room
is 5 yards wide and 6^ yards long, what will the painting of
it come to ?
40. What will the painting of an entry cost, at the same
rate ; that is 1^ yards wide and 7 yards in length ?
41. A stone-cutter agrees to lay a hammered stone door-step
for 50 cents for every square foot of hammered surface. The
stone is 5 feet long, 3^ feet wide, and 9 inches thick ; what is
the surface of the top, the two ends, and the front edge, added
together ? What will be the cost of the stone ?
42. How many men could stand on |- of a mile square,
allowing each man 1 square yard to stand upon ?
There are various ways of finding the answer to the above
questions. To encourage the student's invention, some of
them will be here 'suggested.
First Method. As there are dO^ square yards in one square
rod, multiply 80 (the number of rods in one fourth of a mile,)
by itself; and this product by 30^. 80X80=6400 ; 6400 X
30^=180,000-fl2,000+1600=:193,600, answer.
Second Method. Multiply 80 by 5J-, which will give the
number of men in one line one fourth of a mile long ; multiply
this product by itself. 80X5^=440; 4402=160,000+32,000
+1600=193,600, answer.
There are still other ways of solving the question, which
the student may discover for himself.
Note. For those students who have not time to go through
the hook, the course of instruction in Mental Arithmetic may
properly close at this place. With a similar view, the Second
Part may be divided at the close of Section XXXVI. The
ground thus gone over will be found to embrace all the princi-
ples and practice needed in the transactions of ordinary
business. Those whose opportunities permit, should have the
advantage of the higher discipline furnished in the remainder
of the book.
CONSTRUCTION OF THE SQUARE. 85
SECTION XVI.
CONSTRUCTION OF THE SQUARE.
If I place three dots in a row, and place three such rows
side by side, this .will represent to the eye the square of the
number 3.
^ In the same way you may represent
Thus, • • * the square of 4, 5, or any number what-
9 # S ever. I will now ask your attention to
the square of 4. We may make it by
making a row of 4 dots, and placing 4
such rows side by side. But there is another way of coming
at the square of 4. We will take the square of 3, as shown
above, and see what additions we must make to it, in order to
make it the square of 4* You observe that it must be wider by
one row and longer by one row than it is now. We will then
add a row above the others, and also a row on the right hand.
I have made the additions by stars
Thus, • # • >fc to distinguish them from the dots.
You now see there is something
wanting to complete the square, — a
sinojle star in the corner.
*
* *
•
• •
*
•
• •
*
•
• •
*
*
* *
*
•
• •
*
e
• •
*
•
• •
*
You observe, therefore, that you
obtain the square of 4 by adding to
Thus, ^ "^ ^ -^ the square of 3, twice 3 plus 1. We
will now take the square of 4, and by
additions to it obtain the square of 5.
Adding a row of 4 at the top, and a
row of 4 at the right hand, there will be one \yanting at the
corner to complete the square. Adding this, which makes
twice 4-f-l, we have the square, complete. If therefore we
have the square of any number, we can find the square of a
number one greater by adding twice the first number plus 1.
The square of 5 is 25 ; what must you add to this square to
make the square of 6 ?
What must you add to the square of 6 to make the square
of 7 ? What must you add to the square of 7 to make the
square of 8 ?
S
86, MENTAL ARITHMETIC.
What must you add to the square of 9 to make the squaro
of 10?
The square of 15 is 225 ; what is the square of 16, by the
above method ?
The square of 20 is 400 ; what is the square of 21 ?
The square of 30 is 900 ; what is the square of 31 ?
The square of 40 is 1600 ; what is the square of 41 ?
The square of 50 is 2500 ; what is the square of 51 ?
What is the square of 60 ? of 61 ? of 70 ? of 71 ? of 80 ?
of 81 ? of 90 ? of 91 ?
We will now return to the square of 3, and I ask your close
attention once more. Supposing we have the square of 3
l^efore us, and we wish to make such additions to it as shall
make the square of 5. As 5 is two greater than 3, we must
add two rows instead of one. If we add 2 rows of 3 at the
top, and 2 rows of 3 at the right hand, the figure will stand
thus, _ _ _ . . . .
Here you see there are four stars
wanting to complete the square. I
have marked their places by the
circle (O). If you suppose these
four to be added the square will be
complete, and will be the square of
5. The question is now, what has
been added to the square of 3 in order to make the square of
5 ? You observe there are added 6 stars or two rows of
three at the top, 6 on the right hand, and 4 in the corner, to
make the square of 5. But we can express this in a different
way. We may consider 5 as consisting of two parts, 3 and
2 added together. We will call 3 the first part, and 2 the
second part of 5. Now by the figure you perceive that the
square of 5 is made up, first, of the square of the first part,
that is, the nine dots ; then the stars at the top are the product
of the first part multiplied by the second, and adding to these
the stars on the right hand, we have twice the product of the
first part into the second ; and, last, we have, in the corner,
the square of the second part.
To state it briefly once more : regarding 5 as made up of
the two parts, 3 and 2, the square of 5 we find is equal to the
square of the first part-f-twice the product of the two parts+
the square of the second part.
*
*
*
O O
*
*
*
o o
•
•
•
* *
•
9
•
* *
•
•
•
* *
THE SQUARE OP FRACTIONAL NUMBERS. 87
This is called expressing the amount of a square in the
terms of its parts.
Examine and answer the following questions :
1. If we regard the number 6 as made up of two parts, 4
and 2, how will you express the square of 6 in the terms of
its parts ?
2. Regard the number 7 as consisting of two parts, 5 and
2 ; what is the square of 7 in the terms of its parts ?
You can draw the figure for yourself and see the applica-
tion of the principle in the above cases.
It is of no consequence in what way the number is divided ;
the operation will bring out the exact square of the whole
number in all cases. To show this we will take the number
10, the square of which is 100. We will first divide 10 into
the parts, 7 and 3 ; then, by the formula given above, the sq.
of 7-|-twice the product of 7 into 3-f- sq. of 3, will be the sq.
of the whole number. The sq. of 7=49, twice 7 X 3=42, the
sq. of 3=9 ; 49+42+9=100.
We will now divide 10 into the parts 6 and 4, proceeding as
above ; we find 36+48+16=100.
Again, we will divide 10 into the equal parts, 5 and 5,
25+50+25=100. *
Finally divide 10 into the parts 8 and 2. 64+32+4=100.
We will now apply the above method to the purpose of
finding some squares of larger numbers.
3. What is the sq. of 25 ? dividing into 20+5 ; Ans. 400+
200+25=625 ?
4. What is the sq. of 35, or 30+5 ? Ans. 900+300+
25=1225.
5. What is the sq. of- 46 or 40+6? Ans. 1600+480+
36=2116.
6. What is the sq. of 55 ? of 64? of 75 ? of 83 ? of 92 ?
7. What is the sq. of 125 ? divide into 100+25. 100 sq.
=10000, twice 100X25=5000, 25 sq.=625. Ans. 15,625.
8. What is the square of 150 ? of 230 ? of 510.
The same formula will embrace the examples mentioned in
the first part of this section, when the second part of the num-
ber is 1, for example,
9. What is the sq. of 5 or 4+1 ? Here twice the product
of the two parts is merely twice the first part, inasmuch as
multiplying by 1 does not increase the number ; and the sq.
of 1 is only 1. The answer, therefore, by the formula, is 16
+8+1=25.
88. ' MENTAL ARITHMETIC.
10. This metliod will apply to the squaring a whole num-
ber and a fraction, as follows : What is the sq. of 1+^. Ans.
1+1-|-^=2;|, for twice the product of ^ into 1 is 1, and the
sq. of 1^ is :^.
To test the correctness of this answer, we will perform the
operation another way, 1^=|, | sq.=|=2:^.
11. Whatisthesq. of 21-? 3^? 4^? 5^? 6^? 7^?
The answer in each of these cases may be tested by chang-
ing the mixed number to an improper fraction, as 2|=f, &c.
We may in the same way square the sum of two fractions.
12. What is the sq. of ^+^? Ans. i+i+i=l.
Now i-|-i=l, the sq. of which is 1.
13. What is the sq. of f+J ? Ans. tV+i^+TV=i|- ov 1.
Now |-(-i=l, the sq. of which is 1.
14. What is the sq. of f+J? The sum of these is 2, the
sq. of which is 4 ; the answer, therefore, should be 4. Apply-
ing the formula, the operation is as follows ; 1+1+1= ^=4.
PE ACTIO AL QUESTIONS.
We will now apply the above principle to the solution of
some questions which appear at first a little difficult.
1. A boy had some apples ; he placed part of them in rows
making a square, and found he had 6 apples left. He placed
another row on two sides, and found he had enough to com-
plete the square except one apple at the corner. How many
apples were there in the first square ? How many apples
had he ?
2. Three boys were playing at marbles ; the first says, I
have just marbles enough to make a square ; and he placed
them in rows on the floor, forming a square ; the second boy
says, I have 1 2 marbles, and I will put a row on two sides of
yours, and make your square larger ; but on placing his mar-
bles he found he wanted 3 more to complete the square ; then
the third boy says, I have just three, and that will make the
square complete.
How many had the first boy ? How large was the square
which all the marbles made ?
3. The boys of a school thought, one day, at recess, they
would form themselves into a square. A part of them first
formed a square, when itrwas found that there were 15 boys
left ; these 15 then placed themselves in a row on two sides of
COMPLETING OF THE DEFECTIVE SQUARE. 80-
the square, when it was found that it required 2 boys more to
complete the square. How many boys were there in the first
square ? How many in all ?
4. A general, drawing up his soldiers in a square body, with
the same number in rank and file, found he had 55 men over
and above. He placed these in a row on two sides, and found
that he now wanted 30 men to complete the square. How
many men were there on a side of the first square ? How
many men in the first square ? How many men had he in all ?
5. There is a certain square number expressed in the terms
of its parts, that is, it is expressed in three terms, the first of
which is the square of the first part, the second is twice the
product of the two parts, and the third is the square of the
second part. Now the first two terms are 16-|-24, what is
the third term ? What is the number ?
6. There is a square number expressed in the terms of its
parts ; the first two terms are 9-|-24 ; what is the third ?
7. The first and last terms of a square are 4-(- 25 ; what^
must be the middle term ? What is the number ?
8. The first and last terms of a sq. are 9+4 ; what is the
second term ? What is the square ?
9. Complete the square whose first two terms are 16-|-40
+ D.
10. Complete the sq. 36+24+ Q.
11. Complete the sq. 4+4+ D .
12. Complete the sq. 9+6+ □.
13. Complete the sq. 16+8+IZ].
14. Complete the sq. 25+10+a.
15. Complete the sq. 25+20+CI3.
16. Complete the sq. 36+72+a.
17. Complete the sq. 25+30+a.
18. Complete the sq. 25+40+a.
The square root is the number which multiplied into itself
produces the square. Thus 3 is the sq. root of 9, 2 is the sq.
root of 4, 5 is the sq. root of 25.
The square root of a square of three terms, like those
given above, is the sq. root of the first term, plus the square
root of the third, for these multiplied by themselves will pro-
duce the square. Thus the square root of the square 9+12
+4 is 3+2, or 5. 3 is the sq. root of 9, and 2 the sq. root of
4. This number, 3+2, multiplied by itself, will produce the
square of 9+12+4.
8*
90 MENTAL ARITHMETIC.
19. Complete the sq. 36+60+n. What is its root?
20. Complete the sq. 36-f-12-(-a. What is its root?
21. Complete the sq. 16-|-44-n. What is its root?
22. Complete the sq. 9+12=|-[Z3. What is its root?
23. What is the square root of 169 ?
Divide the number into three terms, 100+60-|-9. We
divide it so because 100 is a square, and 60 is twice the pro-
duct of 10, the root of the first term, into the root of what this
way of dividing loaves us for the third term. That is, if we
take 60 for the 2d term, we leave 9 for the 3d term, and this
is as it should be, for 60 is twice the product of 10 into 3.
The root is .therefore 10+3, or 13.
24. What is the sq. root of 196 ?
We will take for the first term 100, whose root is 10. Now
as the 2d term is twice the product of the two terms of the
root, if we divide half of it by the 1st, it will give the 2d term
of the root ; or what is the same thing, if we divide the 2d
term of the square by twice the first term of the root, it
will give the 2d of the root. Now 96 contains the 2d and
third terms of the square ; we must separate it into two parts,
such that the first part, divided by twice 10, or 20, will give
for quotient the root of the second part.
Let us first try by dividing it into 60 and 36. Now 60
divided by 20 gives three, which is not the root of 36; our
division, therefore, was wrong. The 2d term was too small,
and the 3d too great. We will try again. By dividing it
into 80 and 16, we find that 80, divided by 20, gives 4, which
is the root of 16. The number 196, when arranged in the
three terms of the square, will be lOO-j-80-1-16, and the root
is 10+4, or 14.
25. What is the root of 225 ? Here we must not take for
our first term 200, for this is not a square. We must take the
largest square whose root is an even 10. This is 100. We
have remaining 125 ; this we must divide into two terms,
such that the first divided by twice 10 will give the root of the
second term. We will first divide into 80 and 45 ; 80 di-
vided by 20 gives 4, which is not the square root of 45. We
will divide into 100 and .25 ; 100 divided by 20 gives 5,
which is the exact root of 25. The number 225, therefore,
when arranged in the three terms of a square, stands 100+
100+25, and its square root is 10+5, or 15.
26. What is the square root of 256 ? Taking for the first
MENSURATION. 91
term 100, it remains to divide the remainder, 156, according
to the principle stated above. Now 120 will contain 20, 6
times, which is the root of the remainder, 36 ; the number
stands, therefore, 100-|-120-(-36 ; square root lO-f-6, or 16.
27V What is the square root of 484 ? Here we take for the
first tel-m 400, for that is the largest square whose root is in
even tens ; its root is 20. The remainder we may divide into
80 and 4 ; dividing 80 by twice 20, or 40, we have for the quo-
tient 2, which is the root of the third term. The square,
therefore, is 400-f-80+4. The root, 20+2, or 22.
28. Whatisthesquarerootof 529? of576? of 625? of 676?
29. The following is a ready method of squaring a mixed
number whose fraction is ^ ; as 2^, 3|, 4^. The fraction will
be ^ ; for the other number, multiply the whole number by a
number greater than itself by one. Thus the square of 2^;
the fraction is ^^ the whole number 2X3 or 6, 6^. The
square of 3^, 3X4, or 12, and ^. The sq. of 4^ is 4X5, or 20,
and I. What is the sq. of 5^? 6^? 7^? 8|? 9^? lOi
The same principle will apply to the square of whole num-
bers whose last figure is 5 ; as 25, 45, 55, &c., for such a
number consists of a certain number of tens, and half of 10.
As the right hand figure is 5, the two right hand figures of
the square must be 25 ; then multiply the number at the left
of the 5 by itself increased by 1, and this, read at the left
hand of the 25, will be the square. Thus, for the sq. of 25,
the two right hand figures will be 25 ; for the rest, multiply 2
by 3, which is 6 ; ans. 625.
30. What is the sq. of 35? 3X4=12; to this annex 25;
1225, ans.
What is the sq. of 45 ? of 55 ? of 65 ? of 75 ? of 85 ? of 95 ?
SECTION XVII.
PRACTICAL QUESTIONS IN SQUAEE MEASURE.
1. How many square rods are there in a square mile ?
2. How many acres are there in a square mile ?
3. Divide a square mile into 4 equal farms ; how many acres
would there be in each ?
4. How many acres are there in 1 fourth of a mile square ?
5. How many acres are there in a town 6 miles long and
5 miles broad ?
^2 MENTAL ARITHMETIC.
6. If half of the town is unfit for improvement in conse-
quence of water and mountains, how many farms of 100 acres
might be made from the other half?
7. A man bought a rectangular piece of land containing 40-
acres ; on going out with his son to measure round it, to as-
certain how much fence it would require to enclose it, they
found the first side they measured to be 160 rods.
We need not measure any more, said the son, for I can tell
all the rest in my head.
How wide was the piece ? and how many rods of fence
would it take to go round it ?
8. A man bought 7 acres of land, in rectangular form ; the
width of it was 28 rods ? what was its length ?
If a four sided piece of land is rectangular, its contents may
be found by multiplying two adjacent sides, or sides that meet
and form a corner.
12 c
Thus, ' ~
4 if a piece is 12 rods long
and 4 rods wide, the two boundaries, 12 and 4, which meet
and form the right angle at c, will, when multiplied together,
give the contents in square rods ; 12X4=48.
If the opposite sides are parallel, but the angles are not
right angles, the distance between the two sides must be
measured by a perpendicular line, thus :
— y the length, 16 rods, multiplied
/ by the width, 4 rods, will give
the contents.
If the piece is a triangle, there must be a perpendicular
Tine drawn to the longest side from the angle opposite to it.
This perpendicular we may call the height of the triangle,
and the longest side, its length; and the height multiplied
into the length will give double the area ; dividing this by 2
we get the area.
To show the reason of this, take the following figure : by
16 examining this you will see
that there are in it two trian-
gles just alike ; the length of
each is 1 6 rods, and the height
4 rods. Now 16X4: will give, as in the case above, the area of
MENSURATION.
93
the whole figure ; that is, of both the triangles ; therefore it
will give twice the area of one of them.
9. What is the area of a triangle whose longest side is 16
rods, and the perpendicular, from the opposite angle to this
side, 12 rods ?
10. What is the area of a triangle whose longest side is 24
rods, and its height 9 rods ?
11. What is the area of a triangle whose longest side is 18
rods, and height 1 6 rods ?
12. A triangle contains just one acre : its longest side is 20
rods ; how long must the perpendicular be from the opposite
angle to that side ? ■^
13. A triangle contains 2 acres ; its longest side is 32 rods ;
how long is the perpendicular, from the opposite angle to this
side?
In a right angled triangle the longest side is called the
hypotenuse ; the sides containing the right angle are called
the legs, or one the base, and the other the perpendicular. In
all right angled triangles, the square of the hypotenuse is just
equal to the sum of the squares of the two other sides. This
important principle is exhibited to the eye in the following
figure.
The hypotenuse is
divided into 5 equal
parts, and its square
is therefore 25. The
base has 4 equal parts
of the same length,
making a square of
1 6 ; the perpendicu-
lar is divided into 3
equal parts, of the
same length as the
others, which makes
a square of 9. The
square of the perpen-
dicular and of the
base added together,
16+9=25, which is
the square of the hy-
potenuse.
If we know the square of the hypotenuse, we know the sum
of the squares of the two legs. If we know the sum of the
94
MENTAL ARITHMETIC.
squares of the two legs, we know the square of the hypote-
nuse. If we know the square of the hypotenuse and of one
leg we can find the square of the other leg. And if we know
the square of any one of these sides we can obtain the length
of the side by extracting the square root.
14. In a certain right angled triangle the square of the
hypotenuse is 100 feet; what is the length of the hypotenuse ?
In the same triangle the square of the base is 64 feet ; what
is the length of the base ?
In the same triangle, what must be the square of the per-
pendicular ? What is the length of the perpendicular ?
15. A and B set out from* the same place ; A travels east
6 miles, B travels north till he is 10 miles in a straight line
from A ; how far north has B traveled ?
16. There is a triangle, the perpendicular of which is 3
feet, the hypotenuse is 5 feet ; how long is the base ?
17. A man had a piece of land in the form of a right-
angled triangle, the two legs of which were equal to each
other, and the square of the hypotenuse was 128 rods; how
many rods were there in the piece ?
The circumference of a cir-
cle is 3 times and ^ greater
than the diameter. If the di-
ameter is 1 foot, the circum-
ference will be 34- feet ; if the
diameter is 2 feet, the circum-
ference will be 6f feet.
18. If the diameter of a circle is 3 feet, what will be the
circumference ? If the diameter is 4 feet ? If 5 feet ? If 6
feet? If 7 feet?
If the diameter of a water wheel is 16 feet, what is the
circumference ?
19. If the diameter of the earth is 8,000 miles, what is
the circumference ?
To find the area of a sector of a circle, as a, e, b, multiply
the arc by half the radius. This figure may be regarded as
a triangle, the base of which is the arc, and the radius, the
ANALYSIS OF PROBLEMS. 95
height ; and you have seen before, that in a triangle, the base
multiplied by half the height gives the area. From this, we
may obtain a method of obtaining the area of the whole
circle.
Multiply the circumference by half the radius. For we
may regard the circle as made up of a great number of small
triangles, whose bases added together are the circumference
of the circle, and whose height is equal to radius ; being in
each case the distance from the circumference to the center.
20. What is the circumference of a circle whose diameter
is 14 feet?
What is its area ?
21. What is the circumference of a circle whose diameter
is 12 feet ?
What is its area, ?
22. What is the circumference of a circle whose diameter
is 20 feet?
What is its area ?
23. What is the circumference of a circle whose diameter
is 28 feet?
What is the area ?
SECTION XVIII.
ANALYSIS OF PEOBLEMS.
1. A boy spent one half the money he had, and had 1 dol-
lar left ; how much had he at first ?
2. A boy spent one third of the money he had, and had
1 dollar left ; how much had he at first ?
Ans. If he lost 1 third, he had 2 thirds left ; if 1 dollar
was two thirds, half a dollar must be one third, and f of a
dollar the whole. He had 1 dollar and a half.
3. A boy spent ^ of his money, and had 1 dollar left ; how
much had he at first ?
Let this and the following answers be given in form of a
fraction, like the preceding answer.
4. A boy spent ^ of his money, and had 1 dollar left ; how
much had he at first ?
96 MENTAL ARITHMETIC.
5. A hoj spent J of his money, and had 1 dollar left ; how
much had he at first ?
6. A boy spent | of his money, and had 1 dollar left ; how
much had he at first ?
7. A boy spent ^ of his money, and had 1 dollar left ; how
much had he at first ?
8. A boy spent ^ of his money, and had 1 dollar left ; how
much had he at first ?
9. A boy spent -^ of his money, and had 1 dollar left ;
how much had he at first?
10. A man carried some corn to mill; the miller took -^
of it for toll, and then there was just a bushel; how much
did the man carry to mill ?
11. A man carried some cloth to be fulled; it shrank two
sevenths in its length, and was then just a yard long; how
long was it at first ?
12. A man drew a prize in a lottery ; ^ of the prize was
retained, and then the drawer received just 100 dollars ; how
much was the prize ?
13. If a stick of timber shrink ^ in weight in seasoning,
and then weigh 100 pounds, how much did it weigh at first ?
14. A teamster sold f of a cord of wood, and then had
half a cord left ; how much had he at first ?
15. A man had an estate left him by his father; he lost J
of it; he then received 1000 dollars, and then he had 3500
dollars ; how much had he at first ?
16. A merchant began trade with a sum of money, and
gained so as to increase his original stock by ^ of itself; he
then lost 500 dollars, and had 4,500 dollars left ; how much
did he begin with ?
17. A man set out on a journey, and spent half the money
he had for a dinner ; he then paid half of what he had left for
provender for his horse ; then, half of what now remained
for toll in crossing a bridge, and had 10 cents left; how much
had he at first ?
18. A boy spent f of his money for a book, and i^ of it for
some paper, and had 8 cents left ; how much had he at first ?
19. A boy playing at marbles lost, in the first game, ^ of
what he had ; in the second game, I of what he then had ;
in the third, ^ of what he then had; in the fourth 11, and
then he had 16 marbles left; how many had he at first ?
20. A boy playing at marbles, wins in the first game so as
ANALYSIS OF PROBLEMS. 97
to double the number of marbles he had ; in the second game,
he loses ^ of what he then had ; in the third game, he loses
5, and then finds he has just as many as at first ; how many
had he at first ?
21. A man had his sheep in three pens ; in the first ther^
were 10 sheep, in the second there were as many as in the
first, and half the number in the third ; in the third there
were as many as in the first and second; how many had he
in all ? ■
22. In an orchard ^ of the trees are plumb trees ; there
are 20 cherry trees ; and the apple trees, which constitute the
remainder, are half as many as the plumb and cherry trees
added together ; how many trees are there in the orchard ?
23. John and William were talking of their ages, John
says, I am twielve years old ; William says, if half my age,
were multiplied by one fourth of yours, and half your age plus
one subtracted from the product, that would give my age.
How old was he ?
24. A man talking of the age of his two children, said
the youngest was three years old ; the age of the eldest was
^ his own age; if his own age was divided by that of his
youngest, and once and one third the age of the youngest
subtracted from the quotient, that would give the age of the
eldest; how old was the eldest?
25. The number of pupils in a school is such that if you
take half of them, and increase that by 2 ; then take one
third of this last number ; and increase it by 3, and from this
number subtract 6, the remainder will be 7 ; how many are
there in the school ?
26. A boy plays three games at marbles ; in the first, he
loses a certain number ; in the second, he gains 8 ; in the
third, he loses 4, and then he finds he has two more than he
began with ; how many did he lose in the first game ?
27. A boy playing at marbles first lost one third of what
he had ; he then doubled his number, when he had 5 marbles
more than he had at first ; how many had he at first ?
28. There is a certain number, one third of which exceeds
one fourth of it by 2 ; what is the number ?
29. There is a certain number, one fourth of which exceeds
one fifth of it by 1 ; what is the number ?
30. There is a certain number, one third of which added
to one fifth of it amounts to 16 ; what is the number?
9
98 MENTAL ARITHMETIC.
31. There is a number, one third, one fourth, and one fifth
of which added together are 94; what is the number?
32. What is that number a fifth of which exceeds a sixth
of it by 4?
33. What number is that of which a fourth part exceeds
a seventh part by 9 ?
34. In a certain orchard there are apple, peach, and pear
trees ; the apple trees are two more than half the whole ; the
peach trees are one third of the whole, and are 14 less than
the apple trees ; the rest are pear trees ; how many are there
of each kind, and how many in all ?
SECTION XIX.
SOLID MEASURE.
Whatever has length and breadth, and thickness is a solid.
A block of wood 1 inch long, 1 inch high, and 1 inch wide,
is a solid inch. A block 1 foot long, 1 foot wide, and 1 foot
high, is a solid foot. A block 1 yard long, 1 yard wide, 1
yard high, is a solid yard.
1. How many solid inches are there in a block 3 in. long, 2
in. wide, and 1 in. high ?
2. How many in a block 4 in. long, 3 in. wide, and 2 in.
high?
3. How many solid feet are there in a block 5 feet long, 3
feet wide, and 2 feet high ?
4. How many solid feet in a block 7 jTeet long, 2 feet wide,
and 2 feet high ?
When a solid has its length, height, and breadth, equal to
each other, it is called a cube ; and the linear measure of its
length, height, or breadth is called the root of the cube. We
have seen what is a cubic inch, a cubic foot, and a cubic yard.
Suppose now we have a pile of cubic inch blocks, and we
wish to construct from them a cube, each of whose dimensions
shall be 2 inches ; we will first take 2 blocks, and place them
down side by side ; this will be as long as the required figure,
but it will not be wide enough, nor high enough j to make it
SOLID MEASURE. 99
wide enough, we will place 2 more blocks-down by the side
of the former ; the figure now contains 4 cubic inches, and is
2 inches long, and 2 inches wide, but it is only 1 inch high.
To make it 2 inches- high, we must place upon this another
layer of 4 blocks arranged just like the former. The figure
will then be 2 in. long, 2 in. wide, and 2 in. high ; it contains
8 cubic inches, and is the cube of 2.
5. How many blocks will you require, and how will you
arrange them to make the cube of 3 ? '9
6. How many blocks will you re(|uire, and how will you
arrange them to make the cube of 4 ?
7. How many blocks will you require, and how will you
arrange them to make the cube of 5 ?
The cube when expressed in numbers is the same as the
3d power of the root. It is found by taking the root 3 times
as a factor. Thus the 3d power of 2 is 2X2X2=8. The
3d power of3 is 3X3X3=27. Of 4, is 4X4X4=64. Of
5, is 5X5X5=125.
In this way we may find the 3d power of any number.
8. How many blocks, of a cubic foot each, will it take to
form a cubic solid, 6 ft. on a side ?
9. How many blocks of a cubic foot each will it take to form
a cubic solid of 7 feet each way ?
10. How many cubic feet will it take to form a cube of 8
feet?
11. How many cubic feet will it take to form a cube of 9
feet?
12. How many cubic feet will it take to form a cube of 10
feet?
13. How many cubic inches are there in a cubic foot ?
• 14. A pile of wood 8 feet long, 4 feet high, and 4 feet wide,
makes a cord ; how many cubic feet are there in a cord ?
15. A pile of wood, 4 feet loog, 4 feet high, and 1 foot
thick, makes what is called a cord foot ; how many cubic feet
are there in a cord foot ?
16. How many cord feet are there in a cord?
17. There is a pile of wood 40 feet long, 4 feet wide, and 5
feet high ; how many cords does it contain ?
18. There is a stick of hewn timber 25 feet long, 1 foot
wide, and 1 foot thick ; how many cubic feet does it contain ?
19. -There is a tree from the but-end of which a stick may
Se hewn 13 feet long, 2 feet wide, and 2 feet thick ; how many
•rnbic feet will it contain ?
100 MENTAL ARITHMETIC.
20. It is estimated that 50 feet of hewn timber weigh a
ton ; if 50 cubic feet weigh 20 cwt. net weight, what will 1
foot weigh ?
21. If you divide a cubic inch into blocks measuring ^ an
inch each way, h.ow many such will there be in a cubic inch ?
22. How many cubic half inches are in a cubic inch ?
23. If you divide a cubic inch into cubes of ^ of an inch
each, how many such will there be ?
24. How many cubic quarter inches are there in a cubic
inch?
25. How many cubic inches are there in a cube of one inch
and a half?
26. K a man digs a cellar at the rate of f of a dollar for
a cubic yard, what^ill the job come to, if the cellar is 18 feet
long, 12 feet wide, and 6 feet deep ?
27. A stone-layer agreed to build a solid wall 30 feet long,
4^ feet thick, and 6 feet high, for 2^ dollars a cubic yard ,
what did the wall cost ?
CONSTRUCTION OF THE CUBE.
We have seen that the third power, or cube of any number,
is obtained by taking the number three times as a factor ; the
product is the cube, or third power.
In this way the cube of any number whatever may be ob-
tained. There is another way, however, of constructing the
cube, the knowledge of which is very important in the ope-
ration of extracting the cube root.
Suppose we wish to find the cube of 5. Instead of taking
5 three times as a factor, thus, 5X5X5=125, we wiU re-
gard the number 5 as consisting of two parts, 3 and 2. We
will call 3 the first part, and 2 the second part of 5.
We will begin by making the cube of the first part 3, thus,
3X3X3=27.
We will regard this as a cube of 3 inches, that is, 3 in. long,
3 in. wide, and 3 in. high ; and represent it by the following
figure.
The question now is, how shall we enlarge this cube of 3, so
as to make it the cube of 5 ? It is evident, it must be 2 in.
longer, 2 in. broader, and 2 in. higher than it now is. We
CONSTRUCTION O^ TEE CUBE. 101
y^Hf begin* <ten by putling '2 layers
of inch blocks on the front side, 2
layers on the right side, and 2 layers
on the top. The figure thus en-
larged is not a cube. There are sev-
eral places not filled up. It is nearer
the cube of 5 than it was before;
but something more must be added.
Before making that addition, how-
ever, let us see what we have done. The figure is the cube
of 3, which is the first part of 5. To this there are 3 equal
additions made. Each of these additions is 3 in. square, and
2 in. thick. Now 3 is the first part of 5 ; each addition,
therefore, contains the square of the first part, 3, multiplied
by the second part, 2, or 3^X2; therefore the three addi-
tions will be 3 times the square of the first part multiplied
by the second.
The whole figure, therefore, after these three additions are
made, contains S^-j-S times 3^X2.
We will now see what additions must next be made to the
figure.
There are 3 places that need filling up, each 3 in. long,
2 in. wide, and 2 in. high. Each of these new additions is
2 in. square, and 3^ in. long. It consists, therefore, of the first
part of 5 multiplied by the sq. of the second ; and the three
together are 3 times the first part multiplied by the sq. of
the second. There is one addition wanting to complete the
cube ; that is at the corner. It must be 2 in. long 2 in.
wide, and 2 in. high ; that is, the cube of 2 ; or, in other
words, the cube of the 2d part.
Remembering that the two parts of 5, as here divided, are
3 and 2 ; the printed figure is the cube of the first part, the
first addition is 3 times the square of the 1st part, multiplied
by the 2d ; the 2d addition is 3 times the first part, multiplied
by the sq. of the 2d ; the 3d addition is the cube of the 2d
part. Let the letter a stand for the first part, 3 ; and the
letter b, for the 2d part, 2. The printed figure will then be a^ ;
the first addition, 3a2b ; the second addition, Sab^ ; the third
addition, b^. The whole cube, therefore, will be a^-j-Sa^b-j-
3ab2-|-b3. Observe that the letters and numbers are to be
multiplied together though there is no sign of multiplicatiou
9*
102 MENTAL AKITaMETIC.
betweenthem/^ da^b i«*;t]liree. iimes the square of a, multi-
plied by b\ ' '- .
These are called the four terms of the cube, when the root
is in two parts.
If we express the above in the numbers for the cube of 5,
it will stand thus
-~r^ 1st. 2d. 3d. 4th.
y^y 33+3x32x2+3X3X22+2^.
1. What number makes the 1st term of this cube?
2. What number forms the 2d term ?
3. What number forms the 3d term ?
4. What number forms the 4th term ?
5. What do all the 4 terms amount to ?
6. Which of the four terms contains the third power of the
first part ? Which contains the 2d power of the first part ?
7. Which contains the first power of the first part ? Which
term contains the 1st power of the second part ? Which the
2d power ? Which the 3d ?
8. If the 4th term of the above cube were not given, how
could you determine from the others what it must be ?
9. If the 3d term were gone, how could you restore it ? If
the 2d was gone how could you restore it ?
10. If you divide the number 5 into the two parts, 4 and 1,
and express the cube according to the above rule, what will
the 1st term be ? What will be the 2d term ? What will be
the 3d term? What the 4th?
Remember here, that all powers of one are one, — neither
more nor less.
Divide the number 6 into 4+2, and form the cube accord-
ing to the above rule.
11. What will the 1st term be? The 2d? The 3d?
The 4th? ,
12. What will they all amount to
Multiply 6 into itself 3 times, thus, 6X^X6, and see if it
amounts to the same.
13. Divide 6 into the parts 5+1, and form the cube.
What will be the 1st teym ? The 2d ? The 3d ? The 4th ?
What do they all amount to ?
1 4. There is a cube in 4 terms, the first two terms of which
are 33+3X3^X1 ; what must be the 3d term? AYhat the
4th term ? What is the number of the cube ? What is the
THE CUBE. 108
root of the cube ? This root is the cube root of the first
term, added to the cube root of the last.
15. There is a cube in 4 terms, the first two of which are
23-|-3X22X2; what is the third term? What the fourth?
What is the whole cube ?^ What is the first part of the root ?
What is the second part ? What is the whole root ?
16. Complete the cube 43-f3x42x2-H:ZI+23.
What is the number of the cube ? What the root ?
17. Complete the cube 33+3x32 x3+3X3x32-fa.
What is the number of the cube ?
18. There is a cube in 4 terms,. the first of which is 1000;
what is the first part of the root ?
19. The second term of the same cube is 300X6, or 1800 ;
what is the third term ?
20. What is the fourth term of the same^cube ?
21. AVhat is ihe root of the above cube ?
22. The first term of a cube is 1000, the second is 300X8,
or 2400 ; what is the third term ?
23. What is the fourth terra of the abov6 cube ?
24. The first term of a cube is 8000, the second is 1200X3 ;
what is the third term ?
Observe that 1200 is 3 times the square of the first term;
consequently, one third of it is the square of the first term.
25. What is the fourth term in the above cube ? What is
the root ?
SECTION XX.
RATIO. —PROPORTION.
If we compare the two numbers, 3 and 9, in order to ascer-
tain their relative magnitude, we may subtract 3 from 9 ; we
find the difference to be 6.
There is another way of comparing4he two numbers. We
may see how many times 3 will go in 9 ; we shall find the
quotient to be 3.
The numbers we obtain in each of these comparisons is
called the Ratio of the two numbers ; but they differ in kind ;
104 MENTAL ARITBMETIC.
the former. is called Ai'ithmetical ratio; the latter, Geometri-
cal ratio.
Arithmetical ratio, then, expresses the difference of two
numbers ; Geometrical ratio expresses the quotient of one
of the numbers, divided by the otlier. As we shall speak
only of geometrical ratio in what follows here, the word
ratio whenever it is used, may be understood to mean geo-
metrical ratio. The ratio of 4 to 2, written 4 : 2, is 2 ; for 2
will go in 4 twice. The ratio of 12 to 3, written 12 ; 3, is 4;
for 3 will go in 12, 4 times.
The two numbers compared, are together called the terms
of the ratio, or simply, the ratio ; the first is called the Ante-
cedent, the second is called the Consequent. These two terms,
you will perceive, correspond exactly to the numerator and
denominator of a fraction ; for in a fraction, the numerator is
divided by the denominator. A ratio is then another way of
expressing a fraction. The antecedent is the numerator ; the
consequent the denominator. 4 : 2 is the same as f ; 6 : 3, the
same as §.
As a ratio is essentially the same as a fraction, everything
is true of a ratio which is true of a fraction.
1. What effect will it have on the value of the ratio, if
you increase the antecedent ? if you diminish the antecedent ?
if you double the antecedent ? if you divide the antecedent
by 2?
2. What effect will it have on the value of the ratio, if
you increase the consequent? if you diminish the conse-
quent ? if you multiply the consequent ? if you divide the
consequent ?
3. Take the ratio 4:2; how can you multiply it by 2 ? In
what other way ?
4. How can you divide it by 2 ? In what other way ?
5. Take the ratio 6 ; 3 ; how can you multiply it by 2 ?
Can you do it in more than one way ? If you cannot, why ?
6. How can you divide it by 4 ? Can you do it in more
than one way ? If not, why ?
Take the ratio 4:2; multiply both terms by the same num-
ber, 3, for example ; it will be 12 : 6 ; you see the value is not
altered ; divide both terms 4:2 by 2 ; it will be 2 : 1 ; tho
value is not altered ; it is still 2.
PROPORTION. 105
PROPORTION.
If. there are four numbers, and the first has the same ratio
to the second that the third has to the fourth, the four num-
bers are said to be in proportion. Thus "the numbers 2:1;:
12:6, are in proportion. The first has the same ratio to the
second, that the third has to the fourth. The ratio is 2.
A Proportion then is the equality of two Ratios.
The four dots : : between the two ratios, are the same as
the sign of equality, =.
In order to preserve the proportion, the two ratios must al-
ways be equal to each other. You may make any change
you please in the terms, provided you do not destroy this
equality.
Let us take the proportion 4 : 2 : : 12 : 6 ; the value of the
two ratios is now equal.
1st. Multiply the antecedents by 2 ; 8 : 2 :: 24 : 6 ; the
numbers are still in proportion, for the value of the two ratios
is equal.
2d. Divide the antecedents by 2 ; 2 : 2 : : 6 : 6 ; the value
of the two ratios is equal.
3d. Multiply the consequents by 2 ; 4 : 4 : : 12:12; the
value of the two ratios is equal.
4th. Divide the consequents by2;4:l::12:3; the value
of the ratios is still equal.
5th. Multiply both terms of the first ratio by 2 ; 8 : 4 : : 12 :
6 ; or multiply the 2 terms of the second ratio by 2 ; 4:2::
24 : 12 ; the ratios are still equal. In the same way we might
take any other number for our operations instead of 2 ; the
same operations might be performed without destroying the
proportion.
The two middle terms of a proportion are called the means ;
the first and last terms are called the extremes.
In a proportion the product of the two means is equal to
the product of the two extremes.
Take the proportion 4 : 2 : : 6 : 3 ; the product of the means
2X6 is 12 ; and the product of the extremes, 4X3, is 12.
Take the proportion 6 : 2 : : 9 : 3 ; 2X9=18, 3X6=18.
Take the proportion 10 .: 2 : : 30 : 6 ; 2X30=6X10.
If we know, then, the product of the means, we know the
product of the extremes.
106 MENTAL ARITHMETIC.
In a certain proportion the product of the means is 30 ;
what must be the product of the extremes ?
6. Further, if we know the product of the means, and if
we know one of the extremes, we can find the other. If, as
in the above case, the product of the means is 30, and if one
of the extremes is 3, what must be the other ?
How do you find that number ?
7. If the product of the means is 30, and one of the ex-
tremes is 10, what must the other be ?
How do you find the number?
8. If the product of the means is 30, and one of the ex-
tremes is 5, what is the other ?
If one of the extremes is 6, what is the other ?
If one of the extremes is 15, what is the other?
You see, therefore, that if you muUiply the means togeth-
er, and divide the product by one extreme, the quotient will
be the other extreme.
9. If the product of the means is 72, and one of the ex-
tremes is 24, what will the other be ?
10. If the first 3 terms of a proportion are 9:6:: 12,
what must the fourth term be ?
11. "What is the fourth term of the proportion 5 : 3 : : 15?
12. Complete the proportion 8 : 6 : : 12.
13. Complete the proportion 14 : 8 : : 7.
14. Complete the proportion 10 : 4 : : 15.
By means of this rule, many interesting questions may be
solved.
15. If 8 yards of cloth cost 6 dollars, what will 20 yards
of the same cloth cost?
It is evident, that the length of the shorter piece is to the
length of the longer, as the cost of the shorter, 'is to the cost
of the longer. Now we know all these numbers except the
last, and can express them in a form of a proportion, thus,
Yd«. Yds. Dolls.
8 : 20 : : 6. 8 is the length of the shorter piece ; 20, the
length of the longer ; 6 is the cost of the shorter piece. The
fourth term, that is, the cost of the longer piece, we have not
yet found ; you must discover that yourself; how can you do
it?
16. If 18 yards of cloth cost 15 dollars, what will 12 yards
of the same cloth cost ?
What do you here seek, — the quantity of clothe or tho
price ? Is it the price of the longer, or of the shorter piece ?
PROPORTION. 107
How can you make a proportion with the two quantities
of cloth, and the two sums they cost ? State this proportion
in general terms, putting the thing sought as the fourth term.
State the proportion in figures all except the fourth term.
How will you find the 'fourth term ?
17. If 5 yards of cloth cost 2 dollars, what will 7 yards of
the same cloth cost ?
State the proportion in general terms.
State the first three terms in figures, and find the fourth.
18. If a horse travels 16 miles in 3 hours, how far will
he travel in 2 hours ?
As the longer time is to the shorter time, so is the greater
distance to the smaller distance.
Remember that things of the same kind should stand in
the same ratio ; and that the quantity sought must be the
fourth term. Then inquire what the true proportion must be,
and state it in general terms, repeating the trial, if neces-
sary, till you perceive that you are right. This is far better
than any special rule, for it leads you to reason on what you
do.
19. If a certain number of cubic feet of timber weighs a
certain number of hundred weight, and if we wish to know,
without weighing, how many hundred weight a certain small-
er number of cubic feet will weigh,' what will be the propor-
tion in general terms ?
20. If 16 cubic feet of wood weigh 5 cwt, what will 6
cubic feet weigh ?
21. If 3 barrels of flour last a family 7 months, how many
barrels will last them 12 months ?
22. If an iron rod of equal size throughout and of a certain
length, Aveighs a certain number of pounds, and is broken into
two parts, not in the middle, how can you find the weight of
one of the parts without weighing it ?
23. If an iron rod 7 feet long weighs 20 pounds, what will
5 feet of it weisrh ?
COMPARISON OF SBHLAR SURFACES.
As all the above questions may be answered by analysis aa
well as by proportion, the rule of proportion might be dis-
pensed with for the purposes of solving this kind of questions
108 MENTAL ARITHMETIC.
It has, however, very important and interesting applications
in the measurement of similar surfaces and solids. To pre-
pare for this, you must attend carefully to a few introductory
statements.
Two surfaces are similar to each other when they are
shaped alike, though they may be unequal in size. Thus a
large circle is similar to a small circle, for they are both
shaped alike.
So one square is similar to another, though they may be
unequal in size. One equilateral triangle is similar to anoth-
er equilateral triangle.
If a rectangle is twice as long as it is wide, and a larger
or a smaller rectangle is twice as long as it is wide, the two
are similar. In the same way any two surfaces, however
irregular their shape, are similar, provided they are shaped
alike. .
We will now come to a stricter definition of similar sur-
faces. Similar surfaces are such as have their corresponding
dimensions proportional. Take the circle : the dimensions of
the circle are the diameter and the circumference ; the "diam-
eter of one circle is to its circumference, as the diameter of a
larger or a smaller circle is to its circumference. For you
have learned before that the circumference of a circle is 3^
times its diameter.
Take the square ; one side of a square is to another side
of it, as one side of a larger or smaller square is to another
side of it.
If a rectangle is twice as long as it is wide, another rec-
tangle in order to be similar, whatever be its size, must be
twice as long as it is wide.
1 . There are two similar rectangles ; one is 8 feet long
and 6 feet wide ; the other is 6 feet long ; how wide is it ?
2. There are two similar rectangles ; one is 5 feet long and
2 feet wide ; the other is 11 feet long ; how wide is it ?
3. There are two similar right-angled triangles ; in the
largest the base is 9 ; the perpendicular 4 ; in the smallest the
base is 8 ; how long is the perpendicular ?
We will now come to the comparison of the areas of
similar surfaces. .
4. There are two circles ; one is 1 foot in diameter, the
other 2 feet ; how much greater is the area of the larger, than
the area of the smaller ?
SIMILAR FIGURES. 109
It is clearly more than twice as large, for you could lay
two of the smaller circles on the larger, and still leave a
considerable space uncovered. Before answering this ques-
tion,, we will take the simple case of two squares, one of
which measures 1 foot on a side, and the other 2 feet. You
perceive the larger one is 4 times as great as the smaller.
Let one square measure 2, feet on a side ; the other, 4 feet ;
how much greater is the larger than the smaller ? The
smaller contains 4 square feet ; the larger, 16 ; it is therefore
4 times as large.
5. If one square measures 3 times as much on a side as
another, how much greater is its area than that of the smaller ?
Let one square measure 1 foot on a side ; the other, 3 feet ;
in what ratio are their areas ? Let one measure 2 feet on a
side ; the other, 6 ; in what ratio are their areas ?
The area of the larger you find is 9 times as great as that
of the smaller. This may serve to suggest the principle by
which the areas of all similar surfaces may be compared.
The areas of similar surfaces are to each other as the squares
of their corresponding dimensions.
Let one square measure 1 foot on a side ; another, 2 feet.
12; 22:: 1:4.
Let one square measure 2 feet, and another, 4 feet on a side.
22: 42:: 4: 16.
Let one square measure 1 foot on a side ; another, 3 feet.
12: 32:: 1:9.
Let one square measure 2 feet on a side ; another, 6 feet.
22 : 62 ::4:36.
This principle applies to circles, triangles, and all s^^ilar
surfaces whatever. You can now recur to question 4, and
find the answer to it.
6. There are two circles ; the diameter of the greater is 3
times that of the smaller ; the area of the smaller is 1 acre ;
what is tlie area of the greater ?
7. There are two circles ; the diameter of the smaller is 2
thirds that of the greater ; and the area of the smaller is 4
acres ; what is the area of the greater ?
8. There are two similar triangles ; «the corresponding
dimensions ar^ as 3 to 4, and the greater contains 1 2 acres ;
what does the smaller contain ?
< ]0
no • MENTAL ARITHMETIC.
9. A farmer fenced a triangular piece of ground for a field,
but finding it not large enough, he enlarged it, making each
side 1 third greater than before, and it then contained 5 acres ;
how much did it contain at first ?
10. There is an irregular field containing 8 acres ; one of
the sides measures 20 rods ; if the field be enlarged, retain-
ing the same form, so that the above named side measures 25
rods, how much land will it contain ?
11. There are 2 circles ; the smaller is 3 rods, the latter 7
rods in diameter ; how much greater in proportion is the area
of the latter than that of the former ?
12. There are 2 circles, one with a diameter of 3 feet, the
other of 8 ; how much greater is the area of the larger than
that of the smaller ?
COMPARISON OF SIMILAR SOLIDS.
We now come to the comparison of similar solids.
1. Let there be 2 cubes; one of them measuring 1 inch on
a side, the other 2 inches ; how much greater is one than the
other?
You will perceive, by thinking of the construction of the
cube, that the cube measuring 2 inches has in it 8 cubic
inches, and is therefore 8 times as great as the one measuring
only 1 inch.
2. Take cubes measuring 1 inch and 3 inches; how much
greater is the latter than the former ?
3. Let one measure 1 inch, the other 4 inches ; how much
greater will the larger cube be ?
These examples may suggest the principle on which all
similar solids are compared.
/Similar solids are to each other as the cubes of their correS'
ponding dimensions.
Take now the first of the above three examples ; the ratio
of the corresponding dimensions is 2 : 1, the cubes of these
terms, or, 2^:13 are 8 : 1, and this is the proportion of the
one solid to the other.
Li the second example, the ratio of th^ corresponding di-
mensions is 3:1, the cubes of these 3^ : 1^, are 27 : 1, and
this is the ratio of the two solids.
COMPARISON OF SIMILAR SOLIDS. Ill
In the third example the ratio of the corresponding dimen-
sions is 4:1; the cubes of these terms, 4^ : 13 are 64 : 1,
which is the ratio of the two solids to each other.
4. There are 2 iron balls ; the smaller is 1 inch, the other 5
inches in diameter ; how much does the larger weigh more
than the smaller?
5. There are 2 iron balls ; their diameters are 2 inches
and 3 inches ; what is the ratio of their weight ?
6. If the diameter of 2 balls is respectively 3 inches and 4
inches, what is the ratio of their weight ?
7. If a cubic inch of stone weigh 1 ounce, how many ounces
would a cubic stone, measuring 10 inches, weigh ?
8. How many ounces, if the cube measured 11 inches?
9. How many ouncps, if the cube measured 12 inches?
10. If there were a smaller pyramid, of the same material
and shape with the great pyramid of Egypt, and of ^ its
height, how many such would it take to equal in solid contents
the great pyramid ?
11. A common brick weighs 4 pounds, and is 8 inches in
length ; how much will a similarly -shaped brick weigh, that
measures 16 inches in length?
12. If an axe 4 inches wide weighs 4J pounds, what will
be the weight of a similar axe 5 inches wide ?
13. If a blacksmith's anvil, 1 foot long, weighs 200 pounds,
how much will a similar anvil weigh that is 2 feet long ?
14. A farmer sells 2 stacks of hay of the same shape and
solidity ; the smaller is 10 feet high, and is found to weigh 3
tons ; the larger is 15 feet high ; how can its weight be deter-
mined without weighing it, and what will the weight be ?
15. There are 2 similar cisterns, the smaller is 6 feet deep
and holds 500 gallons ; the lai'ger is 8 feet deep ; how many
gallons will it contain ?
These operations will be rendered more easy, if, in every
case where the ratios may be reduced, you reduce them to
their lowest terms.
16. If a coal-pit 8 feet high has required 10 cords of wood,
how much wood would be required, for a coal-pit of similar
shape 10 feet high ?
17. If there are two trees shaped alike, the smaller meas-
uring 4 feet in circumference, the larger 5,^et, how will the
amount of wood in the one compare with that in the other ?
112 MENTAL ARITHMETIC.
The principle given above applies to all similar solids,
whether bounded by plain surfaces, or by curved surfaces.
18. If a dwarf measures 2 feet in height, and a man of the
aame form and solidity, 6 feet high, weighs 180 pounds, how
many such dwarfs would equal the weight of the man ? What
would the dwarf weigh ?
19. If a man 6 feet 2 inches in height weighs 200 pounds,
what would be the weight of a giant of equal solidity and
similar form, 9 feet 3 inches in height ?
20. If an animal 4 feet high weighs 600 pounds, what will
an animal of the same form and equal solidity weigh, whose
height is 5 feet ?
21. An artist in Europe has made a perfect model of St.
Peter's Church at Rome, representing every part in exact
proportion, on a scale of 1 foot to 100 feet ; if the material of
the model is of the same solidity with that of the church, how
many times greater is the solid contents of the church than
that of the model ?
22. If a granite obelisk were constructed in the precise
form of the Bunker Hill monument, of one tenth its height,
how many such obelisks would the monument furnish material
to constrtict ?
The comparison of similar surfaces and solids by proportion
has various interesting applications in determining the com-
parative strength of timbers and materials used in building,
and in other arts.
Case First. — The strength of materials to resist a strain
lengthwise.
1. If an iron rod half an inch in diameter will hold a cer-
tain weight suspended by it, how much greater weight will a
rod hold that is 1 inch in diameter ?
Here the strength is in proportion to the size, without re-
gard to the length ; that is, as the square of the diameters.
2. If an iron rod half an inch in diameter will suspend 2
tons, what weight will a rod susp^id that is three fourths of
an inch in diameter ?
3. A builder finds that an iron rod 1 inch in diameter will
suspend a certain weight ; he wishes, however, to add to the
weight half as much more, and, in order to support it, substi-
STRENGTH OF MATERIALS. 113
tiites for thfi inch rod another rod l^^^ inches in diameter j will
it sustain the required weight ?
4. There are two ropes of the same material ; one, 1^ inches
__ in diameter ; the other, 2 inches ; what is the ratio of their
strength ?
Case Second. — The strength of beams to resist fracture
crosswise. In beams of the same material, length and width,
but of diflferent depth, the strength varies, as the square of the
depth,
1. There are two beams of equal length, but the depth of
one is 10 inches ; of the other, 12 inches; what is the ratio of
their strength ? . '
2. There is a stick of timber 4 inches thick and 12 inches
deep ; if sawed into three 4 inch joists, what part of the former
strength of the whole stick, when placed edgewise, will each
part possess, allowing nothing for waste in sawing ?
3. There is a stick of timber 10 inches in depth ; if 4 inches
of its depth be removed, what will be its strength compared
to what it was before ?
4. There are two sticks of timber, equal in length and
width ; one, 7 inches deep ; the other, 5 ; what is the ratio of
their strength ?
5. If a stick of timber 6 inches deep have 2 inches of the
depth removed, will it be weakened more than one half?
What is the exact ratio of its present, compared with its
former strength ?
6. A builder went to a lumber-yard, wishing to obtain an
oak beam 5 inches wide and 10 inches deep ; the lumber-
merchant said, " I have not such a stick ; but I have two oak
sticks of the right length and width, and 7 inches deep ; they
will both, placed side by side, be stronger than one beam 10
inches deep." " Not so strong," said the builder.
Which was right ? and what is the ratio of strength in the
two cases ?
10*
NOTES TO PART FIRST.
Note 1. — Page 15.
This exercise should be often reviewed till the pupils can gi)
through it with ease, and without mistake. No exercise can be
devised that will more rapidly increase the learner's powers in Ad-
dition.
Note 2. — Page 16. To the Instructor.
The word complement means, something to fill up. In arithme-
tic, the complement of a number, strictly speaking, is that number
which must be added to it, to make it up to the next higher order.
The complement of a number consisting of units only, as 3, 7, 9,
is the number that must be added to make it up to 10, and con-
sists of units only. If the number consist of tens, as 20, 50, its
complement is the number that must be added to make a hundred,
and consist of tens. If the number is hundreds, its complement is
so many hundreds as will make up a thousand.
If the number consist of several orders, its full complement
will consist of the same orders, of such an amount as to raise the
-gum to the next order above the highest named in it. The com-
plement of 745 is 255, for 745-[-255— 1000, which is the order
next above the highest named In the given sum.
The more restricted use of the word, as employed in the text, is
sufficient for the purposes here had in view.
A few suggestions will here be made in reference to the best
mode of conducting the accompanying recitation. The object of
the lesson is to cultivate the po^ver of instantly associating a num-
ber and its complement together. In conducting tha recitation, the
answer to each question as it is given out, should be required sim-
ultaneously by the whole class. The teacher should stand before
them, and require that every eye be fixed on him. The questions
should not be hurried, but the class should be encouraged to an-
swer instantly on hearing the question. This will be easy in the
first class of numbers given, which are even tens. In regard to
NOTES TO PART FIRST. 115
the remaining numbers, however, which are not even ti?ns, some-
thing more will be necessary. Suppose the question is, what is the
complement of 37 ? it may be conducted as follows :
Teacher. What is the complement of — 30 ?
Class^ 70. ^
Teacher. Now listen to me without speaking ; what is the com-
plement of 30 ? you observe, I am going to say something
more ; what will it be ?
Class. Something between 30 and 40.
Teacher. "Well then, whereabouts will the complement be
found ?
Class. Between 60 and 70.
Teacher. Very good ! Now when I say 30, and keep my voice
suspended, showing that that is not all, what number can you think
of, that you know will be a part of the complement ?
Class. 60.
Teacher, Very well. Now listen ; what is the complement of
80 ? what have you now in your mind ?
Class. 60.
Teacher. Well, now once more listen, and all answer as soom
as you hear the question ; what is the complement of 3 7 ?
Class. 63.
In the following questions, let the teacher always make a short
pause between pronouncing the tens, and the units ; and if the
class hesitate or disagree in their answer, let the question be re-
solved into its elements, and each one presented separately. Thus,
if 64 is the number, and the class have not answered promptly
and alike, say thus, — what is the complement of 60 ?
Class. 40.
Teacher. What is the complement of 60 ■ ? what do you
think of?
Class. 30.
Teacher. Now answer all together ; what is the complement of
64 ?
Class. 36.
In the examples of addition that follow, the teacher should make
% pause between the two numbers, and see that every member of
the class is intent and eager to catch the second number, and an-
swer instantly. A few questions answered by the whole class in
this way, will benefit them more than whole pages recited in an
indolent and listless manner.
Note 3. — Page 17.
In these and all other examples, the large numbers should be
taken first. If the pupil begins with the units, as in written
arithmetic, he should be checked at once. Such a method would
116 NOTKS TO PART FIRST.
only lead to a laborious imitation of the j)rocess of written arith-
metic, which is not the natural one, and could give no new power
to the pupil, nor awaken any new interest in the study. Only a
small portion of these questions should be recited at one lesson.
Note 4. — Page 23.
Care must be taken here that the pupil does not imitate the
process of written arithmetic, but be required to regard every
number in its true value. Thus in the question, what is one fifth
of 250 1 he must not say 5 In 25 is contained 5 times: and 5 in 0, no
times ; but one fifth of 25 is 5 ; therefore, one fifth of 250 is 50.
Note 5. — Page 25.
In the higher as well as the lower numbers, let the pupil grapple
at once with the number as It stands. In this way his interest will
be very much increased. He will see, throughout, the progress he
is making; whereas, in written arithmetic as usually studied, the -
pupil has no sooner begun an operation than he loses sight of the
process, and goes on in blind bondage to his rule, till he comes out
at the end, and then looks to the book, as to an oracle, for the
answer.
Let the oldest class in arithmetic In a school be called up, and
one of them be required to perform on the board the question,
*' what is one sixth of 43,248 ? " and when he has obtained the first
quotient figure, stop him, and ask him, what he has now done ; he
will most likely be unable to tell. The answer he will give- will
probably be, that he has divided 43 by G ; and no one of his class
will probably have a better answer to offer. If he - says he has
divided 43 thousand, he is still wrong ; for he has divided only 42
thousand, leaving one thousand undivided.
In some of the examples given in this section, the large num-
bers may be separated in diiferent ways preparatory to division.
Thus, in the last example, 92,648 may be divided 80,000, 12,000,
600, 48 ; or 88,000, 4000, 640, 8, and in still other ways.
Pupils should be encouraged to exhibit more methods than one
for obtaining the answer. If a scholar has two methods he should
be allowed to give them both, and if another has a different one
still, it should be brought forward, and the most lucid and easy one
should receive the commendation of the teacher.
Note 6. — Page 81. Strictly speaking, there is no relation in
quantity between a line and a surface, but only between a line and
the dimensions of a surface. By the square of a line is meant a
square surface, each of whose sides has th^ same length as the
given line.
PART SECONDi
CONTAINING
RULES AND EXAMPLES FOR PRACTICE ,
IN
WRITTEN ARITHMETIC.
NUMERATION OF WHOLE NUMBERS.
In common Arithmetic there are 9 figures used for the ex-
pression of numbers. 1, one; 2, two; 3, three ; 4, four; 5,
five ; 6, six ; 7, seven ; 8, eight ; 9, nine. jpSVhen one of these
figures stands alone, it signifies so many units, or ones ; when
two figures stand side by side, the left hand figure signifies so
many tens ; when three stand side by side, the left hand figure
signifies so many hundreds ; and universally, as you advance
to the left, the figures increase in value tenfold at each step,
as will be seen in the table on the next page.
The right hand place is always that of units. When there
are tens, and no units, a cipher, 0, must stand in the unit's
place, thus, 20 ; this merely serves to occupy the unit's place,
and shows that the figure, 2, is in the place of tens. When there
are hundreds, and no tens nor units, two ciphers are wanted ;
one in the unit's place, and one in the place of tens ; as, 200 ;
and so of all higher numbers.
To annex a cipher to a figure, therefore, is the same as to
multiply the number by ten, for it removes the figure from
the unit's place to the place of tens. To annex two ciphers
is the same as to multiply the number by a hundred, for it
removes the figure from the unit's place to that of hundreds.
118
TABLE OF NUMERATION.
two.
two tens, that is, twenty.
two hundred.
enumerate. •
enumerate.
f two tens of thousands,
( that is, twenty thousand.
two hundred thousand.
enumerate.
twenty-two million.
enumerate.
enumerate.
In writing numbers, every place not occupied by a figure
must be occupied by a cipher ; otherwise the true value of the
NUMERATION.
119
figures at the left hand of that place woqld not be preserved.
Thus, if you wish to write in figures the number, three hun-
dred and four, as there are no tens, a cipher must stand in the
place of tens, 304. Should you omit the cipher, and write 34,
the 3 would have slid into the ten's place, and it would not
express three hundred and four.
As in advancing to the left, figures increase their value
tenfold at each step, so if you begin at any place in a line of
figures, and move towards the right, the figures will diminish
in value tenfold at each step. That is, each figure will sig-
nify but a tenth part of what it would, if it stood in the next
left hand place. This will prepare you to look at the
Whole Numbers.
NUMERATIOlil OF DECIMALS.
Decimals.
^S
I
3
two, and two tenths.
I four, and two tenths, and fivo
I hundredths, or 25 hundredths.
( twenty-two and twenty-
( two hundredths.
two thousandths,
enumerate.
{four hundred and seventeen hun-
dred thousandths. .
enumerate.
120
ADDITION.
SECTION I.
ADDITION.
Addition is the uniting of several sums into one, to show
their amount.
Rule. Set down the numbers, units under units, tens under
tens, and so on. Add the column of units, set down the units
of the amount, and carry the tens, if there are any, to the
column of tens ; add the column of tens, and set down the unit
figure of the amount, carrying the figure of tens to the next
column ; and so on. In adding the last column set down the
whole amount.
To prove the work, repeat the operation, beginning at the
top and adding downwards.
1. 472+842.
2. 376+421-}-645.
3. 431-}-843+794.
4. 821+954+359.
5. 267+549+121.
6. 834+682+762.
7. 468+912+683.
8. 871+934+340.
9. 516+617+713.
10. 685+937+742.
11. 840+931+672.
12. 963+847+784.
13. 421+317+844.
Examples.
14. 6342+1896+4741+8962.
15. 3249+856+8007+4990.
16. 3819+42+906+1728.
17. 1645+2718+92+1807.
18. 1543+1899+3054+26.
19. 1854+1962+2168+666.
20. 1062+6300+9071+7001.
21. 2593+1801+9201+2113.
22. 9064+2118+1802+3076.
23. 1001+9016+7990+26.
24. 106+2307+9436+108.
25. 1214+6403+7113+4009.
26. In 1840, the population of the New England States
was as follows : Maine, 501,793 ; New Hampshire, 284,574 ;
Vermont, 291,948 ; Massachusetts, 737,699 ; Connecticut,
309,978 ; Rhode Island, 108,850. What was the population
of all the New England States ?
27. The population of the Middle States, in 1840, was as
follows : New York, 2,428,921 ; New Jersey, 373,306 ; Penn-
sylvania, 1,724,033 ; Delaware, 78,085 ; Maryland, 469,232 ;
Virginia, 1,239,797. What w^as the total population of the
Middle States ?
ADDITION. 121
28. The population of the Southern States, in 1840, was :
North Carolina, 753,419 ; South Carolina, 594,398 ; Georgia,
691,392 ; Alabama, ,590,756 ; Tennessee, 829,210 ; Mississippi,
375,651 ; Ai'kansas, 97,574 ; Louisiana, 352,411. What was
the total population of these States ?
29. In 1840, the population of the Western States was as
follows : Ohio, 1,519,467 ; Indiana, 685,866 ; Illinois, 476,183 ;
Michigan, 212,267 ; Kentucky, 777,828 ; Missouri, 383,702.
What was the total population of these States ?
30. What is the total population of all the United States,
as set down in the four preceding examples ?
Whenever, in adding a column, two figures occur together,
which amount to 10, as 8 and 2, 7 and 3, take them both to-
gether and cal? them 10. This will make the addition more
rapid and easy.
When you have become familiar with the operations in ad-
dition, you may occasionally vary your method, by taking two
columns of figures at a time. If you have been thorough in
the mental part of this wock, you will be able to do this. It
will furnish an agreeable variation in your method of work,
and greatly increase your power of rapid calculation.
31. This method is seen in the following example :
3124
7681
4942
L 15747
42 and 81 are 123, and 24 are 147 ; set down
the 47, and carry the 1 hundred to the column
of hundreds ; 50 and 76 are 126, and 31 are
157.
It will be well often to adopt this as your method of proof.
After performing the work by taking one column at a time,
prove it by taking two columns ; or perform it first in the latter
way, and prove it in the other.
32. 1467+894+1721+8396.
33. 9461+8134+2016+4317.
34. 84161+9632+78167+43180.
35. 109761+20671+437674+963.
36. 26431+184097+467124+84321.
37. 43126+91434+237210+127.
38. 1235467+1096+34271+4081.
39. 10467+31762+10921+9634.
40. 37193+10634+206721+104367.
11 -
122
SUBTRACTION.
SECTION II.
SUBTRACTION.
Subtraction is the taking of a smaller number from a
larger, to show the difference. The larger number is called
the minuend ; the smaller, the subtrahend ; the diflference is
called the remainder.
Rude. Set down the numbers, the larger number upper-
most, units under units, tens under tens. Subtract the units
of the lower number from the' unit figure above, and set down
the difference. Proceed in the ' same way, with the tens and
higher orders, to the close. If, in any case, the figure of the
minuend is less than the figure below it, increase it by ten, by
borrowing one from the next higher figure of the minuend,
remembering at the next step, that the figure in the minuend
has already been diminished by 1.
To prove the work, add the remainder and the subtrahend
together, and, if the work is correct, the sum will agree with
the minuend.
Examples.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
748—365.
674—582.
849—634.
347—267.
431—249.
867—312.
419—224.
519—499.
318—201.
856—106.
3416—2999.
4162—4091.
7089—3007.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
8990—7096.
8243—6492.
784—96.
210—100.
681—504.
901—75.
16432—14968.
195864—137461.
228476—13962.
740016—116799.
86400—199.
10006—4364.
26. America was discovered in 1492 ; Plymouth was settled
in 1620 ; how long was that after the discovery of America?
27. The Independence of the United States was declared
in 1776 ; how long was that after the settlement of Plymouth ?
28.' George Washington was born in 1732; betook com-
mand of the American armies in 1776 ; how old was he then ?
29. Gen. Washington became President of the United
States in 1789 ; how old was he then ?
SUBTRACTION. - 123
30. In 1820, the population of Maine was 298,335 ; in 1830
it was 399,955 ; what was the increase in 10 years ?
dl. The population of Maine in 1840, was 501,973 ; what
was the increase fi'om 1830 to 1840 ?
32. The population of Massachusetts in 1810, was 472,040,
in 1820, 523,487 ; how much had it increased from 1810 to
1820 ?
33. The population of Massachusetts in 1830, was 610,408 ;
how much had it increased from 1820 to 1830 ?
34. In 1840, the "population of Massachusetts was 737,699 ;
how much had it increased from 1830 to 1840 ?
35. The population of the §tate of New York in 1810, was
959,949 ; in 1820, it was 1,372,812 ; what was the gain ?
36. The population of New York in 1830, was 1,918,608 ;
what was the gain from 1820 to 1830 ?
37. In 1840, it was 2,428,921 ; what was the gain from
1830 to 1840 ?
38. The population of Ohio in 1810, was 230,760 ; in 1820
it was 581,434 ; what was the gain from 1810 to 18*20 ?
39. In 1830, the population of Ohio was 937,903 ; what was
the increase from 1820 to 1830 ?
40. In 1840, the population of Ohio was 1,519,467 ; what
was the increase from 1830 to 1840 ?
Another method of performing subtraction, often more con-
venient than the former, is the following :
Regard the subtrahend as a round number, one greater than
the figure of its highest order ; that is, if the subtrahend is 43,
call it 50 ; if 251, call it 300 ; subtract this round number from
the minuend, and then to the remainder add the complement
required to make up the subtrahend to the round number ; as
follows :
41. 674—381 ; 400 from 674 leaves 274 ; add 19, the com-
plement of 381 ; 274+19=293, ans.
Apply this method to example 11 above.
42. 3416 — 2999 ; the first remainder you see is 416, and
the complement is 1. Ans. 417.
This example shows how much shorter the work often be-
comes by adopting this method.
One of the above methods may be used as a proof of the other.
43. 384—219. I 45. 1679—291.
44. 1260—984. 46. 2496—954.
124
MULTIPLICATION.
SECTION III
MULTIPLICATION.
In multiplication a number is repeated a certain number of
times, and the result thus obtained is called the Product.
Rule.
Set down the smaller factor under the larger, units under
units, tens under tens. Begin with the unit figure of the
multiplier ; multiply by it, first, the units of the multiplicand,
setting down the units of the product, and reserving the tens
to be added to the next product. Proceed thus through all
the figures of the multiplicand. If there are more figures
than one in the multiplier, take, next, the tens, and multiply
the figures of the multiplicand as before, setting the figures of
the product one degree farther to the left than before.
Add the several partial products, and the amount Avill be
the whole product.
Examples.
342] Proof,
64
The best practical meth-
od of proof is carefully
to repeat the opera-
tion.
1. 342X64. Thus,
1368
.2052
21888 Ans.
2. 346X34.
3. 579X82.
4. 976X38.
5. 826X91.
6. 376X121.
7. 345X243.
8. 798X114.
9. 6181X35.
10. 6821X82.
11. 7413X96.
12. 7921X22.
13. 8964X85.
14. 9056X43.
15. 8007X41.
16. 4559X741.
17. 9642X864.
18. 8721X317.
19. 1841X134.
20. 13763X26.
21. 97623X318.
22. 1172671X216.
23. 1874215X341.
24. 742634X912.
25. 189423X62.
26. 14376281X194.
27. 17284265X36.
28. 671234X427.
29. 1895453X28.
30. 3469528X672.
31. 906421384X923.
MtJLTlt^LlCATlON. 125
35. 443754262X916.
36. 1123496113X413.
32. 713489605X84.
33. 843469537X906.
34. 236749024X516.
37. The average length of the State of Massachusetts ia
150 miles its breadth, 50 miles ; how many square miles does
it contain ?
38. The average length of Pennsylvania is 275 miles ; its
breadth, 1 65 miles ; how many square miles does it contain ?
39. The State of Ohio averages 223 miles in length, 180 in
breadth ; how many square miles does it contain ?
40. The StaM of Illinois averages 245 miles in length, 147
in breadth ; how many square miles does it contain ?
41. If there are 365 days in one year, how many days are
there in 25 years ?
42. If the wages of a soldier is 8 dollars a month, what will
be the wages of 7867 soldiers for 12 months ?
43. There are 320 rods in 1 mile ; how many rods are there
in 278 miles ?
44. 741X84. 47. 946734X496.
45. 19643X892. 48. 1623X198.
46. 246731X9210. 49. 9336X1998.
When the multiplier is a composite number, you may mul-
tiply first by one of its factors, and the product thus obtained
by the other factor, or by the others in succession if there are
more than two.
Apply this method to the following examples : —
50. 8476X45. 51. 1371X125. 52. 7465X108.
If a figure in the multiplier is a factor of the figure in the
next higher place, you may shorten the operation by multi-
plying the partial product of the lower figure by the other
factor of the higher: thus, in Ex. 44, above, having found
4 times 741, you know that 8 times the same is twice as many,
and 80 times is 20 times as many ; you need, therefore, only
double the line of the first partial product, setting it one de-
gree farther to the left, to express the tenfold higher value.
The same may be done if the right hand figure is a factor of
the number expressed by the next two higher figures.
Apply the process to the following examples : —
53. 947X639. 56. 27934x369.
• 54. 13674X4812. 57. 67514x64164.
55. 19742X568. 58. 259385X13212.
11*
126 DIVISION.
If the multiplier is 10, or any power of 10, annex to the
multiplicand, for the answer, as many ciphers as there are in
the multiplier.
If the multiplier consists of 9s, add as many ciphers to the
multiplicand as there are 9s in the multiplier, and from the
product subtract the multiplicand. The remainder will be
the product sought ; for, by adding the ciphers, you multiply
by a number greater by one than the multiplier. The multi-
plicand, therefore, will be found in the product once too many
times. So if the multij^lier is 2 or 3 less than some power
pf 10, you may do the same, remembering to take the multi-
plicand out as many times as the multiplier is units less than
a power of 10.
In this way perform the following examples : —
59. 3847X99. 61. 54327X98.
60. 4572X999. 62. 45314X997.
SECTION IV.
DIVISION.
In Division, two numbers are given, in order to find how
many times one contains the other ; or, in order to separate one
number into as many equal parts as there are units in the other.
The number to be divided is the dividend ; the number it
is divided by is the divisor ; the answer is the quotient.
To perform the operation, set down the divisor • at the left
of the dividend. Take as many figures on the left of the
dividend as will contain the divisor one or more times. See
how many times the divisor is contained in these figures, and
set down the number as the first figure of the quotient.
Multiply the divisor by the quotient figure, and subtract the
product from the number taken. To the remainder bring
down another figure of the dividend, and proceed as before.
1. 13276-r-122 ; thus, 122)13276(108 quotient.
122
1076
976
100 remainder.
DIVISION.
127
To prove the work, multiply the divisor and the quotient
together, and add the remainder, if there be any, and the
amount, if the worjc be right, will be equal to the dividend.
Thus in the above example, 122
108
976
122
100
13276
2.
11764-^-34.
11. 3059-r-214.
3.
47478-^-82.
12. 700601-r-34.
4.
37088-^38.
13. 643817-r-150.
5.
75116-^-91.
14. 300796-^145.
6.
45496^121.
15. 3264291-i-27.
7.
83835-^-243.
16. 18947633 : 181
8.
90972H-114.
17. 384628910-^-26.
9.
89743-^-17.
18. 137900-^-62.
10.
7426831H-141.
19. 3946908-^172.
20. A man divided 35,785 dollars equally among five chil-
dren ; how much did each receive ?
21. In one barrel of flour there are 196 lbs. ; how many
barrels of flour are there in 13,916 lbs. ?
22. In 1840 the population of Maine was 501,793 ; the
State contained then 30,000 sq. miles ; how many inhabitants
were there on an average to a sq. mile ?
23. The State of Massachusetts contained, in 1840, 737,699
inhabitants ; its territory is 7500 square miles ; how many
inhabitants are there to a square mile ?
24. The population of Ohio in 1840 was 1,519,467; its
territory is 40,000 square miles ; how many inhabitants to
a sq. mile ?
If the divisor is less than 12 the multiplication and subtrac-
tion may be carried on in the mind, and only the quotient set
down. This may most conveniently be written directly under
the dividend.
25. 7846-r-3. Operation, 3)7846
26. 964385-f-5.
2615-J-l Rem.
128 DIVISION.
27. 346218-7-7.
28. 214681-^9.
29. 684219-i-8.
30. 9640279-^-4.
31. 146710063-^-6.
32. 1143762-i-ll.
33. 1964217-i-12.
34. 4691382-^4.
It is welt to adopt the method of short division sometimes,
when the divisor is larger than 12.
35. 33467-H15.
36. 46943-f-15.
37. 81743-T-16.
38. 91674-r-21.
39. 673845-T-22.
Miscellaneous Examples on the foregoing Rules.
1. A merchant began to trade with 4325 dollars; he gained
in one year 784 dollars ; what was he then worth. ?
2. A man's income is 948 dollars a year ; his expenses are
762 dollars ; how much does he save of his income in one
year?
3. How much will he save in 9 years ?
4. A man bequeathed his property, 3882 dollars, one third
to his wife, and the remainder in equal shares to his four
children ; what was each child's share.
5. A merchant buys 643 barrels of flour, at 5 dollars a
barrel; he pays in addition — for freight, 65 dollars, for insur-
ance, 17 dollars ; what does the whole cost him then ? What
does each barrel cost him ?
6. A drover bought 7 oxen for 46 dollars a head, 12 cows
for 32 dollars a head, 96 sheep for 3 dollars a head ; how
much did they all come to ?
7. A drover buys 48 head of cattle at 32 dollars a head ;
the whole expense of driving them to market and selling them
is 72 dollars ; he sells them for 38 dollars a head ; what does
he gain ?
8. A laborer receives 16 dollars for every four weeks'
labor ; he works 48 weeks ; what will his earnings amount to ?
9. A man buys 7 tons of hay in the field for 13 dollars a
ton ; the cost of carrying it all to market is 48 dollars ; he
sells it for 15 dollars a ton ; does he gain or lose, and how
much? "^
10. A man receives a salary of 950 dollars ; he spends for
groceries 154 dollars, for milk 21 dollars, for meat 75 dollars ;
for wood 67 dollars, for clothing 184 dollars, for horse hire
38 dollars, for journeying 93 dollars, for repairs 19 dollars,
REDUCTION. 1^
for hired help 132 dollars, for attendance of the physician
26 dollars, for furniture 51 dollars, for house-rent 184 dollars,
and 86 dollars in churity and other incidental expenses ; has
he spent more than his salary, or less, and how much ?
SECTION V.
REDUCTION.
The object in Reduction is to change a quantity, in one
denomination, to another, which shall have the same value.
See Sec. VI., Part I. Higher denominations are reduced to
lower by multiplication.
Examples.
1. Reduce 3 yds. to feet.
2. Reduce 42 yds. to feet.
3. Reduce 4 feet to inches. . ^^
4. Reduce 17 feet to inches. r -
5. Reduce 132 feet to inches. ^ .
6. Reduce 16 yds. to inches.
7. Reduce 21 yds. to inches.
8. In 112 feet 7 inches, how many inches?
9. In 165 feet 4 inches, how many inches'?
10. In 5 yds., 2 feet, 9 inches, how many inches ?
11. In 24 rods, how many feet ?
12. In 87 rods, how many feet ?
13. In 567 rods, how many inches ?
14. In 7 rods, 4 feet, how many feet?
15. In 31 rods, 2 feet, 6 inches, how many inches ?
16. In 131£ how many shillings?
17. Reduce 781£ to shillings.
18.-Reduce 758£ to shillings.
19. Reduce 19 shillings to pence.
20. Reduce 7£ 11 shillings to pence.
21. Reduce 141£ 16 shillings, 4 pence, to pence.
22. Reduce 4£ 7 shillings, 3 pence, to farthings.
23. Reduce 14 lbs. 8 oz. Av. to oz.
24. Reduce 3 qrs. 9 lbs. 13 oz. to oz. - "*
130 REDUCTION.
25. Reduce 44 cwt. 3 qrs. 19 lbs. to lbs.
26. Reduce 13 T. 12 cwt. 2 qrs. to lbs.
27. Reduce 3 lbs. 6 oz. 17 dwt. Troy, to dwt.
28. Reduce 13 lbs. 2 oz. 14 dwt. to dwt.
29. Reduce 4 oz. 16 dwt. to grs.
30. Reduce 6 lb. 7 oz. 9 dwt. 4 grs. to grs.
31. Reduce 27 gallons, wine measure, to pints.
32. Reduce 7 hhd. 13 galls. 2 qts. to qts.
33. Reduce 1 hhd. to gills.
34. Reduce 174 bushels to qts.
35. Reduce 73 bushels to pints.
36. Reduce 231 bushels to qts.
37. In 13 sq. feet, how manj sq. inches?
38. In 84 sq. rods, how many sq. feet?
39. Reduce 13 sq. rods to inches.
40. Reduce 3 R. 17 rods, to feet.
41. Reduce 5 A. 2 R. 14 rods, to feet.
42. Reduce 1 7 solid feet to inches.
43. Reduce 19 s. yds. 14 feet, to inches.
44. Reduce 24 s. yds. 8 feet, 504 inches, to inches.
45. Reduce 6 cords, 13 s. feet, to feet.
46. Reduce 27 cords, 28 s. feet, to feet.
47. Reduce 45 cords, 13 s. feet, to feet.
48. In 75 E. E. of cloth, how many qrs. ?
49. Reduce 78 yds. 3 qrs. to qrs.
50. Reduce 194 yds. 1 qr. to nails.
51. Reduce 11 yds. 3 qrs. 2 nails, to inches.
52. Reduce 174 E. Fr. to nails.
53. Reduce 4 m. 5 fur. 13 rods, to feet.
54. Reduce 17 m. 6 fur. 20 rods, 8 feet, to inches.
55. Reduce 21£ 17s. 3d. to pence.
56. Reduce 24£ to sixpences.
57. Reduce 95£ 3s. to sixpences.
. 58. Reduce 45£ 5 s. to threepences.
59. Reduce 84£ to fourpences.
60. In 1 cwt. 3 qrs., how many times 7 lbs. ?
61. In 8 bis. of flour at 7 qrs. each, how many parcels of
14 lb. each ?
62. In 13 bis. of cider, at 31^ galls, each, how many timef
8 gallons ?
63. In. a town 5 miles wide, and 6 long, how many acres ?
64. How many acres in 7500 sq. miles ?
REDUCTION. 13^
SECTION VI.
KEDUCTION.
Lower denominations are reduced to higher by Division.
Examples.
1. In 348 shillings how many £ ?
2. Keduce 5000 shillings to £.
3. Reduce 13680 shillings to £.
4. Reduce 11040 pence to £.
5. Reduce 11292 pence to £.
6. Reduce 20220 pence to £. .
7. Reduce 1405 pence to £.
8. In 678 sixpences how many £.
9. Reduce 549 threepences to £. ''
10. Reduce 974 threepences to shillings.
11. Reduce 1776 hours to days.
12. Reduce 13841 hours to days.
13. Reduce 1964210 minutes to days.
14. Reduce 3742196 seconds to hours.
15. Reduce 964 gills to gallons.
16. Reduce 84672 gills to gallons.
17. Reduce 6794 gallons of wine to bis.
18. Reduce 3469 qts. to pecks.
19. Reduce 96431 pints to bushels.
20. Reduce 3846 qts. to bushels.
21. Reduce 5674 rods to furlongs.
22. Reduce 38961 rods to miles.
23. Reduce 76381 feet to rods.
24. Reduce 7960 inches to rods.
25. Reduce 7126734 inches to miles.
26. How many steps of 2^ feet each, are there in 1 mile?
27. A man walks 30 miles ; how many steps does he take,
2| feet each ?
28. Reduce 179 lb. av. to cwt.
, 29. Reduce 413 lb. to cwt.
30. Reduce 1048 oz. to quarters.
31. Reduce 4352 drams to lb.
32. Reduce 6130 oz. to cwt.
33. Reduce 1280 dwt., Troy, to Ibs^
132 REDUCTION.
34. Reduce 1511 dwt. to lbs.
35. Reduce 17812 grs. to lbs.
36. Reduce 720 square inches to square feet.
37. Reduce 1029 square feet to yards.
38. Reduce 2203 square inches to yards.
39. Reduce 3267 square feet to rods.
40. Reduce 5631 square feet to rods. _
41. Reduce 86 solid feet to yds.
42. Reduce 191934 solid inches to feet.
43. Reduce 2333 solid inches to feet.
44. Reduce 876 solid feet to yds.
45. Reduce 2293 solid inches to yds.
46. In 92 qrs. cloth, how many E. E?
47. Reduce 361 nails to qrs.
48. Reduce 467 nails to yds.
49. Reduce 3,741 inches to E. E.
50. Reduce 467 yds. to E. E.
51. In 27 acres, 2 roods, 17 rods, how many lots of 32 rods
52. How many times does a carriage wheel 11^ feet in
circumference, go round in one mile ?
53. In 35 tons weight, how many wagon loads of 22
cwt. each ?
54. In 186£ 12s., how many guineas of 21 shillings each?
55. In 75 yds. how many E. E ?
56. How many cannon balls at 24 lbs. each, will it take to
weigh 1 ton gross weight?
57. How many times must you apply a pole 12 feet long,
to the ground, to measure 1 mile ?
58. How many 10 gallon kegs may be filled from 17 hhd.,
wine measure ?
SECTION VII.
COMPOUND ADDITION.
When numbers are used without being applied to any par-
ticular kind of quantity, as 67, 84, they are called Abstract
Numbers ; when they are applied to some particular quantity,
as 67 yards, they are called Denominate Numbers.
COMPOUND ADDITION.
133
When several numbers of different denominations are to
be added, as 3£ 7s.4-7£ 4s., it is called Compound Addition.
Examples.
1. 3£ 14s. 9d+14£ lis. 6d.
Operation.
£ s. d.
3 14 9
14 11 6
18 6 3
Ans.
Set down numbers of the same denomination under each
other. Add first the numbers of the lowest denomination ;
if the sum amounts to more than one of the next higher, set
down what is over, and carry the number of the higher to
the next column. So proceed through the whole ; in adding
the last column set down the whole amount.
2. 5£ 15s. 4d.+14£ 17s. lld.+2£ 6s. 5d.
3. 13£ 14s. 6d. 2qr.+65£ 17s. lOd. Iqr.
4. 48£ 16s.+73£ 10s.+91£ 15s.+16£ 17s.
5. 17s. 3d.+15s. 7d. 2qr.+14s. Od. 2qr.
6. Troy weight. 12 lbs. 1 oz. 16 dwt. 14 grs.-}-3 dwt. 17 grs.
7. 17 lbs. 2 oz. 16 dwt. 5 gr.4-6 lbs. 14 dwt. 17 grs.
8. 21 lbs. 0 oz. 0 dwt. 3 grs.-t-19 dwt. 19 grs.+13 dwt.
9. 21 lbs. 3 oz. 16 dwt. 15 grs.4-4 lb. 4 oz. 17 dwt. 13 grs.
10. Av. wt. gross. 3 cwt. 0 qr. 17 lb. 14 oz.-}-12 cwt.
8 qrs. 13 lbs. 12 oz.
11. 1 T. 14 cwt. 3 qrs. 17 lb.+3 T. 17 cwt. 1 qrs. 21 lbs.
12. 3 T. 16 cwt. 1 qr. 20 lbs. 6 oz.+5 cwt. 3 qrs. 19 lbs.
13 oz.
13. 14 cwt. 1 qr. 20 lbs.+18 cwt. 1 qr. 16 lbs.+17 cwt.
1 qr. 11 lbs.
14. 1 m. 3 fur. 17 r. 6 ft.+3 m. 5 fur. 36 r. 12 ft.
15. 65 m. 7 fur. 31 r.+18 m. 19 fur. 23 r.+19 m. 4 fur.
17 r.
16.
17.
18.
19.
20.
21.
22.
5 r. 15 ft.4-27 r. 14 ft.+16 r. 11 ft.+21 r. 12 ft.
15 r. 9 ft. 6 in.+17 r. 3 ft. 4 in.+25 r. 15 ft. 11 in.
3 fur. 17 r. 4 ft. 5 in.+5 fur. 16 r. 14 ft. 9 in.
7 fur. 16 r. 3 ft. 2 in.4-6 fur. 34 r. 12 ft. 10 in.
13 m. 7 fur. 31 r.+6 m. 3 fur. 22 r.-j-ll m. 5 fur. 8 r.
Square measure. 2 A. 3 R. 6 p.+15 A. 1 R. 17 p.
16 A. 2 R. 21 p.+8 A. 3 R. 33 p.+9 A. 2 R. 9 p.
12
134 COMPOUND SUBTRACTION.
23. 2 R. 15 p. 63 ft. 29 in.+l R. 17 p. 31 in.4-37 p. 18
inches.
24. 13 p. 45 ft. 18 in.+19 p. 3 ft. 23 in.+17 p. 64 ft. 71
inches.
25. Solid measure, 3 yds. 17 ft. 126 in.+4 yds. 23 ft. 64 in.
26. 19 yards, 3 feet, 61 inches4-2 yards, 26 feet, 1650
inches-f-4 yards, 18 feet, 91 inches.
27. 16 gals. 3 qts. 1 pt.+84 gals. 2 qts. 1 pt.
28. 13 bushels, 3 pks. 4 qts.-f-76 bushels, 3 pks. 5 qts.
29. 19 bu. 1 pk.+76 bu. 3 pks.+18 bu. 2 pks.
30. 14 yds. 3 qrs. 1 n.+21 yds. 2 qrs. 3 n.
31. 3 days, 16 hours, 23 minutes-[-17 days, 13 h. 51 m.
32. 1 year, 11 weeks, 4 days-}-3 y. 14 w. 2 d.
33. 7 deg. 14 min. 34 sec.+lS deg. 20' 30".
34. 21 deg. 1> 11"+14 deg. 18' 19".
35. 61 deg. 26' 14"+34 deg. 1' 8".
SECTION VIII.
COMPOUND SUBTRACTION.
Compound Subtraction is the subtraction of numbers
of different denominations.
Rule, Set numbers of the same denomination under each
other. Begin at the right hand, setting down the remainder
found by subtraction, under its own denomination. If, in any
case, the minuend is less than the subtrahend, borrow one
from the next higher denomination of the minuend.
Examples.
1. 15£ 8s. 9d.— 11£ lis. 4d.
2. 22£ 19s. 8d.— 18£ 15s. 9d.
3. 13£ 4s. 6d.— 9£ 15s. lOd.
4. 18 bushels, 3 pecks, 4 quarts, — 16 bushels, 2 p^s. 5 qts.
5. 44 bushels, 1 peck, 3 quarts, — 20 bushels, 2 pks. 6 qts.
6. 4 years, 3 months,* 14 days, — 2 years, 4 months, 18 d.
I — —'t
* Allow 30 days to a month.
Operation.
£
s.
d.
15
8
9
11
11
4
3 17 5An8.
COMPOUND MULTIPLICATION. 135
7. 28 years, 8 months, 5 days, — 19 years, 11 months, 2 days.
8. 18 gallons, 3 quarts, 1 pint, — 10 gallons, 1 quart, 1 pint.
9v 14 gallons, 1 quart, — 2 quarts, 1 pint.
10. 4 miles, 3 furlongs, 17 rods, — 3 miles, 4 furlongs, 21 rods.
11. 19 miles, 7 furlongs, 11 rods, — 9 miles, 6 furlongs, 13 r.
12. 5 cwt. 3 quarters, 14 pounds, — 4 cwt. 1 quarter, 20 lbs.
13. 12 cwt. 2 qrs. 21 lb.,— 9 cwt. 3 qrs. 23 lbs.
14. The battle of Bunker Hill was on June 17, 1775 ; the
battle of Long Island, August 27, 1776 ; what was the length
of time between them ?
15. The battle of the Brandy wine was September 11, 1777 ;
how long was that after the battle of Long Island ?
16. The battle of Monmouth was June 28, 1778 ; how long
was that after the battle of the Brandy wine ?
17. The army of Burgoyne was captured October 17, 1777 ;
that of Cornwallis, October 9, 1781 ; how long between these
events ?
18. If I give a note on interest, June 5,. 1839, and pay it
March 10, 1841, for how long a time must the interest be cast ?
19. If I give a note on interest, August 17, 1841, and pay it
June 9, 1843, for what time must the interest be cast ?
20. How long is it from Dec. 17, 1843, to June 6, 1844 ?
21. How long from September 9; 1842, to Aug. 3, 1844 ?
22. How long from January 16, 1840, to July 17, 1843 ?
23. How long from Nov. 14, 1841, to August 21, 1844?
24. Boston is in longitude 71° 4' W. ; New York, 74° 1' ;
what is the difference of longitude ?
25. Cincinnati is in longitude 84° 27' ; how many degrees
W. from Boston ?
26. How many degrees of longitude is Cincinnati west from
New York ?
27. How many degrees of longitude is Cincinnati west from
Philadelphia, whose longitude is 75° 11'?
SECTION IX.
COMPOUND MULTIPLICATION.
Multiply, first, the lowest denomination ; if the product
amounts to more than one of the next higher, set down what
136
COMPOUND DIVISION.
is over, and carry the number of the next higher to the next
product. Multiply the next denomination in the same way,
and so on.
Examples.
Opferatlon.
T. cwt. qr. ^
3 7 3
7
U 1 Ans.
23
1. 3 T. 7 cwt. 3 qr. multiplied by 7.
2. Multiply 4£ 5s. 6d. by 2.
3. 6£4s. 3d. lqr.X3.
4. 6 hours, 43 min. 15 sec.X4:.
5. 9h. 11m. 41 sec.X6.
6. 14 days, 17 hours, 15 minutes, 3 seconds X 8.
7. 8 bushels, 3 pecks, 1 quart X 16.
8. 15 bushels, 2 pecks, 3 quarts, 1 pintXl2.
9. 19 bushels, 1 peck, 2 quarts X 5.
10. 1 mile, 5 furlongs, 13 rods, 12 feetX2.
11. 13 miles, 2 furlongs, 4 rods, 6 feet, 3 inches X 8.
12. 2cwt..3qrs. 16 lbs. X 7.
13. 14 cwt. 3 qrs. 14 lbs.X4.
14. What is the weight of 12 casks of lime, each weighing
3 cwt. 1 qr. 17 lbs.?
15. How many yards in 9 pieces of calico, each measuring
23 yards, 3 qrs. ?
16. Troy weight. 6 lbs. 11 oz. 5 dwt.X7.
17. 9 lbs. 8 oz. 16 dwt. 4 grs.X9.
18. Square measure. 7 acres, 2 roods, 17 rods X. 9-
19. 15 acres, 1 rood, 34 rodsXl4.
SECTION X,
COMPOUND DIVISION.
Divide the highest denomination first, and set the quotient
under it; reduce the remainder, if any, to the next lower
denomination, add it to those of the same in the dividend, and
divide again ; and so on to the end.
Examples.
1. Divide 7 bu. 3 pks. 5 qts. by 2.
Operation,
bu. pk. qt.
2)7 3 5
3 3 ejAns.
MISCELLANEOUS EXAMPLES. 1S7
2. 3£ lis. 6d.-r-2.
3. 18£ 12s. 9d.^3.
4; 26 hours, 53 miiautes-H4.
5. 55 hours, 10 minutes, 6 seconds-r-6.
6. 114 days, 18 hours, 0 minutes, 24 seconds-^8.
7. 140 bushels, 2 pecks-Hl6.
8. 187 bushels, 1 peck, 2 quartsH-12.
9. Seven men are entitled to equal shares of 67£ 13s. 4d. ;
what is each man's share ?
10. Three men are to receive equal shares of 114£ 19s. 9d. ;
what is each man's share ?
11. What is 1 fourth of 13 lbs. 6 oz. 17 dwt., Troy?
12. A teamster has 7 T. 11 cwt. 3 qrs. of merchandise,
which he loads on three wagons, giving an equal load to
each ; how much was each load ?
13. If you divide 7 bushels and 3 pecks of oats equally
among 5 horses, how much will each receive ?
14. If a piece of land containing 35 acres, 3 roods, 14 rods,
be divided into 4 equal parts, how much will each part be ?
SECTION XI.
MISCELLANEOUS EXAMPLES.
1. A teamster loads a quantity of merchandise equally on
8 wagons, putting on each 1 T. 11 cwt. 2 qrs. ; finding these
loads too heavy, he takes a fourth wagon ; how much must he
load on each, to divide the whole equally among the four ?
2. A man's estate amounts to 784£ 10s. ; his wife is to
receive 214£ 15s. and the remainder is to be divided equally
among 4 children ; what will be each child's share ?
3. Three men have equal shares in a scaffold of hay, the
whole of which weighs 5 T. 11 cwt. ; what is each man'^ share ?
For the following examples see p. 40, Part I.
4. Rome is in longitude 12° 28' E. from London ; what
time is it at Rome when it is noon in London ?
5. Petersburg is in longitude 29° 48' E. ; what time is it at
Petersburg when it is noon in London ?
12*
138 DIVISIBILITY OF NUMBERS.
6. Paris is in longitude 2° 20' E. ; what time is it at Paris
when it is noon in London ?
7. Boston is in longitude 71° 4' "W. ; what time is it in
Boston when it is noon at London ?
8. New York is in longitude 74° 1/ W. ; what time is it in
New York when it is noon at London ?
9. What time is it in Cincinnati, 84° 27' W., when it is
noon in Boston, which is 71° 4' W. ?
SECTION XII.
DIVISIBILITY OF NUMBERS.
In order to ascertain if a number is divisible by either of
the following numbers, 2, 3, 4, 5, 6, 8, 9, 10, or any combina-
tion of these, see Sec. VIII, Part I.
To ascertain if a number is divisible by any other number
than the above, make trial of other prime divisors, as 7, 11,
13, 17, &c., beginning with the smallest, tiU you find one that
will divide the given number, or find that it is indivisible.
Remember, that in making trial by these numbers, you
need not go higher than the square root of the given number,
for if a number is divisible, one of the factors will certainly
be as small as the square root. Let us take the number 1079 ;
what are its prime factors ? By inspection you may see it is
not divisible by 2, 3, 5, or 11, consequently not by 4, 6, 8, 9,
10, or 12. On trying it by 7, it is found not divisible by 7 ; the
next number is 13 ; this divides it, giving a quotient, 83, which
is prime. Its only factors, therefore, are 13 and 83.
Examples.
1. What are the prime factors of 667 ?
2. What are the prime factors of 406 ?
3. What are the prime factors of 419 ? of 361 ? of 742 r
of 281? of 316?
4. Prime factors of 941 ? 812? 749? 1116? 246? 8104.
5. Prime factors of 266? 884? 1917? 376?
REDUCTION OF FRACTIONS. 139
SECTION XIII.
REDUCTION OF FKACTIONS.
[See Section VIII. Part I.]
1. Reduce f f to its lowest terrfis. Ans. ^.
2. Reduce |§ to its lowest terms.
3. Reduce -^x^ to its lowest terms.
4. Reduce ff to its lowest terms.
5. Reduce ^V^ to its lowest terms.
6. Reduce ^|^ to its lowest terms. In this example it is
not evident on inspection whether the two terms of the frac-
tion have any common divisor. In such cases you may adopt
the following Rule to find
27ie Greatest Oommon Divisor.
Divide the greater number by the less, and then take the
divisor for a new dividend, and divide it by the remainder,
and so on, till there is no remainder ; the last divisor will be
the greatest common divisor.
Apply the above rule to the sixth example.
187)22]_(1 '
187
~o7\1Qj/K The greatest common divisor is,
, -1 70 therefore, 17, and, dividing the terms
of the fraction by this, we have for
17)34(2 the lowest terms, U.
34 ^^
00
Demonstration of the Rule.
If the larger number is a multiple of the smaller, it is evi-
dent that the smaller is a common divisor of the two numbers ;
it is also the gi'eatest common divisor ; for a number cannot
be divided by any number greater than itself; the answer,
therefore, is found by the first division. But if there is a
remainder, next find whether the remainder will exactly
divide the divisor. If it will, it will divide both the original
numbers, for if it will divide the divisor, it will divide any
multiple of the divisor ; and, as it will of course divide itself,
it will divide any multiple of the divisor, plus itself. Now the
\
140 THE GREATEST COMMON DIVISOR.
larger of the original numbers is a certain multiple of tLe
smaller, plus the remainder. If, therefore, after the first
division, the remainder will divide the divisor, it is a common
divisor, or measure, of the two numbers.
It is also the greatest common divisor ; for, as^ it will ex-
actly measure the smaller of the two numbers, it\vill exactly
measure any multiple of the smaller. Now the greater num-
ber, is a certain multiple of the smaller, plus the remainder.
The remainder, therefore, in measuring the larger number, is
obliged to measure itself. No number greater than itself can
do this ; therefore the remainder is the greatest common divi-
sor. If the work has to be carried on farther than the second
division, the same reasoning in the demonstration will apply.
JUxamples.
7. What is the greatest common divisor of 874 and 437 ?
8. What is the greatest common divisor of 497 and 451 ?
9. What is the greatest common divisor of 817 and 913 ?
10. What is the greatest common divisor of 1007 and
1219?
11. What is the greatest common divisor of 608 and 192 ?
12. What is the greatest common divisor of 869 and 1343 ?
When there are more than two numbers, first find the
greatest common divisor of two of them, and then, of that
divisor, and the third number.
13. What is the greatest common divisor of 608, 941 and
451?
Whenever it is possible, by inspection, to separate the
numbers into their prime factors, this method should be
adopted.
14. What is the greatest common divisor of 94, 804 and
126?
15. What is the greatest common divisor of 1274, 896 and
580?
Apply the above Rules to the reduction of the following
fractions.
16. Reduce ^|-f- to its lowest terms.
17. Reduce to their lowest terms f §^, ^ff, ^Hf.
18. Reduce to their lowest terms |^|, ^f |, §|i.
19-. Reduce to thciis^lowest terms t^^, ^^, 1^.
CHANGE OF NUMBERS TO HIGHER TERMS. 141
To reduce an improper fraction to a whole or mixed number.
Perform the division indicated by the fraction as far as
possible ; if there is a .remainder, express that part of the
division by placing the denominator under the remainder.
20. Reduce \^ to a whole, or mixed number. Ans. l-^^.
21. Reduce ^- to a whole, or mixed number. Ans. 3|.
22. Reduce to a whole, or mixed number, 3^, f f, ff .
23. Reduce to a whole, or mixed number, ^, ^|, y^.,
24. Reduce the improper fractions, Vif j ^f ^> Wj ^%^'
SECTION XIV.
CHANGE OF NUMBERS AND FRACTIONS TO HIGHER TERMS.
It is sometimes convenient to express whole numbers in the
form of fractions, and to express fractions in higher terms
without altering the value. Thus 3=|, or ^. 10=^,
or V*- '
Examples.
1. In 4 how many fifths? Ans. 20.
2. Express the value of 4 m fifths. Ans. ^.
3. Express 7 in thirds.
4. Express 19 in the form of sevenths.
5. In 13 how many eighths ?
6. Express 21 in thirds.
7. Express 7 in eighteenths.
8. Express 41 in fourths.
9. In 3^ how many halves ?
10. Change 4^ to an improper fraction.
11. Change 17^ to an improper fraction.
12. Change 24^ to an improper fraction.
13. Change to an improper fraction 18|. 112f. 318^.
14. Change f to eighths, without altering its value.
15. Change f to fifteenths.
16. Change I to 24th3. ^3^ to 80ths. f to 99ths.
17. Change /^ to 26ths. ^ to 56ths. -ff to 60ths.
18. Change | to 7ths.
This example presents a difficulty, because the required
denominator, 7, is not, as in the preceding examples, a multi-
142 MULTIPLICATION AND DIVISION OP FRACTIONS.
pie of the given denominator, 4. We have seen, however,
that if we multiply or divide both terms of a fraction by the
same number, the value will not be altered. We must then
multiply and divide both terms by such numbers as will give
us, in the end, 7 for the denominator. The question then is,
how can we, by multiplication and division, change 4 into 7 ?
We can multiply it by 7, which will give 28, and then divide
by 4, giving 7 for the quotient. Thus the denominator has
been changed, by multiplication and division, from 4 to 7.
Now whatever has been done to the denominator must be done
to the numerator to preserve the value of the fraction. Multi-
plying 3 by 7, we have 21 ; dividing this by 4 we have 5^ for
the required numerator. The answer, therefore, is -3. This
fraction, as one of its terms contains a fraction in itself, is
called a complex fraction.
19. Change f to 8ths. | to 9ths. ^j to 7th3.
20. In f , how many 4ths ? how many 5ths ? 6ths ?
21. Change 4| to 5ths. 8^ to llths. 7^- to 4ths.
22. Change 22^ to 4ths. 18^ to 7ths. 31^ to 5ths.
23. In 8^, how many 3ds ? 4ths? 5ths? 9ths?
24. In 19 J how many 5ths ? 4ths? 7ths?
25. In 9^ how many 3ds? 5ths ? 8ths ?
26. In 13^ how many 14ths ? 15ths?
27. In 8^ how many 17ths? 13ths?
28. In 20^ how many 7ths ? 8ths ?
29. In 16^ how many 4ths ? 5ths ?
30. In ll| how many 37ths ? 19ths ?
SECTION XV.
MULTIPLICATION AND DIVISION OF FRACTIONS.
[Sec Section VIII. Part I.]
1. A man worked 72 days for f of a dollar a day ; what
did his wages amount to ?
2. Multiply I by 46.
3. A man bought 139 bush, of apples for | of a dollar a
bush. ; what did they come to ?
4. Multiply f by 341. ^X127.
MULTIPLICATION AND DIVISION OF FRACTIONS. 143
5. A garrison of 700 soldiers are allowed f of a pound of
flour a day, for each man ; how much would they consume in
1 day ? How much in 7 days ?
6. If a horse eats -^^ of a bush, of oats in a day, how many
bush, will he eat in 365 days?
7. If a horse eats 2^ cwt. of hay in a week, what part of a
cwt. will he eat in one day ? How many cwt. will he eat in
a year ?
8. Three men gain by an adventure 56^ dollars, which
they are to share equally ; what is each man's share ?
9. What is ^ of 187^ ? What is ^ of 91f ?
10. f^X 141? AX 97? 1X140?
11. If 3:^ cwt. of flour be divided equally into 5 equal parts,
what part of a cwt. will each share be ?
12. Divide 17^ by 8. 37^ by 14. 18| by 9.
13. Divide 74^-T-5. 811^-7. 381^-^-9.
SECTION XVI.
MULTIPLICATION AND DIVISION OF FRACTIONS.
[See Section IX. Part I.]
1. In 63 gallons how many bottles of j^ of a gal. each ?
2. From 7 lbs. of flour how many loaves of bread may be
made, each containing § of a lb. of flour ?
3. Divide f f^f . il^. {^-i-U-
4. A man left 5846 dollars ; f of the whole to go to his wife,
and the remainder to be equally divided among four children;
what was each child*s share ?
5. From a piece of cloth 32 yds. long, how many coats can
be made, each requiring 2| yds. ?
6. From a stick of timber 2Q^ feet long, how many blocks
can be cut, each -^^ of a foot long ?
7. If a family consume 22^ lbs. of flour- in a week, how
much is that a day ? How much will they use in a year ?
8. Divide 13i-H^. 18f-r-lU. 86^9^.
9 Multiply 25^X16^. 23|X2f. 31fXl9^.
A fraction of a fraction, as ^ of ^, is called a Compound
Fraction. This is reduced to a simple fraction by multiply-
144 ADDITION AND SUBTRACTION OF FRACl?IONS.
ing the numerators together for a new numerator, and the de-
nominators for a new denominator. It is in fact the same as
the multiplication of two fractions together.
10. What is i off of 76?
11. What is ^ of ^ of 12 ? What is | of f of 18^ ?
12. What is ^^ of f of 8J ? What is f of | of 34 ?
13. Divide f of | by ^. Divide f of f by 8^.
14. Multiply ^ of 37 by 19^. Mult. ^ of 18 by 19^.
15. What is the value of 32:^ yds. of cloth at 4^ dollars a
yard?
16. What do 17|^ tons of hay come to at 11|^ dolls, a ton?
17. What is the value of 21^ cords of wood at 4^ dolls, a
cord?
18. What is the value of 24^ ban-els of apples at If dolls,
a barrel ?
19. What is the amount of 12|^ shares Bank stock, at 641iJ-
dollars a share ?
SECTION XVII.
ADDITION AND SUBTRACTION OF FRACTIONS.
If the fractions have a common denominator, perform the
required operation on the numerators, and place the result
over the common denominator.
1. Add A+A+l- tV+i^+if 1^+^+1%-
2. Subtract f — |. ^— ^. « — H- M — H-
If the fractions have not a common denominator, reduce
them to a common denominator, (Sec. IX. Pt. 1.) and then
add or subtract, as the question requires.
s. Add m+f . f +A+if . T^F+t+f .
4. Add f +1+1^, i+A+i§. I+A+A-
5. Subtract 5i^ — f S^\ — i' | — iV
6. Add 3U+57i+18i. 19^+211+3/^.
In cases like the above, it is easiest to add the whole num-
bers first.
7. Add ei^+l- of 7^+^ of 22. 4f\+^ of 18^+| of 1 3.
8. A man spent for various articles ^ of a dollar, f of a
dollar, ^ o£ & dollar, ^ of a dollar ; what part of a dollar
did he spend in all ?
REDUCTION OP DENOMINATE FRACTIONS. 145
9. From 56^ bushels, llf bushels were taken ; how much
remained ?
10. From a ^firkin* of butter containing 4^2^ lbs., 18];^ lbs.
were taken ; how much remained ?
11. Add ^+4|. 1U+I|. 15i+7f+g.
12. Add 2H+^. n+^1, 13^+^+11^.
13. Add 19I+21H. 8i+H. 26i+H+g.
14. Add4f+^. 8^+^, 12^+3^+18^.
15. Subtract 19i—^. 22^ — ?|. 18^ — 13/^.
16 4d
16. Add 3i+8f+Hi_LA. 4U4ii-fi^.
SECTION XVIII.
BEDUCTION OF DENOMINATE FRACTIONS.
Denominate fractions are fractions of numbers when ap-
plied to a particular denomination. See Sec. XI. Part I.
Mcamples.
1. What part of a bushel is ^ of a quart? f of a quart?
2. What part of 2 bushels is ^ of a quart ? f of a peck ?
3. What part of 7 bushels is f of a peck ? | of a peck ?
4. What part of 3£ is ^ of a shilling ? y^ of a shilling ?
5. What part of a shilling is | of a penny ? ^^3^ of a d. ?
6. What part of a mile is ^ a rod ? ^ of a rod ?
7. What part of 3 furlongs is | of a rod ? f of a rod ?
8. What part of a mile is 3 furlongs 19 rods?
9. What part of a mile is 1 foot ? 2 feet ?
10. What part of a ton is 6 cwt. 3 qrs. 7 lbs. ?
11. What part of a ton is 18 cwt. 1 qr. 6 lbs. ?
12. What part of a square rod is 17 feet? 31 1 feet?
13. What part of a cord is 12^ cubic feet? 25^ cubic feet?
14. What part of a cord is 18;^ cubi« f«et ? 84^ €ubi« f«et ?
18
146 CHANGE OF DENOMINATE INTEGERS.
15. What part of a bushel is ^ of a quart ? f of a qt. ?
16. What part of a bushel is 7^ qts. ? 11^ qts. ?
17. What part of a week is 4|^ hours ? 5^ hours ?
18. What part of a week is 7^ hours ? 8^ hours ?
19. What part of 3 hours is 12 minutes? 15^ minutes?
20. What is the value in shillings and pence of ^ of a £ ?
If it were 3£, there would be 60 shillings ; but it is not 3£,
but one seventh of that ; therefore, it is } of 60 shillings=8f
shillings. To find the value in pence of ^ of a shilling, pursue -
the same reasoning ; if it was 4 shillings, it would be 48 pence ;
but it is not 4 shillings, but ^ of thatz=:6f pence. The answer,
then, is 8s. 6f d.
21. What is the value in, shillings and pence of f of a £?
22. Value in hours, minutes, and seconds, of | of a week ?
23. How many minutes in ^i^ of a week ?
24. How many minutes and seconds in ^ of an hour ?
25. Value of f of a gal. ? Value of A ^^ a bl. of wine ?
26. How many oz., dwt. and grs.in /^ of a lb. Troy?
27. What is the value in oz., dwt. and grs. of ^j of a lb. Troy ?
28. How many square rods in f of an acre ?
29. How many square feet and inches in ^ of a sq. yard ?
SECTION XIX.
CHANGE OF DENOMINATE INTEGERS TO FRACTIONS.
[See Section XI. Part I.J
Examples.
1. What part of a furlong is 5 feet ? 1^ feet ?
2. What part of a mile is 17 feet ? 28:^ feet ?
3. What part of a mile-ts 31^ feet ? 47§ feet?
4. What part of a ton is 25^ lbs. ? 82| lbs. ?
5. What part of a ton is 107^ lbs ? 130^ lbs.
6. What part of a ton is 3 qrs. 9 lbs. 8 oz. ? 17^ lbs.
7. What part of a ton is 4 lbs. 13 oz. ? 19^^ lbs.
8. What part of an acre is 4 rods 17 feet? 139^^ feet?
9. What part of an acre is 113 rods 51^ feet?
10. What part of a hhd. of wine is 17 gals. 3 qts. 1 pt. ?
11. What part of 10 gallons is 3^ pints? 8^ pints?
12. What part of a guinea is 3s. 7d. ? 14s. 3^ ?
PEACTIONS. 147
13. What part of a £ is 8s. 5id. ? 10s. 9f d. ?
14. What part of a week is 3| hours? 1^ hours?
15. What part of 5 days is 1 hour, 41 m. 15 sec. ?
16. What part of a month of 31 days, is 17 h. 18^ m.?
17. What part of an ell Eng. is 2 n. 1 in. ?
18. What part of 21 yards is 3^ qrs.? 9^. yards?
19. What part of 37^ yards is 1^ ell English ? -
20. What part of 7 cords of wood is 21^ cubic feet?
21. What part of 31£ is 5s. S^d. ? 14s. 2|d. ?
SECTION XX.
PKACTICAL EXAMPLES.
I. What is the cost of If yds. broadcloth at 4|^ dolls, a yd. ?
^^ 2. What is the cost of 2\ yds. cloth at ^ of a doll, a yd. ?
3. What is the value of 12 bbls. of flour at 4| dolls, a bbl. ?
4. What is the value of 31 casks of lime at ^ of a dollar a
cask?
5. What is 5^ cwt. of beef worth at 4f dolls, per cwt. ?
6. What is the cost of 43^ bush, corn at f of a doU. a bush. ?
7. If a horse eat 1^ bush, of oats in a week, how much will
he eat in 52 weeks ?
8. What will be the cost of the oats for 52 weeks at f of a
doll, a bushel ?
9. What part of a cwt. is ^ of a bbl. of flour containing
196 lbs.?
10. What is the weight of f of a lot of hay weighing 4^
tons?
II. How many cubic feet are there in f of J of a cord of
wood?
12. A man sold f of a lot of wood, the whole of which was
17f cords ; how much did he sell ?
13. What part of 9 rods in length is 10 feet ?
14. What part of a sq. rod is 3 sq. yards ?
15. What part of a sq. rod is 5 sq. feet ?
16. What is 36 sq. feet of land worth at 9^ dolls, a sq. rod ?
17. How many gallons are there is \^ of a barrel of wine ?
18. Two men bought a lot of hay for 11^ dollars ; one took
13 cwt. ; the other, the remainder, which was 8^ cwt. ; what
•?ught each to pay ? _
148 DECIMAL FRACTIONS.
19. Two men divided a lot of wood wliicli they purchased
together for 27|^ dollars; one took 5^ cords; the other 8
cords ; what ought each to pay ?
20. The main-spring of a watch weighs about 1 dwt. 12
grs., Troy weight ; estimating its worth at ^ of a dollar, what
would a pound Troy of steel be worth after it was manufac-
tured into watch main-springs, allowing nothing for waste in
manufacturing ?
21. A hair-spring of a watch weighs | of a grain, Troy;
estimating its value at 3 cents, what would be the value of 1
lb. Troy, of steel, made into hair-springs, allowing nothing for
waste ?
22. Two men hired a horse one week for 6^ dollars ; one
rode him 70 miles ; the other, 84 ; how much ought each to
pay?
23. A stack of hay is bought by two men for 76^ dollars,
to be paid for in proportion to the amount of hay each one
takes ; one takes 3| tons, the other the remainder, which was
2^ tons ; how much ought each to pay?
SECTION XXI.
DECIMAL F.RACTIONS.
Addition and Subtraction. Sec. XII. Pt. I.
Examples.
1. 24.5+68.3+17.14+87.96+3.125.
2. 165.3+96.45+8.431+.641+9412.5.
3. 450.61+27.134+89.4216+984.
4. 64.25+3.125+87.25+181.7.
5. 125.17+34.27+.125+3761.5.
6. 186.4—27.31; 800.4—21.67.
7. 34.21—18.525; 94.31—81.167.
8. 167.51—35.125; 204.5—31.09.
9. 20.41—3.817; 601.4—517.24.
10. 648.62— .541; 346.4—91.324.
11. 5.1—1.324; .5— .0067.
12. .81— .126; .94— .3816.
EEDUCTION OF FRACTIONS TO DECIMALS. 149
Multiplication and Division.
13. 1243X87; 321.67X24.3. 31. 35H-.36; 48-f-.47.
14. 97.125X6; 31.4X.125. " 32. .17-^-31; .26-^.013.
15. 37.5X.94; 18.4X64. 33. 43-H.06 ; 45-^.003.
16. 21X-106; 312X.05. 34. 75H-.125 ; .95-i-.04.
17. 31.1X.004; 18.61X.03. 35. .18-i-.0045 ; 11H-.34.
18. 641X.41; 843.5X.95. 36. 9-^.0225; .7-^-.035.
19. 184.2X.121; 35.6X.025. 37. 80-i-.18 ; 51-V-.031.
20. .625X71; .875X31.5. 88. .55X.031; 71.4X.ia.
21. 84H-.012; 965-f-.15. 39. 8.44-.021 ; .65H-.8.
22. 1.65-H15; 846^3.4. 40. 1.21X.09 ;' .14X.03.
23. 1640^.96; 425H-.055. 41. .64X.31; .08X.009.
24. l-T-.OOl; 2-^.0002. 42. 36-J-.13 ; 28-^-11.4.
25. .OOl-f-2; 384-^-.0012. 43. 40.1-f-8; 64-^.9.
26. 96-^-.024; 64-H.016. 44. 81.4^.03; 7-f-.4.
27. 1827^.9; 34-^.17. 45. 9-^.5; 15H-.7.
28. .63-^-8; .15-H-14. 46. 80.2X.03 ; 16-r-.9.
29. .48^.9; .33—16. 47. 105.4-f-37.15.
30. 181-H-.41; 41-H.6. 48. 118.75~.0044.
SECTION XXII.
EEDUCTION OF VULGAB FRACTIONS TO DECIMALS.
[See Section XITI. Part 1. 3
Examples.
1. Reduce f to a decimal.
2. Reduce f to a decimal.
3. Reduce to decimals \ ; f .
4. Reduce to decimals iV ; -^ 5 i^e*
5. Reduce to decimals t^ ; i^ ; M*
If the fractions are reducible to decimals without a remainder,
obtain the answer exactly ; if they are irreducible, obtain the
proximate answer to four places, and annex the fractional
remainder. In order to know if a fraction is exactly expres-
sible in decimals, see Section XIII. Part I, as directed above.
6. Reduce to decimals ^ ; §| ; §|-.
7. Reduce to decimals ^ ; ^t 5 ^*
8. Reduce to decimals \^ ; -^ ; jf 5.
13*
150 REDUCTION OF FRACTIONS TO DECIMALS.
9. Reduce to decimals :^ ; ^|f ; |^.
10. Reduce to decimals y^ij ; y^^ ; ^f x*
In ordinary transactions it is usual to carry the decimal
answer to three or four places ; the remainder is then so
email in value that it may be dropped as of no importance.
At whatever place you stop, however, the decimal obtained,
and the fractional remainder, when added together, will ex-
actly equal the original fraction.
11. In order to show this, we will take \. Reducing it,
')_ we obtain at the first step 1 tenth-M- of 1 tenth;
1+^
adding these, 7^+^x7==^^=^' which is the original faction.
We now carry the reduction one step further, ^ — -'
14-|-^
we obtain 14 hundredths,-f-f of a hundredth. Adding these,
^+Y%7f=^%=h the original faction.
We will carry the reduction one step further ; I
^ ^ 142+f.
We obtain 142 thousandths+f of a thousandth. Adding these
by using the common denominator 7000, TVA+Tif(T^=^§^^
s=:^, the original fraction.
12. Reduce -^ to a decimal, of one figure, with the remain-
der ; carried to 2 places, with the remainder ; carried to 3
places, with the remainder.
13. Reduce -j^ to a decimal of 7 places.
14. Reduce ^^ to a decimal of 9 places.
15. Reduce ^ to a decimal of 10 places.
Repeating and Oirculating Decimals.
When a fraction is irreducible, the decimal figure will either
repeat, as ^=.333-[- ; or the decimal figures obtained by the
partial reduction will, after a time, recur again, in the same
order as at first. Thus, ^ gives .090909-}- and so on, with-
out end. When the same figure is repeated continually, it is
called a repeating decimal ; when the same series of different
figures recurs, it is called a circulating decimal.
▲NALTSIS OF PBOBLEMS. 15t
SECTION XXIII.
BEDUCTION OF DENOMINATE INTEGERS TO DECIMALS.
1. Reduce 5s. lid. to the decimal of a £. First, reduce
the quantity to the vulgar fraction of a £ ; then reduce that
vulgar fraction to a decimal.
2. Reduce Ss. 2^. to the decimal of a £.
3. Reduce 5d. to the decimal of a guinea.
4. Reduce 3 qts. to the decimal of a bushel.
5. Reduce 2^ pints to the decimal of a gallon.
6. Reduce 3 feet 5 inches to the decimal of a rod,
7. Reduce 7 feet 8 inches to the decimal of a rod.
8. Reduce 15 rods 9^ feet to the decimal of a furlong.
9. Reduce 23 rods 13 feet to the decimal of a mile.
10. Reduce 5 hours 18 m. to the decimal of a day.
11. Reduce 21 hours 6 m. to the decimal of a week.
12. Reduce 12^ sq. rods to the decimal of an acre.
SECTION XXIV.
TO FIND THE INTEGRAL VALUE OF DENOMINATE DECIMALS.
1. Wliat is the value of .7 of a rod ?
Supposing the quantity was 7 rods, its value in feet would
be found by multiplying it by 16^ ; 16^X7=115^, or 115.5 ;
but it was not 7 rods, but 7 tenths, of a rod, whose value we
wish to find ; the answer obtained, therefore, is 10 times too
large; dividing by 10, it is 11.55, — 11 feet and 55 hundredths.
In order to find the value in inches of 55 hundredths of a
foot, we will call it 55 feet; the answer is, 55X12=660 —
660 feet ; but, as we regarded the 55 as 100 times greater
in value than it is, the answer is 100 times too large ; divid-
ing it by 100, the answer is 6.60 inches, = 6 inches and 60
hundredths, or 6 tenths.
The above analysis shows the nature of the operation in
all cases.
152 PRACTICAL EXAMPLE^.
' 2. What is the value, in feet and inches, of .3 of a rod ?
3. What is the value of .94 of a rod ?
4. What is the value of .26 of a rod ?
5. How many shillings and pence are there in .65 of a £?
6. How many shillings and pence are there in .8 of a £ ?
7. How many pence are there in .7 of a shilling ?
8. How many pence are there in .16 of a shilling?
9. What is the value of .19 of a £?
10. What is the value of .74 of a bushel ?
11. What is the value of .9 of a bushel?
12. What is the value, in rods and feet, of .7 of an acre ?
13. What is the value of .9 of an acre ?
14. What is the value of .12 of an hour?
15. How many minutes and seconds in .15 of an hour?
16. Find the value of .34 of a week.
17. Find the value of .162 of a week.
18. Find the value of .84 of a minute.
19. How many feet in .761 of a cord?
20. How many feet and inches in .2 of cord ?
21. How many feet in .74 of a cord ?
22. How many feet in .13 of a cord?
SECTION XXV.
PRACTICAL EXAMPLES.
1. Add $1.50+$.3754-$.0625+$.1875+$5.00.
2. Add $34.75-[-^6.00-4-$.3754-$.08.
3. A man had S50, and spent $.375 of it ; how much had
he left?
4. A man had $10.00, and spent $.875 of it ; how much had
he left?
5. A watch cost $45.675 ; the chain and key, $4.845 ; what
did the whole cost ?
6. The owner then sold the watch, chain, and key, for
$48,375 ; how much did he lose?
7. A man set out on a journey with $10.00; the first day
he spent $1,125 ; how much had he left?
DECIMAL FRACTIONS. 158
8. The second day he spent $1.425 ; how much had he left ?
9. The third day he spent S1.67 ; how much h^d he left ?
10. The fourth day he spent $.875; how much had he
left?
11. What is the cost of 21 lbs. of flour at $.05 per lb.?
Why do you point off two decimals in the answer ?
12. What is the cost of 35 lbs. of flour at $.045 per lb.?
Why do you point off three decimals ?
13. What is the cost of 12.5 lbs. of flour at $.05 a lb. ?
14. What is the cost of 15.5 lbs. of flour at $.045 a lb.?
15. What is the cost of 26.25 lbs. of flour at $.0375 a lb.?
16. What is the cost of 13.75 lbs. of flour at $.0425 a lb. ?
17. What is the cost of 15 barrels of flour at $4.75 a bar-
rel?
18. What is the cost of 17.5 barrels of flour at $5.25 a
barrel ?
19. What is the cost of 3 tons of hay at $7.56 a ton ?
20. What is the cost of 13.5 tons of hay at $9.00 a ton ?
21. What are 17 barrels of cider worth at $1.75 a barrel?
22. What cost 16 gallons of molasses at $.345 a gallon?
23. Divide $1.05 into 21 equal parts; what will each part
be?
24. How many lbs. of flour will $15.75 buy, at $.045 a lb. ?
25. How many times is $.05 contained in $.625 ?
26. How many lbs. of flour can be bought for $.6975, at
$.045 per lb. ?
27. How many times is $.0375 contained in $984,375 ?
28. How many times is $.0425 contained in $584,375 ?
29. How many barrels of flour will $71.25 buy, at $4.75
per barrel ?
30. How many barrels of flour will $91,875 buy, at $5.25
per barrel ?
31. How many tons of hay can be bought for $22.68, at
$7.56 per ton?
32. How many times is $9.00 contained in $121.50 ?
33. A shipmaster paid $29.75 for ballast, giving $1.75 a
ton ; how many tons did he buy ?
34. How many times is $.345 contained in $5.52 ?
35. What cost 14 lbs. of flour at $.045 a lb., .and 28 lbs.
of sugar at $.095 a lb. ?
154 FRACTIONS.
SECTION XXVI.
PRACTICAL QUESTIONS IN VULGAR AND DECIMAL
FRACTIONS.
1. Bought 7 cwt. 15 lbs. sugar at S6.62| per cwt., and sold
it at 7 cents per lb. What was the gain ?
2. Bought 156 gallons of wine at 93 cents per gallon, and
sold it at 34 cents per quart. What was the gain ?
3. Bought 7 cwt. 1 qr. 11 lbs. coffee at $12.50 per cwt., and
sold it at 14 cents per lb. What gain ?
4. Bought 37 yards broadcloth at $5.25 per yard ; sold 20
yards of it at S7.00 per yard, and the remainder at $6.31 per
yard. What was the gain ?
5. Bought 24 yards broadcloth at $6.40 per yard; sold 22f
yards at $7.25 per yard, and the remnant for 5 dollars. What
was the gain ?
6. Bought 87 E. E. calico at 17 cents per E. E., and sold
it at 21 cents per yard. What gain ?
7. Bought 4 dozen books at $1.50 per dozen, and sold
them at 16 cents each. What gain?
8. Bought 13 dozen brooms at $1.04 per dozen, and sold
them at 15 cents each. What gain?
9. Bought 5^ dozen mats at $3.40 per dozen, and sold them
at 36 cents each. What gain ?
10. Bought 17 bushels of salt at 65 cents per bushel, and
sold it at 21 cents per peck. What gain ?
11. Bought one barrel of wine at 78 cents per gallon, and
sold it at 16 cents per pint. What gain?
12. Bought 3 dozen baskets, at $2.05 per dozen, and sold
1 dozen at 31 cents ; 1 dozen at 37 cents ; and 1 dozen at 42
cents each. What gain ?
13. Bought 48 yards broadcloth at $5.62 per yard ; lost
17 yards by fire, and sold the remainder at $6.25 per yard.
How much gain or loss ?
14. Bought a hhd. molasses containing 131 gallons, at 34
cents per gallon ; 1 6 gallons leaked out ; sold the remainder
at 37 cents per gallon. What gain or loss ?
15. Bought 3^ dozen axes at $6.80 per dozen, and sold
them at 92 cents each. What gain ?
16. Bought 7 dozen pails at $1.42 per dozen, and sold them
at 21 cents each. What gain ?
REDUCTION OF CURRENCIES. 155
17. Bought 8J dozen shovels at S9.25 per dozen, and sold
thera at $1.00 each. What gain ?
18. Bought 74 yards carpeting at 73 cents per yard, and
sold it at 87^ cents pei* yard. What gain?
19. Bought 164 bushels corn at 54 cents per bushel; sold
93 bushels at 67 cents, and the remainder at 50 cents per
bushel. How much loss or gain ?
20. Bought 75 barrels apples at $1.37 per barrel ; lost 15
barrels by decay, and sold what remained at $ 2.12 per bar-
rel. What loss or gain ?
21. Bought 13 dozen oranges at 7 cents per dozen ; lost by
decay 2^ dozen, and sold the remainder at 2^ cents each.
What gain ?
22. Bought 15 dozen pairs of shoes at $4.87 per dozen,
and sold them at 63 cents per pair. What gain ?
23. Bought 18^ thousand of boards at $9.50 per thousand;
sold 6 thousand at $12.25 per thousand, and the remainder at
$8.42 per thousand. What gain ?.
24. Bought 21^ cords wood at $4.75 per cord ; sold 8 cords
at $5.50 per cord, and the remainder at $4.25 per cord.
What gain or loss ?
25. Bought 209 bushels apples at 27 cents per bushel ; sold
46 bushels at 49 cents per bushel, and the remainder at 25
cents per bushel. What gain ?
SECTION XXVII.
REDUCTION OF CUEEENCIES.
English Currency.
1. Reduce 67£ to dollars and cents.
As 4s. 6d. or 54d.=$1.00 (see Table, p. 35,) and 20s. or
240d.=l£, 1 dollar is t^-^q of a £. Reducing this fraction to
its lowest terms, it is ^^. The question therefore is this ; in
67£ how many ^ of a £ ? Dividing 67 by the fraction, we
have 297^ dollars, for the answer. The fraction ^ gives 77
cents and 7 mills.
156 REDUCTION OP CURRENCIES.
2. Reduce 87£ to dollars and cents.
3. Reduce 104£ to dollars and cents.
4. Reduce 64£ to dollars and cents.
5. Reduce 167£ to dollars and cents.
6. Reduce 520£ to dollars and cents.
7. Reduce 84£ 6s. to dollars and cents.
First reduce the 6s. to the decimal of a £, ^j=.3 ; the
sum then is, 84.3£. Reduce it in the same way as the cases
above.
8. Reduce 124£ 13s. to Federal money.
9. Reduce 36£ 9s. 6d. to Federal money.
10. Reduce 71£ 18s. 4d. to Federal money.
To Reduce Federal money to Sterling.
11. In 684 dollars, how many £, s. and d. ?
As $1.00=^^ of a £, 1£=^ of $1.00. The question
therefore is, in 684 dollars, how many ^ of Sl.OO ? Divid-
ing by the fraction we have for the answer, £153.9, or 153£
18s.
12. In $74.25, how many £, s. and d.?
13. Reduce $186.40 to Sterling money.
14. Reduce $564.35 to Sterling money.
15. Reduce $640.15 to Sterling nioney.
The comparative value of the dollar and the pound sterling,
as given above, is called the nominal par value. The actual
value of the £, is higher than is here given. This difference
is usually estimated in trade by adopting the nominal par
value, given above, as the basis of the calculation, and then
adding or subtracting a certain per cent, as 8 or 10 per cent.,
to compensate for the inequality of value.
Canada Currency.
5s.=60d.=$1.00.
16. In 74£ 15s., how many dollars and cents.
As $1.00=60d., and l£=240d, $1.00 is ^, or ^ of a £ ;
multiplying by 4, the answer is $299.00.
1 7. In 126£ 12s. Canada currency, how many dollars and
cents ?
18. . Reduce $841.50 to Canada currency.
INTEREST. 157
New England Currency.
6s.=72d.=$1.00.
19. In 64£ 8s. how many dollars and cents ?
$1.00=3^,=^^^ of a £. Eeduce the 8s. to a decimal of
a £, and divide by the fraction ; we have $214.66f .
20. Reduce 120£ 12s. 6d. to Federal money.
New York Currency.
8s.=96d.==$1.00.
21. Reduce 146£ 6s. 4:d. to Federal money.
' As $1.00=T^j=-^ of a £, reducing the shillings and
pence to the decimal of a £, and dividing by the fraction, we
have $365.75.
22. Reduce 54£ 10s. 6d. to Federal money.
PennsylvaniaT Currency.
7s. 6d.=90d.=S1.00.
23. Reduce 16£ 5s. 6d. to Federal money.
$1.00=/^=! of a £.
24. Reduce 7£ 8s. 9d. to Federal money.
SECTION XXVIII.
INTEREST.
[See Section XTV. Part I.J-
Ruh. Find the interest of 1 dollar for the given time ;
multiply the principal by it, and point off as in the multipli-
cation of decimals.
1. What is the interest of S156.34 for 11 months and 20
days?
As the interest of 1 dollar for 2
months is 1 cent, for 10 months, it
will be 5 cents, .05. As the interest
of 1 dollar for 6 days is 1 mill, for
30 days it will be 5 mills, and for 20
days, 3 mills and ^, making 8 mills
and \. Set down the 8 at the right
14
3)156.34
.058
125072
78170
5211
$9.11983 Ans.
158 INTEREST.
hand of the .05, and for the ^ divide by 3. Observe, the 0
before the 5 must be retained, otherwise it would be 5 tenths
of a dollar, or 50 cents, and the answer would be 10 times
too great. If there are no cents, there must be two ciphers
at the left hand of the mills. The number of cents for the
multiplier is always equal to half the greatest even number
of months, — the number of mills is one sixth of all the days
over and above the greatest even number of months.
2. Interest of $384.18 for 7 months and 10 days ?
3. Interest of $147.19 for 5 months 15 days?
4. Interest of $568.25 for 9 months 13 days?
5. Interest of $81.40 for 10 months 14 days ?
6. Interest of $56.32 for 12 months 24 days?
7. Interest of $75.30 for 14 months 18 days ?
8. Interest of $644.46 for 15 months 24 days?
9. Interest of $831.00 for 1 year 4 months 12 days ?
10. Interest of $380.00 for 1 year 7 months ?
11. Interest of $500.00 for 1 year 5 months 6 days?
12. Interest of $27.42 for 4 months 17 days ?
13. Interest of $13.18 for 6 months 23 days ?
14. Interest of $1000.00 for 5 months 4 days ?
15. Interest of $65.48 for 30 days, or 1 month?
16. Interest of $94.00 for 30 days ?
17. Interest of $840.60 for 18 days ?
18. Interest of $632.00 for 18 days?
19. Interest of $349.40 for 12 days ?
20. Interest of $267.62 for 12 days ?
21. Interest of $384.92 for 15 days?
22. Interest of $811.19 for 20 days?
23. Interest of $673.94 for 5 months 11 days ?
24. Interest of $460.00 for 8 months 18 days ?
25. Interest of $460.00 for 8 months 18 days, at 12 per
cent. ?
26. Interest of $460.00 for 8 months 18 days, at 8 per
cent. ?
27. Interest of $460.00 for 8 months 18 days, ^t 7 per
cent. ?
28. Interest of $460.00 for 8 months 18 days, at 5 per
cent. ?
29. Interest of $1500.00 for 15 months, at 4 per cent. ?
30. Interest of $145.80 for 7 months 11 days, at 8 per
cent. ?
INTEREST. 159
31. Interest of $341.18 for 2 years 9 months IS days?
Asjhe interest of a dollar for 30 days, is 5 mills, for ^ of
30 days, or b days, it is one mill. As 1 mill is txAtttj ^^^ thou
sandth of a dollar, it follows that the interest of 1 dollar for
1 day, is one sixth of a thousandth, or ^^Vtt ^^ ^ dollar. For
two days, therefore, it wiU be ^^^^j^ ; for 15 days, s^^j^ of a
dollar.
A convenient rule, therefore, when the time is short, is the
following :
Mvltiply the sum hy the number of days, and divide the
product hy 6000.
This is often the shortest method. You divide by 1,000,
by removing the decimal point three places to the left. It
only remains then, after doing this, to multiply by the num-
ber of days, and divide by 6.
32. What is the interest of $348.25 for 18 days?
Dividing by 1000, you have $0,348:^, — thirty four cents,
eight mills and a quarter. Instead now of multiplying by 18
and dividing by 6, you may multiply by 3, for 18 is 3 times 6.
3 times 0.348^ is $1.044| — Ans.
33. Interest of $725.80 for 24 days ?
34. Interest of $341.18 for 36 days?
35. Interest of $67.45 for 54 days ?
36. Interest of $641.18 for 42 days?
37. Interest of $84.16 for 15 days?
To find the amount, add the interest to the principal ; or,
find the amount of $1.00 for the given time, and multiply
the principal by .it.
38. What is the amount of $560.50 for 8 months 12 days?
39. Amount of $964.25 for 15 months 18 days?
40. Amount of $460.00 for 1 year 6 months ?
41. Amount of $120.50 for 2 yeai-s 4 months?
42. Amount of $68.40 for 1 year 6 months 24 days?
43. Amount of $500.00 for 2 years 3 months ?
44. Amount of $730.50 for 6 months 12 days?
45. Amount of $840.25 for 4 months 18 days ?
46. Amount of $40.50 for 8 months 12 days ?
160 PARTIAL PAYMENTS.
SECTION XXIX,
PARTIAL PAYMENTS.
\
When Partial Payments are made on a note, the amount
due on the final payment of the note, may be found by the
following rule.
Find the interest on the note up to the time of the first
payment ; if the payment exceeds the interest, deduct it from
the amount, regarding the remainder as a new principal ; on
this, calculate the interest to the time of the next payment,
and so on. If any payment is less than the interest then due,
reserve it, and compute the interest on to the time when the
payments, added together, shall exceed the interest due ; then
subtract the sum of the payments from the amount then due,
and proceed as before.
1. A note of 200 dollars is given July 1, 1834, on which
are the following partial payments ;
Dec. 15, 1834, $25.00.
March 1, 1835, 2.50.
Aug. 10, 1835, 45.00.
What was due Dec. 31, 1835 ?
2. A note of $340.25 is given Aug. 1, 1840.
Endorsements, — Jan. 10, 1841, $28.40.
July 1, 1841, 9.00.
March 14, 1842, 74.00.
What was due Jan. 1, 1843 ?
8. A note of $480.00 is given June 9, 1841.
Endorsements, — Sept. 11, 1842, $60.00.
Jan. 3, 1843, 95.00.
March 12, 1844, 100.00.
What was due Dec. 1, 1844?
4. A note of $675.40 is given July 3, 1843.
Endorsements, — Jan. 4, 1844, $65.00.
April 17, 1844, 29.50.
Nov. 18, 1844, 74.00.
What is due Jan. 1,1845?
PARTIAL PAYMENTS. — ANNUAL INTEREST. 161
5. A note of $345.40 is given Aprin, 1843.
Endorsements, — Dec. 1, 1843, S40.00.
. : .June 10, 1844, 90.00.
f ^ Oct. 4, 1844, 31.50.
Feb. 6, 1845, 17.00.
What is due Jan. 1, 1846 ? ^
The rule given above is the legal rule. When, ho;i^ver,
the note is paid within a year from the time when it was
given, the following rule is usually employed.
Find the amount of principal and interest of the whole
note, from the time it was given, till the final payment.
Find the amount of each payment, from the time it was
paid, till the final payment ; and the sum of these amounts
subtract from the amount of the whole note ; the remainder
will be the balance due.
6. A note of $525.00 is given Sept. 1, 1844.
Endorsements, — Dec. 30, 1844, $58.75.
March 4, 1845, 104.20.
June 8, 1845, 63.40.
What is due Aug. 21, 1845 ?
7. A note of $784.50, given July 7, 1844, has the follow-
ing endorsements, —
Sept. 5, 1844, $54.00.
Nov. 10, 1844, 60.00.
Jan. 12, 1845, 75.00.
March 17, 1845, 100.00.
What is due May 1, 1845 ?
ANNUAL INTEREST.
When a note is given, payable at a longer period than a
year from the date, it is usual to express in the note, that the
interest shall be paid annually. At the end of a year, the
holder of the note may compel the payment of the interest.
In such cases the debtor, instead of paying the interest that
is due, sometimes renews the note, adding the interest to the
principal. Thus, at the end of each year the interest due
is added in, and goes to make a new principal for the follow-
ing year. This is called Compound Interest ; but the com-
14*
162 ANNUAL INTEREST.
putation of it, is the same as in simple interest ; for, if the
interest is not computed every year, and either paid or put
into the note by renewal, that interest cannot draw interest.*
The law regards it the duty of the creditor to remind tj^e
debtor of his debt, by exacting the payment of the interest^
every year. If he does not do this, he can derive no ad-
vantage from the promise in the note to pay the interest
annually.
ILLUSTRATION.
Boston, March 1, 1845.
$100.00.
For value received, I promise to pay to John Jones, or
order, one hundred dollars, in five years, with interest an-
^"^1^- Samuel Barton.
If John Jones does not exact the interest till the end
of the five years, and, if he obtains no renewal of it, the
amount of the note will be only $130.00 ; for the interest of
100 dollars, for five years, is 30 dollars.
If, however, he obtains a renewal of the note at the end
of each year, the principal of the note, for the second year,
will be $106.00.
8. What will the principal of the note for the third year
be?
9. What will the principal of the note for the fourth year
be?
10. What will the principal of the note for the fifth year
be?
11. What will be due, principal and interest, at the end of
the fifth year ?
12. How much would the holder of the above note lose by
omitting to obtain any renewal of it, or any payment of an-
nual interest ?
$250.00 ITew York, July 1, 1845.
For value received, I promise to pay John Foss or order,
two hundred and fifty dollars, in four years, with interest
annually. Amos Carr.
• In some of the States, the interest, alter it falls due, draws simple interest tiU it is paid.
DISCOtTNT. 163
13. If no interest is paid on this note till the principal is
due, and if no renewal of the note is made, what will be the
amount of the note at the time of payment ?
14. If the note is renewed each year, what 'will be the
principal of the note, for the second year ?
15. What will be the principal of the note for the third
year?
16. What will be the principal of the note for the fourth
year?
17. What will be the amount of the last note at the time
of payment?
18. How much would the holder, John Foss, lose, by neg-
lecting to obtain any annual payment of interest, or any re-
newal of the above note ?
SECTION XXX.
DISCOUNT.
CSee Section XIV. Part I.]
Examples.
1. What is the present worth of $475.50, payable in 3
months ?
2. What is the present worth of $341.00, payable in 65
days?
3. Present worth of $940.25, payable in 4 months ?
4. Present worth of $156.30, payable in 96 days?
5. Present worth of $312.60, payable in 35 days?
6. Present worth of $500.00, payable in 41 days ?
7. Present worth of $814.67, payable in 65 days ?
8. Present worth of $46.30, payable in 20 days ?
9. Present worth of $124.45, payable in 5 months ?
10. Present worth of $360.20, payable in 4^ months ?
164 BANKING. ^
SECTION XXXI.
BANKING.
[See Section XIV- Part I.]
To find the present worth of a note given to a bank, pay-
able at some future time, find the present worth of 1 dollar
for the given time, and multiply the sum named in the note
by it.
1. What is the present worth of a note for 100 dollars, dis-
counted at a bank, for 60 days ?
Interest of 1 dollar for 63 days is .0105 ; this subtracted
from 1 dollar, leaves for the present worth .9895.
2. What is the present worth of a note for $450.00, dis-
counted at a Bank, for 90 days ?
3. I give my note to a bank for $250.00, for 60 days ;
what do I receive ?
4. I give my note to a bank for $520.00, for 120 days ;
what do I receive ?
5. Present worth of a bank note for $600.00, discounted
for 60 days ?
6. Present worth of a bank note for $150.00, discounted
for 120 days ?
7. Present worth of a bank note for $75,00, discounted for
30 days ?
8. Present worth of a bank note for $1000.00, discounted
for 60 days ?
9. Present worth of a bank note for $560,00, discounted
for 120 days?
10. Present worth of a bank note for $150.00, discounted
for 30 days ?
To find what must be the face of a note given to a bank,
in order to obtain a certain sum — find the present worth of 1
dollar for the given time, and divide the sum you wish to obtain
hy it ; the quotient, will express the sum that must he named in
the note. This, you observe, is just the reverse of the pre-
ceding case.
11. For what sum must I give my note to a bank, payable
in 60 days, in order to receive $98.95 ?
LOSS AND GAIN. — PER CENTAGE. 165
12. For what sum must I give my note to a bank, payable
in 120 days, in order to receive $509.34 ?
13. For what sum must I give my note to a bank, payable
in 60 days, in order to receive $593.70 ?
14. For what sum must I give my note to a bank, payable
in 30 days, in order to receive $10000.00 ?
SECTION XXXII.
LOSS AND GAIN. — PER CENTAGE.
[See Section XIV. Part I.]
1. A man bought a horse for 75 dollars, and sold him for
$82.50 ; what did he gain per cent. ?
2. A man bought a chaise for $178.00, and sold it for
$154.50 ; what did he lose per cent. ?
3. A merchant bought a lot of flour at $4.62 a barrel, and
sold it at $5.15 a barrel ; what was his gain per cent. ?
4. A merchant bought a piece of broadcloth for $4.30 per
yard ; what must he sell it for to gain 12 per cent. ?
5. A man has $1200.00 invested in a manufactory; he
receives, for his half-yearly dividend, 30 dollars ; what per
cent, is that on his stock ?
6. A merchant fails, owing $8540.00, and can pay but
$2700.00 ; how much will that be on a dollar ?
7. A man failing in business, agrees to pay his creditors
87 cents on a dollar ; what must a creditor receive, whose
claim is $740.30 ?
The pupil should be encouraged habitually to reason upon
the operations he performs ; so that his method of procedure
may be suggested by the relations of the numbers, and not
dictated by a special rule. To aid in this important habit, a
few remarks will be made on some of the foregoing examples.
These may serve as specimens of analysis, and suggest to the
student a similar course of reasoning in other cases.
Example 1. The whole gain is $7.50 ; if this gain were
made on an outlay of one dollar, the gain would be • seven
hundred and fifty per cent. ; but the gain is made on an outlay
of 75 dollars ; the gain per cent., therefore, is one seventy-fifth
of the whole gain.
166 PER CE^TTAGfe.
Example 4. If the cost was 1 dollar a yard, lie must add
12 cents ; if 2 dollars, he must add 24 cents, &c.
Example 5. If 30 dollars had been the gain upon 1 dollar,
it would have been 30 hundred per cent. ; but the gain was
upon 1200 dollars ; the per cent., therefore, must be one
twelve hundredth of 30 dollars.
8. I invest in a factory 1260 dollars^ and receive for my
yearly dividend 86 dollars ; what is that per cent. ?
9. I purchase flour at $4.75 per barrel ; what must I sell it
for to gain 12 per cent. ?
10. A merchant bought a ship for 11475 dollars, and sold
her for $13680 ; what did he gain per cent. ?
11. The population of the State of New York in 1810, was
959949 ; in 1820, it was 1372812 ; what was the gain per,
cent, in that term of 10 years ?
12. In 1830, it was 1918604; what was the gain per cent,
from 1820 to 1830 ?
13. In 1840, it was 2428921 ; what was the gain per cent,
from 1830 to 1840 ?
14. The population of Ohio in 1810, was 230760 ; in 1820,
it was 581434 ; what was the gain per cent, ?
15. The population of Ohio in 1830, was 937903 ; what
was the gain per cent, from 1820 to 1830 ?
16. In 1840, it was 1519467 ; what was the gain per cent,
from 1830, to 1840 ?
17. Massachusetts had, in 1810, 472040 inhabitants; in
1820, it had 523287 ; what was the gain per cent, in 10 years ?
18. Massachusetts had, in 1830, 610408 inhabitants ; what
was her gain per cent, from 1820 to 1830 ?
19. In 1840, Massachusetts had 737699 inhabitants ; what
was the gain per cent, from 1830 to 1840 ?
20. An agent sells 12000 ' dollars' worth of cloth for a
factory, charging 2^ per cent, commission ; what will be his
remuneration ?
21. If I buy for a merchant, at a commission of 4 per cent.,
500 barrels of flour, at $4.40 per barrel, what am I entitled
to for my commission ?
22. What is 3 per cent, on $674.54?
23. What is 2 per cent, on $781.50 ?
24. What is the value of five 100 dollar shares in a bank,
at 4^ per cent, advance ?
25. What is the value of seven 100 dollar shares, at 6 per
cent, discount ?
FEB CENTAGE, 167
26. "What is the value of 18 shares bank stock, 60 dollars
a share, at 4 per cent, discount .''
27. What is the duty on a quantity of broadcloth, whose
value is 1735 dollars, at 15 per cent. ?
28. What is the duty on a quantity of iron, whose value is
3456 dollars, at 18 per cent.?
29. What is the commission on the sale of 1246 dollai's'
worth of cloth, at 3 per cent. ?
30. A man bought a lot of hay for 13 dollars a ton ; he
sold it for $14.25 a ton ; what did he gain per cent. ?
31. Bought tea for 46 cents a pound ; what must I sell it
for a pound to gain 12 per cent. ?
32. What is the worth of 750 dollars, bank stock, at 7^ per
cent, advance ?
33. What is the worth of 8500 dollars, bank stock, at 9 per
cent, discount ? '
34. I sell flour at $5.32 per barrel, and thereby gain 12
per cent, on my outlay, what did the flour cost ?
Every $1.00 laid out in the purchase has brought me a
return of $1.12*; the number of dollars I paid out on a barrel
must therefore equal the number of times $1.12 will go in
$5.32.
35. A merchant sells a ship for 13680 dollars, gaining
thereby 14^^ per cent, on what she coBt him ; what did the
ship cost ?
36. 300 dollars is 2^ per cent, on what sum ?
37. $15.63 is 2 per cent, on what sum ? .
38. Bought 12 barrels of flour, each containing 196 pounds,
at $5.42 per barrel, and sold it at 26 cents for 7 pounds ; how
much gain in the whole, and how much gain per cent. ?
39. Bought 43 dozen pairs of shoes, at $4.30 per dozen,
and sold them at 62 cents per pair ; what gain in all? what
gain per cent. ?
40. Bought 20 barrels of apples, each containing 2f bushels,
at $2.10 per barrel, and sold them at $1.25 per bushel ;
What gain in all ? what gain per cent. ?
41. Bought 375 barrels of flour, at $5.20 per barrel, and
sold 200 barrels at $6.10, the remainder at $6.42 per barrel ;
What gain in all ? what gain per cent. ? ,.
. 42. Bought 34 acres of land, at 41 4ifollars per acre ; sold
ifpit $1700.00 ; how much gain in all ? what gain per qent. ?
168 ALLIGATION.
SECTION XXXIII.
ALLIGATION.*
The operations under this rule show the method of finding
the value of a mixture, when the price and quantity of each
of its ingredients are given ; also, to find the quantity of each
ingredient, when its price is given, and it is required to unite
them so as to form a mixture of a given value.
Case 1. To find the value of the mixture, when the quantity
and 'price of each of the ingredients are given.
1. Mix 15 bushels of oats, at 40 cents per bushel; 12
bushels of barley, at 60 cents ; and 24 bushels of corn at 83
cents ; what will the mixture be worth per bushel ?
It is evident that if you find the value of the whole, and
divide the sum by the number of bushels, the quotient will be
the value per bushel.
2. Mix 20 pounds of tea, at 43 cents per pound ; 18 lbs. at
61 cents ; and 11 lbs. at 74 cents per pound; what will the
mixture be worth ?
3. If 41 lbs. of coffee, at 13 cents per lb. be mixed with 45
lbs. at 9^ cents ; and 27 lbs. at 15 cents ; what will the mix-
ture be worth per pound ?
Gase 2. To find the quantity of each ingredient, when its
price and that of the required mixture are given.
4. K I mix oats worth 2s. per bushel, with rye worth 5s.,
so as to make the mixture worth 3s. per bushel, in what pro-
portion must I mix them ?
It is evident, that if I put in 1 bushel of oats, I gain 1 shil-
ling, f Now I must put in rye enough with this bushel of oats
to lose (I i shilling. On every bushel of rye put in, I lose 2
shillings l! therefore, in order to lose 1 shilling, I must put in
^ a bushel. ^ I must therefore put in 1 bushel of oats to ^ a
* The word'-^j^gWifion sigftifiBi^i^^j/???^ torjether ; and has reference to a
particular way O'^VWl'*^ ^^^"^ ^&rs toget;iier, by means of which operations
of this kind have'^^HM^lform'W^. T^^name is retained as a matter of
convenience; but'^r^M^&bn;:]!! it J>^s^or the proj^rcss of tlie pupil that
fce should pursue a sin^PJPl^glyticat liit^if^^n all tlic operations.
ALLIGATION. 169
bushel of rje. It is evident that, if I double the quantity thus
found of each ingredient, the value of the mixture will be the
same ; or I may take any equal multiples of the quantities, as
4 bushels of oats, and 2 bushels of rye ; 6 bushels of oats, and 3
bushels of rye ; 20 bushels of oats, and 10 bushels of rye, &c.
5. If I mix oats, worth 2s. per bushel, with rye, worth 6s.,
BO as to make the mixture worth 3s. per bushel ; in what pro-
portion must they be mixed ?
6. Mix oats, worth 3s. per bushel, with wheat, worth 7s.,
so as to make the mixture worth 5s. per bushel ; in what pro-
portion must they be mixed ?
7. Mix the same ingredients, at the same price, so as to
make the mixture worth 6s. per bushel j in what proportion
must they be mixed ?
8. In what proportion must oats, worth 2s., and wheat, worth
8s., be mixed, to make the mixture worth 4s. per bushel ?
9. How can you mix corn, worth 80 cents per bushel, and
rye, worth 85 cents, with barley, worth 46 cents, so as to
make a mixture worth 60 cents per bushel ?
Here you have three ingredients. First, mix barley with
one of the dearer ingredients, so as to make a mixture of the
required value ; then mix barley with the other ingredient,
and see how much you have taken of each.
10. Mix 3 sorts of tea, at 25 cents, 33 cents, and 40 cents
per lb., so as to make a mixture worth 30 cents per lb.
11. Mix tea at 20 cents, with tea at 45 cents, and tea at 54
cents per lb., so as to make a mixture worth 38 cents per lb.
12. If you mix sugar, at 6 cents, 8 cents, 10 cents, and 11
cents per lb., in what quantities may they be taken, so as to
make a mixture worth 9 cents per lb. ?
First, take two of the ingredients, one cheaper and one
dearer than the mixture ; form a mixture of these ; then take
the two remaining ingredients in the same way.
13. If three sorts of spirit, worth 60 cents, 75 cents, and 80
cents per gallon, are mixed with water costing nothing, what
must be the proportion to make^a mixture worth 70 cents per
gallon?
It is immaterial in what way you sele^the pairs of ingre-
dients, provided, in each j)air, one of the,ingredients be cheaper
and the other dearer than the reqiured mixture. Thus a great
15
170 ALLIGATION.
Tariety of answers may be obtained whenever there is more
than one pair of ingredients. In all cases, however, the cor-
rectness of the operation may be proved in the following way :
Find the total value of all the ingredients ; if this is equal to
the value of the whole mixture at the required price, the work
is right.
14. Mix 5 sorts of grain, at 25 cents, 30 cents, 33 cents,
45 cents, and 50 cents, so as to make a mixture worth 40
cents per bushel.
Case 3. When the quantity of one ingredient is given.
15. Mix brandy, at 74 cents per gallon, with 24 gallons of
brandy, at 1 dollar per gallon, so that the mixture may be
worth 80 cents per gallon.
Here you observe that the quantity of one of the ingredients
is given. We will first make a mixture of the two, without
regard to this circumstance. If I put in 1 gallon at 1 dollar,
I lose 20 cents. For every gallon at 74 cents, which is put
in, I gain 6 cents. In' order to gain 20 cents, I must, there-
fore, put in 3^ gallons. The quantities stand, then, 1 gallon
at 1 dollar, 3^ gallons ,at 80 cents. But I wish to put in 24
gallons at 1 dollar. To balance this, I must, therefore, put in
24 times 3^ gallons, at 74 cents ; that is 80 gallons.
16. Mix sugar, at 8 cents, 11 cents, and 12 cents, with 100
lbs. of sugar, at 7 cents, so as to make the mixture worth 10
cents per lb.
Case 4. When the quantity of the required mixture is given.
17. Mix oats, at 40 cents, with corn, at 60 cents, so as to
form a mixture of 100 bushels, worth 48 cents per bushel.
If I put in 1 bushel, at 40 cents, I gain 8 cents ; if I put in
1 bushel, at 60 cents, I lose 12 cents. To lose 8 cents, there-
fore, I must put in only § of a bushel. The qiiantities are,
then, 1 bushel at 40 cents, f of a bushel at 60 cents ; making,
when added. If bushels. But 100 bushels is the quantity
required. 100-^-^=60. Each ingredient, therefore, must be
multiplied by 60 ; 60X1=60 ; 60 Xf=40.- The quantities,
then, are 60 bushels at 40 cents, and 40 bushels at 60 cents.
EQUATION OP PAYMENTS. 17-1
SECTION XXXIV.
EQUATION OF PAYMENTS.
If A owes B several sums of money to be paid at differ-
ent times, he may desire to pay the -^hole at once, and con-
sequently to know at what time the whole becomes due.
This time is found by making an equation of the paymerUs,
multiplied by the time, as follows.
1. A owes B 200 dollars; 100 due Jan. 1, 1844; 100 due
Jan. 1, 1846 ; he wishes to pay it all at once. At what time
should he pay it ?
Now, on Jan. 1, 1844, A is entitled to the use of 100 dol-
lars for 2 years longer; 100X2=200 ; equal to the use of 1
dollar for 200 years. If he is to pay the whole together, he
must keep the 200 dollars long enough to balance that claim ;
200)200(1 year, — the answer. The whole should be paid
one year from Jan. 1, 1844.
2. A owes B 100 dollars, due in 6 months ; 200 dollars
due in 12 months ; in how many months should the whole be
paid together ?
100 X 6= 600
200X12=2400
300: 300)3000
10 months ; the answer.
The above is the method usually employed, and is suffi-
ciently exact for the necessities of business ; but it gives a
result a little in favor .of the debtor; that is, it makes the
equated time a little later than it should be. To find the
exact equated time is a problem far too difficult to be used in
ordinary business.
Rule. Multiply each payment by the length of time be-
fore it becomes due. Divide the sum of the products by the
sum of all the payments ; the quotient will express the length
of time in which the whole is due.
3. A owes B several sums, due at different times, as fol-
lows ; $600 in 2 months, $150 in 3 months, $75 in 6 months;
what is the equated time for the whole ? •
4. A man owes $1000 ; of which, 200 are to be paid in 3
172 SQUARE MEASURE.
montlis, 400 In 6 montlis, and the remainder in 8 months ;
what is the equated time for the payment of the whole ?
5. If I owe $1200, one half to be paid in 3 months, one
third in 6 months, and the remainder in 9 months ; in what
time should the whole be paid ?
6. A owes B $640 ; 150 due in 30 days, 200 due in 60
days, and the remainder in 90 days ; what is the equated
time for the whole ?
7. A merchant buys goods to the amount of $1800 ; one
third to be paid in 30 days, one third in 45 days, and the re-
mainder in 90 days ; what is the equated time for the whole ?^
8. If I owe $1000, half to be paid in 60 days, and half
in 120 days, and if I pay $100 down, what will be the
equated time for the remainder ?
SECTION XXXV.
SQUAEE MEASURE.
[See Section XV. Part I.]
1. There is a field in the form of a square, 15 rods on a
side ; how many square rods does it contain ?
2. If the square be 15^ rods on a side, how many square
rods will it contain ?
3. How many square rods are there in a square field meas-
uring 17 rods on a side ?
4. If the field measure 17^ rods on a side, how many
square rods will it contain ?
5. What is the contents of a square field measuring 21^
rods on a side ?
6. What is the area of a rectangular field, its length being
64 rods, and its breadth, 12| rods?
7. There is a rectangular field, its dimensions being 24^
rods, and 76^ rods ; what is the area ?
8. How many acres- are there in a rectangular field, its di-
mensions being 94 rods and 76^ rods?
9. There is a rectangular field containing 7 acres ; its
length is 35 rods ; what is its breadth ?
SQUARE MEASURE. 178
10. There is a rectangular farm ; its length being 132 rods j
its breadth, 86 J; how many acres does it contain?
11. There is a rectangular lot of land containing 325
acres ; it measures on one side 176 rods ; what will it meas-
ure on the other ? .
12. There is a board containing 12 square fpet; it is 13
inches wide ; how long is it ?
13. A table contains 15 square feet ; it is 4 feet long ; how
wide is it?
14. A certain room contains 30 square yards; it is 16 feet
wide ; how long is it ?
15. A piece of cloth is If yards wide ; how much in length
will it require to make 8 square yards ?
16. There is a room 15 feet by 18; how many yards of
carpeting, ^ of a yard wide, will it require to cover it ?
17. How many feet of boards will it require to cover the
sides and ends of a barn, as high as to the eaves, — its
length is 42 feet, width 34, and height 18, — allowing one
fifth of the boards to be wasted in cutting ?
18. What will the above-named amount of boards cost at
$11.50 a thousand feet?
19. A road 3^ rods wide, passes through a man's land 1
mile ; how much of his land does it take ?
20. To what damages will he be entitled, allowing him 28
dollars an acre ?
21. There is a right angled triangle ; its base is 64 rods,
and perpendicular 20 rods ; how many acres does it contain ?
(See Sec. XVII. Pt. I.)
22. There is a right angled triangle ; its base is 84 rods,
and perpendicular 26 rods ; how many acres does it contain?
23. There is a right angled triangle ; its base is 49 rods,
perpendicular 34 rods ; how many acres does it contain ?
24. There is a right angled triangle ; its area is 640 rods ;
the base is 64 rods ; what is the perpendicular ?
25. A right angled triangle has an area of 1092 rods ; its
base is 60 rods ; what is the perpendicular ?
15*
174
DUODECIMALS.
SECTION XXXVI
DUODECIMALS.
In measuring wood and lumber, the dimensions are taken
in feet and inches. As one inch is -ji^- of a foot, the multipli-
cation of feet and inches by feet and inches, is the same as
multiplying integers and twelfths by integers and twelfths.
Take the following example :
Operation.
2 A
1. What is the contents of a
board 3 feet 7 inches long, and
2 feet 4 inches wide ?
Ans. 6 ft T^.
This answer may be reduced to more simple terms. -^^^
=i^+tI¥ ; adding T^j-f l=f l» and this again = 2 feet-|-
■^ ; adding the 2 feet to the 6 feet, the answer stands 8 feet
As the fractions decrease in value at a twelve fold rate,
whenever the numerator exceeds 12, the excess may be set
down, and the one or more carried to the next higher fraction.
2. Multiply 5 feet, 2 inches,
by 11 feet, 9 inches.
5
11
T^
rW
3
56
it
if
tU
Ans. 60 ^ tI^.
To render the operation more simple, call the 12ths or
inches, primes, (marked ') and the 144ths or fractions of the
second order, seconds, (marked " ;) then begin with the lowest
order and multiply, setting each product in its own place,
with the mark appropriate to express its value.
3. Multiply 13 ft. 5 in., by 2 ft.
11 in.
ft.
/
13
5
2
11
12
3'
7"
26
10
Ans. 39 1' 7"
EXTRACTION OF THE SQUARE ROOT. 175
4. Multiply 3 ft. 9 in. by 7 ft. 4 in.
5. Multiply 9 ft. 8 in. by 4 ft. 9 in.
6. Multiply 15 ft. 2 in. by 9 ft. 1 in.
7. Multiply 8 ft. 6 in. by 2 ft. 4 in.
8. What is the contents of a board, 14 ft. 5 in. long, and
1 ft. 1 in. wide ?
9. How many feet in a load of wood, 8 ft. 6 in. long, 4 ft.
2 in. wide, and 3 ft. 7 in. high ?
Multiply two of the dimensions together, and that product
by the third dimension.
10. How much wood in a load 11 ft. 3 in. long, 4 ft. 4 in.
wide, 3 ft. 11 in. high?
Divide the cubic feet by 128 for cords, and the remainder
by 16 for cord feet, or eighths of a cord.
11. How much wood in a pile 38 ft. 6 in. long, 4 ft. 2 in.
wide, and 4 ft. high ?
12. How much wood in a load 9 ft. 4 in. long, 4 ft. 3 in.
'wide, 3 ft. 8 in. high ?
13. How much wood in a load 7 ft. 8 in. long, 4 ft. 2 in.
wide, 3 ft. 4 in. high ?
14. How much wood in a load 8 ft 2 in. long, 4 ft. wide,
4 ft. 3 in. high?
15. How many cords of wood will a shed contain, whose
dimensions inside are 22 ft. 6 in. long, 10 ft. 6 in wide, 7 ft.
8 in. high ?
16. Three men own equal shares in a lot of wood lying in
two piles ; one pile is 13 ft. 4 in long, 4 ft. 3 in. wide, 4 ft. 4
in. high ; the other pile is 17 ft. long, 4 ft. wide, 3 ft. 10 in.
high ; how much wood is each man share ?
See note on page 84.
SECTION XXXVII.
EXTRACTION OF THE SQUARE ROOT,
CSee Section XVI. Part I.]
This operation will be best understood, by talking first the
simplest case, where the number is an exact square, and the
root containing only two figures.
176
EXTRACTION OF THE SQUARE ROOT.
What is the square
root of 196?
Operation.
196
100
20)'
96
80
16
96
00
10 1st part of the root.
4 2d part of the root.
Ans.
14
Place a period over the unit figure ; another OA^er that of
hundreds. This will show how many figures there will be in
the root ; for the square of a number has always either twice
as many figures as the number, or one less than twice ai
many. Find the greatest square of tens in the first period,
(in the given example, 100,) and set its root (10) in the
quotient. This will be the first part of the root. Square
the root ; subtract the square from the first period, and bring
down the figures of the next period for a dividend. To the
left hand, place double the part of the root already found for
a trial divisor. Find by trial, what the next figure of the
root must be, and set it down under the first part of the root.
This is the second part, or unit figure of the root. [In try-
ing for this figure, remember, that it must be so small that
when the divisor shall be multiplied by it, and the square of
itself shall be added to the product, the sum shall not ex-
ceed the dividend.] Multiply the divisor by the new figure
of the root; to this add the square of the same figure, and
subtract the sum from the dividend. If the number is an
exact square of two periods, as in the above example, there
M'ill be no remainder; and the two parts of the root thus
found, when added together, will give the whole root.
2. What is the square root of 225 ? Of 324 ?
3. What is the square root of 289 ?
4. What is the square root of 361 ?
5. Wliat is the square root of 625 ?
6. What is the square root of 784 ?
7. Wliat is the square root of 841 ?
8. Wliat is the square root of 961 ?
If there are more than two periods, first consider only the
two left hand periods, and find their root as above directed ;
then consider the part of the root expressed by these two
Of 529?
Of 729 ?
Of 1024?
Of 1296?
Of 1849?
Of 2601 ?
EXTRACTION OF THE SQUARE ROOT. 177
figures as the first part with reference to the next figure, (to
indicate this, you must annex a cipher,) and work for the
next ; and so on.
9. What is the square root of 15625 ?
10. What is the square root of 60516? Of 104976?
Of 2134M ?
Square Root of a Decimal.
If there are decimals in the number, point oflp each way
from the place of units ; adding a cipher, if necessary, to
make the right hand period complete.
11. What is the square root of 2.56 ? Of 12.25 ?
12. What is the square root of 2.25 ? Of 20.25 ?
13. What is the square root of 156.25 ? Of 132.25 ?
14. What is the square root of 13.6^? Of 21.16?
15. What is the square root of 88.36? Of 53.29 ?
16. What is the square root of 1.69 ? Of 1.44?
17. What is the square root of .81 ? Of .64?
18. What is the square root of .01 ? Of 6.25 ?
Square Root of a Vulgar Fraction.
To obtain the square root of a vulgar fraction, find the
square root of the numerator, and of the denominator, and
write the former over the latter.
19. What is the square root of f ? Of if ?
20. What is the square root of ^ ? Of ^ ?
21. What is the square root of i|f ? Of || ?
22. What is the square root of ||| ? Of ||| ?
The correctness of the answer may always be tested by
multiplying the answer found, by itself; if correct, it will re-
produce the original square.
23. What is the square root of ^ ? Of ^ ?
^ 24. What is the square root of if? Of t^V?
Another Method of finding the Root of a Fraction. Re-
duce the fraction to a decimal, and proceed as already directed
in the case of decimals.
25. What is the square root of ^ ? The square root of 1
is 1, the square root of 4 is 2 ; ans. ^ ; or reduce |^ to a deci-
mal, = .25 ; square root, .5, answer.
If the number is not a complete square, annex periods of
ciphers, as decimals, and carry the operation as far as desired.
178 EXTRA.CTION OF THE SQUARE ROdT.
26. What ia^flie square root of 70 ? Of 80 ?
27. What is the square root of 90 ? Of 45 ?
28. What is the square root of 60? Of 84?
29. What is the square root of 200 ? Of 120 ?
30. There is a field in the form of a square, containing 1
acre ; how many rods does it measure on a side ?
31. There is a right angled triangle ; its hypotenuse meas-
uring 60 rods. What is the sum of the squares of the two
legs ? (See Sec. XVII. Part I.)
32. There is a right angled triangfe ; the squares of its
legs added together are 81 rods ; what is the length of the
hypotenuse ?
33. There is a right angled* triangle ; its legs measure, —
one 25, the other 30 rods ; how long is the hypotenuse ?
34. Two men start from the same place; one travels 8
miles east ; the other, 15 miles north ; how far are they then
apart?
35. A ladder 40 feet long stands against a house, the foot
resting on the ground, on a level with the foundation of the
house, and 20 feet distant from it ; how far up will it reach ?
36. The floor of a room measures 16 feet in length, and 14
feet in width ; how long a line will reach diagonally from
corner to corner ?
37. The two parts of a carpenter's square, one 12, the other
24 inches long, may be regarded as the legs of a right angled
triangle ; how long would be the hypotenuse connecting their
extremities ?
38. There is a room 16 feet long, 14 feet wide, and 10 feet
high ; how long must a straight line be, reaching from a
corner of the room, at the bottom, to the diagonal corner, at
the top ?
39. There is a room, the length, breadth, and height of which
are each 10 feet ; how far is it from a corner of the room at
the bottom, to the diagonal corner at the top ?
40. There is a room, the length, breadth, and height of which
are equal ; the distance from a corner, at the bottom, to the
diagonal corner at the top, is 18 feet ; what is the size of the
room?
41. I have a cubic block, measuring 4 inches each way ;
how far apart are its diagonal corners ?
42. How large a cube can be cut from a sphere which is 1
foot in diameter ?
\
EXTRACTION OF THE CUBE ROOT.
179
SECTION XXXVIII
EXTRACTION OF THE CUBE EOOT.
[See Section XIX. Part I.]
We will first consider those numbers tlie cube root of which
is expressed by a single figure. Every exact cube of not more
than three figures, will have for its root some number less
than 10, and, consequently, it will be expressed by a single
figure. This root can be found by successive trials.
Examples.
1. What is the cube root of 125 ? . ,
2. What is the cube root of 216 ?
3. What is the cube root of 512 ?
4. What is the cube root of 729 ?
We will next take perfect cubes, the root of wluch consists
of two figures. operation.
300
30
4096
1000
10, 1st part of the root.
6, 2d part of tlie root.
5. What is the cube
330
3096
16, Answer.
root of 4096?
1800
1(?80
216
3096
0000
Rule. Place a .period over the unit figure, and another
over that of thousands. Find the greatest cube in the first
period, whose root is expressible in tens. Set down this root
as a quotient in division ; find the cube of the root, and sub-
tract it from the first period, and bring down the second period
as a dividend. At the left hand of this set down three times
the square of the root, and under this three times the root ;
add these together, for a trial* divisor. Find, by trial, what the
next figure of the root will be, and set it under the first part
already found. Multiply, by this figure, three times the square
of the first part of the root, setting the product under the
dividend. Multiply, by the square of this figure, three times
the first part of the root, setting the product underneath the
180 EXTRACTION OF THE CUBE KOOT.
other ; under these set the cube of the root figure last found.
Add these three numbers together, and subtract their sum
from the dividend. If the work be correct, there will be no
remainder. Add together the two parts of the root for the
answer.
6. What is the cube root of 2744? Of 205379 ?
7. What is the cube root of 3375 ? Of -5832 ?
8. What is the cube root of 4913 ? Of 10648 ?
9. What is the cube root of 9261 ? Of 15625 ?
10. What is the cube root of 13824? Of 19683 ?
11. What is the cube root of 46656 ? Of 39304 ?
We will next consider the case where there are more than
two figures in the root. The number of figures in the root
can always be determined by the number of periods placed
over the sum, beginning with units, and placing a period over
every third place. If there are more than three periods in
the cube, regard, first, only the two left hand periods, obtain-
ing the first and second figures of the root, just as if they
constituted the whole root. Then, after bringing down the
figures of another period, add the two parts of the root, and
consider their sum as the first part of the root, and proceed to
find the next part. To indicate this, you must annex a cipher
to the figures of the root already found.
12. What is the cube root of 1953125 ?
13. What is the cube root of 2406104 ?
14. What is the cube root of 3796416 ?
If there are decimals in the given sum, point ofi* both ways
from the units' place, adding ciphers, if necessary, to the deci-
mal, in order to make the period complete.
1.5. What is the cube root of 15.625 ?
16. What is the cube root of 35.937 ?
' If the number given is not a perfect cube, add periods of
ciphers, and carry out the root in decimals as far as may be
desired.
17. What is the cube root of 10 ?
18. What is the cube root of 20 ?
19. What is the cube root of 50 ?
20. What is the cube root of 100 ?
21. A bushel, even measure, contains 2152 solid inches ;
what would be the inside measure of a cubic box containing
12 bushels ?
PROrORTION. 181
22. A gallon, wine measure, contains 231 cubic inches ;
what must be the inside measure of a cubic cistern containing
10 barrels ?
23. What would be the measure of a cubic pile of wood,
containing one cord ?
SECTION XXXIX.
' PEOPORTION.
[See Section XX. Part I.]
Several changes that may be made in the terms of a Pro-
portion, are exhibited in page 105. In continuing the subject,
we will first state some further changes that may be made in
the terms without destroying the proportion.
1. Multiply all the terms by the same number.
2. Divide all the terms by the same number.
3. Add the terms of the first ratio for the first antecedent,
and~the terms of the second ratio for the second antecedent.
4. Add the terms of the first ratio for the first consequent,
and the terms of the second ratio for the second consequent.
5. Instead of the sum of the terms in the third case above,
take the difference of the terms.
6. Instead of the sum of the terms in the fourth case' above,
take the difference of the terms.
7. Raise each term to the same power, as second or third
power.
8. Extract of each term the same root.
The result, after each of these operations, will still be a
proportion, and may be proved to be so, by multiplying the
extremes together, and finding the product, equal to that of
the means.
Take the proportion, 4 : 16 : : 9 : 36, and perform on it the
first change, using any number you please for a multiplier,
and then prove the proportion.
Perform on the same proportion the second change.
Perform the third change.
Perform the fourth change.
Perform the fifth change.
16
182 ^ PROrORTION. '
Perform tlie sixth change.
Perform the seventh change, raising to the second power.
Perform the eighth change, extracting the square root.
Finally, you may, iu any case, invert the whole proportion;
or, invert the terms of each ratio ; or invert the means, or the
extremes.
Practical Questions.
1. If 7 lbs. of flour cost 31 cents, what will 196 lbs. cost ?
As the smaller quantity is to the larger quantity, so is the
price of the smaller quantity to the^rice of the larger.
2. If 3 cwt. of hay cost 2 dollars, what will 35 cwt. cost ?
3. If 4 qts. of molasses cost 38 cents, what will 10 qts. cost?
4. If a horse travels 19 miles in 3 hours, how far will he
travel in 11 hours ?
5. ir^he freight of 7 cwt. cost 2 dollars, what will the
freight of 20 cwt. cost ?
6. If 11 dollars buy 3 cords of wood, how many cords will
50 dolla;:s buy ?
7. If 7 bushels of oats last a horse 2 months, how long will
23 bushels last him, at the same rate ?
8. A man bought a horse for 84 dollars, and sold him for
$93 ; what did he gain per cent. ?
As the whole outlay is to 1 dollar, so is the whole" gain to
the gain on a dollar.
9. A merchant buys flour at $4.35 a barrel, and sells it for
$4.63 ; what is his gain per cent.?
10. A and B form a partnership in trade ; A puts in $500,
and B $300, for the same time ; they gain $180 ; what ought
each to share ?
As, the whole stock is to each one's share, so is the whole
gain to each one's gain.
. 11. C and D trade in company; C puts in 750 dollars, and
D $450, for the same time ; they gain 240 dollars ; how much
gain ought each to receive ?
12. Two men buy a lot of wood in company for 340 dollars ;
one takes away 42 cords, the other the remainder, which was
34 cords ; what ought each to pay ?
13. T'wo men hire a sheep-pasture in company for 20,
dollars ; one keeps 30 sheep in it 14 weeks ; the other 24
sheep, 16 weeks ; what ought each to pay ?
Find how many weeks' pasturing for a single sheep each
one had. *
PROPORTION. SIMILAR SURFACES. 183
14. Two men purchase a lot of standing grass for S3 6.50 ;
one takes 3^ tons, the other If tons ; what ought each to pay ?
Reduce the quantity of hay to fourths of a ton, and then
state the proportion.
15. There is a circular piece of ground, whose diameter is
14 rods ; what will be the diameter of a circle containing
twice as much?
16. There is a circular piece of ground containing 2.5
acres ; what will be the area of a circle, the diameter of
which is 3 times as great ?
17. There are two similar triangular fields* ; the smaller
contains 3 acres, the larger 4 ; the base of the smaller is 44
rods ; how long is the base of the larger ?
18. There are two similar rectangular fields ; the smaller is
34 rods wide, and 60 rods long; the other has twice as great
an area ; what are its dimensions ?
19. There is a grindstone 4 feet in diameter; what will be
its diameter after half of it is ground ofi"?
20. There are two similar triangular pieces of land ; the
base of one measures 44 rods ; the other piece has an area 7
times as large as the first ; what is the length of its base ?
21. There are two cisterns of the same shape ; one is 5
' feet deep ; the other has a capacity three times as great ; how
deep is it ?
22. If a ball 5 inches in diameter, weighs 14 lbs., what
will be the weight of one of the same material 6 inches in
diameter ?
23. What, on the same supposition, will be the weight of a
ball of 7 inches diameter ?
24. There are two marble statues of the same form, but
differing in size ; one is 5 feet high, and weighs 740 lbs. ; the
other is 7 feet high ; what will it weigh ?
25. If a tree 2^ feet in diameter at the ground, contains 3
cords of wood ; how much will there be in a tree of the same
-shape, S^ feet in diameter?
26. There are two similar stacks of hay; the smaller is
11^ feet high, and contains 4^ tons of hay; the larger is 14
feet- high ; how much hay does it contain, supposing both to
be of the same solidity ?
27. If an iron field piece 5^ feet long, weighs 1140 lbs.,
184 PROPORTION.
how many lbs. will an iron cannon of the same shape weigh,
that is lOf feet long?
28. There are two anchors of similar form ; the smaller
weighs 1100 lbs., the larger is 2} times as long; what is its
weight ?
When a cause and an effect are connected together, the
increase of the one is always connected with an increase of
the other. If 6 horses eat 20 bushels of oats, we may regard
the horses as the cause, and the consumption of the oats the
effect ; or, if we please, we may regard the oats as the cause,
and the support of the horses as the effect. But in either
case, an increase of one would require an increase of the
other. When numbers are connected in this way, in a pro-
portion, having the relation of cause and effect to each other,
the proportion is said to be Direct.
But it often happens, that quantities are connected togeth-
er, not as cause and effect, but as limitations of each other ;
where an increase of one quantity requires a diminution of
the other.
Thus, if the provisions of a ship's company are sufficient to
last 17 weeks at the rate of 13 oz. of bread per day for each
man, it is evident that these quantities, 17 and 13, are not
cause and effect, but limitations of each other. If one is in-
creased, the other must be diminished. So, if, with a speed
of 11 miles per hour, a journey be performed in 31 hours, it
is evident that an increase of one term must diminish tbe
other. When quantities mutually limiting each other enter
into a proportion, it is called Indirect proportion. No special
rule however is needed for the statement of such questions ;
for you can always determine by strict attention, whether the
statement you make is reasonable.
29. If, with a speed of 11 miles per hour, a journey is-
performed in 37 hours, how long will it take to perform the
same journey with a speed of 15 miles per hour ?
30. If a stable keeper has grain for the supply of 29
horses 43 days, how long will the supply last, if he buys 6
horses more ?
31. If . a barrel of flour last a family of 7 persons 6 weeks,
how long will it last 15 persons ?
32. If 42 men can do a job of work in 60 days, how long
will it take 53 men to perform the same work ?
ir
COMPOUND TROPORTION. 185
83. A ship builder employs 50 men to complete a ship,
which they can do in 45 days ; if 7 of the men fail to engage
in the work, how long will it take the others to perform it ?
34. If 8 yds. of cloth, 7 qrs. wide, cost 54 dollars, what
will be the cost of 15 yds. of cloth, of the same quality, 9
qrs. wide ?
Tn this example, the length of the two pieces of cloth will
not represent the ratio of their values, for they are of differ-
ent widths. The answer can be found by two statements.
First, regarding the two pieces as of the same width,
8 : 15 : : 54 : lOlJ dollars, first ans.
JSText, taking the width into view,
7:9:: lOlJ : 130^\ dollars, final ans.
If we examine the above question, we shaU see, that in
neither of thein is the quantity of cloth expressed ; but, in ^
the first statement, its length, and in the second, its width.
Now the quantity of cloth is expressed by the length multi-
plied by the breadth. In the smaller piece, it is 7X8=56 y
qrs. of a sq. yard ; in the larger piece, it is 15X9^135 qrs. f
of a sq. yard. These numbers, 56 and 135, express the
quantities of cloth ; and taking these, instead of the dimen-
sions, a single proportion gives us the answer.
56: 135 :: 54: 130/^. - V',
As the question is first stated, you observe, that, instead ^
of the numbers which form the ratio, 56 : 135, you have
only the factors of those numbers given. This is called a
Compound Proportion, *
A Compound Proportion, then, is one in which two or more
of the terms of the simple proportion are expressed in the form
of their factors.
Every question containing a Compound Proportion may
be solved by means of two or more simple proportions ; or, it
may be reduced to one simple proportion, as is seen in Ex.
34, above. This method, however, often requires calculations
in large numbers ; it may therefore be desirable to have a
method given by which the process may be made less tedious.
The following rule is offered, as applicable to all cases of
Proportion, Simple, or Compound, — Direct, or Inverse. It is
16*
186 COMPOUND PROPORTION. >
sliort, and the attention required in applying it will afford a
good discipline for the reasoning powers.
Mule of Proportion.
Draw a horizontal line ; then examine the conditions of
the question, and consider, in the case of each, whether its
increase would make the answer greater, or smaller ; if it
would make it greater, set it above the line ; if smaller, set it
below.
N Regard the numbers thus set down, as the terms of a com-
^^^ pound fraction ; cancel common factors ; multiply together the
pr\ terms that remain, for the answer.
^' £ixample.
k •» 35. If 8 men build a wall, 36 ft. long, 12 ft. high, and 4
J ft. thick, in 72 days, when the days are 9 hours long, how
many men will build a wall 100 ft. long, 10 ft. high, and 3 ft.
thick, in 24 days, when the days are 10 hours long?
Cancelling like factors above and below the line,
\ Operation, and multiplying the remaining terms.
8 72 9 100 10 3
.=3L§<i=37i men. Answer.
86 12 4 24 10 ^ ^
^ Explanation,
The question is, ," how many men ? " " If 8 men will
\ build," &c. Now, if it took 80 men to build the first wall
instead of 8, it would require more men to build the second ;
then put 8 above the line. " Build a wall 36 ft. long." If it
were 360 feet long, instead of 36, it would take fewer men to
build the second wall ; therefore, put 36 below the line. Pur-
* Bue the same reasoning with all the other conditions.
y 36. in 5 horses consume 40 tons of hay in 30 weeks,
how many horses will consume 6Q tons of hay in 32 weeks ?
^ 37. If 1 dollar gain .06 of a dollar interest, in 12 months,
how much will 740 dollars gain in 8 months ?
38. If a crew of 75 men have provisions for 5 months,
allowing each man 30 oz. per day, what must be the allow-
ance per day, to make the provisions last 6^ months ?
39. If 18 brick layers, in 12 days, of 9 hours each, build a
wall 175 feet long, 2 feet thick, and. 18 feet high, in how
PROPORTION. 187
many days will 6 men, working 10 hours each day, build a
wall 100 feet long, l^- feet thick, and 16 feet high ?
40. If 10 masons lay 160 thousand of bricks in 12 days,
working 8 hours per day, how many men will lay 224 thou-
sand in 15 days, of 10 hours each?
Partnership,
41. Two men trade in company ; one puts in 1000 dollars ;
the other 2000, for the same length of time ; they gain 600
dollars ; what is each one's share of the gain ?
It is evident that each man's share ought to bd in propor-
tion to the sum he put in. '
As the whole investment is to each partner's investment, so
is the whole gain or loss, to each partner's gain or loss.
42. Two men trade in company ; one puts in 3500 dollars ;
the other 4000 ; they gain 600 doUai-s ; what is each one's
share of the gain ?
43. Three men trade in company ; the first puts in 3400
dollars; the second, 800; and the third, 1200 dollai's; they
gain 475 dollars ; what is each partner's share ?
44. Two men purchase a ship for 11000 dollars; one pays
8000 dollars; 'the other 3000; they sell the ship for 9500
dollars ; what was each one's loss ?
45. Two men trade in company ; one puts in 1000 dollars
for 6 months, the other puts in 1000 for 18 months ; they
gain 600 dollars ; what was each one's share of the gain ?
Here, though the money was equal, it is evident the gain
of one ought to be three times as great as the other, because
his money was in three times as long.
Where the investments are made for different times, each
partner's interest will be expressed by multiplying his money
by the time it was in trade. Then, as the sum of all the in-
terests is to each partner's interest, so is the whole gain or
loss, to each partner's gain or loss.
46. Three men trade in company ; the first puts in 400
dollars for 8 months; the second, 1100 dollars for 6 months;
the third, 1000 dollars for 7 months ; they gain 840 dollars ;
what is each man's share ?
47. Four men trade in company ; the first puts in 1200 for
2 years ; the second, 1500 dollars for 18 months ; the third, 600
for 8 months ; the fourth, 900 dollars for 6^ months ; they gain
1340 dollars ; what was each man's share ?
188 PROGRESSION.
SECTION XL..
PROGKESSION.
When a series of numbers is given, each one of wliich has
the same ratio to the number which follows it, the series is
called a progression.
Progression is Arithmetical or Geometrical. Arithmetical
Progression is made by the successive addition or subtraction
of a common difference. When the common difference is
added to each term in order to make the succeeding one, the
series is called an ascending series; as 1, 3, 5, 7, 9, 11, &c.
When the common difference is subtracted, the series is
called a descending series; as 11, 9, 7, 5, 3, 1.
If you know the first term, and the common difference of
an Arithmetical Progression, you can write the whole series,
for to do this, you have only to add or subtract the common
difference for each succeeding term. If the whole series is
written out, it is evident you can find by inspection, any par-
ticular term, as the 7th, the 15 th, the 20th, &c. But, if the
series be a long one, this may be a very tedious operation.
Suppose the series given above, 1, 3, 5, 7, &c., were con-
tinued to 87 terms, and you were required to find what was
the last term.
By examining the series, you will see that the 2d term=the
Ist-f-the common difference ; the 3d=lst-[-twice the com-
mon difference ; the 4th=lst-|-3 times the common differ-
ence; the 5th=lst-|-4 times the common difference, &c.
Any term whatever equals the lU term,~\-the common difference,
multiplied by a number one less than that which expresses the
place of the term. The 87th term in the above series, there-
fore, is 1+2X86=173.
1. What is the 38th term of the series 1, 3, 5, 7, &c. ?
2. What is the 53d term of the same series ? the 91st
term? the 89th term ? the 107th term?
3. In an arithmetical series, the first term of which is 1,
and the common differrnce 3, what is the 64th term ? the
75th term ? the 81st term ?
4. In the series 1, 5, 9, 13, &c., what is the 40th term? the
67th term? the 80th term ?
ARITHMETICAL PROGRESSION. 189
5. In the series 2, 4, 6, &c., what is the 45th term ? What
is the lOQth term ? What is the 200th term ?
Hence, if you know the number and place of any term, and
the common difference, you may find the first or any other term.
6. If the 5th term of an arithmetical series is 13, and the
common difference 3, what is the 1st term ? What is the
24th term ? What is the 191st term ?
7. If the 6th term of a series is 77, and the common differ-
ence 15, what is the 2d term ? What is the 14th term ?
8. If the 22d term in a series is 89, and the common differ-
ence 4, what is the 10th term ? What is the 43d term ?
By knowing the number and the place of any two terms, we
may find the common difference.
9. In a certain series the 4th term is 10, and the 7th term
is 19 ; what is the common difference ?
19 — 10=9 ; now this difference, 9, is made by the addition
of the common difference three times ; for 7 — 4=3 ; the com-
mon difference, therefore, is 9-^3=3.
10. In a certain series the 5th term is 9, and the 1 1th term
is 21 ; what is the common difference ?
11. In a certain series the 4th term is 13, and the 9th term
is 33 ; what is the common difference ?
If we know the 1st term, the common difference, and the
number of terms, we can find the sum of all the terms.
12. How many strokes does a clock strike in 24 hours, from
noon to noon ?
We might write down the series, 1, 2, 3, &c., up to 12,
which would express the number of strokes in 12 hours, from
noon till midnight ; we might write the same series again, for
the time from midnight tiU noon ; and by adding these num-
bers together, might obtain the answer. But a much shorter
way may be found. To exhibit it we wdll write the two series
thus :
1st series, 12345678 9 10 11 12 from noon tm midnight.
2d series, 121110 9 8 7 6 5 4 3 2 1 from midniglit tiU noon.
13 13 13 13 13 13 13 13 13 13 13 13
Sum of both series equal to 12 times 13. But 13 is the
sum of the first and last terms ; and 12 is the number of terms.
Therefore, the sum of the first and last terms, multiplied by the
number of terms, gives the sum of all the terms of both series
Half this number will be the sum of one series.
190 ARITHMETICAL PROGRESSION.
13. What is the sum of the series 1, 4, 7, to 20 terms ?
First find the 20th term.
14. What is the sum of 50 terms of the series 2, G, 10 ?
15. A farmer instructed his boy to carry fencing-posts from
a pile to the holes in the ground where they were to be in-
serted, taking one post at a time ; the holes are 1 2 feet apart,
in a straight line, and the pile of posts 30 feet from the first
hole ; how far must he travel, in carrying to their places 100
posts ?
1 6. If the hours in a whole week were numbered in regular
progression, and were struck in this way by the clock, how
many strokes must the clock strike for the last hour of the
week ?
What would be the whole number of strokes in the week ?
If we know the first and last terms and* the common differ-
ence, we can find the number of terms.
17. The first term of a series is 4; the last term is 19; the
common difference is 3 ; what is the number of terms ?
The difference between the extremes 19 — 4, is 15 ; this,
you know, is the common difference 3 ; taken a certain number
of times ; 15-i-3=5 ; there are then 5 additions of the common
difference ; now the number of terms is 1 more than the num-
ber of times the common difference has been added. To find
the number of terms, then
Find the difference of the extremes ; divide it hy the common
difference ; increase the quotient hy Ifor the number of terms,
DESCENT OF FALLING BODIES.
18.- A body falling through the air falls, the 1st second,
16.1 feet ;* in the 2d second, 48.3 feet ; in the 3d second, 80.5
feet ; how many feet farther does it fall each second than it
fell the second before ?
19. Taking the answer to the preceding question as the
common difference, and 16.1 as the first term of a series, how
far will a body fall in 4 seconds ?
^ This is a more exact statement than that made in Part I. See Olm-
Btcad's Natural Philosophy. It should be remarked, also, that no allow-
ance, in these examples, is made for the resistance of the atmosphere,
which always diminishes the speed somewhat, and becomes greater and
greater as the speed increases.
GEOMETRICAL PROGRESSION. 191
20. How far will a body fall in 5 seconds ?
21. How far will a body fall in 6 seconds?
22. A stone in falling to the ground, falls the last second
209.3 feet ; how many seconds has it fallen ; and from what
height?
23. A stone in falling to the ground, falls the last second
241.5 'feet ; how many seconds has it fallen ; and from what
height ?
24. If a stone dropped into a well strikes the water in 3
seconds, how far is it to the surface of the water ?
SECTION XLI.
GEOMETKICAL PROGEESSION.
A series of numbers such that each is the same part or the
same multiple of the number that follows it, is called a geomet-
rical series. The ascending series, 1, 3, 9, 27, is of this kind,
for each term is one third of that which succeeds it. So, in
the descending series, 64, 16, 4, 1, each term is 4 times the
following term.
The number obtained by dividing any term by the term
before it, is called the ratio of the progression. Thus, in the
first of the above examples, the ratio is 3 ; in the second exam-
ple it is :^.
Let us take the series, 2, 6, 18, 54, and observe by what
law it is formed. The ratio is 3 ; the first term, 2. The
second term is 2X3, or the first term X the ratio ; the third
term is 2X3^, o; the first term X the second power of the
ratio ; the fourth term is 2X3^, or the flrst term X the third
power of the ratio.
Thus each term consists of the first term multiplied hy the
ratio, raised to a power whose index is one less than the number
expressing the place of the term.
1. What is the 7th term in the series 1, 4, 16, &c.?
2. What is the 10th term in the series 3, 6, 12, &c.?
3. A glazier agrees to insert a window of 16 lights for what
the last light would come to, allowing 1 cent for the first light,
2 for the second, and so on ; what did the wiudow cost ?
^ Wg
192 MENSURATION OF SURFACES.
4. If, in tlie year 1850, the population of the United States
shall be 20000000, and if it shall thenceforward double once
in every thirty years, what would be the population in 1970 ?
To obtain the sum of the terms, when the first and last
terms are given, and the ratio,
Rule. — Multiply the last term by the ratio, suhtract^the first
term from this product, and divide the remainder hy the ratio
diminished hy one.
5. A gentleman promises his son, 11 years old, one miU
when he shall be 12 years old, and, on each succeeding birth-
day till he is 21 years old, ten times as much as on the pre-
ceding birth-day ; what will the son's fortune be, without
interest, when he is 21 years old ?
SECTION XLII.
MENSURATION OF SURFACES.
For the mensuration of the triangle and the parallelogram,
when the base and height are known, see Sec. XVII. Part I.
To find the area of an equilateral triangle when the sides
only are known.
Square one side ; multiply that product hy the decimal .433.
To find the circumference of a circle, when the diameter is
.known.
Multiply the diameter hy 3.1416.
1. What is thejcircumference of a circle the diameter of,
which is 36 feet^'i?^
2. What is the circumference of a circular race-course
whose diameter is 1;^ miles ?
3. What is the circumference of a wheel the diameter of
which is 24 feet, 6 inches ?
4. What is the <^^^^mference of the earth on the line of the
equator, its diameter being 7925.65 miles ?
To find the dia^ieter of a circle, when the circumference is
known.
Divide the circumference Jy 3.1416.
MENSURATION OP SOLIDS. 193
5. How far is it across a circular pond, the circumference
of which is 231 rods?
To find the area of a circle, when the diameter and circum-
ference are known,
Multiply the circumference hy one fourth of the diameter; or,
multiply the square of the diameter by the decimal .7854.
6. What is the area of a circle whose diameter is 34 rods ?
7. What is the area of a circle whose diameter is -24 feet ?
When a circle is given, to find>a square, which shall have
an equal area,
Find the area of the circle, extract the square root, which
will be one side of the square,
8. There is a circular piece of land, 40 rods in diameter ;
what will be the side of a square of equal area ?
9. There is a. circular green, containing 8 acres ; what will
be the side of a square of equal area ?
10. There is a circle 35 rods in diameter, and a square 31
rods ; which is the greater ; and how much ?
SECTION XLIII.
MENSUKATION OF SOLIDS.
A plane solid is bounded by flat surfaces ; a round solid is
bounded by curved siu-taces.
To find the surface of a solid bounded by plane surfaces,
Find the area of each plane surface, and add the sums to-
gether for the whole surface,
A prism is a solid, whose ends are any equal, parallel, and
similar rectileneal figures, and whose sides are parallelograms.
To find the solidity of a prism,
Multiply the area of the base by the height.
1. What is the solidity of a prism whose ends are equilat-
eral triangles, 14 inches on a side, and whose height is 8 feet ?
A cylinder is a round solid, whose ends are equal, and
parallel circles.
To find the solidity — the same ride as for a prism,
17
194 MENSURATION OF SOLIDS.
2. "What is the solid contents of a cylinder, whose ends are
circles 18 inches in diameter, and whose height is 12 feet?
A regular pyramid is a solid, whose sides are equal and
similar triangles, meeting in a point at the top. The slant-
height is the distance from the point, at the top, to the middle
of the base of one of the triangles.
To find the solid contents of a pyramid,
Multiply the area of the base by vne third the peiyendicular
height.
3. What is the solid contents of a four-sided pyramid, whose
J)ase measures 40 feet on a side, and whose height is 42 feet ?
A cone is a round solid, standing on a circular base, and
terminating in a point at the top.
To find the solidity of a cone — the same rule a^ for a
pyramid.
4. What is the solidity of a cone, whose base is 13 feet in
diameter, and whose height is 22 feet ?
5. What is the solidity of a cone, whose base is 4 feet in
diameter, and height 18 feet ?
To find the surface of a sphere,
Multiply the diameter by the circumference.
6. How many square miles of surface has the earth, regard-
ing it as a sphere the diameter of which is 7925.65 miles ?
To find the solidity of a sphere.
Multiply the surface by | o/ the diameter ; or, multiply the
cube of the diameter by the decimal .5236.
To find the measure of a sphere, when the solidity is given,
Divide the solidity by the decimal .5236 ; extract the cube
root of the quotient, which will be the diameter of the sphere.
7. What is the solid contents of a sphere the diameter of
which is 14 inches ?
8. What will be the solidity of the largest sphere that can
be cut from a cubic block 1 foot on a side ?
MISCELLANEOUS THEOREMS AND QUESTIONS. 195
SECTION XLIV.
MISCELLANEOUS THEOREMS AND QUESTIONS.
Given the sum and the difference of two numbers, to find
the larger and the smaller numbers,
Add half the difference to half the sum, for the larger ; sub-
tract half the dfference from half the sum, for the smaller.
1. The sum of two numbers is 140, the difference 32; what
are the numbers ?
2. The sum of two numbers is 572, the difference 94; what
are the numbers ?
3. The sum of two numbers is 187, the difference 44; what
are the numbers ?
4. The sum of two numbers is 190, the difference 57; what
are the numbers ?
Given the sum and the product of two numbers, to find the
larger and the smaller number.
5. There are two numbers ; their sum is 80, and their pro-
duct 1551 ; what are the numbers ?
The theorem on which the solution of this question depends
is this, If a number be divided into two equal parts, and also
into two unequal parts, the product of the two equal parts,
that is, the square of half the number, will equal the product
of the two unequal parts, plus the square of the difference
between one of the equal and "one of the unequal parts.
Take 16; divide it equally, 8-1-8, and unequally, 9-|-7 ;
the difference between an equal and an unequal part, 1 ; 82=
64 : 9X7=63 ; loss 1, which is the square of the difference.
Divide unequally into 10-|-6 ; difference 2 ; 82=64 : lOX
6=60 ; loss 4, which is the square of the difference.
Divide unequally into ll-f-5 ; difference 3 ; 8^=64 : 11 X
5=55 ; loss 9, which is the square of the difference.
Hence, to solve the question, — subtract the product of the
unequal parts from the square of half the number, find the
square root of the difference, add it to half the number for the
greater, sttbtract it from half the number for the less.
6. There are two numbers, the sum of which is 100 ; their
product is 2419 ; what are the numbers ?
196 MISCELLANEOUS QUESTIONS.
. 7. There is a rectangular piece of land, the two contiguous
boundaries of which measure together 120 rods ; the area of
the piece is 2975 square rods ; what is it§ length ? What is
its width ?
8. A rectangular piece of land is surrounded by 480 rods
of fence ; the area is 13104 square rods ; what is its length
and breadth ?
Given the sura of two numbers, and the difference of their
squares, to find the greater and the less number,
The 'product of the sum and the difference of two numbers
is equal to the difference of their squares.
Take the two numbers, 6 and 9 ; their sum is 15 ; their dif-
ference 3 ; 15 X 3=45. The square of 6 is 36 ; the square of
9 is 81 ; 81—36=45.
Hence, if we divide the difference of the squares by the
sum, the quotient :^'ill be the difference, and from this we may
find the gi-eater and the less.
9. There is a triangle, the hypotenuse and one leg of which
measure together 90 feet ; the other side measures 40 feet ;
what are the lengths of the two first-named sides respectively ?
10. There is a triangle ; the hypotenuse and base measure
together 120 feet ; the perpendicular measures 64 feet ; what
is the length of each of the first-named sides ?
This principle enables you to multiply readily any two
numbers, one of which exceeds a certain number of tens by
as many units as the other falls short of it ; as 63 X-57 ; the
first exceeds 60 by 3 ; the second falls short of it by 3.
Square the tens, — 3600 ; subtract from this the square of
the units — 9 ; 3591, answer.
11. Multiply 64X56, 3600—16=3584, answer.
12. Multiply 82X78; 47X33; 92X88.
Theorem of Parallel Sections.
If a line is drawn in a triangle, parallel to one of the .sides,
and meeting the other two sides, it divides those sides propor-
tionally ; and the small triangle cut off, is similar to the whole
undivided triangle.
If a plane pass through a pyramid or cone, parallel to the
base, it divides allUhe lines it meets proportionally; and the
siTiall solid cut off at the top is similar to the whole undivided
{ Dlid.
MISCELLANEOUS QUESTIONS. 197
13. There is a triangular field, containing 7 acres ; a line is
drawn through it, parallel to one side, cutting the other two
sides f of the distance from the apex to the base ; how much
land does it cut off?
14. There is a board in the form of a right angled triangle,
8 feet in perpendicular height ; how far from the top must a
line be drawn parallel to the base to cut off f of the board ?
15. A man had a field of 3 acres, in the form of a right
angled triangle, with the base equal to the perpendicular ; he
sells one acre, to be cut off bj a line running parallel to the
base ; he then sells another acre, to be cut off by another line
parallel to the base ; how far from the base must the first line
be ? How far from the base must the second line be ?
16. There was a cone, 20 feet high; but the upper part
being defective, 11 feet in height of the top was taken down ;
how much of the cone has been removed ?
17. There was a square pyramid, the base of which meas-
ured 48 feet on a side ; when it was partly completed, its slant
height, measuring from the middle of a side at the bottom to
the middle of the same side at the top, was 40 feet, and the
width of a side at the top was 12 feet; how high was the apex
of the pyramid, when completed ? and what part of the pyra-
mid remained to be built ?
18. In a right angled triangle, whose base and perpendicu-
lar are equal, what is the ratio of the square of the hypotenuse
to the square of the base ? What is the ratio of the hypote-
nuse to the base ?
19. If a man travel on Monday 6 miles, due north, and on
Tuesday 8 miles, due east ; how far is he then from where he
set out on Monday ?
If on Wednesday he travels 12 miles, due south, how far
will he then be from w^here he was on Monday morning ?
How far from where he was on Monday night ?
20. In repairing a meeting-house, it Avas thought desirable
to alter the form of the posts, which were one foot square.
It was proposed to cut away the corners, so as to make them
regular eight-sided prisms. HoW wide must each face be, so
as to have all the eight faces of exactly the same width ?
21. Sound moves through the air at the rate of 1090 feet
in a second ; how long would it be in passing 100 miles ?
22. At the above rate, how long would it require for a
wave of sound to compass one half the circuit of the globe,
17*
198 SPECIFIC GRAVITY.
on the line of the equator, the circumference being 24899
miles ?
23. Two men purchase in equal shares, a stick of hewn
timber, 40 feet long, 2 feet square at the larger end, and 1
foot square at the smaller end ; how far from the larger end
shall they cut it in two, so that each may have exactly one
half?
24. A surveyor, in laying out a lot of land, first runs a
line due North, to a certain tree ; from the tree he runs be-
tween South and West till he comes to a point due West
from the place he started from ; the whole of these two lines
is 212 rods, but those who measured it neglected to note how
far the tree was from the starting point ; on measuring a
third line, connecting the extremities of the two first lines,
they find it 98 rods ; how many acres does the triangle con-
tain ?
Specific Gravity.
The specific gravity of a body, is its weight compared with
the weight of an equal bulk of water. To find the specific
gravity of a body heavier than water.
Weigh the body in water, and out of water, and find the
difference in the weight ; then, as the difference in the weight
is to the weight out of water, so is 1 to the specific gravity.
The weight of a cubic foot of water is 62|^ lbs. av. The
specific gravity of the. most important of the metals is as
follows :
Iron, 7.78 Tin, . 7.2 Copper, 8.895
Lead, 11.325 Mercury, 13.568 Silver, 10.51
Gold, 19.257 Platinum, 21.25
From the above table, we may find the weight of any mass
dF one of the above metals the magnitude of which is known.
25. What is the weight of a cubic foot of iron ?
26. What is the weight of an iron ball six inches in diam-
eter?
27. What is the diameter of a 24 lb. cannon ball ?
28. What is the diameter of a 48 lb. cannon ball ?
29. What is the weight of a cannon ball one foot in di-
ameter ? .
30. If the column of mercury in the barometer be 29^
MECHANICAL POWERS. 199
inches liigh, what would be the weight of a column of mer-
cury of that height, one inch square ?
31. As the weight of mercury in the barometer equals the
weight of the atmosphere on the same base, what is the pres-
sure of the atmosphere on a square foot, when the mercury
in the barometer is 29 inches high ?
The height to which water will rise in a suction pump,
and the height of the mercury in the barometer, are in in-
verse proportion to the specific weight of those two bodies ;
that is, the water is as much higher than the mercury, as
mercury is heavier than water.
32. How high will water rise in a suction pump, when the
mercury in the barometer is 29^ inches high ?
33. What is the weight of a copper prism, its base being
an equilateral triangle, 3 inches on a side, and its height 15
inches ?
Mechanical Powers.
The object to be gained by the application of mechanical
powers, is to overcome a large weight or resistance, by means
of a comparatively small .power.
In doing this, however, the power must move through a
space as much larger than the space which the weight moves
through, as the weight is heavier than the power.
Or, the distancey^weightf of the power^==distance'X.weight,
of the weight, or mass to be moved.
This is the great 'law of mechanical powers, and applies
to them all, without exception. In the practical application
of them, a certain allowance must be made on account of
the friction in the machine. The amount of friction differs
in different powers. No account of this will be taken in the
examples which follow, unless it is particularly mentioned ;
nor will any difference be made between the power when in
motion, and when in equilibrium, or at rest.
The Lever.
The lever is a straight bar used to support or raise heavy
weights. It is supported by a jt)rc»p or fulcrum, placed near
the weight, and the power is applied at the other end of the
lever. The distance from the fulcrum to the weight, is called
200 THE LEVER.
the shorter arm ; the distance from the fulcinim to the power,
the longer arm of the lever. If the lever were to turn over
the fulcrum as a centre, the longer arm would describe a
larger circle, and the shorter arm would describe a smaller
circle. The circumferences of these two circles, or an arc
of the same number of degrees in both, would be the dis-
tances passed through by the power and the weight respec-
tively. But we may take the arms themselves as represent-
ing these distances, for they are the radii of the two circles ;
and the radii of different circles have the same ratio to each
other as the circumferences.
We have therefore this proportion :
The longer krm is to the shorter arm, as the weight to the
power. Or, let 1, a. stand for the longer arm ; S. a., for the
shorter ; w. for the weight, and p. for the power.
1. a. : S. a. : : W. : p. ; and any change admissible in the
terms of a proportion, may be made in these terms.
34. If a lever 10 feet long have its fulcrum one foot from
the weight, how great must the power be, to raise a weight
of 1640 lbs.?
35. If a lever IQ feet long have its fulcrum 18 inches
from the weight, how great a weight will be raised by a pow-
er of 160 lbs.?
36. A lever 18 feet long rests on a fulcrum 2 feet from the
end ; how large a weight can two m*en raise, — one weighing
164 lbs., the other 172 lbs., — by applying their weight at the
longer arm ?
37. If a lever 7\ feet long rest on a fulcrum 15 inches
from the end, how heavy must the power be to support a ton,
gross weight ?
38. If the weight be 3600 lbs., and the power 140 lbs.,
how far from the weight must the fulcrum be placed under a
lever 12 feet long, so as to have the weight and power bal-
ance?
39. If the^ weight be 6480 lbs., the power 312, and the
lever 16^ feet long, how far from the weight must the ful-
crum be, to have the weight and power balance ?
40. In a certain machine, it is necessary to adjust a lever
3 feet long, so that a power of 1^ lbs. shall balance 13^
lbs. ; how far from the weight must the fulcrum be placed ?
THE WHEEL AND AXLE. 201
The Wheel and Axle.
In this case the power is applied at the circumference of
the wheel, and the weight is drawn up by a rope passing
round the axle, which is a smaller wheeh The principle,
therefere, is the same as in the lever ; the semi-diameter of
Ihe wheel is the longer arm ; the semi-diameter of the axle,
ihe shorter arm.
41. In a grocery store the wheel and axle used in raising
heavy articles, are of the following dimensions, viz. : the wheel
5 feet in diameter the axle 7 inches in diameter ; what pow-
er must be applied to the rope passing over the wheel, to
balance a barrel of flour weighing 205 lbs., suspended by a
rope passing over the axle ?
42. With the same wheel and axle, what power will raise
a box of sugar weighing 431 lbs., adding ^ to the^ power, to
overcome the friction ?
43. In digging a well, the wheel employed in raising stones
and earth, is 6 feet in diameter ; the axle 6^ in. in diam. \ what
power will raise a rock weighing 640 lbs., adding ^ to the
power, to overcome the friction,?
44. If a wheel is 14 feet in diameter, what must be the
diameter of the axle, in order that a power of 140 lbs. may
balance 5760 lbs. ?
45. If an axle is 16| inches in diameter, what must be the
diameter of the wheel in order that a power of 56 lbs., may
balance a weight of 1344 lbs. ?
The Screw.
In this case, the distance passed through by the power in
one revolution, is equal to the circumference of the circle
described by the lever which turns the screw ; the distance
passed by the weight, is the distance between two threads of
the screw, measured in the direction of its axis.
In the practical application of this power, a large allow-
ance must be made to compensate for the friction.
46. If the lever of a screw is 11 feet in length, and the
distance of the threads 1^ inches, what power will raise a
weight of 6431 lbs., making no allowance for friction ?
202 STRENGTH OF BEAMS TO RESIST FRACTURE.
47. With the same conditions as in the last example, what
weight will be raised by a power of 1 24 lbs. ?
48. What must be the length of the lever of a screw-
the threads of which are 1 inch asunder, in -order that a pow-
er of 3 lbs. may balance a weight of 1640 lbs., making no
allowance for friction ?
49. How far asunder must the threads of a screw be, so
that, with a lever of 8^ feet in length, 26 lbs. will balance
6590 lbs. ?
Strength of Beams to resist Fracture.
[See Section XX. Part I.]
In addition to the principles that have already been stated
in estimating the strength of timbers, the following are among
the most important. It is understood in all cases, when tim-
bers are compared, that they are of the same wood, and
equally good in quality.
When the depth of two beams is the same, and the thick-
ness the same, the strength is inversely as the length.
50. There are two beams of the same depth and thickness ;
one 18 feet in length, the other 13 ; the longer beam will
sustain a weight of 68 cwt. ; what weight will the shorter
beam sustain ?
51. Two beams of the same size, measure in length 22 and
17^ feet ; the shorter beam will sustain 76 cwt. ; how much
will the longer beam sustain ?
52. Two beams of equal thickness have a depth of 14 and
16 inches respectively; the deeper beam is 20 feet long,
and will sustain 84 cwt. ; the other is 17 feet in length ; what
weight will it sustain ?
First take into view the length ; then, in a second propor-
tion, the depth.
53. If a beam 25 feet in length and 9 in. in depth, will sus-
tain a weight of 12 cwt., what weight will be sustained by a
beam of the same thickness 18 feet long, and 10 in. in depth ?
When beatns are of the same length and depth, the strength
varies directly as the width.
54. There are two beams of equal length and depth ; one
9 inches in width, the other 7\ inches ; the wider beam will
sustain 47 cwt. ; what weight will the narrower beam sustain ?
STIFFNESS OF BEAMS TO RESIST FLEXURE. 203
55. There are two beams of equal depth ; one measures 20
feet in length, and 11 inches in width, and will sustain 94
cwt. ; the other beam is 14 feet in length and 10 inches in
width ; Avhat weight will it sustain ?
56. There are two. beams of the same width, one measures
16 feet in length, and 10 inches in depth, and Avill sustain 66
cwt. ; the other is 18 feet long, and 12 inches in depth; what
weight will it sustain ?
It is sometimes desirable to know how the strength of a
beam will vary by removing the point on which the pressure
is made, as in the following example.
57. A beam 20 feet in length will sustain, at its centre, a
weight of 44 cwt. ;' what weight will it sustain applied 7 feet
from one end ?
The following formula will give the variation in the
strength.
As the product of the two unequal sections of the beam,
(in this case 13X7,) is to the square of half the length, so is
the weight which the beam will sustain at the centre, to the
w^eight it will sustain at the other given point.
58. A beam 24 feet in^length, will sustain at its centre 56
cwt. ; what weight will it sustain at the dist^ce of 9 feet
from one end ?
59. A beam 28 feet in length will sustain at its centre 33
cwt. ; what weight will a beam of the same width and length,
and of f the depth of the former, sustain at the distance of
10 feet from the end ?
Stiffness of Beams to Resist Flexure.
The stiffness of beams of the same length and width varies,
as the cuhe of the depth. If the depth is the same, the stiff-'
ness varies as the width.
60. There are two beams of equal length and width, one
is 8 inches in depth, the other 11 inches ; if it require 30
cwt. to bend the former one inch, what weight will it require
to bend the latter one inch ?
61. There is a stick of timber 8 inches by 6; if it require
24 cwt. to bend it 2 inches when lying flat, what weight will
bend it 2 inches when turned up on the edge ?
204 BUSINESS FORMS AND INSTRUMENTS.
G2. If 10 cwt. will bend the stick just described 1|- inches
when it lies flat, what weight will be requisite to bend it 1^
inches, when turned up on the edge ?
63. There is a board 12 inches wide, and 1 inch in thick-
ness ; what is the ratio of its strength when lying flat, sup-
ported at the ends, to its strength when turned edgewise ?
64. If it require 12 lbs. to bend the same board ^ an inch,
when lying flat, how much will it require to bend it ^ an inch
when turned edgewise ?
SECTION XLV.
BUSINESS FORMS AND INSTRUMENTS.
Promissory Notes.
1. On Demand, with Interest.
$500. — Boston, March 1, 1846. For value received, I
promise A. B., to pay him, or his order, five hundred dol-
lars, on demand, with interest. T. M.
2. On Time, with Interest,
$200. — Boston, March 1, 1846. For value received, I
promise A. B., to pay him, or his order, two hundred dol-
lars, in three months, with interest. T. M.
3. On Time, without Interest.
$400. — Boston, March 1, 1846. For value received,
I promise A. B., to pay him, or his order, four hundred dol-
lars, in sixty days from date. I. M.
4. Payable by Instalments, with Periodical Interest.
$1000. — Boston, March 1, 1846. For value received, I
promise A. B. to pay him, or his order, one thousand dollars,
as follows, viz. ; — two hundred dollars in one year ; two hun-
dred dollars in two years ; and six hundred dollars in three
years, from this date, with interest semi-annually. I. M.
BUSINESS FORMS AND INSTRUMENTS. 205
Remarks on Promissory Notes.
Tyiien the words " or order," are inserted in a note, the
holder of the note may endorse it, that is, write his name on
the back of it, and pass it to a third person, who can collect
it in the same manner as if he were the original holder. If
the maker of the note neglects to pay, the holder may collect
it of the endorser.
If the words "or bearer," are inserted instead of "or
order," any person who has possession of the note may collect
It of the maker. Such a note would be like a bank note,
which passes from hand to hand, without endorsement.
A note, in order to be legal in the first holder's hands, must
be for value received. A note, therefore, given to pay a debt
incurred in gambling or betting, cannot be collected by law,
unless it has passed into the hands of an innocent holder.
When a note contains the promise to pay interest annually,
and the interest is not collected annually, the law does not
permit the holder to draw compound interest. The holder
may compel the payment of the interest when it' becomes
due, but if he neglect to do this, he can recover only Simple
Interest.
When a note is given to pay in a certain commodity, as
wood, grain, &c., if the note is not paid when due, the holder
may compel the payment of the equitable value of the com-
modity in money. The reason of this is, that it is supposed
that the commodity may have a value to the holder at the
time when it is promised, which it will lose, if not paid then.
Receipts. -
1. — A general Form.
$500. — Boston, March 1, 1846. Received of O. P. the
sum of five hundred dollars, in full of all demands against
him. A. B.
2. — For Money paid hy another Person.
$300. — Boston, March 1, 1846. Received of O. P., by
the hand of Y. Z., three hundred dollars, in full payment for
a chaise by me sold and delivered to the said O. P.
A. B.
18
206 BUSINESS FORMS AND INSTRUMENTS.
3. — For Money received for Another,
$700. — Boston, March 1, 1846. Received of O. P. seven
hundred dollars, it being for the balance of account due from
said'O. P. to Y. Z. A. B.
4. — In Part of a Bond.
$3000. — Boston, March 1, 1846. Received of O. P. the
sum of three thousand dollars, being a part of the sum
of five thousand dollars due from said 0. P. to me on the
day of . A. B.
5. — For Interest due on a Bond,
$600. — Boston, March 1, 1846. Received of O. P.
six hundred dollars, due this day from him to me as the an-
nual interest on a bond, given by said O. P. to me on the 1st
of May, 1831, for the payment to me of ten thousand dollars
in three years, with interest annually. A. B.
6. — On Account.
$50. — Boston, March 1, 1846. Received of O. P. fifty
dollars, for which I promise to account to him on a settlement
between us. , A. B.
7. — Of Papers,
Boston, March 1, 1846. Received of O. P. several
contracts and papers, which are described as follows ; — [^de-
scribe the papers ;] which I promise ta return to the said O.
P. on demand. , A. B.
Order at Sight.
Boston, April 18, 1846. At sight, pay to the order of
John Brown, one thousand dollars, value received, which
place to account of your obedient servants, A. W. & Ck).
Jacob Smith, Esq., New York.
Order on Time.
Boston, April 18, 1846. Six months after date, pay to
the order of John iBrown, one thousand dollars, value re-
BUSINESS FORMS AND INSTRUMENTS. 207
ceived, which place to the account of your obedient ser-
vants, A. W. & Co.
Jacob Smith, Esq., New York.
Eemarks. If J B. present this order to'' J. S., and J. S. write his name
across the face of it, it J)ecomes what is called an acceptance. J. S.
'agrees to pay it at the date named.
If J. B. writes his name on the back of the acceptance, it becomes ne-
gotiable ; he may pass it to a third person, who may endorse it, and pass
it to a fourth. All those whose signatures are on the order are bound for
its payment ; the acceptor to the drawer ; the acceptor and di-awer to the
first endorsor; and they and each endorser to the one succeeding him,
and the last endorser, and all previous parties, to the holder.
Award by Referees.
We, the undersigned, appointed by agreement of the par-
ties herein named, having met the parties, and heard their
several allegations, arguments and proofs, and duly considered
the same, do award and determine that A. B. shall recover
of C. D. the sum of together with all the costs of this
reference, which are to the amount of ; and that this
shall be final and in full of all claims and dues of the parties
on matters herein referred to us. I. M.
R. N.
L. S.
Letter op Credit for (xOODs.
Boston, March 1, 1846.
Messrs. Y. & Z., Merchants, Baltimore.
Gentlemen, — Please to deliver Mr. C. D., of , or
to his order, goods and merchandize to an amount not exceed-
ing in value in the whole, one hundred dollars ; and on your
so doing, I hereby hold myself accountable to you for the pay-
ment of the same, in case Mr. C. D. should not be able so to
do, or should make default, of which default you are required
to give me reasonable and proper notice.
Your obedient servant, A. B.
A letter of credit for money may be given in the general form of the
above ; specifying, in the letter, the amount of credit granted.
Power op Attorney.
Know all men by these presents, that I, A. B., of
do hereby appoint C. D., of , to be my sufficient and
lawful attorney, to act for me, and in my name, [here state
208 THE STANDARD OF WEIGHTS AND MEASURES.
the objects for which he is to act.] And for the purposes
aforesaid, I hereby grant unto my said attorney full power
to execute all needed legal instruments, to institute and pros-
ecute all claims in my behalf, to defend all suits against me,
to submit to arbitration, or settle all matters in dispute, and
to do all such acts as he shall think expedient for the full ac-
complishment of the objects for which he is appointed my
attorney, as fully as I might myself do them if present ; and
all acts done by the said C. D.~my attorney, under authority
of this appointment, I will ratify and confirm.
In testimony whereof, I hereby set my hand and seal, this
' day of , in the year —
Signed, sealed, and delivered,
in the presence of
a N.
W. F.
A. B. [l. s.]
SECTION XLVI.
ON THE STANDARD OF WEIGHTS AND MEASURES.
In the earlier states of society, tl e standard of weights and
measures was, of necessity, very indefinite and fluctuating.
In one nation, it was one thing, — in another nation, another;
and in no case was it deserving of a very high degree of con-
fidence.
Sometimes the length of the king's foot was the standard
for all measures of length ; again, the length of the king's
arm from the elbow to the extremity of the fingers, was made
the standard.
The length t)f journeys was measured by the hours or
days employed in performing them, or by the number of
steps taken.
In land measure the standard was, what a yoke of oxen
could plough in a day, when, in fact, one yoke might plough
twice as much as another.
In dry measure, it was as much as a man could conve-
niently carry, without first deciding how strong the man
should be.
THE STANDARD OF WEIGHTS AND MEASURES. 209
In weight the standard was, what a man could hold and
swing in his hand.
Sometimes vegetables were taken as measures, as " thi-ee
barley corns make one inch." But barley corns do not all
grow of exactly equa.l length, any more than the feet and
arms of kings.
As science advanced, and commerce became farther ex-
tended among diiferent nations, the mischiefs of these vague
and fluctuating methods of measurement became more and
more deeply felt.
But it was far easier to see the faults of the old system,
than to devise a new one that should be perfect. "What ob-
ject could be selected as an ultimate standard for all weights
and measures ?
We have seen that the parts of animals or of vegetables,
are too liable . to change to deserve any confidence. If some
arbitrary standard should be adopted, as a foot, or yard, and
this measure should be kept as the standard, by which all
others should be tried, what security could there be that it
would never be altered by fraud, or destroyed by accident ?
Or, if some natural distance were taken, as the distance be-
tween two points of some well known rock or cliff, this dis-
tance might vary with a change of temperature, or be altered
by some convulsion of nature.
We will proceed to give a short account of the English
system of weights and measures, adopted by their Act of
Uniformity/, which took effect Jan. 1, 1826. To begin with
measures of capacity ; all English measures of capacity,
whether for liquors or grain, are referred to the standard im-
perial gallon. This gallon contains 217^ cubic inches. From
this gallon, quarts, pints, and gills are obtained by subdivi-
sion ; and pecks and bushels by multiplication. Hence, you
can find the number of cubic inches in an English quart, pint,
peck, or bushel. Thus the adoption of the imperial gallon
introduces entire uniformity into all English measures of
capacity. It refers them all ultimately to the cubic inch.
We must now inquire, what has been done to fix the measure
of the inch ? for, if there is any error or variation here, it
will render false all the measures of capacity which depend
upon it.
To determine the measure of the inch, it is made by law
g^ of the standard yard. That standard yard is a straight
18* _
210 THE STANDARD OF WEIGHTS AND MEASURES.
brass rod in the custody of the Clerk of the House of Com-
mons. The yard is the distance on that rod between the
centres of the points in the two gold studs or pins in the rod.
And as heat would make the yard longer, and cold would
make it shorter, the law requires that it shall be used when
it is of the temperature of 62° (Fahrenheit.) This standard
yard, however, may be destroyed by accident. We must
then inquire for a still more permanent standard. To effect
this, the law declares that the standard yard, if destroyed,
may be restored, by making it f ff §§§ of the length of a pen-
dulum, that vibrates seconds in the latitude of London, in a
vacuum, at the level of the sea. If all these conditions are
fulfilled, a pendulum that vibrates seconds must have an ab-
solutely invariable length.
Thus we have brought the whole system of measures back
to seconds, as the standard. The whole scheme now depends
upon seconds being of an invariable length.
Seconds are parts of a year ; the year is not made up by
multiplying seconds, but seconds are obtained by dividing the
year. If, then, the year is of a fixed length, seconds are so.
Now the year is the time of the revolution of the earth round
the sun. It is the same, without change, from one year to
another, and from century to century.
Thus the whole system of measures has been brought, for
its ultimate standard, to the unalterable period of the earth's
revolution round the sun. —
We will now retrace the steps of this investigation, be-
ginning with the primary standard, the earth's yearly revo-
lution.
The time of the earth's revolution round the sun is always
the same ; therefore, a second, which is a certain part of this
time, is an exact measure. If the second is a fixed measure,
then the pendulum which, under the same circumstances,
vibrates seconds, is of a fixed length. If the length of the
pendulum vibrating seconds is fixed, the length of the stand-
ard yard is fixed, for it is ff ^§§f of the pendulum. If the
standard yard is fixed, the inch is fixed, consequently the
cubic inch, the gallon, quart, pint, gill, and bushel.
In the preceding investigation no mention has been made
of the standard of weight. It is obtained by making a cubic
inch of distilled water equal to 252.458 grs., of which 5760
make a pound troy, and 7000 make a pound avoirdupois.
THE STANDARD OP WEIGHTS AND MEASURES. 211
Thus weights and measures are alike brought to an unal-
terable standard.
The imperial gallon contains 277^ cubic inches.
The Winchester* gal., wine measure, • • • -231 "
" " « • beer measure, 282 "
The imperial gallon of water weighs 10 lbs. avoirdupois.
The system of weights and measures established by law
in the United States, is very nearly the same as the English.
The gallon, United States measure, contains 9 lbs. 14 oz. of
water. This is the legal standard for all measures, dry and
liquid. In many parts of the country, however, especially in
the interior, the legal -standard has not supplanted the sys-
tem derived in earlier times from the English.
In France, where the system of weights has been carried
to greater perfection than in any other country, the decimal
ratio is adopted in all denominations. In some cases, how-
ever, there is still retained some part of the old system,
combined with the decimal.
In obtaining an ultimate standard of measure, the French
measured one quarter of a meridian line of longitude.
One ten millionth part of this arc they made the basis of
their system of measures. This standard, the metre, is 3.28
feet. The lower denominations are made by successive
divisions of this, by 10, 100, &c. ; and the higher by multi-
plication.
The following table presents the French decimal weights
and measures, with the English equivalents.
French Long Measure,
feet.
10 mellimetres make • • 1 centimetre, .0328
10 centimetres 1 decimetre, .328
10 decimetres • 1 metre, 3.28
10 metres 1 decametre, 32.8
10 decametres 1 hectometre, 328
10 hectometres 1 kilometre, 3280
10 kilometres 1 myreametre 32800
* Winchester, so called because the standard measures were kept at
Winchester.
212 THE STANDARD OF WEIGHTS AND MEASURES.
French Square Measure.
The unit square measure is the are, which is the square of
the decametre ; consequently, it is the square of 32.8 feet, — a
little less than 4 square rods.
This ^unit is multiplied for the 'higher denominations, and
divided for the lower, in the same way as the metre.
French Decimal Weight.
10 milligrammes make 1
10 centigrammes 1
10 decigrammes 1
10 grammes 1
10 decagrammes 1
10 hectogrammes • • • • 1
10 kilogrammes 1
10 myriagrammes • • • • 1
10 quintals • • 1
centigramme, • •
decigramme, • •
gramme,
decagramme, • • <
hectogramme, •
kilogramme, • •
myriagramme, •
quintal,
million,
gre. Troy.
.1543402
1.543402
15.43402
154.3402
1543.402
15434.02
154340.2
1543402.
5434020.
APPENDIX.
The examples that follow are miscellaneous, and designed to carry still farther
the practice in Written Arithmetic.
1. James Ball bought of Amos SeAvall three pieces broadcloth, measur-
ing 12^, 13, and 24 yards, at $4.87|- per yard; five pieces kerseymere,
measuring 24|-,.25, 27, 265^, and 26 yards, at 67 cts.per yard ; eight pieces
cotton sheeting, measuring 33 yards each, at 9\ cents per yard, ^ per cent
off. What was the amount of the bill 1
2. Bought of John Jones, on six months' credit, 47 yards broadcloth,
at f4.3l per yard ; 16 yards vestings, at $1.15 per yard; 63^ yards satinet,
at 62^ cents per yard ; 5^ pieces sheeting, containing 33 yards a piece, at
8| cents per yard. John Jones agrees, if paid in cash, to deduct 4 per
cent, from the bill ; what is the cash amount of the bill 1
3. Bought of Asa Wood, on six months' credit, 45 ban-els flour, at $5.37
per barrel; four hhds. molasses, containing 124^, 131, 134, and 136 gal-
lons, at 27^ cents per gallon ; five bags coffee, containing 541^, 56, 49^, 62,
and 65j lbs. at 8;^ cents per lb. Gave, in payment of the above, a note
payable in six months ; what was the cash value of the note when it was
given, reckoning interest at 6 per cent. 1
4. A bought of B, on six months' credit, goods with the amount aa
follows,
April 3, 1845. $254.75
June 8, " 135.00
Aug. 1, " 200.00
Sept. 14, " 168.25
At what date shall A makcan equated payment of the whole amount ?
5. A gives B a note for $600, payable in six months ; what is the cash
value of the note, two months after date, reckoning the interest at 6 per
cent. 1
6. A man agrees to dig and stone a well, on the following terms ; $1.00
per foot for the first ten feet, $2.50 per foot for the second 10 feet, and
$4.00 per foot for the remainder, till he finds a supply of water ; for every
foot of rock through which he digs he shall receive double pay. He digs
. 42 feet in all, and through rock from 17 to 31|- feet from the surface ; what
pay is he entitled to for the whole ?
7. A man engages to build 160 rods of road, one half for $1.42 per
rod, the other half for $1.83 per rod; he hires 93 days of men's labor at
84 cents per day, pays for board of the same at $1.50 per week of six
days; pays for tools and repairs $11.60; he works himself, with 4 oxen
214
APPENDIX.
and his son, 34 days. "What wages will he receive per day for himself,
for his oxen, and for his son, allowing for the 4 oxen as much as for him-
self, and for his son half as much 1
8. A bought a lot of standing wood for 105 dollars, and agreed with
B to cut and haul it to market for three-fifths of the proceeds; there
were 54 cords of pine which was sold for $2.84 per cord, and 61 cords of
hard wood which sold for $4.75 per cord. Did A gain or lose, and how
much ? B labored himself 35 days, employed one yoke of oxen 24 days,
and hired sixty-eight days of men's labor at 85 cents per day 5 what pay
does he receive per day for himself and for his oxen, allowing for his
oxen, per day, two-thirds as much as for himself *?
9. A and B bought a quantity of grass, ready for cutting, for 26 dol-
lar, for which they paid 13 dollars apiece. In cutting and curing it A
fiirnished 3 days of hired men's labor and* worked himself 2^ days, B
hired 7 days of men's labor, a team 1^ days, and worked himself 2^ days.
There were 62- tons of hay ; what is each one's share of the hay, allowing
for the labor $1.00 per day, and for the whole work of the team $1.50 ?
10. A agrees to dig and stone a cellar, 7 feet deep; it is to be 13 feet'
wide and 16 feet long inside the walls, which are to be 2 feet in thickness.
He is to receive for digging 22 cents a cubic yard for earth, and $1.55 a
cubic yard for rock, and for stoning 87 cents for every perch of 25 cubic
feet, for which the stone is found in digging the cellar, and $1.50 a perch
when he^ has to bring the stone from another place. In digging the cel-
lar he digs 7^ cubic yards of rock, which furnishes stone for 7^ cubic
yards of wall ; what is he to receive for the whole job '?
11. A man agrees to dig a canal 10 rods long, 6 feet in depth, 33 feet
wide at the top, and 24 feet wide at the .bottom, for 10 cents per cubic
yard ; what sum will the work amount to ?
12. What is the value of six loads of wood at $4.67 per cord, measur-
ing as follows ; 1st, 8 ft., 4 ft. 3 in. ; 3 ft.. 10 in. ; 2d, 8 ft. 4 in., 4 ft. 1 in.,
4 ft. ; 3d, 8 ft. 2 in., 4 ft. 7 in., 3 ft. 11 in. ; 4th. 8 ft, 4 ft. 2 in., 4 ft; 5th,
8 ft. 2 in., 3 ft. 10 in., 3 ft. 9 in. ; 6th, 8 ft. 6 in., 4 ft., 4 ft. 4 in. 1
13. What will be the dimension, at the two ends, of the largest square
stick of timber that can be hewn from a round log 3 feet in diameter at
the larger end, and 2 feet in diameter at the smaller ?
14. What will be the solid contents of such a stick, if it is 16 feet in
length ?
15. What will be the width at the two ends of the largest stick of tim-
ber that can be hewn from a log 3 feet in diameter at the larger end, and
2 feet in diameter at the smaller,, if the stick is hewn 10 inches in thick-
ness through its whole length 1
16. What will be the solid contents of such a stick, if it is 20 feet in
length 1
17. There are three pieces of cloth; the length of the first is to that of
the second as 3 to 2, the length of the second is to that of the third as 4
to 15, and the length of the three added together is 50 yards ; what is the
length of each piece "?
18. A man bought a chaise and paid 20 dollars for repairing it, he then
Bold it for one fifth more than he gave, and found that, allowing one dol-
lar for his own trouble in the business, he had lost thirteen dollars ; what
did he give for the chaise ?
19. A gentleman began his preparation for college at a certain age,
and spent in school and at college half as many years as he had lived
APPENDIX. 215
before ; he then went to Europe, and after spending there one ninth as
many years as his age amounted to when he left Europe, spent in his pro-
fession one thu-d as many years as he had lived when he entered it, and
was then 36 years old ; at what age did he begin his preparation for
college ?
20. One half Of three fourths of A's age equals one sixth of B's age,
and the sum of their ages* is 78 ; what is the age of each ?
21. Three fourths of the liquor in a cask equals five sixths of what has
leaked out, and the whole before any leaked out was sixty gallons ; how
much is there in the cask 7
22. Reduce 5 pence to the decimal of a shilling, carrying the decimal
to the sixth figure.
23. Reduce 9^ gallons to the decimal of a barrel, wine measure, carry-
ing the decimal to the tenth figure.
24. Reduce 7^ quarts to the decimal of a bushel.
25. What is the least common multiple of 784, 1386, and 12351
26. What are the prime numbers between 1010 and 1020 1
27. What are the prime factors of 2326 ?
28. Reduce — r — - and — - to simple fractions, with a common
. 22^ 19 30i ^
denominator.
29. What is the value in shillings and pence of the following decimals
when added together, £.0431, .67142s., .73462c?.?
30. What is the interest of $642.25 for 1 year and 3 months at 7^
per cent. 1
31. What is the interest of $954.30 for 2 years, 4 months, and 21 days,
at 5 per cent. 1
32. What is the present worth of a note of $640.50 due in 3 months 1
33. What is the present worth of a note of $1263.00 due in 3^ months
at 5 per cent, interest ?
34. Boston, June 14, 1836. Eor value received I promise to pay
John Ball, or order, three hundred and sixty^five dollars in four years
with interest annually. James Fkost.
If no interest is paid, and the note is renewed annually, what will be
the amount of the note four years after date ?
35. New York, Jiily 1, 1840. For value received I promise to pay
Abel Jones or order five hundred and forty dollars, on demand, with in-
terest. . John Fkost.
Endorsements,
Oct. 3, 1840, $63 00
Feb. 4, 1841, 120.00
June 1, 1841, 60.00
Sept. 15, 1841, 200.00
What will be due Feb. 14, 1842, at seven per cent, interest?
36. A owes B $400.00, due in 2 months ; $320.00, due in 3 months ;
$600.00, due in 4^ months ; what is the equated time for the payment of
the whole 1
37. What is the present worth of- a bank note for 800 dollars payable
in three months ?
38. What is the present worth of a bank note for 346 dollars payable
in six months ?
39. For what sum must I give a note to a bank payable in three months
in order to obtain 674 dollars ?
216 APPENDIX.
40. Bought seven 100 dollar shares of Bank stock at 6^ per cent, ad-
■ vance, gave in payment nine 60 dollar shares of Rail Road Stock at 3
per cent, discount, and a bank note payable in 60 days ; what was the faco
of the note ?
41. How many square inches are there in 15|- square rods ?
42. What is the cost of plastering the sides, ends, and ceiling of a room
22^ feet long, 17| feet wide, and 11 feet 1 inch in height, at 18 cents per
square yard, making no deduction for windows, doors, or wood work ?
43. There is a house 40 feet in length, and 26 feet in breadth ; from
the beam to the ridgepole is 1 1 feet ; the roof projects 7 inches beyond
the walls at each end, and the line of the eaves is 8 inches, measured
horizontally, from the side walls ; how many square feet are there in the
roof?
44. A painter agrees to paint tlie outside of a house which is 44 feet
long, 28 feet wide, 22^ feet in height to the top of the beam, and 11 feet
8 inches from the beam to the ridgepole, for 44 cents per square yard ;
what is he entitled to for the job, making no deduction for windows, and
no addition for cornices or other projections 1
45. What is the square root of 9743 to three places of decimals?
46. What is the square root of 17431 to two places of decimals ?
47. The base of a right angled triangle is 744 feet, the hypotenuse 834
feet ; what is the perpendicular, to two places of decimals ?
48. The base of a ti-iangle is 76 rods, the sum of the hypotenuse and
perpendicular is 186 rods ; what is the length of the hypotenuse ?
49. From a cylinder, twelve inches in diameter, it is desired to cut the
largest possible four sided prism, whose opposite sides .shall be parallel,
and whose width shall be to its thickness as 2 to 1 ; what will be its width
and what its thickness 1
50. What are the dimensions of the largest prism, with parallel sides,
that can be cut from a cylinder 12 inches in diameter, making the width
to the thickness as 3 to 1 ?
51. What are the dimensions of the largest prism, with parallel sides,
that can be cut from a sphere 15 inches in diameter, making the length
and breadth equal, and each of them double of the thickness 1
52. What is the cube root of 674 to three places of decimals i
53. What is the cube root of 1736 to two places of decimals ?
54. What is the cube ro'Ot of 31 to three places of decimals ?
55. Two men purchase a lot of land for 750 dollars, one pays $406.50,
the other the remainder; they expend $341 in equal shares on its im-
provement, and sell the land for $1430.00; what is each one's sham of
the. gain?
56. 1841 are how many times four-fifths of 76j1
57. How many bottles, each containing 1| puits, can be filled from a
hogshead containing 63 gallons, allowing a loss of one eleventh in the
process 1
58. What is the value, in Federal money, of £456 at 9^ per cent, ad-
vance 1
59. How many gallons, each containing 231 cubic inches, Avill fill a
cylindrical cistern four feet in diameter and five feet deep ?
60. There is a cylindrical cistern 6 feet deep, eontaining 10 barrels of
31|^ gallons each, each gallon containing 231 cubic inches ; what is the
diameter of the cistern 1
61 . There is a cistern, in the form of an inverted cone, eight feet deep,
APPENDIX. 217
and of the fame capacity as the cistern last named ; what is its diameter
at the top ?
62. Bought 74 barrels of flour at $4.56 per barrel, and after keeping it
35 days sold it at $5.16 per barrel; what per cent, did I gain, allowing
6 per cent, interest on the money invested 1
63. The first and fourteenth terms of an arithmetical series are 3 and
19; what is the common increase?
64. The tifth term of an arithmetical series is 18, the 16th term is 39;
what is the first term ?
65. What is the sum of an arithmetical series the extremes of which
are 9 and 1 64, and the number of terms forty 1
66. Find the ninth term of a geometrical series whose first term is 2,
and ratio |.
67. What is the fifth term of a geometrical series whose second term
is 4, and ratio § 1
68. James Wildes bought of John Good,
' 45A bushels Salt at 39^ cents per bushel, ^^
143| lbs. Rice at 3| cents per lb.
43i lbs. Tea at 39 cents per lb.
94| lbs. Coffee at 10^ cents per lb.
12 bbls. Flour at $5.87 per bbl.
What is the amount of the bill 1
69. If he pays the above bill by a bank note, discounted for 60 days,
what must be the sum named in the note ?
70. Add 23 A +1814-^+ -^-f^-
71. Reduce to a common denominator
2^ and | of j\ of 12^ and | of ^.
72. Bought 3 boxes of sugar, containing 3 cwt. 2 qrs. 17 lbs.; 3 cwt.
3 qrs. 1 U lbs : 3 cwt. 2 qrs. 22| lbs., at G^ cts. per lb. Paid in corn at
57^ cts. per bushel ; how many bushels did it take?
73. What is the solid contents of a wall 5 ft. high; 2 ft. 3 in. wide at
the bottom, and 1 foot 10 in. wide at the top. and 34 ft. in length 1
74. Three men agreed to build a wall 6 ft. high, 3 ft. wide at the bot-
tom, and 2 ft. wide at the top ; the first man built the wall to the hCight
of two feet from the ground, the second raised it 2 feet more, and the
third finished it; what proportional share of the pay ought each to re-
ceive ?
75. There is a lever, 13 ft. in length, which is supported by a fulcrum
14 in. from the end; how many pounds applied to the longer end will
balance a weiglit of 17 cwt. 2 qrs. at the shorter end ?
76. The axle of a wheel is 13 in. in diameter, the wheel is 11^ ft. in
diameter ; what power applied to the ch-cumfercncc will balance 3"^ tons
suspended at the axle ?
77. If tlic threads of a screw are l^- in. apart, what power applied at
the end of a lever 9^ ft. in length will support 7 tons, allowing nothing
for friction ?
78. With the same screw, what power would support 7 tons, making an
allowance of one fourth for friction? Reflect whether the friction in thi.s
case is in favor of the weight or in favor of the power.
79. With the same screw, what power would raise 7 tons, allowing one
fourth for friction ? Notice, in this case, in which way the friction will
operate. 19
218
ArPKNDlX.
6
1
80. There are two right angled triangles U])on ahase of 21 ft. in length,-
the perpendicular of the larger is 9 ft. in length, that of the smaller 8^ ft.,
^rhat is the difference in the length of the hypotenuse of the two triangles'?
81. Divide the number 78 in two such numbers that the first shall be
6 more than one fifth part of the second.
v^ 82. Divide the number 82 into two such parts that the first diminished
)y 5 shall equal one sixth part of the second.
83. What is the value in Federal money of £194 16s. at 8^ per cent,
advance ?
84. How many Winchester gallons, Avine measure, would be contaihed
\ in a cubical vat measuring 4 ft. each way ?
\\ 85. Divide the number 1520 into three such parts that twice the first
shall be 40 less than the second, and the second shall be half as great as
the third.
, 86. How many yards of lining | of a yard wide will line 13| yds. of
^ cloth l^v yds. wide ?
87. There is a rectangular field containing 7j acres, the width of which
is one half as great as its length ; what is the length of a diagonal line
connecting its opposite corners ?
88. Around a rectangular common containing 18 acres, the length of
which is to its breadth as 6 to 5, a road runs 40 ft. in breadth ; how many
rods would be saved in travel by crossing the common diagonally, rather
than going round on two sides of it, supposing the traveller to begin and
end in both cases in the middle of the road in range with the diagonal
line?
89. What is the difference between the square root of half of 4^ and
half the square root of 4, carried to three decimal places 1
90. What is the difference between the square root of one third of 12,
and one third the square root of 1 2, carried to three places of decimals 1
91. How many times | of 171 are equal to 13^ times 15^ ?
92 When it is noon in Boston, Lon. 71° 4' W. what time is it at Liver-
pool, Lon. 2' 59' W.: at Greenwich, Lon. 0; at Havre, Lon. 0 16' E.;
and at Paris, Lon. 2" 20' E. ?
93. Wlien it is noon at London, Lon. 0 5' W., what time is it at New
York, Lon. 74° 1' W.; at Washington, Lon. 77' 2' W. ; and at Cincin-
nati, Lon. 84° 27' W. ?
94. A field in the form of an equilateral triangle contains 8| acres;
what is the length of one of its sides ?
95. Reduce f of — to fifths. Reduce 9 of — - to fifteenths.
11 ISi
96. What is the amount in Federal money of 6s. 7d,, 5s. 3d., 14s. gd^
and lis. e^d.l
97. What is the cube root of 144 to three places of decimals ?
98. Wliat is the weight of a cylinder of lead 3 ft. long and 4 inches in
diameter 1
99. Multiply 7^ times Hi by | of ^
100. How many square feet in a triangle whose base measures 45 feet,
and whose height is 17 fcef?
101. What is the solid contents of a triangular pyramid whose base
measures 18 ft. on each side and whose height is 23 ft. "?
102. What is the interest of $546.25 for 23 mo. 13 days, at 7^ per ct. ?
APPENDIX. 219
103. What number is that of which ^ and ^ of it added together ex-
ceed ^ of it by 2 ?
104. What is the cube root of 197 to two decimal places ?
105. What is the cube root of 501 to four decimal places ?
106. What is tlie 43d term of an arithmetical series whose first term
is 7^ and common difference 3§ "?
107. What is the suni of an arithmetical series of 57 terms whose 4th
term is 15 and common difference 2^1
108. How many lbs. of coffee at 11 cents per lb. can be mixed with
56 lbs. at 8 cents, and 96 lbs. at 9^ cents, so as to make the mixture worth
10^ cents ?
109. If a sphere of gold weigh 36 oz. how many oz. will a sphere of
silver weigh of equal size, the specific gravity being as given on p. 198 ?
110. What will be the weight of a ball of iron 6 in. in diameter?
111. What will be the weight of the largest cube that can be cut from
a ball of iron 6 in. in diameter ?
112. What is the value in Federal money of £13 6s., 2 guineas, 3d.,
7:|^d., added together 1
113. How many times will a wheel 4 ft. 3 in. in diameter go round in
traversing the circumference of a circle containing 5 acres ?
114. If a lever is 16 ft. in length, the weight 13 cwt. 3 qrs. 11 lbs., and
the power 94 lbs., what must be the distance of the fulcrum from the
weight in order that the weight and power may balance 1
115. If the longer arm of a lever be 10 ft. and the shorter arm 2 feet in
length, how must 480 pounds be divided so that one part shall be the
weight and the other the power that will balance it on the lever 1
116. Divide f of J of 16^ by 18^ times | of 7|.
117. Reduce 1 pk. 3 qts. 1 pt. 1 gill to the decimal of a bushel.
118. How many shillings and pence in .4562 of & £.
119. A general drew up his army in a square, with the number in rank
and file equal, and had 576 men left; he then increased the square by
placing two lines of men in front, and two files on one side from front to
rear, when he found he wanted 12 men to complete the square; how
many men had he ?
120. What number is that one third of which exceeds two sevenths of
it by 19 ?
51 9
121. What number is that, — of which exceeds — • of it by 31 ?
13 23 -^
122. How many tiles each 8 inches square will it require to cover one
acre 1
123. If the hypotenuse of a right angled triangle measure 34 ft. and
the base 19^, what is the measure of the perpendicular?
124. If the sum of the base and hypotenuse is 63 ft. and the perpen-
dicular 14 ft. how long is the base ?
125. A man travels South 20 miles, then East 15 miles, then South 2^
miles, and East 7 miles ; how far is he then from where he set out ?
126. What is the difference between the cube root of one third of twelve,
and one third of the cube root of twelve ?
127. What is the value in dollars and cents of £94 16s. 3d.-f-^3 19s.
7d.-f-jei4 13s. 9d.?
128. If a note of $1000 promising annual interest is renewed at the
end of each year for five years, without the payment of any interest, what
is the amount, principal, and interest, at the end of the fifth year ?
220
APPENDIX.
129. What is the difference in avoirdupois weight between a ball of
silver, and a ball of gold, each 3 in. in diameter?
1.30. There are three numbers, the first, plus 4, is equal to one sixth of
the second, the second is one half as great as the third, and the sum of
the three is 186; what are the numbers?
131. There are two numbers, the first increased by 4 equals one sixth
of the second, and the second diminished by 6 is eight times the fii-st ;
what are the numbers ?
132. What is the value in Federal money of 13 shares of Bank stock,
par value $125 per share, and sold at 11^ per cent, advance ?
133. What is the square root of 14734?
134. What is the interest of $1974.36, for 3 yrs. 2 mo. 17 days, at 5|
per cent. ?
135. What is the 14th term of a geometrical series the first term of
which is 4 and the ratio | ?
136. What is the 16th term of a geometrical series the first tenn of
which is 7 and the ratio 1| ?
137. A sells to B 5 loads of wood, measuring, 1st, 8 ft. 6 in.. 4 ft., 3 ft.
9 in.; 2d, 9 ft., 4 ft. 2 in., 3 ft. 11 in.; 3d, 9 ft, 2 in., 4 ft. 4 in., 4 ft. 1 in.;
4th, 8 ft. 2 in., 3 ft. 7 in., 4 ft. 2 in. ; 5th, 7 ft. 11 in., 4 ft., 3 ft. 6 in ; at
$4.75 per cord. He receives in payment, 47 bushels of oats, at 38 cents
per busliel ; 56 lbs. cheese, at 8^ cents per lb. ; and the balance in butter,
at 15| cents per lb. ; how much butter does he receive ?
138. What is the weight of one rod of lead pipe one fourth of an inch
in thickness, if the inner diameter measures l^- inches ?
139. What is the weight of a plate of iron half an inch in thickness,
4 feet long, and 2 feet 3 inches in breadth '{
140. How many feet of silver wire, one tenth of an inch in diameter,
can be made from one pound avoirdupois of silver 1
141. How many gallons, imperial measure, will a cylindrical cistern
hold, 3 feet in diameter and 4^ feet deep ?
142. What is the cost of transporting 64 barrels of flour, each contain-
ing 7 qrs. gross, 100 miles, at $3.12|^ per ton, allowing for the weight of
each cask 16 lbs.?
143. If freight by rail road is $3.12^ per ton, for one hundred miles, and
freight by wagon road is $20 dollars per ton for 80 miles, how much is
saved in the freight of a barrel of flour 100 miles by rail road, allowing
its weight to be as in the preceding example ?
144. At the rate named above, how far could a barrel of flour be car-
ried by wagon road, before the freight should amount to as much as the
flour was worth, when the price is $5.62 per barrel ?
145. If the distance from New York to Liverpool is 3000 miles, what
would be the cost of transporting a barrel of flour that distance, at the
rate of $20 per ton for every 80 miles ?
146. If a barrel of flour can be transported from New York to Liver-
pool for 65 cents, what would that give for the transport of one ton 80
miles ?
147. The summit of the Rocky mountains, visited by Freemont, is in
Longitude 110 8'; what time is it at Greenwich when it is noon there ?
148. There is a field 20 rods long and 8 rods broad, with a path 3^ feet
wide running round it; how many square feet are there in the path ?
149. What is the cost of excavating a cubical pit, measuring 7^ feet in
each direction, at 31^ cents per cubic yardl
APPENDIX. 221
150. What is the Aveight of an iron cannon 9^ feet in length, 26 inches
in diameter at the larger end, and 18 inches at the smaller, with a bore
9 feet long and 10 inches in diameter, allowing for no inequalities in the
surface ?
151. What will be the duties on an invoice of goods amounting to
$1 156.80, at 30 per cent. 1 •
152. Bought an invoice of imported goods, amounting to £564 153.
I agree to give 18 per cent, in advance of the invoiced price; what is the
amount paid in Federal money, reckoning $4,444 to the pound ?
153. I buy for cash an invoice of imported goods, amounting to £1146
16s.; what is the invoiced price in Federal money, allowing Sterling
money to be 9 per cent, in advance of the nominal par value?
154. A merchant owes $16472.50; he fails, his means of payment amount-
ing to only $4345.62 ; how much is a creditor entitled to, who holds a
note against him of $100, dated 7^ months previous to the final settle*
ment, and promising interest 60 days after date 1
155. A road three and a half rods wide is laid out 1 mile and 14^ rods
in length, for which damages are awarded to the land-owners, as follows :
for 80 rods of the road, at the rate of 37 dollars per acre ; for 110 rods, at
the rate of 22 dollais per acre; and for the remainder, at the rate of 30^
dollars per acre ; what is the whole amount of the damages 1
156. What is due for the freight of 12 barrels of flour 65 miles, at f of
a cent per lb., allowing each barrel to contain 7 qrs. gross, of flour, and
the cask to weigh 18 lbs. ?
157. If 5 barrels of flour suffice for a family of 11 persons 7 months,
how many barrels will suffice for a family of 15 persons 4^ months 1
158. 13^ times 144 is 7^ times what number?
7i - 5
159. 384 is — of how many times 1^ •
160. Reduce 62^ cents to the decimal of a £, at nominal par value.
161. How many seconds were there in the year 18441
162. The report of a signal gun fired on the equator, at 12 o'clock, is
heard at a place due west, distant 16 miles ; at what time is the report heard
at the latter place, allowing 69^ miles to a degree of longitude, and sound
to move 1 mile and 10 rods in 5 seconds 1
163. A communication is made by the magnetic telegraph from Boston
to Washington, at 1 o'clock, P. M. ; at what time will it be received at
Washington, allowing no time for the transmission of the fluid ; longi-
tude of Boston being 71° 4' 9"; that of Washington. 77° 1' 24" ?
164. A engaging in partnership with B and C, puts in 1600 dollars, for
9^ months: B puts in 3100 dollars, for 14 months; C puts in 2200 dol-
lars, for 12^ months. They gain 1046 dollars; what is each one's share
of the gain?
165. If 100 dollars in one year gain 6^ dollars interest, what will 467
dollars gain in 9^ months, at the same rate 1
166. What is the value of 17 shares of Bank stock, par value 60 dollars
per share, and sold at 6^^ per cent, advance ?
167. A man buys 640 barrels of flour at $5| per bbl.; pays for freight,
$42.75; for storage, $11.17; and sells it for 6^^. per baiTcl, allowing a
commission of 2^ per cent. ; what was his loss or gain per cent. ?
168. The pipe of an aqueduct, 12 inches in diameter, is divided into
two branches, such that thek united capacity is equal to that of the main
19*
222 APPENDIX.
pipe, and the diameter of one of the branches is 9 inches ; what is the
diameter of the other?
169. The pipe of an aqueduct is 3 feet in diameter; what number of
pipes, each 2^ inches in diameter, would have a capacity equal to that of
the main pipe 1
17u. If a tree measures at the distance of 2 feet from the ground 12
feet in circumference, and at 14 feet from the ground divides into two
branches, measuring respectively 9 feet and 7 feet in circumference, how
many square inches more of surface would the lower horizontal section of
the tree contain than the upper one ?
171. A certain tree measures at the distance of 3 feet from the ground
22^ feet in circumference ; at some distance above, its four branches meas-
ure respectively, 8 ft. 4 in., 9 ft., 7 ft. 6 in., and 6 ft. 5 in. in circumference ;
what is the ratio of the magnitude of the tree at the lower, compared with
its magnitude at the higher place of measurement ?
172. The shadow of a certain tree, as cast upon the ground, measures
102 feet in diameter; allowing the shadow to be circular, how many rods
of ground does it cover ?
173. How many cubic inches does a wine glass contain, measuring 3 J
inches in depth, and 2 inches in diameter at the top, the form being that
of an inverted cone ?
174. How many yards of lining, f of a yard wide, will line 37^ yards
of cloth 1^ yards wide 1
175. What is the 3d term of the square 900-f-4204-n ?
176. What is the 3d term of the square 1600+480-f-n ?
177. Complete the square 4900-f-n-f-64.
178. What is the 4th term of the cube 27000+5400+360+0 ?
179. What is the 4th term of the cube 64000+4800+1 20+ Q ?
180. What is the 4th term of th^ cube 125000+22500+1350+C ?
181. What is the cube root of 68437 1
182. What is the cube foot of 954326 1
183. What is the solid contents of the largest sphere that can be cut
from a cubic block 13| inches on a side 1
184. From a sphere measuring 10 inches in diameter, the largest pos-
sible cubic block has been cut ; and from this block again the largest pos-
sible sphere has been cut ; what is the diameter of the last named sphere 1
185. From a sphere 20 inches in diameter what are the dimensions of
the largest parallel prism that can be cut, making the length to the
breadth as 4 to 3, and the breadth to the thickness as 3 to 2 ?
186. How many cubic feet are there in the walls of a brick house, the
length and breadth of which are each 44 feet outside, and the height 24
feet, supposing the walls to be perpendicular outside, and the thickness to
be 2 feet at the bottom, and 1 foot at the top, making no deduction for
doors or windows 1
187. What are the prime factors of 3746 ? of 9862 ?
188. What is the least common multiple of 684, 963, and 8416 1
189. What is the greatest common divisor of 94620, 3642, and 1646 ?
190. How many times will the wheel of a rail road car, if it be 2^ feet
in diameter, revolve in going 40 miles 1
191. How many times will such a wheel revolve in a minute, if the
speed of the car be 20 miles an hour 1
192. What is the cost in Federal money of the freight of 940 bales of
cotton, averaging 340 lbs. per bale, at | of a penny per lb, ?
ArpEN'Dix. 223
193. Sold 34 pieces cotton goods, averaging 31|- yards each in length,
at 9| cents per yard, 1^ per cent, off; what is the amount of the bill 1
194. Bought 840 barrels of flour on six months' credit, at 4| dollars per
ban-el ; sold the flour the same day on three months' credit, at 4i^ dollars
per barrel ; did I gain or lose, and how much, estimating the present
worth of the debts at the date of the transaction 7
195. A merchant buys for me, on commission, 400 barrels of flour, for
$4.34 per barrel, cash ; he sells the flour on the same day for cash, at $4.46
per barrel ; how much do I gain by the operation, allowing 1^ per cent,
commission on the i)urchases, and 2 per cent, on the sales ?
196. A sets out on a journey, travelling 24 miles a day; B sets out 2
days after, travelling 31§ miles a day; A, after travelling 3 days, goes 25
miles a day ; and B, after travelling 4 days, goes 33^ miles a day ; in how
many days after A sets out, will B overtake him ?
197. If a parallel prism is 4^ inches thick, 5 inches wide and 6 inches
long, what must be the diameter of the hollow sphere that would enclose it ?
198. If 18 horses in 16 weeks consume 204 bushels of oats, how many
horses will it require to consume 413 bushels in 18 weeks ?
199. If 100 dollars gain 6 dollars interest in 12 months, what will be
the interest of 840 dollars for 5^^ months ?
200. If a field containing 17 acres measures 81 rods on one side, what
must be the length of the corresponding side of a similar field containing
26^ acres '? JT
201. The contents of two similar fields fft-e as 4^ to 7, and the smaller
measures on one side 63 rods : what must;be the corresponding dimen-
sion of the larger field ?
202. There are two circles ; their areas are as 14 to 19, and the diame-
ter of the smaller is 16 rods ; what is the diameter of the larger ?
'203. What is the area of a circle ■v\^j||pediameter'is 44 feet ?
204. "What is the circumference of J^im^\^j|P^ diameter is 67 inches?
205. Given the circumference of a circle 60 rods to find its area.
206. There are three equilateral triangles whose areas are to each oth-
er as the numbers 3, 4, and 7 ; a side of the smallest measures 40 rods;
what is the sum of their areas ?
207. A grindstone is 3 feet in diameter ; allowing the hole in the mid-
dle to be 2 inches in diameter, how many inches must be ground off to
grind away half of the stone ?
208. The fore wheels of a waggon are 3 feet 10 inches in diameter, the
hind wheels 4 feet 2 inches in diameter, how many times more does one
of the fore wheels turn round than one of the hind wheels in going one
mile?
209. What is the weight of a cast iron cylinder 6 feet long and 4^
inches in diameter, the specific gravity being as already given 1
210. The frustum of a cone 7 feet long is 14 inches in diameter at the
larger end, and 10 inches in diameter at the smaller ; how faf from the
base must it be cut in two to divide it into equal parts ?
211. What is the first prime number above 901 ?
212. What is the first prime number below 10000?
213. What is the greatest common divisor of 1846, 3105, 684 and 1006 ?
214. What are all the prime factors of 801, of 3042, of 586, of 908 ?
215. In what proportion may corn at 80 cents, be mixed with rye at
86 cents, and with oats at 43 cents per bushel, to make the mixture worth
50 cents per bushel ?
224 . Kt'E!>L>JX.
S-l 4.9 1 Q
216. Add the fractions ^4-^+-if .
7 61:5^ S/g-
217. What is the value of ^ of ^ £ f of ^tj S, expressed in the de-
cimal of a £ ?
218. What is the present value of a note of 584 dollars, payable in
three months ?
219. What is the bank discount on a note of 150 dollars, payable in
three months 1
220. What sum will be paid on a note of 240 dollars, discounted at a
bank, for 90 days 1
221. What is the interest of 1200 dollars for 10 days, at 7 per cent. ?
222. Divide the sum of the decimals 2016 + 9172 + 0064, by f of f
reduced to a decimal.
223. Divide 3 T. 17 cwt. 3 qrs. 19 lbs. by 6.
224. Divide 9 m. 3 fur. 21 r. 14 ft. bv 8.
225. Multiply 31 d. 14 h. 37 m. 15 sec. by 19.
226. Multiply 83 A. 3 R. 22 r. by 12.
"227. What is one fifth of 16 T. 11 cwt. 2 qrs. 20 lbs. ?
228. What is the value at 4 dollars a cord, of three loads of wood
measuring as follows : 1st, 8 ft. 6 in., 4 ft 2 in., 3 ft. 9 in. ; 2d, 9 ft., 4 ft.
1 in., 3 ft. 10 in. ; 3d, 8 ft. 1 in., 4 ft., 4 ft. 2 in. ?
229. What is the 15th term of an arithmetical series, the first term of
which is 3, and the commSflKifFerence ^ ?
230. If the 9th term of a^ arithmetical series is 23, and the common
diiference §, ^vhat is the 3d ^rm ?
231. What is the sum of an arithmetical series of 40 terms, if the first
term is 2 and the common difference 31. ?
232. What is thersum of an aathmetical series of 72 terms, if the first
term is 1 and the conmon'jM^TOce 1^ ?
233. A man engageS to walk 1000 miles in 1000 hours, on condition
of receiving 1 cent for the first mile, and for each mile after, 5- of a cent
more than he had for the mile preceding it; what will he be entitled to
on the fulfillment of his contract?
234. There are two grindstones the thickness of which is to the diam-
eter as 2 to 11 ; the smaller one is 2 feet in diameter, the other is three
times as heavy ; what is its diameter ?
235. A man has a triangular field containing 7 acres, tlie vertex or
point of the triangle is 54 rods from the base, at what distance from the
base must a line be drawn parallel to it so as to cut off one half the Held ?
236. The hypotenuse and perpendicular of a triangle measure togeth-
er 816 rods, the base measures 61 rods ; what is the length of the ])erpen-
dicular 1
237. A rope 100 feet long passes straight from the ground at flic dis-
tance of 10 feet from a perpendicular pole, over the top of the polo, which
is 25 feet in height, and thence is drawn so as to reach the ground at tli<j
farthest possible point ; allowing the ground to be level, how fiir is the
last named point from the foot of the pol
238. A stick' of timber in the form of a truncated wedge is 10 feet
long, 2 feet wide through its whole extent, 20 inches in thickness at one
end, and 14 inches thick at the other; how far from the thicker end must
it be cut in two so as to divide it into two equal parts ?
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