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INTELLECTUAL
ABIT II ME TIC,
INDUCTIVE METHOD
INSTRUCTION.
BY WARil^N COLBURN, A. M.
IIIIBOTYPED AT THE BOSTON TYPE AND STEREOTYPE For
BOSTON:
HILLIALD, GRAY, «fc CO"JF
ism ,
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'Tiss Sue Dunbar
Educ • Library
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COLBURN'S FIRST LESSONS.
INTELLECTUAL
ARITHMETIC,
UPON THE
INDUCTIVE METHOD
INSTRUCTION.
BY WARREN COLBURN, A.M.
STEREOTYPED AT THE BOSTON TYPE AND STEREOTYPE FOUNDRY.
BOSTON:
•
PUBLISHED BY
HILLIARD, GRAY, AND CO*
1836.
DISTRICT OF MASSACHUSETTS, to trft .«
District Clerk's Office.
BB IT REMEMBERED, That on the twenty-third day of March, A. D. J82fi,
in the fiftieth year of the Independence of the United States of America,
CUMMIXOS, MILLIARD, AND COMPANY, of the said district, have deposited
in this office the title of a book, the right whereof they claim as proprie-
tors, in the words following, to wit:
" Intellectual Arithmetic, upon the Inductive Method of Instruction. By
Warren Colburn, A. M."
In conformity to the act of the Congress of the United States, entitled,
u An Act for the encouragement of learning, by securing the copies of maps,
charts, and books, to the authors and proprietors of such copies during the
times therein mentioned;" and also to an act, entitled, " An Act supple-
mentary to an act, entitled, ' An Act for the encouragement of learning, by
securing the copies of maps, charts, and books, to the authors and proprie-
tors of such copies during the times therein mentioned,' and extending tho
benefits thereof to the arts of designing, engraving, and etching historical
and other prints." JOHN W. DAVIS,
Clerk of the District tf Massachusetts.
RECOMMENDATIONS.
Sir, BOSTON, 15 November, 1821.
I have made use of the Arithmetic and Tables, which you some-
time since prepared, on the system of Pestalozzi ; and have been much
f ratified with the improved edition of it; which you have shown me.
am satisfied, from experiment, that it is the most effectual and in-
teresting mode of teaching the science of numbers with which I am
acquainted. Respectfully,
Your obedient servant,
HENRY COLMAN.
Mr. WARREN COLBURN.
Having been made acquainted with Mr. Colburn's treatise on Arith-
metic, ajid having attended an examination of his scholars, who had been
taught according to this system, I am well satisfied that it is the most
easy, simple, and natural way of introducing young persons to the first
principles in the science of numbers. The method here proposed is
the fruit of much study and reflection. The author has had considera-
ble experience as a teacher, added to a strong interest in the subject,
and a thorough knowledge not only of this but of many of the higher
branches of mathematics. This little work is therefore earnestly
recommended to the notice of those who are employed in this branch
of early instruction, with the belief that it only requires a fair trial in
order to be fully approved and adopted.
J. FARRAR,
Prof. Math. Harvard University.
CAMBRIDGE, Nov. 1C, 182L
Ex
PREFACE. /,,
As soon as a child begins to use his senses, nature contin-
ually presents to his eyes a variety of objects ; and one of the
first properties which he discovers, is the relation of number.
He intuitively fixes upon unity as a measure, and from this
he forms the idea of more and less; which is the idea of
quantity.
The names of a few of the first numbers are usually learned
very early ; and children frequently learn to count as far as a
hundred before they learn their letters.
As soon as children have the idea of more and less, and the
names of a few of the first numbers, they are able to make small
calculations. And this we see them do every day about their
playthings, and about the little affairs which they are called
upon to attend to. The idea of more and less implies addition ;
hence they will often perform these operations without any
previous instruction. If. for example, one child has three
apples, and another five, they will readily tell how many they
both have ; and how many one has more than the other. If a
child be requested to bring three apples for each person in
the room, he will calculate very readily how many to bring,
if the number does not exceed those he has learnt. Again,
if a child be requested to divide a number of apples among
a certain number of persons, he will contrive a way to do it,
and will tell how many each must have. The method which
children take to do these things, though always correct, is not .
always the most expeditious.
The fondness which children usually manifest for these
exercises, and the facility with which they perform them,
seem to indicate that the science of numbers, to a certain
extent, should be among the first lessons taught to them.*
To succeed in this, however, it is necessary rather to fur-
nish occasions for them to exercise their own skill in per-
forming examples, than to give them rules. They should be
allowed to pursue their own method first, and then they
should be ma*de to observe and explain it; and, if it was not
* See, on this subject, two essays, entitled Juvenile Studies, in the
Prize Book of the Latin School, Nos. I. and IP., published by Cum-
mings & Hilliard, 1820 and 1821.
&291992!
4 PREFACE.
the beat, some improvement should be suggested. By follow-
ing this mode, and making the examples gradually increase
in difficulty, experience proves, that, at an early age, children
may be taught a great variety of the most useful combinations
of numbers.
Few exercises strengthen and mature the mind so much as
arithmetical calculations, if the examples are made sufficiently
simple to be understood by the pupil; because a regular,
though simple process of reasoning, is requisite to perform
them, and the results are attended with certainty.
The idea of number is first acquired by observing sensible
objects. Having observed that this quality is common to all
things with which we are acquainted, we obtain an abstract
idea of number. We first make calculations about sensible
objects ; and we soon observe, that the same calculations
will apply to things very dissimilar ; and, finally, that they
may be made without reference to any particular things.
Hence from particulars we establish general principles, which
serve as the basis of our reasonings, and enable us to proceed,
step by step, from the most simple to the more complex opera-
tions. It appears, therefore, that mathematical reasoning
proceeds as much upon the principle of analytic induction, as
that of any other science.
Examples of any kind upon abstract numbers, are of very
little use, until the learner has discovered the principle from
practical examples. They are more difficult in themselves,
for the learner does not see their use ; and therefore does
not so readily understand the question. But questions of a
practical kind, if judiciously chosen, show at once what the
combination is, and what is to be effected by it. Plence the
pupil will much more readily discover the means by which
the result is to be obtained. The mind is also greatly assisted
%in the operations by reference to sensible objects. When the
pupil learns a new combination by means of abstract examples,
it very seldom happens that he understands practical examples
more easily for it. because he does not discover the connection
until he has performed several practical examples, and begins
to generalize them.
After the pupil comprehends an operation, abstract examples
are useful to exercise him, and make him familiar with it.
And they serve better to fix the principle, because they teach
the learner to generalize.
From the above observations, and from his own* experience,
the author bos been induced to publish this treatise ; in which
he has pursued the following plan, which seemed to him the
most agreeable to the natural progress of the mind.
PREFACE.
GENERAL VIEW OF THE PLAN.
EVERY combination commences with practical examples.
Care has been taken to select such as will aptly illustrate the
combination, and assist the imagination of the pupil in per-
forming it. In most instances, immediately after the prac-
tical, abstract examples are placed, containing the same
numbers and the same operations, that the pupil may the
more easily observe the connection. The instructer should
be careful to make the pupil observe the connection. After
these, are a few abstract examples, and then practical questions
again.
The numbers are small, and the questions so simple, that
almost any child of five or six years old is capable of under-
standing more than half the book, and those of seven or eight
years old can understand the whole of it.
The examples are to be performed in the mind, or by
means of sensible objects, such as beans, nuts, &c., or by
means of the plate at the end of the book. The pupil should
first perform the examples in his own way, and then be
made to observe and tell how he did them, and why he did
them so.*
* It is remarkable that a child, although he is able to perform a va-
riety of examples which involve addition, subtraction, multiplication,
and division, recognizes no operation but addition. Indeed, if we
analyze these operations when we perform them in our minds, we
shall find that they all reduce themselves to addition. They are only
different ways of applying the same principle. And it is only when
we use an artificial method of performing them, that they take a
different form.
If the following questions were proposed to a child, his answers
would be, in substance, like those annexed to the questions : — How
much is five less than eight ? Ans. Three. Why ? Because five and
three are eight. What is the difference between five and eight ? Ans.
Three. Why 1 Because five and three are eight. If you divide eight
into two parts, such that one of the parts may be five, what will the
other be ? Ans. Three. Why ? Because five and three are eight.
How much must you give for four apples, at two cents apiece ?
Ans. Eight cents. Why ? Because two and two are four, and two
are six, and two are eight.
How many apples, at two cents apiece, can you buy for eight
cents ? Ans. Four. Why ? Because two and two are four, and two
are six, and two are eight.
We shall be further convinced of this, if we observe that the same
table serves for addition and subtraction j and another table, which is
0 PREFACE.
The use* of the plates is explained in the Key at the end
of the book. Several examples in each section are performed
in the Key, to show the method of solving them. No answers
are given in the book, except where it is necessary to explain
something to the pupil. Most of the explanations are given
in the Key ; because pupils generally will not understand any
explanation given in a book, especially at so early an age.
The instructor must, therefore, give the explanations viva voce.
These, however, will occupy the instructor but a very short
time.
The first section contains addition and subtraction, the sec-
ond multiplication. The third section contains division. In
tlii.- section, the pupil learns the first principles of fractions, and
the terms which are applied to them. This is done by making
him observe that one is the half of two, the third of three, the
fourth of four, &c., and that two is two thirds of three, two
fourths of four, two fifths of five, &c.
The fourth section commences with multiplication. In this
the pupil is taught to repeat a number a certain number of
times, and a part of another time. In the second part of this
section the pupil is taught to change a certain number of twos
into threes, threes into fours, &c.
In the fifth section, the pupil is taught to find .J, £, £, &c.,
*nd}, 1,-f, &c., of numbers which are exactly divisible into
these parts. This is only an extension of the principle of
fractions, which is contained in the third section.
In the sixth section, the pupil learns to tell of what number
any number, as 2, 3, 4, &c., is one half, one third, one fourth,
&c. ; and also, knowing f , $, ^, &c., of a number, to find that
number.
Tiiese combinations contain all the most common and most
useful operations of vulgar fractions. But being applied only
to numbers which are exactly divisible into these fractional
parts, the pupil will observe no principles but multiplication
and division, unless he is told of it. In fact, fractions contain
no other principle. The examples are so arranged, that
almost any child of six or seven years old will readily compre-
hend them. And the questions are asked in such a manner,
that, if the instructer pursues the method explained in the.
Key, it will be almost impossible for the pupil to perform any
example without understanding the reason of it. Indeed, in
formed by addition, serves both for multiplication and division. la
this treatise, the same plate serves for the four operations.
This remark shows the necessity of making the pupil attend to
his manner of performing the examples, and of explaining to him the
difference between the.rn.
PREFACE. 7
every example which he performs, he is obliged to go through
a complete demonstration of the principle by which he
does it ; and at the same time he does it in the simplest way
passible. These observations apply to the remaining part of
the book.
These principles are sufficient to enable the pupil to per-
form almost all kinds of examples that ever occur. He will
not, however, be able to solve questions in which it is neces-
sary to take fractional parts of unity, though the principles
are the same.
After section sixth, there is a collection of miscellaneous
examples, in which are contained almost all the kinds that
usually occur. There are none, however, which the princi-
ples explained are not sufficient to solve.
In section eighth and the following, fractions of unity are
explained, and, it is believed, so simply as to be intelligible to
most pupils of seven or eight years of age. The operations
do not diiier materially from those in the preceding sections.
There are some operations, however, peculiar to fractions.
The two last plates are used to illustrate fractions.
When the pupil is made familiar with all the principles con-
tained in this book, he will be able to perform all examples
in which the numbers are so small, that the operations may
be performed in the mind. Afterwards, he has only to learn
the application of figures to these operations, and his knowl-
edge of arithmetic will be complete.
The Rule of Three, and all the other rules which are
usually contained in our arithmetics, will be found useless.
The examples under these rules will be performed upon
general principles with much greater facility, and with a
greater degree of certainty.
The following are some of the principal difficulties which
a child has to encounter in learning arithmetic in the usual
way, and which are seldom overcome : — First, the exam-
ples are so large, that the pupil can form no conception of
the numbers themselves ; therefore it is impossible for him
to comprehend the reasoning upon them. — Secondly, the
first examples are usually abstract numbers. This increases
the difficulty very much ; for, even if the numbers were so
small that the pupil could comprehend them, he would dis-
cover but very little connection between them and practical
examples. Abstract numbers, and the operations upon them,
must be learned from practical examples; there is no such
thing as deriving practical examples from those which are
abstract, unless the abstract have been first derived from
those which are practical. — Thirdly, the numbers are ex-
pressed by figures, which, if they were used only as a con-
8 PREFACE.
tracted way of writing numbers, would be much more diffi-
cult to be understood at first than the numbers written at
length in words. But they are not used merely as words ;
they require operations peculiar to themselves. They are,
in fact, a new language, which the pupil has to learn. The
pupil, therefore, when he commences arithmetic, is presented
with a set of abstract numbers, written with figures, and
so large that he has not the least conception of them even
when expressed in words. From these he is expected to
learn what the figures signify, and what is meant by addition,
subtraction, multiplication, and division ; and, at the same
time, how to perform these operations with figures. The
consequence is, that he learns only one of all these things, and
that is, how to perform these operations on figures. He can,
perhaps, translate the figures into words ; but this is useless,
since he does not understand the words themselves. Of the
effect produced by the four fundamental operations he has
not the least conception.
After the abstract examples, a few practical examples are
usually given; but these again are so large that the pupil
cannot reason upon them, and consequently he could not
tell whether he must add, subtract, multiply, or divide,
even if he had an adequate idea of what these operations
are.
The common method, therefore, entirely reverses the natu-
ral process ; for the pupil is expected to learn general princi-
ples, before he has obtained the particular ideas of which they
are composed.
The usual mode of proceeding is as follows : — The pupil
learns a rule, which, to the man that made it, was a general
principle ; but with respect to him, and oftentimes to the
instructer himself, it is so far from it, that it hardly deserves
to be called even a mechanical principle. He performs the
examples, and makes the answers agree with those in the
book, and so presumes they are right. He is soon able to do
this with considerable facility, and is then supposed to be
master of the rule. He is next to apply his rule to practical
examples ; but if he did not find the examples under the rule,
he would never so much as mistrust they belonged to it. But,
finding them there, he applies his rule to them, and obtains
the answers, which are in the book, and this satisfies him that
they are right. In this manner he proceeds from rule to rule
through the book.
When an example is proposed to him, which is not in the
book, his sagacity is exercised, not in discovering the opera-
tions necessary to solve it, but in comparing it with the ex-
amples which he has performed before, and endeavoring to dis-
PREFACE. 9
cover some analogy between it and them, either in the sound,
or in something else. If he is fortunate enough to discover
any such analogy, he finds what rule to apply, and if he has
not been deceived in tracing the analogy, he will probably
solve the question. His knowledge of the principles of his
rule is so imperfect, that he would never discover to which
of them the example belongs, if he did not trace it, by some
analogy, to the examples which he had found under it.
These observations do not apply equally to all ; for some
will find the right course themselves, whatever obstacles be
thrown in their way. But they apply to the greater part ;
and it is probable that there are very few who have not expe-
rienced more or less inconvenience from this mode of proceed-
ing. Almost all, who have ever fully understood arithmetic,
have been obliged to learn it over again in their own way.
And it is not too bold an assertion to say, that no man ever
actually learned mathematics in any other method, than by
analytic induction ; that is, by learning the principles by the
examples he performs ; and not by learning principles first,
and then discovering by them how the examples are to be
performed.
In forming and arranging the several combinations, the
author has received considerable assistance from the system of
Pestalozzi. He has not, however, had an opportunity of seeing
Pestalozzi's own work on this subject, but only a brief outline
of it by another. The plates, also, are from Pestalozzi. In
selecting and arranging the examples to illustrate these com-
binations, and in the manner of solving questions generally,
he has received no assistance from Pestalozzi.
THE BOY WITHOUT A GENIUS.
Mr. Wiseman, the schoolmaster, at the end of his sum-
mer vacation, received a new scholar with the following
letter :—
Sir, — This will be delivered to you by my son Samuel, whom I be?
leave to commit to your care, hoping that, by your well-known skill
and attention, you will he able to make something1 of him, which, I am
sorry to say. none of his masters have hitherto done. He is now elev-
en, and yet can do nothing but read his mother tongue, and that but
indifferently. We sent him at seven to a grammar school in our
neighborhood ; but his master soon found that his genius was not
turned to learning languages. He was then put to writing, but he
set about it so awkwardly that he made nothing1 of it. He was tried
at accounts, but it appeared that he had no genius for that either.
He could clo nothing in geography for want of memory. In short, if
10 PREFACE.
he has any genius at all, it does not yet show itself. But I trust to
your experience, in cases of this nature, to discover what he is fit for,
and to instruct him accordingly. I beg to be favored shortly with
your opinion about him; and remain, sir,
Your most obedient servant,
HUMPHREY ACRES.
When Mr. Wiseman had read this letter, he shook his head, and said
to his assistant, A pretty subject they have sent us here ! a lad that has
a great genius for nothing at all. But perhaps my friend Mr. Acres
expects that a boy should show a genius for a thing before he knows
any thing about it — no uncommon error ! Let us see, however, what
the youth looks like. I suppose he is a human creature at least.
Master Samuel Acres was now called in. He came, hanging down
his head, and looking as if he was going to be flogged.
Come hither, my dear ! said Mr. Wiseman. Stand by me, and do
not be afraid. Nobody will hurt you. How old are you ?
Eleven last May, sir.
A well-grown boy of your age, indeed. You love play, I dare say 7
Yes, sir.
What, are you a good hand at marbles ?
Pretty good, sir.
And can spin a top and drive a hoop, I suppose ?
Yes, sir.
Then you have the full use of your hands and fingers ?
Yes, sir.
Can you write, Samuel ?
I learned it a little, sir, but I left it off again.
And why so ?
Because I could not make the letters.
No ! Why, how do you think other boys do ? Have they more
fingers than you 1
No, sir.
Are you not able to hold a pen as well as a marble ?
Samuel was silent.
Let me look at your hand.
Samuel held out both his paws, like a dancing bear.
I see nothing here to hinder you from writing as well as any boy
in the school. You can read, I suppose ?
Yes, sir.
Tell me, then, what is written over the school-room door.
Samuel, with some hesitation, read, WHATEVER MAN HAS
DONE MAN MAY DO.
Pray how did you learn to read ? Was it not with taking pains ?
Yes, sir.
Well— taking more pains will enable you to read better. Do you
know any thing of the Latin Grammar ?
No, sir.
Have you never learned it ?
I tried, sir, but I could not get it by heart.
Why, you can say some things by heart. I dare say you can tell
me the names of the days of the week in their orrW.
PREFACE. 11
Yes. sir, I know them.
And the months in the year, perhaps.
Yes, sir.
And you could probably repeat the names of your brothers and
sisters, and all your father's servants, and half the people in the
village besides.
I believe I could, sir.
Well — and is hie, hcec} hoc, more difficult to remember than these ?
Samuel was silent.
Have you learned any thing of accounts ?
I went into addition, sir, but I did not go on with it.
Why so ?
I could not do it, sir.
How many marbles can you buy for a penny 1
Twelve new ones, sir.
And how many for a half-penny ?
Six.
And how many for two-pence ?
Twenty-four.
If you were to have a penny a day, what would that make in a
week?
Seven-pence.
But if you paid two-pence out of that, what would you have left ?
Samuel studied awhile, and then said, Five-pence.
Right. Why, here you have been practising the four great rules of
arithmetic, — addition, subtraction, multiplication, and division. Learn-
ing accounts is no more than this. Well, Samuel, I see what you are
fit for. I shall set you about nothing but what you are able to do 3
but, observe, you must do it. We have no / can't here. Now go
among your school-fellows.
Samuel went away, glad that his examination was over, and with
more confidence in his powers than he had felt before.
The next day he began business. A boy less than himself was
called out to set him a copy of letters, and another was appointed to
hear him in grammar. He read a few sentences in English, that he
could perfectly understand, to the master himself. 'Thus, by going
on steadily and slowly, he made a sensible progress. He had already
joined his letters, got all the declensions perfectly, and half the mul-
tiplication table, when Mr. Wiseman thought it time to answer his
father's letter 3 which he did as follows :—
Sir,
I now think it right to give you some information concerning
your son. You, perhaps, expected it sooner ; but I always wish to
avoid hasty judgments. You mentioned in your letter that it had not
yet been discovered which way his genius pointed. If by genius
you meant such a decided bent of mind to any one pursuit as will
lead to excel with little or no labor or instruction, I must say that I
have not met with such a quality in more than three or four boys in
my life, and your son is certainly not among the number. But if you
mean only the ability to do some of those things which the greater part
IS) PREFACE.
of mankind can do when properly taught, I can affirm that I find in
him no peculiar deficiency. And, whether you choose to bring him
up to trade or to some practical profession, I see no reason to doubt
that he may in time become sufficiently qualified for it. It is my
favorite maxim, sir, that every thing most valuable in this life may
generally be acquired by taking pains for it. Your son has already
lost much time in the fruitless expectation of finding out what he
would take up of his own accord. Believe me, sir, few boys will
take up any thing of their own accord but a top or a marble. I will
take care, while he is with me, that he loses no more time this way,
but is employed about things that are fit for him; not doubting that
we shall find him fit for them.
I am. sir, yours, &c.
SOLON WISEMAN.
Though the doctrine of this letter did not perfectly agree with Mr.
Acres's notions, yet, being convinced that Mr. Wiseman was more
likely to make something of his son than any of his former preceptors,
he continued him at his school for some years, and had the satisfaction
to find him goinff on in a steady course of gradual improvement. In
due time, a profession was chosen for him, which seemed to suit his
temper and talents, but for which he had no particular turn, having
never thought at all about it. He made a respectable figure in it, and
went through the world with credit and usefulness, though without a
genius.
MRS. BARBAULD.
ARITHMETIC.
PART I.
SECTION I.
A.* 1. How many thumbs have you on your right
hand 1 How many on your left ? How many on
both together 1
2. How many hands have you?
3. If you have two nuts in one hand, and one in
the other, how many have you in both ?
4. How many fingers have you on one hand?
5. If you count the thumb with the fingers, how
many will it make ?
6. If you shut your thumb and one finger, and
leave the rest open, how many will be open ?
7. If you have two cents in one hand, and two in
the other, how many have you in both ?
8. James has two apples, and William has three ;
if James gives his apples to William, how many will
William have?
9. If you count all the fingers on one hand, and
two on the other, how many will there be?
10. George has three cents, and Joseph has four;
how many have they both together?
* For the manner of solving questions, and the explanation of the
plates, see the Key at the end of the book. The first questions in
this section are intended for very young1 children. It will be well
for the instmcter to give a great many more of this kind.— -Older
pupils may omit these.
2
14 ARITHMETIC. [PART 1.
11. Robert gave five cents for an orange, and two
for an apple; how many did he give for both?
12. If a custard cost six cents, and an apple two
cents, how many cents will it take to buy an apple
and a custard 1
13. If you buy a pint of nuts for five cents, and
an orange for three cents, how many cents would
you give for both? How many more for the nuts
than for the orange ?
14. If an ounce of figs is worth six cents, and a
half a pint of cherries is worth three cents, how
much are they both worth ?
15. Dick had five plums, and John gave him four
more ; how many had he then ?
16. How many fingers have you on both hands?
17. How many fingers and thumbs have you on
both hands?
18. If you had six marbles in one hand, and four
in the other, how many would you have in the one
more than in the other? How many would you
have in both hands?
19. David had seven nuts, and gave three of them
to George ; how many had he left ?
20. Two boys, James and Robert, played at mar-
bles ; when they began, they had seven apiece, and
when they had . done, James had won four ; how
many had each then ?
21. A boy, having eleven nuts, gave away three
of them ; how many had he left ?
22. If you had eight cents, and your papa should
give you five more, how many would you have?
23. A man bought a sheep for eight dollars, and
a calf for seven dollars; what did he give for
both?
24. A man bought a barrel of flour for eight dol-
lars, and sold it for four dollars more than he gave
for it ; how much did he sell it for ?
SECT. 1.] ARITHMETIC. 15
25. A man bought a hundred weight of sugar for
nine dollars, and a barrel of flour for seven dollars ;
how much did he give for the whole ?
26. A man bought three barrels of cider for eight
dollars, and ten bushels of apples for nine dollars;
how much did he give for the whole ? ^ I
27. A man bought a firkin of butter for twelve
dollars, but, it being damaged, he sold it again for
eight dollars ; how much did he lose ?
28. A man bought three sheep for fifteen dollars,
but could not sell them again for so much by eight
dollars ; how much did he sell them for ?
29. A man bought sixteen pounds of coffee, and
lost seven pounds of it as he was carrying it home ;
how much had he left ?
30. A man bought nineteen pounds of sugar, and,
having lost a part of it, he found he had nine pounds
left ; how much had he lost ?
31. A man, owing fifteen dollars, paid nine dol-
lars of it ; how much did he then owe ?
32. A man, owing seventeen dollars, paid all but
seven dollars ; how much did he pay ?
B. 1. Two and one are how many?
2. Two and two are how many ?
3. Three and two are how many ?
4. Four and two are how many ?
5. Five and two are how many ?
6. Six and two are how many ?
7. Seven and two are how many?
8. Eight and two are how many ?
9. Nine and two are how many ?
10. Ten and two are how many ?
11. Two and three are how many ?
12. Three and three are how many?
13. Four and three are how many ?
14. Five and three are how many ?
16 ARITHMETIC. [PART 1.
15. Six and throe are how many?
16. Seven and three are how many 1
17. Eight and three are how many ?
18. Nine and three are how many ?
19. Ten and three are how many ]
20. Two and four are how many?
21. Three and four are how many ?
22. Four and four are how many ?
23. Five and four are how many ?
24. Six and four are how many?
25. Seven and four are how many?
28. Eight and four are how many?
27. Nine and four are how many ?
23. Ten and four are how many ?
29. Two and five are how many ?
30. Three and five are how many ?
31. Four and five are how many ?
32. Five and five are how many ?
33. Six and five are how many ?
34. Seven and five are how many ?
35. Eight and five are how many ?
36. Nine and five are how many ?
37. Ten and five are how many ?
33. Two and six are how many?
39. Three and six are how many ?
40. Four and six are how many ?
41. Five and six are how many ?
42. Six and six are how many ?
43. Seven and six are how many ?
44. Eight and six are how many ?
45. Nine and six are how many ?
46. Ten and six are how many ?
47. Two and seven are how many ?
48. Three and seven are how many ?
49. Four and seven are how many ?
50. Five and seven are how many?
51. Six and seven are how many?
SECT. 1.] ARITHMETIC. 17
52. Seven and seven are how many ?
53. Eight and seven are how many 1
54. Nine and seven are how many ?
55. Ten and seven are how many 1
56. Two and eight are how many ?
57. Three and eight are how many ?
58. Four and eight are how many 1
59. Five and eight are how many ?
60. Six and eight are how many 1
61. Seven and eight are how many?
62. Eight and eight are how many 1
63. Nine and eight are how many ?
64. Ten and eight are how many ?
65. Two and nine are how many?
66. Three and nine are how many 1
67. Four and nine are how many ?
68. Five and nine are how many ?
69. Six and nine are how many ?
70. Seven and nine are how many ?
71. Eight and nine are how many?
72. Nine and nine are how many ?
73. Ten and nine are how many ?
74. Two and ten are how many ?
75. Three and ten are how many ?
76. Four and ten are how many 1
77. Five and ten are how many ?
78. Six and ten are how many ?
79. Seven and ten are how many ?
80. Eight and ten are how many ?
81. Nine and ten are how many?
82. Ten and ten are how many ?
C. 1. Two and one are how many ?
2. Two and two are how many ?
3. Three and two are how many ?
4. Five und two are how many 1
5. Four and two are how many ?
2*
18 ARITHMETIC. [PART 1.
6. Six and two are how many ?
7. Eight and two are how many ?
8. Five and three are how many ?
9. Seven and three are how many ?
10. Four and three are how many ?
11. Two and three are how many ?
12. Two and six are how many ?
13. Two and eight are how many ?
14. Six and three are how many ?
15. Three and four are how many ?
16. Three and six are how many ?
17. Two and seven are how many ?
18. Ten and two are how many ?
19. Two and four are how many?
20. Three and seven are how many?
21. Four and four are how many ?
22. Five and four are how many ?
23. Seven and two are how many ?
24. Two and five are how many ?
25. Three and three are how many ?
26. Four and five are how many ?
27. Nine and two are how many ?
28. Three and five are how many ?
29. Two and ten are how many 7
30. Three and eight are how many ?
31. Ten and three are how many ?
32. Two and nine are how many ?
33. Four and six are how many ?
34. Eight and three are how many ?
35. Seven and four are how many ?
36. Nine and three are how many ?
37. Six and four are how many ?
38. Five and five are how many ?
39. Three and nine are how many ?
40. Four and seven are how many ?
41. Six and five are how many?
42. Three and ten are how many ?
SECT. 1.] ARITHMETIC. 19
43. Eight and four are how many 1
44. Five and eight are how many ?
45. Four and nine are how many ?
46. Five and six are how many ?
47. Ten and four are how many ?
48. Seven and five are how many 1
49. Six and six are how many 1
50. Nine and four are how many?
51. Eight and five are how many ?
52. Five and nine are how many ?
53. Four and ten are how many ?
54. Six and seven are how many ?
55. Four and eight are how many ?
56. Nine and five are how many ?
57. Six and eight are how many ?
58. Ten and five are how many ?
59. Seven and six are how many 1
60. Eight and seven are how many ?
61. Six and nine are how many ?
62. Seven and seven are how many ?
63. Eight and six are how many 1
64. Ten and six are how many ?
65. Eight and eight are how many ?
66. Nine and seven are how many ?
67. Ten and eight are how many ?
68. Six and ten are how many ?
69. Five and seven are how many ?
70. Nine and six are how many ?
71. Seven and eight are how many ?
72. Eight and nine are how many ?
73. Nine and nine are how many ?
74. Five and ten are how many ?
75. Seven and nine are how many ?
76. Nine and eight are how many T
77. Eight and ten are how many ?
78. Ten and nine are how many ?
79. Seven and ten are how many ?
20 ARITHMETIC. [PART 1.
80. Nine and ten are how many ?
81, Ten and ten are how many ?
D. 1. Three boys, Peter, John, and Oliver, gave
some money to a beggar ; Peter gave seven cents,
John four cents, and Oliver three cents ; how many
did they all give him ?
2. How many did Peter give more than Oliver ?
3. Frank had nine pears, and gave three of them
to Harry ; how many had he left ? and how many
more than Harry had he then ?
4. Dick had ten peaches, Harry twelve, and
Charles thirteen ; Dick gave three to Stephen,
Harry gave him six, and Charles gave him five ;
how many had Stephen ? and how many had each
left?
5. A boy had twenty apples, and gave them to
his companions as follows: to one he gave three;
to another, two ; to another, four ; and to another,
five : how many did he give away ? and how many
had he left ?
6. A boy gave to one of his companions eight
peaches ; to another, six ; to another, four ; and
kept two himself; how many had he at first ?
7. A boy went to the confectioner's and bought
three cakes of gingerbread, for which he gave a
cent apiece ; two bunns, for which he gave three
cents apiece ; one custard for four cents, and one
orange for six cents ; how many cents did he spend
for the whole ?
8. A boy, having twenty-five cents, bought one
quart of cherries for eight cents, one orange for
six cents, and gave away three cents ; how many
cents had he left ?
9. A boy bought a box for eighteen cents, and gave
eight cents to have it painted, and then sold it for thir-
ty-two cents ; how much did he gain by the bargain ?
SECT. 1.] ARITHMETIC. 21
10. A man bought a sleigh for seventeen dollars,
and gave nine dollars to have it repaired and paint-
ed, and then sold it for twenty-three dollars ; how
much did he lose by the bargain?
11. Eleven and two are how many?
1*2. Eleven and three are how many ?
13. Eleven and four are how many ?
14. Eleven and five are how many ?
15. Eleven and six are how many?
1C. Eleven and seven are how many ?
17. Eleven and eight are how many ?
18. Eleven and nine are how many ?
19. Eleven and ten are how many ?
20. Twelve and two are how many?
21. Twelve and three are how many?
22. Twelve and four are how many ?
23. Twelve and five are how many ?
24. Twelve and six are how many ?
25. Twelve and seven are how many ?
26. Twelve and eight are how many ?
27. Twelve and nine are how many ?
28. Twelve and ten are how many?
29. Thirteen and two are how many ?
30. Thirteen and three are how many ?
31. Thirteen and four are how many ?
32. Thirteen and five are how many?
33. Thirteen and six are how many ?
34. Thirteen and seven are how many ?
35. Fourteen and two are how many ?
36. Fourteen and three are how many ?
37. Fourteen and four are how many?
38. Fourteen and five are how many ?
39. Fourteen and six are how many?
40. Fifteen and two are how many ?
41. Fifteen and three are how many?
42. Fifteen and four are how many?
22 ARITHMETIC. [PART 1.
43. Fifteen and five are how many ?
44. Sixteen and two are how many ?
45. Sixteen and three are how many?
46. Sixteen and four are how many?
47. Seventeen and two are how many ?
48. Seventeen and three are how many ?
49. Eighteen and two are how many ?
E. 1. A man bought a sheep for nine dollars, and
to pay for it he gave five bushels of corn worth four
dollars, and the rest in money ; how much money
did he pay ?
2. If a barrel of flour is worth eight dollars, and
a hundred weight of sugar is worth twelve dollars,
how much more is the sugar worth than the flour ?
3. If a man had eleven dollars, and should buy
three bushels of corn for five dollars, how much
money would he have left ?
4. A man bought a firkin of butter for fifteen dol-
ars, but, it being damaged, he was willing to sell it
again for eight dollars less than he gave for it ; what
did he sell it for ?
5. A man bought three barrels of flour for eighteen
dollars, and sold it again for eleven dollars ; what
did he lose by the bargain ?
6. A man bought a keg of tobacco for thirteen
dollars, and sold it again for eighteen ; what did he
gain by the bargain ?
7. Five less two are how many ?
8. Seven less three are how many ?
9. Three less three are how many ?
10. Nine less three are how many?
11. Six less two are how many?
12. Seven less four are how many ?
13. Eight less three are how many ?
14. Five less four are how many ?
SECT. 1.] ARITHMETIC. 23
15. Seven less five are how many ?
16. Nine less five are how many ?
17. Eight less six are how many?
18. Eleven less two are how many ?
19. Twelve less four are how many 1
20. Ten less seven are how many ?
21. Thirteen less five are how many 1
22. Fourteen less eight are how many ?
23. Twelve less seven are how many ?
24. Seventeen less five are how many ?
25. Eighteen less ten are how many ?
26. Thirteen less seven are how many ?
27. Sixteen less seven are how many ?
28. Fifteen less seven are how many ?
29. Nineteen less six are how many ?
30. Eighteen less five are how many ?
31. Seventeen less eight are how many?
32. Fourteen less nine are how many ?
33. Sixteen less five are how many ?
34. Fifteen Jess eight are how many?
35. Fourteen less nine are how many ?
36. Sixteen less ten are howT many ?
37. Seventeen less nine are how many ?
38. Eighteen less seven are how many ?
F. 1. How many are nine and two? Nineteen
and two 1 Twenty-nine and two ? Thirty-nine and
two ? Forty-nine and two ? Fifty-nine and two ? Six-
ty-nine and two ? Seventy-nine and two ? Eighty-
nine and two ? Ninety-nine and two ?
2. How many are nine and three ? Nineteen
and three ? Twenty-nine and three ? Thirty-nine
and three ? Forty-nine and three 1 Fifty-nine and
three ? Sixty-nine and three ? Seventy-nine and
three ? Eighty-nine and three ? Ninety-nine and
three ?
3. How many are nine and four ? Nineteen and
24 ARITHMETIC. [PART 1.
four ? Twenty-nine and four ? Thirty-nine and four ?
Forty-nine and four? Fifty-nine and four? Sixty-
nine and four ? Seventy-nine and four ? Eighty-nine
and four ? Ninety-nine and four ?
4. How many are nine and five ? Nineteen and
five ? Twenty-nine and five ? Thirty-nine and five ?
Forty-nine and five ? Fifty-nine and five ? Sixty-
nine and five ? Seventy-nine and five ? Eighty-nine
and five ? Ninety-nine and five ?
5. How many are nine and six ? Nineteen and
six? Twenty-nine and six? Thirty-nine and six?
Forty-nine and six? Fifty-nine and six? Sixty-nine
and six ? Seventy-nine and six ? Eighty-nine and
six ? Ninety-nine and six ?
6. How many are nine and seven ? Nineteen and
seven ? Twenty-nine and seven ? Thirty-nine and
seven ? Forty-nine and seven ? Fifty-nine and seven ?
Sixty-nine and seven ? Seventy-nine and seven ?
Eighty-nine and seven ? Ninety-nine and seven ?
7. How many are nine and eight ? Nineteen and
eight ? Twenty-nine and eight ? Thirty-nine and
eight ? Forty-nine and eight ? Fifty-nine and eight ?
Sixty-nine and eight ? Seventy-nine and eight ?
Eighty-nine and eight ? Ninety-nine and eight ?
8. How many are nine and nine ? Nineteen and
nine ? Twenty-nine and nine ? Thirty-nine and
nine? Forty-nine and nine ? Fifty-nine and nine ?'
Sixty-nine and nine ? Seventy-nine and nine ?
Eighty-nine and nine ? Ninety-nine and nine ?
9. How many are nine and ten ? Nineteen and
ten ? Twenty-nine and ten ? Thirty-nine and ten ?
Forty-nine and ten ? Fifty-nine and ten ? Sixty-nine
and ten ? Seventy-nine and ten ? Eighty-nine and
ten ? Ninety-nine and ten ?
10. How many are eight and three ? Eighteen and
three ? Twenty-eight and three ? Thirty-eight and
three ? Forty-eight and three ? Fifty-eight and
SECT. 1.] ARITHMETIC. 25
three 1 Sixty-eight and three ? Seventy-eight and
three ? Eighty-eight and three ? Ninety-eight and
three ?
11. How many are eight and four? Eighteen
and four ? Twenty-eight and four ? Thirty-eight
and four ? Forty-eight and four 1 Fifty-eight and
four ? .Sixty-eight and four ? Seventy-eight and
four? Eighty-eight and four? Ninety-eight and
four?
12. How many are eight and five ? Eighteen and
five ? Twenty-eight and five ? Thirty-eight and
five ? Forty-eight and five ? Fifty-eight and five ?
Sixty-eight and five ?- Seventy-eight and five ?
Eighty-eight and five ? Ninety-eight and five ?
13. How many are eight and six? Eighteen and
six? Twenty-eight and six? Thirty-eight and six?
Forty-eight and six ? Fifty-eight and six ? Sixty-
eight and six ? Seventy-eight and six ? Eighty-eight
and six ? Ninety-eight and six ?.
14. How many are eight and seven ? Eighteen
and seven ? Twenty-eight and seven ? Thirty-eight
and seven ? Forty-eight and seven ? Fifty-eight and
seven ? Sixty-eight and seven ? Seventy-eight and
seven ? Eighty-eight and seven ? Ninety-eight and
seven ?
15. How many are eight and eight? Eighteen
anci eight ? Twenty-eight and eight ? Thirty-eight
and eight? Forty-eight and eight? Fifty-eight
and eight ? Sixty-eight and eight ? Seventy-eight
and eight ? Ninety-eight and eight ?
16. How many are eight and nine ? Eighteen and
nine ? Twenty-eight and nine ? Thirty-eight and
nine ? Forty-eight and nine ? Fifty-eight and nine ?
Sixty-eight and nine ? Seventy-eight and nine ?
Eighty-eight and nine? Ninety-eight and nine?
17. How many are seven and four ? Seventeen
and four ? Twenty-seven and four ? Thirty-seven
3
36 ARITHMETIC. [PART 1.
and four ? Forty-seven and four ? Fifty-seven and
four ? Sixty-seven and four ? Seventy-seven and
four ? Eighty-seven and four 1 Ninety-seven and
four ?
18. How many are seven and five ? Seventeen
and five ? Twenty-seven and five ? Thirty-seven
and five ? Forty-seven and five 1 Fifty-seven and
five ? Sixty-seven and five ? Seventy-seven and five ?
Eighty-seven and five ? Ninety-seven and five ?
19. How many are seven and six? Seventeen
and six ? Twenty-seven and six ? Thirty-seven and
six? Forty-seven and six ? Fifty-seven and six? Six-
ty-seven and six ? Seventy-seven and six ?' Eighty-
seven and six ? Ninety-seven and six ?
20. How many are seven and seven ? Seventeen
and seven ? Twenty-seven and seven ? Thirty-seven
and* seven ? Forty-seven and seven ? Fifty-seven and
% seven ? Sixty-seven and seven ? Seventy-seven and"
seven ? Eighty-seven and seven ? Ninety-seven and
seven ?
21. How many are seven and eight? Seventeen
and eight ? Twenty-seven and eight ? Thirty-seven
and eight ? Forty-seven and eight \ Fifty-seven and
eight ? Sixty-seven and eight ? Seventy-seven and
eight ? Eighty-seven and eight ? Ninety-seven and
eight?
22. How many are seven and nine ? Seventeen
and nine ? Twenty-seven and nine ? Thirty-seven
and nine ? Forty-seven and nine ? Fifty-seven and
nine ? Sixty-seven and nine ? Seventy-seven and
nine ? Eighty-seven and nine ? Ninety-seven and
nine?
23. How many are six and five ? Sixteen and five ?
Twenty-six and five ? Thirty-six and five 1 Forty-,
six and five? Fifty-six arid five? Sixty-six and five?
Seventy-six and five ? Eighty-six and five ? Ninety-
six and five 7
SECT. 1.] ARITHMETIC. 27
24. How many are six and six ? Sixteen and six ?
Twenty-six and six ? Thirty-six and six ? Forty-six
and six ? Fifty-six and six ? Sixty-six and six ? Sev-
enty-six and six? Eighty-six and six? Ninety-six
and six ?
25. How many are six and seven ? Sixteen and
seven ? Twenty-six and seven ? Thirty-six and sev-
en ? Forty-six and seven ? Fifty^six and seven 1 Six-
ty-six and seven ? Seventy-six and seven ? Eighty-
six and seven 1 Ninety-six and seven 1
26. How many are six and eight ? Sixteen and
eight ? Twenty-six and eight ? Thirty-six and eight 1
Forty-six and eight 1 Fifty-six and eight ? Sixty-six
and eight 1 Seventy-six and eight ? Eighty-six and
eight 1 Ninety-six and eight ?
27. How many are six and nine? Sixteen and
nine ? Twenty-six and nine ? Thirty-six and nine ?
Forty-six and nine? Fifty-six and nine? Sixty-six
and nine ? Seventy-six and nine ? Eighty-six and
nine ? Ninety-six and nine ?
28. How many are five and six ? Fifteen and six ?
Twenty-five and six ? Thirty-five and six ? Forty-
five and six? Fifty-five and six ? Sixty-five and six?
Seventy-five and six ? Eighty-five and six ? -Ninety-
five and six ?
29. How many are five and seven ? Fifteen and
seven ? Twenty-five and seven ? Thirty-five and
seven ? Forty-five and seven ? Fifty-five and seven ?
Sixty-five and seven ? Seventy-five and seven ?
Eighty-five and seven ? Ninety-five and seven ?
30. How many are five and eight ? Fifteen and
eight ? Twenty-five and eight ? Thirty-five and
eight? Forty-five and eight? Fifty-five and eight?
Sixty-five and eight ? Seventy-five and eight ? Eighty-
five and eight ? Ninety-five and eight ?
31. How many are five and nine ? Fifteen and
nine ? Twenty-five and nine ? Thirty-five and nine ?
28 ARITHMETIC. [PART 1
Forty-five and nine ? Fifty-five and nine ? Sixty-five
and nine ? Seventy-five and nine ? Eighty -five and
nine ? Ninety-five and nine ?
32. How many are four and seven ? Fourteen and
seven ? Twenty-four and seven ? Thirty-four and
seven ? Forty-four and seven ? Fifty-four and seven ?
Sixty-four arid seven ? Seventy-four and seven ?
Eighty-four and seven ? Ninety-four and seven '?
33. How many are four and eight ? Fourteen and
eight 1 Twenty-four and eight ? Thirty-four and
eight ? Forty-four and eight ? Fifty-four and eight ?
Sixty-four and eight 1 Seventy-four and eight ?
Eighty-four and eight? Ninety-four and eight I
34. How many are four and nine 1 Fourteen and
nine ? Twenty-four and nine ? Thirty-four and nine 1
Forty-four and nine? Fifty-four and nine? Sixty-
four and nine ? Seventy-four and nine ? Eighty-four
and nine ? Ninety-four and nine ?
35. How many are three and eight ? Thirteen
and eight ? Twenty-three and eight ? Thirty-three
and eight ? Forty-three and eight ? Fifty-three and
eight? Sixty-three and eight? Seventy-three and
eight ? Eighty-three and eight ? Ninety-three 'and
eight ?
36. How many are three and nine ? Thirteen and
nine ? Twenty-three and nine ? Thirty-three and
nine ? Forty-three and nine ? Fifty-three and nine ?
Sixty-three and nine ? Seventy-three and nine ?
Eighty-three and nine ? Ninety-three and nine ?
37. How many are two and nine ? Twelve and
nine ? Twenty-two and nine ? Thirty-two and nine ?
Forty-two and nine ? Fifty-two and nine ? Sixty-
two and nine ? Seventy-two and nine ? Eighty-two
and nine ? Ninety-two and nine ?
Cr. 1. A man bought a firkin of butter for nine
dollars, a keg of molasses for six dollars, and five
SECT. 1.] ARITHMETIC. 29
bushels of wheat for seven dollars ; how much did
he give for the whole 1 v
2. .A boy gave some apples to his companions ; to
one he gave seven, to another six, and to another
eight ; how many did he give to the whole ?
3. A man bought a cow for seventeen dollars, a
sheep for nine, and a calf for seven ; how much did
• he give for the whole ?
4. A drover bought sheep as follows ; of one man
he bought twenty-seven, of another eight, of anoth-
er ten, and of another five ; afterwards he sold
nine of them ; how many had he then ?
5. A lady bought a comb for thirty-seven cents,
some tape "for eight cents, some pins for ten cents,
some needles for six cents, and some thread for
six cents ; she gave seventy-five cents ; how much
change ought she to receive back ?
6. Eight, and nine, and six, are how many ?
7. Five, and seven, and three, are how many 1
8. Four, and three, less two, are how many ?
9. Seven, and five, less three, are how many ?
10. Sixteen, and nine, and three, are how many ?
11. Twenty-three and eight are how many?
12. Twenty-seven and five are how many ?
13. Twenty-five, less eight, are how many?
' 14. Thirty-two and seven, less nine, are how
many?
15. Thirty-eight, and six, and four, less seven,
are how many ?
16. Forty-four, and eight, and 'three, and seven,
are how many 1
17. Fifty-two, and six, and four, and five, and
three, are how many ?
18. Fifty-seven, and six, and three, and five, and
two, less eight, are how many ?
19. Sixty-three, and five, and four, and six, and
two, less seven, are how many ?
3*
30 ARITHMETIC. [PART 1.
20. Seventy-five, and six, and eight, and three,
and seven, and four, less nine*, are how many ?
21. Eighty-three, and six, and five, and two, and
seven, and nine, less four, are how many ?
22. Fifty-eight, rand ten, and five, and seven, and
three, and six, and four, less nine, are how many ?
23. Sixty-seven, and five, and e'ight, and nine,
and seven, less six, are how many ?
24. Seventy-four, and nine, and seven, and five,
and two, less six, are how many ?
25. Seventy-eight, and seven, and six, and two,
and five, and eight, less nine, are how many ?
26. Eighty-four, and seven, and six, and eight,
and five, less ten, are how many ?
*27. Forty-seven, and eight, and six, and two, and
four, and eight, and three, and seven, and ten, and
nine, less five, are how many ?
28. Thirty-five, and eight, and four, and six, and
three, and four, less eleven, are how many ?
29. Seventy, and ten, and six, and nine, ^.nd
seven, and two, and five, and eight, and nine, less
three, are how many ? ^*
H. 1. A man bought a cow for twenty-eight
dollars', and a sheep for four dollars, and a pig for
seven dollars ; how much did he give for the whole ?
2. James had twenty-seven cents; John gave
him four more, David seven, and George eleven,
and he bought nine cents' worth of cake; how
many cents had lie left ?
3. A man paid sixteen dollars to A, nine dollars
to B, seven dollars to C, ten dollars to D, six dol-
lars to E, four dollars to F, and had eight dollars
left ; how many had he at first ?
4. From Boston to Roxbury it is three miles ; from
Roxbury to Dedham, six miles ; from Dedham to Wai-
pole, eleven miles ; from Walpole to Wrentham., four
SECT. 2.] ARITHMETIC. 31
miles ; from Wrentham to Attleborough, four miles ;
from Attleborough to Pawtucket, nine miles; from
Pawtucket to Providence, four miles : how many
miles is it from Boston to Providence 1
5. One boy had fifteen nuts ; another boy gave
him seven ; another nine ; and another gave him
enough to make his number forty : how many did
the last boy give him ?
6. A l)oy had thirty-seven apples : he gave five to
one companion, and eight to another ; and when he
had given some to another, he had six left : how
many did he give to the last ?
7. A man owed fifty-six dollars : at one time he
paid sevenfeen dollars ; at another, eight ; at another,
five ; at another, seven ; at last he paid the rest of
the debt, wanting four dollars: how much .was the
last payment ?
8. Six men bought a horse for seventy dollars : the
first gave twenty-three dollars ; the second, fifteen ;
the third, twelve ; the fourth, nine ; the fifth, seven :
how much did the sixth give ?
9. A man bought a horse for forty-five dollars,
and paid fifteen dollars for keeping him ; he let him
enough to receive twenty dollars, and then sold him
for forty-three dollars : did he gain or lose by the
bargain 1 and how much ?
SECTION
A.. 1. What cost three yards of tape, at two cents
a yard ? *
2. What cost four apples, at two cents apiece ?
* The pupil should be made to observe, that three yards will cost
three times as much as one yard ; and say, If one yard cost two
cents, three yards will cost three times two cents. He should be
made to give this reason for the solution of Qfach question, varying
the number according to the question. *
32 ARITHMETIC. [PART 1.
3. What cost five peaches, at three cents apiece ?
4. What must you give for two oranges, at six
cents apiece?
5. What would be the price of three barrels of
cider, at three dollars a barrel ?
6. If one orange is worth three apples, how
many apples are four oranges worth ?
7. What are two barrels of flour worth, at five
dollars a barrel ?
8. What cost three yards of cloth, at- four dollars
a yard ?
9. What cost two pounds of raisins, at eight
cents a pound ?
10. What cost three lemons, at six cents apiece ?
11. If a man travel three miles in an hour, how
many miles will he travel in four hours ?
12. What will five pairs of shoes come to, at two
dollars a pair?
13. What is the price of seven yards of cloth, at
three dollars a yard ?
14. What is the value of two pounds of beef, at
seven cents a pound ?
15. If there are three feet in one yard, how
many feet are there in four yards ?
16. How many feet are there in seven yards ?
17. How many feet are there in six yards and two
feet?
* 18. If a man earn seven dollars in one week, how
much would he earn in five weeks ?
19. What cost seven hundred weight of sugar,
at nine dollars a hundred weight ?
20. What cost seven pounds of sugar, at ten cents
a pound ?
21. If one half yard of cloth cost three dollars,
what would three yards cost ?
22. If one quarter of a yard of cloth cost two
dollars, what is that a yard ?
SECT. 2.] ARITHMETIC. 33
23. How many yards of cloth are there in seren
pieces, each piece containing ten yards ?
24. What will five barrels of flour cost, at six
dollars a barrel 1
25. If a man can travel four miles in an hour,
how far can he travel in eight hours ?
26. If it take four bushels of wheat to make a
barrel of flour, how many bushels will it tak;e to
make seven barrels ?
H. 1. Two times one are how many ?*
2. Two times two are how many ? .
3. .Two times three are how many 1
4. Two times four are how many ?
5. Two times five are how many ?
6. Two times six are how many ?
7. Two times seven are how many ?
8. Two times eight are how many ?
9. Two times nine are how many ?
10. Two times ten are how many?
11. Three times one are how many?
12. Three times two are how many ?
13. Three times three are how many ?
14. Three times four are how many ?
15. Three times five are how many?
16. Three times six are how many ?
17. Three times seven are how many ?
18. Three times eight are how many ?
19. -Three times nine are how many ?
£0. Three times ten are how many ?
21. Four times one are how many ?
22. Four times two are how many ?
23. Four times three are how many ?
24. Four times four are how many?
25. Four times five are how many ?
26. Four times six are how many ?
* See the Key.
34 ARITHMETIC.
27. Four times seven are how many 1
28. Four times eight are how many ?
29. Four times nine are how many ?
30. Four times ten are how many ?
31. Five times one are how many?
32. Five times two are how many ?
33. Five times three are how many?
34. Five times four are how many ?
35. Five times five are how many ?
36. Five times six are how many ?
37. Five times seven are how many ?
38. Five times eight are how many ?
39. Five times nine are how many ?
40. Five times ten are how many?
41. Six times one are how many ?
42. Six times two are how many ?
43. Six times three are how many 1
44. Six times four are how many ?
45. Six times five are how many ?
46. Six times six are how many ?
47. Six times seven are how many ?
48. Six times eight are how many ?
49. Six times nine are how many ?
50. Six times ten are how many ?
51. Seven times one are how many?
52. Seven times two are how many ?
53. Seven times three are how many ?
54. Seven times four are how many?
55. Seven times five are how many ?
56. Seven times six are how many ?
57. Seven times seven are how many ?
58. Seven times eight are how many ?
59. Seven times nine are how many ?
60. Seven times ten are how many ?
61. Eight times one are how many?
62. Eight times two are how many ?
63. Eierht times three are how many ?
SECT. 2.1 ARITHMETIC. 35
64. Eight times four are how many ?
65. Eight times five are how many ?
66. Eight times six are how many?
67. Eight times seven are how many ?
68. Eight times eight are how many ?
69. Eight times nine are how many ?
70. Eight times ten are how many ?
71. Nine times one are how many ?
72. Nine times two are how many ?
73. Nine times three are how many ?
74. Nine times four are how many f
75. Nine times five are how many ?
76. Nine times six are how- many?
77. Nine times seven are how many ?
78. Nine times eight are how many ?
79. Nine times nine are how many?
80. Nine times ten are how many ?
81. Ten times one are how many?
82. Ten times two are how many ?
83. Ten times three are how many ?
84. Ten times four are how many ?
85. Ten times cfive are how many?
86. Ten times six are how many ?
87. Ten times seven are how many ?
88. Ten times eight are how many ?
89. Ten times nine are how many ?
90. Ten times ten are how 'many ?
CJ. 1. Two times two are how many times one?
2. Three times two are how many times one ?
3. Four times two are how many times one ?
4. Five times two are how many times one ?
5. Seven times two are how many ?
6. Nine times two are how many?
7. Six times two are how many ?
8. Eight times two are how many ?
9. Ten times two are how many ?
36 ARITHMETIC. IP*RT 1-
10. Two times three are how many ?
11. Three times three are how many?
12. Four times three are how many?
13. Five times three are how many?
„ 14. Six times three are how many ?
15. Eight times three are how many ?
16. Seven times three are how many ?
17. Ten times three are how many ?
18. Nine times three are how many ?
19. Two times four are how many?
20. Six times four are how many ?
21. Four times four are how many?
22. Seven times four are how many ?
23. Nine times four are how many ?
24. Three times four are how many?
25. Five times four are how many ?
26. Ten times four are how many ?
27. Eight times four are how many ?
28. Two times five are how many ?
29. Five times five are how many ?
30. Three times five are how many ?
31. Six times five are how many?
32. Two times six are how many ?
33. Four times five are how many ?
34. Seven times five are how many ?
35. Three times six are how many?
36. Seven times six are how many ?
37. Seven times seven are how many ?
38. Four times eight are how many ?
39. Six times seven are how many ?
40. Eight times nine are how many ?
41. Six times eight are how many?
42. Three times seven are how many ?
43. Four times nine are how many ?
44. Three times eight are how many ?
45. Six times six are how many ?
46. Six times nine are how many ?
SECT. 2.] ARITHMETIC. 37
47. Nine times five are how many ?
48. Four times six are how many ?
49. Two times nine are how many ?
50. Seven times nine are how many?
51. Nine times eight are how many?
52. Two times eight are how many ?
53. Three times ten are how many ?
54. Eight times seven are how many ?
55. Five times six are how many?
56. Five times eight are how many ?
57. Two times seven are how many ?
58. Two times six are how many ?
59. Eight times six are how many ?
60. Four times seven are how many ?
61. Eight times eight are how many ?
62. Ten times five are how many ?
63. Seven times ten are how many ?
64. Ten times ten are how many ?
65. Nine times six are how many 1
66. Five times nine are how many ?
67. Three times nine are how many ?
68. Nine times seven are how many ?
69. Five times ten are how many ?
70. Seven times eight are how many ?
71. Five times seven are how many ?
72. Ten times eight are how many ?
73. Ten times seven are how many ?
74. Nine times ten are how many ?
75. Eight times five are how many ?
76. Nine times nine are how many ?
77. Four times ten are how many ?
78. Ten times six are how many ?
79. Eight times ten are how many ?
80. Ten times nine are how many ?
JD. 1. What cost three yards of cloth, at five
dollars a yard ?
4
38 ARITHMETIC. [PART 1.
2. What cost four oranges, at six cents apiece ?
3. What cost seven barrels of cider, at three
dollars a barrel 1
4. How much do three barrels of beer come to,
at seven dollars a barrel ?
5. What cost four firkins of butter, at eight
dollars a firkin?
6. What do nine pounds of veal come to, at six
cents a pound ?
7. What cost six reams of paper, at five dollars
per ream ?
8. What cost eight pairs of shoes, at three dollars
a pair ?
"9. What is the value of nine yards of cloth, at
six dollars a yard ?
10. If a man travel five miles in an hour, how
many miles will he travel in nine hours?
11. There is an orchard consisting often rows of
trees, and nine trees in each row ; how many trees
are there in the orchard ?
12. On a chess-board there are eight rows of
squares, and eight squares in each row ; how many
squares are there on the board ?
13. In one penny there are four farthings ; how
many farthings are there in six pence ?
14. How many farthings are there in eight
pence ?
15. How many farthings are there in nine
pence ?
16. How many farthings are there in ten pence ?
17. In one shilling there are twelve pence ; how
many farthings are there in a shilling ?
18. In one pint there are four gills ; how many
gills are there in five pints ?
19. In one quart there are two pints ; how many
pints are there in six quarts ?
20. How many pints are there in three quarts ?
SECT. 2.] ARITHMETIC. 39
21. How many gills are there in six pints ?
22. How many gills are there in one quart ?
23. How many gills are there in three quarts ?
24. In one gallon there are four quarts; how
many quarts are there in three gallons 1
25. How many quarts are there in five gallons ?
26. How many quarts are there in seven gallons ?
27. How many pints are there in one gallon 1
28. How many pints are there in three gallons ?
29. How many gills are there in one gallon ?
30. How many gills are there in five quarts ?
31. How many gills are there in two gallons ?
32. A person bought two oranges, at six cents
apiece ; and seven lemons, at four cents apiece ; and
five pears, at two cents apiece : how much did the
whole come to ?
33. If one pint of gin cost eight cents, what will
one quart cost ?
34. If one gill of brandy cost four cents, what
will one quart cost ?
35. If one gill of beer cost two cents, what will
one gallon cost ?
36. If a stage runs seven miles in an hour, how
far will it run in nine hours ?
37. Two men start from the same place, and
travel different ways; one travels two miles in an
hour ; the other travels three miles in an hour :
how far apart will they be at the end of one hour '?
How far at the end of two hours ? How far at the
end of three hours ? How far at the end of four
hours ]
38. Two men start from the same place, and
travel the same way ; one travels at the rate of two
miles in an hour, the other four : how far apart
will they be in one hour ? How far in two hours ?
How far in four hours 1
39. If three men can do a piece of work in two
40 ARITHMETIC. [PART 1.
days, how many days would it take one 'man to
do it?
40. If four men can do a piece of work in ^five
days, how many days would it take one man to do it ?
41. If six men can do a piece of work in seven
days, how many men would it take to do it in one
day?
42. If a quantity of provisions will serve three
men five days, how many men would it serve one
day?
43. If a quantity of provisions will serve five
men seven days, how many days would it serve one
man ?
44. If fifteen dollars' worth of provisions will serve
eight men five days, how many days will it serve
one man ?
45. A man had a piece of work to perform which
seven men could do in nine days, but it was neces-
sary that the whole should be completed in one
day ; how many men must he employ ?
46. If the interest of one dollar is six cents a year,
what would be the interest of ten dollars for the
same time ?
47. If the interest of one dollar, is six cents for
one year, what would be the interest of it for two
years ? For three years ? For seven years ?
48. If a man can earn seven shillings in a day,
how many shillings will he earn in six days ?
49. If a man can earn eight dollars in a month,
how much can he earn in six months ?
50. At five dollars a week, what will nine weeks'
board come to ?
51. A lady bought three yards of cambric at two
dollars a yard, seven yards of silk for three dollars
a yard, five yards of riband for four dollars, and
some crape for two dollars ; she paid four ten-dollar
bills ; how much must she receive back again ?
SECT. 3.] ARITHMETIC. 41
SECTION III.
A. 1. How many apples, at one cent apiece, can
you buy for four cents ?
2. How many pears, at two cents apiece, can you
buy for four cents ?
3. How many peaches, at three cents apiece, can
you buy for six cents ?
4. How many apples, at two cents apiece, can
you buy for six cents ? How many for eight cents 1
5. How many pears, at three cents apiece, can
you buy for nine cents ? How many for twelve
cents ?
6. If you have eight apples to give to four boys,
how many can you give to each ?
7. If a man travel six miles in two hours, how
many miles does he travel in an hour ?
8. If a man travel three miles in an hour, how
many hours will it take him to travel nine miles 1
9. How many yards of cloth, at three dollars a
yard, can you buy for fifteen dollars?
10. If you had sixteen cents, how many cakes
could you buy at four cents apiece ?
11. If you had ten dollars, how much cloth could
you buy at five dollars a yard ?
12. If you had twelve apples to give to six of
your companions, how many would you give them
apiece 1
13. If a man can travel six miles in an hour, how
long will it take him to travel eighteen miles ?
14. If a man can travel five miles in an hour, how
long will it take him to travel twenty miles ?
15. In a certain orchard there are twenty-four
trees standing in rows, and there are three trees in
each row ; how many rows are there ?
4*
42 ARITHMETIC. [PART 1.
16. In an orcnard there are twenty-one trees, and
there are seven trees in each row ; how many rows
are there ?
17. A man paid twenty-seven dollars for some
sheep, and he gave nine dollars apiece for them ;
how many sheep did he buy ?
18. A man paid twenty-eight dollars for seven
barrels of cider ; how much did he give a barrel ?
19. At five cents apiece, how many oranges can
you buy for thirty cents ?
20. Twenty-five are how many times five 1
21. Thirty-two are how many times four? How
many times eight ?
22. Thirty-five are how many times seven ? How
many times five ?
23. Thirty-six are how many times six? How
many times nine ? How many times four ?
B. Remark. When any thing, or any number, is
divided into two equal parts, one of the parts is
called the half of the thing or the number.
1. If an apple is worth two cents, what is one
half of it worth ?
2. What is one half of two cents ?
Ans. One cent.
Q. Why?
A. Because, if you divide two cents into two
equal parts, one of the parts is one cent.
3. If you can buy a cake for two cents, how
much can you buy for one cent ?
4. One is what part of two?
Ans. One is the half part of two.
5. Two times one are how many times two ?
6. If you can buy one pear for two cents, how
many can you buy for three cents ?
7. Three are how many times two ?
Ans. Once two and one half of two.
SECT. 3.] ARITHMETIC. 43
8. Four are how many times two 1
9. If two shillings will buy one yard of cotton
cloth, how many yards will five shillings buy 1
10. Five are how many times two?
Ans. Two times two and half of two.
11. Six are how many times two ?
12. If two dollars will buy a yard of cloth, how
many yards will seven dollars buy ?
13. How many halves make a whole one?
14. Eight are how many times two ?
15. Nine are how many times two I
16. Ten are how many times two 1
Remark. When any thing, or any number, is di-
vided into three equal parts, one of those parts is
called the third part of the thing or number. When
it is divided into four equal parts, one part is called
the fourth part; and so on.
17. If a yard of cloth be worth three dollars, and
it be cut into three equal pieces, what will one of
the pieces be worth 1 that is, what will one third of
a yard be worth 1
18. What is a third of three?
19. Suppose the yard of cloth to be cut as before,
what will two pieces of it cost ? that is, what will
two thirds of a yard cost ?
20. What is two thirds of three ?
21. If three shillings will buy one bushel of corn,
what part of a bushel will one shilling buy ? What
part of a bushel will two shillings buy ?
22. One is what part of three ?
Ans. One is the third part of three; or one third
of three.
23. Two is what part of three ?
Ans. Two is two times the third part of three
or two thirds of three.*
* The instructor may use either of these expressions ; the former
will be most easily understood by the pupil 5 it would be well; there-
44 ARITHMETIC. [PART 1.
24. Three times one are how many times three ?
25. If you can buy a barrel of cider for three
dollars, how much can you buy for four dollars?
How much for five dollars ?
26. How many thirds make a whole one ?
27. Five are how many times three?
Ans. Once three, and two thirds of three.
28. Six are how many times three ?
29. If you can buy a barrel of fish for three dol-
lars, how much can you buy for seven dollars?
How much for eight dollars?
30. What do you understand by a third, and by
two thirds of any thing ?
For the answer, see remark after example 16th.
31. Eight are how many times three?
32. Nine are how many times three?
33. Ten are how many times three ?
34. Eleven are how many times three?
35. Twelve are how many times three?
36. If a yard of cloth be worth four dollars, and
it be cut into four equal parts, what will one of the
parts be worth ? that is, what is one fourth of it
worth ? What are two fourths of it worth ? What
are three fourths of it worth ?
37. If you can buy a barrel of cider for four dol-
lars, how much can you buy for one dollar ? How
much for two dollars ? How much for three dollars ?
38. What part of four is one ?
Ans. One is the fourth part of four.
39. What part of four is two ?
Ans. Two fourths of four.
40. What part of four is three ?
Ans. Three fourths of four.
41. How many fourths make a whole one?
42. If you can buy a bushel of corn for four shil-
fore, for the instructor to use the former frequently, though the latter
» used for the most part in this treatise.
SECT. 3.] ARITHMETIC. 45
lings, how much can you buy for five shillings ?
How much for six shillings 1 How much for seven
shillings ?
43. Five are how many times four ?
Ans. Once four, and one fourth of four.
44. Six are how many times four 1
Ans. Once four, and two fourths of four.
45. Seven are how many times four ?
Ans. Once four, and three fourths of four.
46. Eight are how many times four?
47. If four bushels of corn will buy one yard of
cloth, how many yards will nine bushels buy ? How
many yards will ten bushels buy ? How many yards
will eleven bushels buy ?
48. What do you understand by one fourth, two
fourths, or three fourths of any thing 1
See remark after example 16th.
49. Ten are how many times four ?
50. Eleven are how many times four 1
51. Twelve are how many times four?
52. Thirteen are how many times four ?
53. Fourteen are how many times four ?
54. Fifteen are how many times four ?
55. Sixteen are how many times four?
56. If a barrel of flour be worth five dollars, and
it be divided equally among five men, what will one
man's share be worth 1 that is, what is one fifth of
a barrel worth ? What are two fifths of it worth ?
What are three fifths of it worth ? What are four
fifths of it worth ?
57. If five dollars will buy one box of butter,
what part of a box will one dollar buy ? What part
will two dollars buy ? What part will three dollars
buy ? What part will four dollars buy ?
58. What part of five is one ?
Ans. One is the fifth part of five.
59. Two is what part of five ?
46 ARITHMETIC. [PART 1
Arts. Two fifths of five.
60. Three is what part of five ?
-4ns. Three fifths of five.
61. Four is what part of five ?
62. How many fifths make a whole one ?
63. If cherries are five cents a quart, how many
quarts can you buy for six cents ? How many for
seven cents 1 How many for eight cents ? How
many for nine cents ? How many for eleven cents ?
How many for thirteen cents ?
64. What do you understand by one fifth, two
fifths, &/c. of any thing ?
See remark after example 16th.
65. Seven are how many times five ?
Ans. Once five and two fifths of five.
66. Eight are how many times five ?
67. Nine are how many times five ?
68. Ten are how many times five ?
69. Eleven are how many times five ?
70. Twelve are how many times five ?
71. Thirteen are how many times five ?
72. Fourteen are how many times five ?
73. Fifteen are how many times five ?
74. If a barrel of beef cost six dollars, and it
were divided into six equal parts, what would one of
the parts be worth 1 that is, what is one sixth of it
worth? What are two sixths of it worth? What
are three sixths of it worth ? Four sixths ? Five
sixths ?
75. If fish is worth six dollars a barrel, what part
of a barrel will one dollar buy ? What part of a bar-
rel will two dollars buy ? Three dollars ? Four dol-
"ars? Five dollars?
76. What part of six is one ?
Ans. One is the sixth part of six.
77. What part of six is two ?
Ans. Two sixths of six.
SECT. 3.] ARITHMETIC. 47
78. Three is what part of six ?
Ans. Three sixths of six.
79. Four is what part of six ?
80. How many sixths make a whole one ?
81. How much rye, at six shillings^ bushel, can
you buy for seven shillings ? How much for eight
shillings? Nine shillings? Ten shillings? Eleven
shillings? Twelve shillings? Thirteen shillings?
Fifteen shillings? Seventeen shillings?
82. What do you understand by one sixth, two
sixths, &c. ?
83. Eight are how many times six?
Ans. One time six and two sixths of six.
84. Nine are how many times six ?
85. Ten are how many times six?
86. Eleven are how many times six ?
87. Twelve are how many times six?
88. Thirteen are how many times six?
89. Fourteen are how many times six?
90. Fifteen are how many times six?
91. If coal is worth seven dollars a chaldron, what
is one seventh of a chaldron worth ? What are two
sevenths of a chaldron worth ? Three sevenths ?
Four sevenths ? Five sevenths ? Six sevenths ?
92. At the rate of seven dollars a yard, how
much broadcloth can you buy for one dollar ? How
much for two dollars ? How much for three dollars ?
How much for four dollars? How much for five dol-
lars ? How much for six dollars ? How much for
eight dollars ? How much for ten dollars ? How
much for twelve dollars ? How much for fifteen
dollars ?
93. What part of seven is one?
Ans. One is one seventh of seven.
94. What part of seven is two?
Ans. Two sevenths of seven.
95. What part of seven is three ?
48 ARITHMETIC. [PART 1.
. 96. Four is what part of seven ?
97. Five is what part of seven ?
98. What do you understand by one seventh, two
sevenths, &c. of any thing ?
99. How many sevenths make a wnole one ?
100. Nine are how many times seven ?
101. Ten are how many times seven?
102. Eleven are how many times seven?
103. Twelve are how many times seven ?
104. Thirteen are how many times seven?
105. Fourteen are how many times seven ?
106. Fifteen are how many times seven?
107. Sixteen are how many times seven ?
108. When wheat is eight shillings a bushel,
what is one eighth of a bushel worth ? What are
two eighths of a bushel worth? What are three
eighths of a bushel worth ? What are four eighths
of a bushel worth ? Five eighths ? Six eighths ?
Seven eighths ?
109. When wood is eight dollars a cord, what
part of a cord can you buy for a dollar ? What part
of a cord can you buy for two dollars ? What part
for three dollars ? What part for four dollars ? What
part for five dollars? What part for six dollars?
What part for seven dollars? How much can you
buy for nine dollars ? How much for ten dollars ?
How much for eleven dollars ? How much for thir-
teen dollars ? How much for fifteen dollars ? How
much for nineteen dollars ?
1 10. What part of eight is one ?
111. What part of eight is two?
112. Three is what part of eight?
113. Four is what part of eight?
114. Five is what part of eight?
115. What do you understand by one eighth, two
eighths, &/c. of any number ?
116. Seven is what part of eight?
SECT. 3.] ARITHMETIC. 49
117. How many eighths make a whole one ?
118. Ten are how many times eight?
119. Eleven are how many times eight?
120. Twelve are how many times eight?
121. Thirteen are how many times eight?
122. Fourteen are how many times eight?
123. When sugar is nine dollars a hundred weight,
what is one ninth of a hundred weight worth ? What
are two ninths of a hundred weight worth ? Three
ninths ? Four ninths ? Five ninths ? Six ninths ?
Seven ninths ? Eight ninths ?
124. When rye is nine shillings a bushel, what
part of a bushel can you buy for one shilling ? What
part for two shillings ? What part for three shil-
lings ? For four shillings ? For five shillings ? For
six shillings? Seven shillings? Eight shillings?
How much for ten shillings ? For thirteen shillings ?
For fourteen shillings ? Sixteen shillings ? Twenty
shillings ?
125. What do you understand by one ninth, two
ninths, three ninths, &c., of any number?
126. Three is what part of nine ?
127. Four is what part of nine ?
128. Five is what part of nine ?
129. Seven is what part of nine ?
130. How many ninths make a whole one ?
131. Thirteen are how many times nine ?
132. Fifteen are how many times nine ?
133. Seventeen are how many times nine ?
134. When hay is ten dollars a ton, what is one
tenth of a ton worth ? What are two tenths of a
ton worth ? What are three tenths of a ton worth ?
Four tenths? Five tenths? Six tenths? Seven
tenths ? Eight tenths ? Nine tenths ?
135. When sugar is ten dollars a hundred weight,
what part of a hundred weight can you buy for one
dollar ? What part for two dollars ? What part for
50 ARITHMETIC. [PART 1
three dollars ? What part for four dollars ? What
part for five dollars'? Six dollars? Seven dollars?
Eight dollars? Nine dollars? How much can you
buy for eleven dollars ? Thirteen dollars ? Fifteen
dollars? Seventeen dollars ?
136. What do you understand by one tenth, two
tenths, three tenths, &c., of any thing?*
137. How many tenths make a whole one ? *
C. Instead of writing the names of numbers, it
is usual to express them by particular characters,
called jigures.
One is written 1
Two is written 2
Three is written 3
Four is written 4
Five is written 5
Six is written 6
Seven is written 7
Eight is written 8
Nine is written 9
Ten is written 10
1. Eleven times one are how many times 2 ?
2. Twelve are how many times 2 ? 3 ? 4 ?
3. Fourteen are how many times 2 ? 4 ? 3 ?
4. If you had fifteen cents, how many cakes could
you buy at 4 cents apiece ? How many at 2 cents
apiece ? How many at 3 cents apiece ? How many
at 5 cents apiece ?
5. Fifteen are how many times 4 ? 2 ? 3 ? 5 ?
6. Sixteen are how many times 5? 3? 6? 2?
7? 4?
7. Seventeen are how many times 6? 2? 7? 3?
5? 4?
8. Eighteen are how many times 4 ? 7 ? 9 ? 6 1
3? 2? 5? 8?
* These questions should frequently be put to the learners.
SECT. 3.] ARITHMETIC. 51
9. Nineteen are how many times 3? 7? 4? 5?
8? 6? 9? 2? 10?
10. Twenty are how many times 6? 2? 8? 3?
9? 4? 10? 5? 7?
11. Twenty-one are how many times 7? 3? 8?
2? 4? 6? 9? 5? 10?
12. Twenty-two are how many times 3? 8? 5?
4? 9? 6? 7? 10? 2?
13. If you had twenty-seven dollars, how much
cloth could you buy at 9 dollars a yard ? How much
at 6 dollars a yard? How much at 4 dollars a yard?
How much at 3 dollars a yard ? How much at 7
dollars a yard? How much at 8 dollars a yard? How
much at 5 dollars a yard ? How much at 10 dollars
a yard ?
14. Twenty-seven are how many times 9 ? 6 ? 4 ?
3? 7? 8? 5? 10?
15. Twenty-four are how many times 6 ? 8 ? 7 ?
5? 2? 10? 3? 4? 9?
16. Twenty-nine are how many times 3 ? 7 ? 5 ?
9? 6? 8? 4? 10?
17. Twenty-three are how many times 4 ? 2 ? 7 ?
8? 3? 9? 6? 5? 10? !
18. Twenty-five are how many times 3 ? 7 ? 2 1
6? 9? 4? 8? 5? 10?
19. Thirty are how many times 10? 2? 3? 7?
9? 6? 5? 4? 8?
20. Thirty-three are how many times 6? 8? 7?
4?- 9? 5? 10? 3?
21. Twenty-six are how many times 9? 4? 7? 3?
8? 5? 6? 10?
22. Thirty-five are how many times 5? 6? 3?
7? 9? 10? 4? 8?
23. Thirty-eight are how many times 8 ? 6? 3?
9? 5? 4? 7? 10?
24. Thirty-four are how many times 7? 3? 9?
10! 6? 8? 4? 5?
53 ARITHMETIC. [PART 1.
25. Thirty-six are how many times 8 1 9 ? 4 ? 57
3? 6? 7? 10?
26. Forty are how many times 8 ? 10? 6? 4? 3?
9? 5? 7?
27. For forty-seven cents, how many pounds of
meat can be bought at 6 cents a pound? How many
pounds at 8 cents? How many at 9 cents? How
many at 3 cents ? How many at 5 cents? How many
at 4 cents ? How many at 7 cents ? How many at 10
cents a pound ?
28. Forty-seven are how many times 6? 8? 9?
3? 5? 4? 7? 10?
29. Forty-three are how many times 9? 8? 7?
6? 4? 3? 5? 10?
30. Forty-five are how many times 10? 8? 3? 6?
4? 7? 5? 9?
31. Forty-nine are how many times 6? 10? 5?
9? 4? 8? 7?
32. Fifty-three are how many times 8 ? 5? 6? 4?
7? 9? 10?
33. Fifty-seven are how many times 9 ? 7? 10?
6? 5? 8? 4?
34. Fifty-five are how many times 6? 4? 8? 10?
9? 7? 5?
35. Forty-eight are how many times 7 ? 5 ? 9 ?
4? 6? 8? 10?
36. Fifty-four are how many times 5 ? 9 ? 6 ? 4 ?
7? 10? 8?
37. Forty-four are how many times 4? 6? 9? 71
5? 8? 10?
38. Fifty-eight are how many times 7? 6? 8? 4?
9? 5? 10?
39. Forty-six are how many times 8? 10? 4? 6?
9? 7? 5?
40. Fifty are how many times 9? 5? 4? 10? 8?
6? 7?
SECT. 3.] ARITHMETIC. 53
41. Fifty-nine are how many times 4 ? 8? 7 ? 6?
10? 9? 5?
42. Sixty-four are how many times 77578? 10?
6? 9?
43. Sixty-eight are how many times 6 ? 8? 9? 7?
10? 5?
44. Fifty-two are how many times 4 ? 6? 8? 10?
5? 7? 9?
45. Sixty-three are how many times 5? 4? 6?
10? 9? 7? 8?
46. Sixty-two are how many times 4? 10? 9? 7?
8? 5? 6?
47. Seventy-three are how many times 10? 7?
8? 6? 5? 9?
48. Seventy-five are how many times 7? 8? 10?
5? 6? 9?
49. If you had sixty-seven dollars, how many
barrels of flour could you buy at 5 dollars a barrel ?
How many at 7 dollars a barrel ? How many at 6
dollars a barrel ? How many at 8 dollars a barrel ?
How many at 10 dollars a barrel ? How many at 9
dollars a barrel ?
50. Sixty-seven are how many times 5? 7? 6?
8? 10? 9?
51. Seventy-four are how many times 10 ? 7? 8?
6? 5? 9?
52. Sixty are how many times 9 ? 10? 6? 4? 7?
5? 8?
53. Seventy-two are how many times 5 ? 7 ? 6 ?
8? 10? 9?
54. Sixty-five are how many times 5 ? 10 ? 8 ? 6 ?
7? 9?
55. Sixty-one are how many times 4? 5? 7? 6?
8? 10? 9?
56. Seventy-nine are how many times 10 ? 9 ? 8 ?
7? 6? 5?
5*
54 ARITHMETIC. [PART 1.
57. Seventy are how many times 9? 5? 6? 8?
7? 10?
58. Eighty-two are how many times 10? 7? 8?
6? 9?
59. Sixty-six are how many times 9 ? 5? 6? 7?
10? 8?
60. Eighty are how many times 10? 7? 6? 8?
9?
61. Sixty-nine are how many times 9 ? 5? 7? 10?
8? 6?
62. Eighty-one are how many times 10 ? 6 ? 8 ?
7? 9?
63. Seventy-six are how many times 9 ? 5 ? 10 ?
6? 7? 8?
64. Eighty-three are how many times 10 ? 6 ? 7 1
9? 8?
65. Seventy-one are how many times 9 ? 5 ? 7 ?
6? 8? 10?
66. Eighty-four are how many times 10 ? 6? 8?
9? 7?
67. Seventy-seven are how many times 9? 7? 5?
10? 8? 6?
68. Eighty-five are how many times 10? 8? 7?
6? 9?
69. Ninety are how many times 9? 10? 6? 7?
8?
70. Eighty-six are how many times 10? 9? 6?
7? 8?
71. Ninety-four are how many times 9 ? 10 ? 8 ?
6? 7?
72. Eighty-seven are how many times 10 ? 9 ? 7 ?
6? 8?
73. Ninety-two are how many times 9 ? 10 ? 6 ?
7? 8?
74. Eighty-eight are how many times 10 ? 9 ? 8 ?
6? 7?
SECT. 3.] ARITHMETIC. 55
75. Ninety-five are Jiow many times 9? 10? 6?
8? 7?
76. Eighty-nine are how many times 10? 9? 6?
7? 8?
77. Ninety-eight are how many times 10 ? 9 ? 8 ?
6? 7?
78. Ninety-four are how many times 9? 10? 6?
8? 7?
79. One hundred are how many times 6 ? 10 ? 8 ?
9? 7?
80. Ninety-three are how many times 10 ? 6 ? 9 ?
8? 7?
81. Ninety-nine are how many times 7? 10? 8?
6? 9?
82. Ninety-six are how many times 9 ? 8 ? 7 ? 6 ?
10?
83. Ninety-seven are how many times 10? 9? 6?
'7? 8?
D. 1. If an orange is worth 3 apples, how many
oranges are 15 apples worth?
2. In 8 pints, how many quarts?
3. In 8 gills, how many pints?
4. If you divide twelve apples equally among
three boys, how many would you give them apiece ?
5. How many hours would it take you to travel
10 miles, if you travel three miles in an hour ?
6. How many pence are there in eight farthings ?
7. How many pence are there in twelve far-
things ?
8. How many pence are there in seventeen far-
things ?
9. How many gallons are there in ten quarts ?
10. How much broadcloth, at 6 dollars a yard,
can you buy for seventeen dollars?
11. How many pounds of raisins, at 8 cents a
pound, can you buy for twenty-five cents?
56 ARITHMETIC. [PART 1.
12. In twenty-eight shillings, how many dollars?*
13. In twenty-eight farthings, how many pence?
14. How many barrels of flour, at 7 dollars a bar-
rel, can you buy for thirty-four dollars?
15. How many reams of paper, at five dollars a
ream, can you buy for thirty-seven dollars?
1(3. In thirty-four gills, how many pints?
17. In twenty-seven quarts, how many gallons?
18. If an orange is worth six apples, how many
oranges can you buy for forty apples?
19. Thirty-six shillings are how many dollars?
20. A man bought thirty apples at the rate of 3
for a cent; how many cents did they come to?
21. A laborer engaged to work 8 months for
ninety-six dollars ; how much did he receive for a
month ? How much a week, allowing 4 weeks to
the month ? How many shillings a day, allowing 6
working days to the week ?
22. If wine is worth twenty cents a pint, what is
1 gill worth ?
23. If you can buy a bushel t of apples for forty
cents, what is the price of a peck ?
24. If you buy a bushel of pears for forty-eight
cents, what will be the price of half a peck, at the
same rate?
25. Four men bought a horse for forty-eight dol-
lars ; what did each man pay ?
26. Five men bought a horse for seventy-five dol-
lars, and sold him again for forty dollars ; what did
each man lose by the bargain ?
27. A man gave sixty-three cents for a horse to
ride nine miles ; what was the price of one mile's
ride?
28. A man hired a horse to ride, and agreed to
give 8 cents a mile ; he paid fifty-six cents ; how
many miles did he ride ?
* 1 dollar is 6 shillings. 1 1 bushel is 4 nerk**
SECT. 3.] ARITHMETIC. 57
29. A man had forty-two dollars, which he paid
for wood at 7 dollars a cord ; how many cords did
he buy ?
30. Two boys are forty-eight rods apart, and fxrth
running the same way ; but the hindermost boy
gains upon the other 3 rods in a minute ; in how
many minutes will he overtake the foremost boy ?
31. There is a vessel containing sixty-three
gallons of wine ; it has a pipe which discharges 7
gallon^ in an hour ; how many hours will it take
to empty the vessel ?
32. There is a vessel containing eighty-seven
gallons, and by a cock ten gallons will run into it in
an hour; in how many hours will the vessel be
filled?
33. If one man can do a piece of work in thirty
days, in how many days can 3 men do it ? In how
many days can 5 men do it ?
34. If you wish to put sixty-four pounds of but-
ter into 8 boxes, how many pounds would you put
into each box?
35. If you had seventy-two pounds of butter, which
you wished to put into boxes containing 8 pounds
each, how many boxes would it take ?
36. If a man can perform a journey in thirty-six
hours, how many days will it take him to do it when
the days are nine hours long ?
37. If a man can do a piece of work in forty-eight
hours, how many days would it take him to do it, if
he works twelve hours in a day ?
58 ARITHMETIC. [PART 1.
SECTION IV.
A. 1. AT two cents a yard, what will 3 yards and
one half of a yard of tape cost ?
2. 3 times 2, and one half of 2 are how many?
3. At three dollars a yard, what will 4 yards and 1
third of a yard of cloth cost ?
4. 4 times 3, and 1 third of three are how many ?
5. At 3 dollars a barrel, what will 3 barrels and 2
thirds of a barrel of cider cost ?
6. 3 times 3, and two thirds of 3 are how many ?
7. If a man earn 4 dollars in a week, how many
dollars will he earn in 3 weeks and 1 fourth of a.
week?
8. 3 times 4, and 1 fourth of 4 are how many ?
9. If a yard of cloth cost 4 dollars, what will 5
yards and 3 fourths of a yard cost ?
10. 5 times 4, and 3 fourths of 4 are how many?
11. If a man spend five dollars in a week, how
many dollars will he spend in 3 weeks and 1 fifth of
a week ? How much in 5 weeks and 2 fifths of a
week ?
12. 3 times 5, and 1 fifth of 5 are how many ?
13. 5 times 5, and 2 fifths of 5 are how many ?
14. 6 times 5, and 3 fifths of 5 are how many ?
15. If beer is worth six dollars a barrel, what
would 4 barrels and 1 sixth of a barrel cost ? How
much would 7 barrels and 5 sixths of a barrel cost ?
16. 4 times 6, and 1 sixth of 6 are how many ?
17. 7 times 6, and 5 sixths of 6 are how many ?
' 18. At 7 dollars a barrel, what will 3 barrels and
1 seventh of a barrel of flour cost? What will 5
barrels and 2 sevenths of a barrel cost ?
19. 3 times 7, and 1 seventh of 7 are how many ?
20. 5 times 7, and 2 sevenths of 7 are how many ?
SECT. 4.] ARITHMETIC. 59
21. 8 times 5, and 4 fifths of 5 are how many ?
22. 8 times 6, and 3 sixths of 6 are how many?
23. At 8 dollars a yard, what will 4 yards and 1
eighth of a yard of broadcloth cost ?
24. 4 times 8, and 1 eighth of 8 are how many ?
25. 2 times 7, and 3 sevenths of 7 are how many 1
26. 8 times 7, and 4 sevenths of 7 are how many 1
27. 9 times 7, and 6 sevenths of 7 are how many ?
28. 3 times 8, and 5 eighths of 8 are how many ?
29. 9 times 8, and 7 eighths of 8 are how
many ?
30. If a hundred weight of sugar cost 9 dollars,
what will 2 hundred weight and 1 ninth of a hun-
dred weight cost 1 What will 5 hundred weight and
2 ninths of a hundred weight cost ?
31. 2 times 9, and 1 ninth of 9 are how many ?
32. 5 times 9, and 2 ninths of 9 are how many ?
33. 6 times 9, and 4 ninths of 9 are how many ?
34. 2 times 10, and 3 tenths of 10 are how many?
35. 7 times 9, and 7 ninths of 9 are how many ?
36. 5 times 10, and 4 tenths of 10 are how many ?
37. 8 times 9, and 5 ninths of 9 are how many ?
38. 4 times 10, and 7 tenths of 10 are how many?
39. 6 times 10, and 9 tenths of 10 are how many?
15. 1. A man bought 2 oranges at 6 cents apiece ;
how many cents did they come to? He paid for
them with cherries at 4 cents a pint; how many
pints did it take ?
2. 2 times 6 are how many times 4?
3. A man bought 3 yards of cloth at 4 dollars
a yard; how many dollars did it come to? How
much flour at 6 dollars a barrel would it take to
pay for it ?
4. 3 times 4 are how many times 6?
5. A man bought 4 peaches at 3 cents apiece ;
how many cents did they come to ? He paid for
60 ARITHMETIC. [PART 1.
them with pears at 2 cents apiece ; how many pears
did it take 1
6. 4 times 3 are how many times 2 ?
7. Bought 2 hundred weight of sugar at 9 dol-
lars a hundred weight, and paid for it with wood
at 6 dollars a cord ; how many cords did it take ?
8. 2 times 9 are how many times 6 ?
9. Bought 3 barrels of flour at 8 dollars a barrel,
and paid for it with cider at 4 dollars a barrel ; how
many barrels did it take ?
10. 3 times 8 are how many times 4?
11. 12 times 3 are how many times 5?
12. 6 times 4 are how many times 8?
13. 3 times 10 are how many times 6 ?
14. 4 times 9 are how many times 6 ?
15. How much flannel, worth 4 shillings a yard,, «
must be given for 3 yards of silk, worth 5 shillings a
yard?
16. 3 times 5 are how many times 4 ?
17. 2 times 7 are how many times 3? 5? 4?
18. 4 times 5 are how many times 3 ? 6? 7?
19. 3 times 7 are how many times 4?5?6?8?
9?
20. Bought 2 kegs and 2 sevenths of a keg of
tobacco at 7 dollars a keg, and paid for it with
wood at 4 dollars a cord ; how many cords did it
take ? How much butter, at 3 dollars a box, would
it take to pay for it ?
21. 2 times 7, and 2 sevenths of 7 are how many
times 4? 3? 5? 6? 8?
22. Bought 3 bushels and 3 fifths of a bushel of
corn at 5 shillings a bushel, and paid for it with
wheat at 6 shillings a bushel ; how many bushels of
wheat did it take ?
23. 3 times 5, and 3 fifths of five are how many
timesG? 9? 4? 7? 3? 8?
24. How much sugar, that is 8 dollars a hundred
SECT. 4.] ARITHMETIC. 61
weight, can be bought for 4 cords and 2 sevenths of
a cord of wood, at 7 dollars a cord ?
25. 4 times 7, and 2 sevenths of 7 are how many
times 6? 8? 5? 3? 9? 10?
26. 5 times 5, and 3 fifths of 5 are how many
times 4? 8? 9? 7? 10? 3? 6?
27. 6 times 7, and 3 sevenths of 7 are how many
times 9? 4? 5? 8? 10?
28. 5 times 8, and 3 eighths of 8 are how many
times 6? 9? 4? 7? 10?
29. 7 times 8, and 5 eighths of 8 are how many
times 9? 6? 10? 4? 5?
30. 5 times 9, and 4 ninths of 9 are how many
times 7? 8? 6? 10? 4?
31. 7 times 9, and 7 ninths of 9 are how many
times 6? 8? 10? 5? 4?
32. 6 times 10, and 3 tenths of 10 are how many
times 7? 5? 4? 9? 8?
33. 8 times 10, and 4 tenths of 10 are how many
times 6? 7? 9?
34. 8 times 9, and 3 ninths of 9 are how many
times 6? 10? 7?
. C. 1. Bought 4 bushels of apples, at 3 shillings a
bushel ; how many dollars did they come to ?
2. How many apples, at 2 cents apiece, must you
give for 2 lemons at 4 cents apiece ?
3. How many pears, at 3 cents apiece, must you
give for 3 oranges at 5 cents apiece ?
4. How many barrels of cider, at 3 dollars a bar-
rel, must be given for 5 boxes of butter, at 4 dollars
a box?
5. A man bought 4 yards of broadcloth, at 7 dol-
lars a yard, and paid for it with flour, at 5 dollars a
barrel ; how many barrels did he give ?
6. If 2 apples cost 4 cents, what would 4 apples
cost?
6
62 ARITHMETIC. [PART 1.
7. If 3 apples are worth 6 cents, how many ap-
ples must you give for 8 pears, that are worth 3
cents apiece ?
8. James had 8 oranges that were worth 5 cents
apiece, and George had 5 quarts of cherries that
were worth 6 cents a quart, which he gave to James
for a part of his oranges ; how many oranges did he
buy, and how many had James left?
9. Bought 8 yards of cloth, at 9 shillings a yard ;
how many dollars did it come to?
10. Bought 5 bushels and 3 sevenths of a bushel
of salt, at 7 shillings a bushel ; how many' dollars did
it come to ?
11. Bought 9 boxes and 2 thirds of a box of rai-
sins for 3 dollars a box, and paid for it with cider
at 4 dollars a barrel ; how many barrels did it take?
12. Bought 8 pounds and 4 sevenths of a pound
of opium at 7 dollars a pound, and paid for it witli
cloth at 5 dollars a yard; how many yards did it
take?
13. Bought 6 chaldrons and 4 ninths of a chaldron
of coal at 9 dollars a chaldron, and paid for it with
oranges at 5 dollars a box ; how many boxes did it
take?
14. Bought 7 cases and 5 sixths of a case of
Florence oil at 6 dollars a case, and paid for it
with sheet lead at 7 dollars a hundred weight ;
how many hundred weight did it take ?
15. How many dozen of eggs, at 8 cents a dozen,
must you give for 7 pounds of sugar, at 10 cents a
pound ?
16. How much barley, at 3 shillings a bushel,
must be given for 8 bushels of wheat, at 7 shillings
a bushel 1
17. How much cloth, at 4 shillings a yard, must
be given for a firkin of butter, worth 8 dollars ?
18. How much cloth, at 5 shillings a yard, can be
SECT. 5.] ARITHMETIC. 63
bought for 2 reams of paper, at 5 dollars a ream'?
19. How much wheat, at 7 shillings a bushel, can
be bought for 2 barrels of cider, at 4 dollars and a
half a barrel ?
20. How long would it take a man to lay up 10
dollars, if he saves 4 shillings a day?
21. If a man earn 8 shillings a day, how many
dollars would he earn in 10 days ?
22. A man bought twenty pears at the rate of 2
for 3 cents ; how much did they come to ?
23. How many eggs, at the rate of 3 for 5 cents,
can you buy for thirty cents 1
24. A man hired a laborer, and agreed to give him
5 dollars for every 3 days' work ; how much did he
give him a week, there being 6 working days in a
week ? How much was it a month, allowing 4 weeks
to the month ?
25. If a man receives 5 dollars for 3 days' work,
how many shillings is that a day ?
26. 5 men bought a horse for sixty-three dollars,
and paid two dollars a week for keeping him; at
the end of 8 weeks, they sold him for fifty-four
dollars ; how much did each man lose by the bar-
gain ?
SECTION V.
A. 1. JAMES had 4 apples, and John had half as
many ; how many had he ?
2. If an orange cost 6 cents, and an apple half as
much, how much does the apple cost?
3. If you divide 8 apples equally between two
boys, what part of them must each have ?
Ans. One half of them.
C4 ARITHMETIC. [PART 1.
4. What is 1 half of 8?
5. If you divide 8 apples equally among 4 boys,
what part of them must each have?
Ans. One fourth of them.
6. What is 1 fourth of 8?
7. If you divide 6 oranges equally among 3 boys,
what part of them must 1 boy have ?
8. What is 1 third of 6?
9. If 4 yards of cloth cost 8 dollars, what part of
8 dollars would 1 yard cost? What part of 8 dollars
would 2 yards cost ? What part of 8 dollars would 3
yards cost?
10. What is 1 fourth of 8 ? What is 2 fourths of
8? What is 3 fourths of 8?
11. If 6 yards of cloth will make 3 coats, what
part of 6 yards will make 1 coat ? What part of 6
yards will make 2 coats ?
12. What is one third of 6 ? What is two thirds
of6?
13. If 3 barrels of cider cost 9 dollars, what part
of 9 dollars will 1 barrel cost ? What part of 9 dol-
lars will 2 barrels cost ?
14. What is 1 third of 9? What is 2 thirds of 9?
15. If 2 yards of cloth cost 10 dollars, what part
of 10 dollars will 1 yard cost? What part of 10 dol-
lars will 3 yards cost ?
16. What is 1 half of 10? What is 3 halves of
10?
17. If 2 barrels of flour cost twelve dollars, what
part of twelve dollars will one barrel cost? What
part of twelve dollars will 3 barrels cost ? What
part of twelve dollars will 5 barrels cost ?
18. What is 1 half of twelve ? What is 3 halves
of 12 ? What is 5 halves of twelve ?
19. If 4 barrels of cider cost twelve dollars, what
part of twelve dollars will 1 barrel cost? What
part of twelve dollars will 3 barrels cost? What
SECT. 5.] ARITHMETIC. 65
part of twelve dollars will 5 barrels cost? What
part of twelve dollars will 7 barrels cost?
20. What is 1 fourth of twelve ? What is 2 fourths
of twelve? 3 fourths? 5 fourths? 7 fourths?
21. If 3 oranges cost twelve cents, what part of
twelve cents wrill 1 orange cost ? What part of twelve
cents will 2 oranges cost? What part of twelve
cents will 4 oranges cost? What part of twelve
cents will 5 oranges cost? 7 oranges? 10 oranges?
22. What is 1 third of twelve ? 2 thirds 1 4 thirds ?
5 thirds? 7 thirds? 10 thirds?
23. If 5 bushels of wheat cost 10 dollars, what
part, of 10 dollars will 1 bushel cost? What part of
10 dollars will 2 bushels cost ? 3 bushels ? 4 bushels ?
6 bushels ? 7 bushels ?
24. What is 1 fifth of 10? 2 fifths? 3 fifths? 4
fifths ? 6 fifths ? 7 fifths ?
25. What is 1 half of fourteen? 1 seventh? 2
sevenths ? 3 sevenths ? 5 sevenths ?
26. What is 1 third of fifteen? 2 thirds? 1 fifth?
2 fifths ? 3 fifths ? 4 fifths ?
27. What is 1 half of sixteen? 3 fourths? 1 eighth ?
3 eighths ? 5 eighths ? 7 eighths ? 1 sixteenth ?
28. What is 1 half of eighteen ? 2 thirds? 1 sixth?
5 sixths ? 1 ninth ? 2 ninths ? 4 ninths ? 5 ninths ?
7 ninths ? 8 ninths ? 1 eighteenth ? 5 eighteenths ?
29. What is 1 half of twenty ? 1 fourth ? 3 fourths ?
1 fifth? 3 fifths? 2 fifths? 4 fifths? 1 twentieth?
3 twentieths ? 7 twentieths ?
30. What is 1 third of twenty-one ? 2 thirds ? 1
seventh ? 5 sevenths ? 3 sevenths ? 6 sevenths ? 4
sevenths ?
31. What is 1 half of twenty-two? 1 eleventh?
3 elevenths? 5 elevenths? 7 elevenths? 10 elev-
enths ?
32. What is 1 half of twenty-four ? 1 third? 2
thirds? 1 fourth? 3 fourths ? 1 sixth ? 5 sixths? 1
6*
66 ARITHMETIC. [PART 1.
eighth? 3 eighths? 7 eighths? 1 twelfth? 5 twelfths?
7 twelfths ?
33. What is 1 half of twenty-six ?
34. What are 2 thirds of twenty-seven ? 2 ninths ?
4 ninths ? 5 ninths ? 8 ninths ? 7 ninths ?
35. What is 1 half of twenty-eight ? 3 fourths ? 2
sevenths? 5 sevenths? 3 sevenths?
36. What is 1 sixth of thirty? 3 fifths? 7 tenths?
2 thirds? 1 half?
37. What is 1 half of thirty-two? 3 fourths,? 5
eighths? 1 sixteenth?
38. What is 1 half of thirty-four ?
39. What is 5 sixths of thirty-six? 2 thirds? 1
half? 7 twelfths? 4 ninths? 3 fourths?
40. What is 1 half of thirty-eight ?
41. What is 1 third of thirty-nine ?
42. What is 3 fourths of forty ? 7 eighths ? 9
tenths? 1 half?
43. What is 3 sevenths of forty-two? 5 sixths?
B. 1. A boy, having twelve apples, kept 1 fourth
of them himself, and divided the other 3 fourths of
them equally among 4 of his companions ; how many
did he give them apiece ?
2. 3 fourths of twelve are how many times 4 ?
3. A man, having fourteen bushels of grain, di-
vided 5 sevenths of it equally among 3 men ; how
much did he give them apiece ?
4. 5 sevenths of fourteen are how many times 3?
5. A man, having fifteen shillings, gave away 4
fifths of it ; how many dollars did he give away ?
6. 4 fifths of fifteen are how many times 6 ?
7. A man, having twenty-one cents, paid away 6
sevenths of them for oranges at 5 cents apiece ; how
many oranges did he buy ?
8. 6 sevenths of twenty-one are how many times
5?
SECT. 5.] ARITHMETIC. 67
9. A man bought a piece of cloth for twenty-four
shillings, and sold it again for 5 thirds of what he
gave for it; how many dollars did he sell it for?
10. 5 thirds of twenty-four are how many times
6?
11. 7 fifths of twenty are how many times 6 ?
12. 5 sevenths of thirty-five are how many times
8?
13. 7 sixths of thirty-six are how many times 5?
14. 9 sevenths of forty two are how many times
8?
15. 7 ninths of forty-five are how many times 6 ?
16. 5 sixths of forty-eight are how many times 7 1
17. 6 fifths of fifty are how many times 9?
18. 8 ninths of fifty- four are how many times 5 ?
19. 9 sevenths of fifty-six are how many times
10?
20. 7 sixths of sixty are how many times 8 ?
21. 9 sevenths of sixty-three are how many times
7?
22. 10 eighths of sixty-four are how many times
9?
23. 6 eighths of seventy-two are how many times
7?
24. 4 sevenths of eighty-four are how many times
9?
25. 7 tenths of ninety are how many times 8 ?
C. 1. Charles had 6 apples, and gave 1 third of
them to John ; how many did he give him ?
2. Albert had 9 cents, and spent 2 thirds of them ;
how many had he left?
3. James had 10 pears, and gave 1 half of them
to one of his companions, and 2 fifths of them to
another ; how many did he give away ?
4. If 3 yards of cloth cost 6 dollars, what is that
a yard 1
68 ARITHMETIC. [PART 1.
5. If 4 yards of cloth cost twelve dollars, what
will 2 yards cost ?
6. If 9 apples cost eighteen cents, what will 3
apples cost ?
7. If 3 oranges cost 18 cents, what will 2 cost ?
8. James had twenty-five cents, and he gave 4
fifths of them for 10 apples ; how much did he give
for all the apples ? how much apiece 1
9. A man had thirty dollars, and gave 5 sixths of
them for 8 yards of cloth ; how much did he give
a yard ?
10. A man had forty yards of cloth, and sold 3
fifths of it for twenty-four dollars ; what was that a
yard?
11. A man had forty-two barrels of flour, and
sold 2 sevenths of it for 6 dollars a barrel ; how
much did it come to ?
12. A boy had sixty-three nuts, and divided 4
sevenths of them equally among six of his compan-
ions ; how many did he give them apiece ?
13. If 4 yards of cloth cost twelve dollars, what
will 3 yards cost ?
14. If 5 oranges cost twenty-five cents, what will
3 cost ?
15. If 3 oranges cost fifteen cents, what will 7 cost ?
16. If 3 barrels of cider cost twelve dollars, what
will 10 barrels cost?
17. If 7 pounds of flour cost thirty-five cents,
what will 9 pounds cost ?
18. If 5 firkins of butter cost forty dollars, what
will 3 firkins cost ?
19. If 2 men can do a piece of work in 6 days,
how long would it take 4 men to do the same
work ?
20. If 6 men can do a piece of work in twelve
days, in how many days will 3 men do the same
work?
SECT. 5.] ARITHMETIC. 69
21. If 3 men can do a piece of work in twelve days,
in how many days will 4 men do the same work ?
22. If 2 cocks of a certain size will empty a cis-
tern in 6 hours, in how long a time will 3 cocks of
the same size empty it ?
23. Three men, setting out on a journey, pur-
chased 5 loaves of bread apiece, but before they had
eaten any of it, two other men joined them, and
they agreed to share the bread equally among the
whole ; how many loaves did they have apiece ?
24. If 4 barrels of flour cost twenty-four dollars,
what would 7 barrels cost ? How much cider, at 3
dollars a barrel, would 7 barrels of flour buy 1
25. A man bought a quantity of flour for fifty-
four dollars ; and another man gave him 9 yards of
cloth for 5 sixths of it ; what was the cloth worth
a yard 1
26. If 9 yards of cloth cost fifty-four dollars, how
many boxes of butter, at 4 dollars a box, would 5
yards of the same cloth buy 1
27. Bought 8 firkins of butter for seventy-two
dollars, and gave 6 of them for 7 yards of cloth ;
what was a yard of the cloth worth ?
28. A man bought 6 barrels of flour for sixty
dollars, and gave 4 barrels of it for cider at 5 dol-
lars a barrel ; how many barrels did he buy ?
Note. The manner of writing numbers with fig-
ures has been explained as far as ten. The numbers
from ten to one hundred are written as follows: — •
Ten is written 10
Eleven 11
Twelve . 12
Thirteen 13
Fourteen 14
Fifteen 15
Sixteen : . 16
Seventeen . ... 17
70 ARITHMETIC. [PART 1
Eighteen 18
Nineteen 19
Twenty 20
Twenty-one 21
Twenty-two 22
Twenty-three 23
Twenty-four 24
Twenty-five 25
Twenty-six 26
Twenty-seven 27
Twenty-eight 28
Twenty-nine 29
Thirty 30
Thirty-one, &c 31
Forty 40
Fifty 50
Sixty 60
Seventy 70
Eighty 80
Ninety 90
One hundred 100
SECTION VI.
A. 1. BOUGHT 1 half of a yard of cloth for 1
shilling ; what would be the price of a yard at the
same rate ?
2. If 1 half pint of cherries cost 2 cents, what
will a pint cost ?
3. If 1 fourth of a barrel of flour cost 2 dollars,
what would a barrel cost ?
4. 2 is 1 half of what number ? 1 fourth of what
number ?
SECT. 6.] ARITHMETIC. 71
5. If 1 third of a yard of cloth cost 2 dollars, what
is that a yard ?
6. 2 is 1 third of what number ?
7. If 1 third of a yard of cloth cost 3 dollars, what
is that a yard ?
<8. 3 is 1 third of what number 1
9. If 1 fourth of a firkin of butter cost 3 dollars,
what is that a firkin ?
10. 3 is 1 fourth of what number 1
11. A man bought 1 third of a bushel of wheat
for 4 shillings ; what would a bushel cost at that
rate?
12. 4 is 1 third of what number ?
13. If a man can ride 2 miles in I fifth of an hour,
how far can he ride in an hour ?
14. 2 is 1 fifth of what number ?
15. A man, being asked the age of his eldest son,
answered that his youngest son, who was 3 years old,
was just 1 fifth of the age of his eldest son ; how old
was the eldest son ?
16. 3 is one fifth of what number?
17. A man bought 1 sixth part of a hundred
weight of sugar for 2 dollars ; what would a hundred
weight cost at the same rate ?
18. 2 is 1 sixth of what number ?
19. Bought 1 fifth of a pound of starch for 5 cents,
what was that a pound ?
20. 5 is 1 fifth of what number ?
21. Bought 1 fourth of a pound of aqua fortis for
6 cents; what was that a pound ?
22. 6 is 1 fourth of what number ?
23. Bought 1 seventh of a hundred weight of co-
coa for 4 dollars ; what would 1 hundred weight cost
at the same rate ?
24. 4 is 1 seventh of what number ?
25. 7 is 1 fifth of what number ?
26. 5 is 1 third of what number ?
72 ARITHMETIC. [?JLRT 1.
27. 4 is 1 eighth of what number ?
28. 6 is 1 sixth of what number ?
29. 8 is 1 third of what number ?
30. 9 is 1 fourth of what number ?
31. 7 is 1 sixth of what number?
32. 8 is 1 seventh of what number ?
33. 9 is 1 eighth of what number 1
34. 8 is 1 tenth of what number ?
35. 7 is 1 ninth of what number?
36. 6 is 1 fifth of what number ?
37. 10 is 1 seventh of what number 1
B. 1. A man bought some linen and some cotton
cloth ; for the linen he gave 4 shillings a yard, which
was twice as much as he gave for the cotton ; what
did he give for a yard of the cotton ?
2. 4 is 2 times what number ?
3. If 2 thirds of a yard of cloth cost 6 dollars,
what would 1 third cost?
4. 6 is 2 times what number ?
5. If 3 fourths of a barrel of flour cost 6 dollars,
what will one fourth of a barrel cost ?
6. 6 is three times what number ?
7. If 2 fifths of a pound of chocolate cost 8 cents,
what would 1 fifth of a pound cost ?
8. 8 is 2 times what number ?
9. If 3 fifths of a pound of candles cost 9 cents,
what will 1 fifth of a pound cost ?
10. 9 is 3 times what number ?
11. If 2 sevenths of a pound of spermaceti can-
dles cost 10 cents, what will 1 seventh of a pound
cost?
12. 10 is 2 times what number ?
13. If 5 eighths of a pound of cotton cost 10
cents, what will 1 eighth cost ?
14. 10 is five times what number ?
15. If 2 thirds of a yard of cloth cost 4 dollars,
SECT. 6.] ARITHMETIC. 73
what will one third cost ? If one third of a yard cost
2 dollars, what will a yard cost ?*
16. If 4 is 2 thirds of some number, what is 1
third of the same number? — 2 is I third of what
number ?* — Then 4 is two thirds of what ?
17. If 2 thirds of a barrel of flour cost 6 dollars,
what will 1 third of a barrel cost ? If 1 third of a
barrel cost 3 dollars^ what will a barrel cost?"
18. If 6 is 2 thirds of some number, what is 1
third of the same number? — 3 is one third of what
number ? — Then 6 is two thirds of what ?
19. If 3 fourths of a bushel of wheat cost 6 shil-
lings, what will 1 fourth of a bushel cost? If 1
fourth of a bushel cost 2 shillings, what will a
bushel cost ?
20. If 6 is 3 fourths of some number, what is 1
fourth of the same number ? — 2 is 1 fourth of what
number? — Then 6 is 3 fourths of what?
21. If 2 fifths of a gallon of wine cost 4 shillings,
what will 1 fifth of a gallon cost ? If one fifth of a
gallon cost 2 shillings, what will a gallon cost ?
22. If 4 is two fifths of some number, what is 1
fifth of the same number ? — 2 is 1 fifth of what
number ?— Then 4 is 2 fifths of what ?
23. If 3 sevenths of a pound of tobacco cost 6
cents, what will 1 seventh of a pound cost ? If 1
seventh of a pound cost 2 cents, what will a pound
cost?
24. If 6 is 3 sevenths of some number, what is 1
seventh of the same number ? — 2 is 1 seventh of
what number ? — Then 6 is 3 sevenths of what ?
25. If 2 sevenths of a barrel of fish cost 4 dol-
lars, what will 1 seventh of a barrel cost? What
will a barrel cost ?
26. 4 is 2 sevenths of what number ?
27. If 3 eighths of a pound of chocolate cost 6
*See this Sect Art. A,
74 ARITHMETIC. [PART 1.
cents, what will 1 eighth of a pound cost ? What
will a pound cost?
28. 6 is 3 eighths of what number ?
29. If eight cents will buy 2 fifths of a pound of
aqua fortis, how many cents will buy a pound ?
30. 8 is 2 fifths of what number ?
31. A man bought 3 fourths of a hundred weight
of yellow ochre for 9 dollars ; what was that a hun-
dred weight ?
32. 9 is three fourths of what number?
33. 8 is 4 ninths of what number ?
34. 9 is 3 tenths of what number ?
35. 10 is 5 sevenths of what number ?
36. 12 is 3 fifths of what number?
37. 12 is 4 ninths of what number?
38. 10 is 2 sevenths of what number ?
39. 14 is 7 fifths of what number?
40. 15 is 3 elevenths of what number?
41. 16 is 2 fifths of what number?
42. 18 is 6 tenths of what number ?
43. 20 is 5 ninths of what number ?
44. 21 is 3 ninths of what number?
45. 24 is 8 ninths of what number?
C» 1 . If 5 eighths of a cask of claret wine cost 15
dollars, what is that a cask ? How much cider, at 4
dollars a barrel, would it take to pay for a cask of
the wine ?
2. 15 is five eighths of how many times 4 ?
3. If 2 thirds of a pound of coffee cost 18 cents,
how much would a pound cost? How many oranges,
at 5 cents apiece, might be bought for a pound ?
4. 18 is 2 thirds of how many times 5 ?
5. A man bought 4 sevenths of a hundred weight
of sugar for 20 shillings; how many dollars would a
hundred weight come to at the same rate ?
6. 20 is 4 seventh? of how many times 6?
SECT. 6.] ARITHMETIC. 75
7. A man sold a cow for 21 dollars, which was
only seven tenths of what she cost him ; how much
did she cost him ? When he bought her, he paid
for her with cloth at 8 dollars a yard ; how many
yards of cloth did he give ?
8. 21 is 7 tenths of how many times 8 ?
9. A man, being asked the age of his youngest
son, answered, that the age of his eldest son was 24
years, which was 3 fifths of his own age ; and that his
own age was 10 times as much as that of his young-
est son ; what was his age ? and what was the age
of his youngest son ?
10. 24 is 3 fifths of how many times 10?
11. 27 is 3 fifths of how many times 7 ?
12. 28 is 7 tenths of how many times 9 ?
13. 30 is 5 eighths of how many times 7 ?
14. 32 is 4 sevenths of how many times 61
15. 36 is 9 eighths of how many times 5?
16. 40 is 8 ninths of how many times 8 ?
17. 42 is 6 fifths of how many times 4 ?
18. 45 is 9 eighths of how many times 6 1
19. 48 is 8 ninths of how many times 7 1
20. 50 is 5 sevenths of how many times 8 ?
21. 54 is 9 sixths of how many times 7 ?
22. 56 is 7 ninths of how many times 10?
23. 60 is 10 sevenths of how many times 4 ?
24. 63 is 9 eighths of how many times 5 ?
25. 64 is 8 ninths of how many times 7 ?
26. 70 is 10 sevenths of how many times 8 ?
27. 72 is 9 fifths of how many times 6 ?
28. SO is 10 thirds of how many times 4?
29. 80 is 8 fifths of how many times 6 ?
I>. 1. A boy gave away 4 cents, which was 1
third of all he had ; how many had he at first ?
2, A boy gave 5 apples to one of his companions,
76 ARITHMETIC. [PART 1.
which was I fourth of what he had ; how many
had he ?
3. A man paid away 4 dollars, which was 2 thirds
of all the money he had; how much had he ?
4. A man sold a watch for 18 dollars, which was
3 fourths of what it cost him ; how much did it cost '?
5. A man sold a cow for 15 dollars, which was 3
fifths of what the cow cost ; how much did he lose
by his bargain ?
6. A man bought 12 yards of cloth, and sold it
for 54 dollars, which was 0 eighths of what it cost
him ; what did it cost him a yard ? and how much
did he gain by his bargain 1
7. There is a pole standing in the water, so that
10 feet of it is above the water, which is 2 thirds of
the whole length of the pole ; how long is the pole ?
8. There is a pole two thirds under water, and 4
feet out ; how long is the pole ?
9. There is a pole two fifths under water, and 6
feet out of the water ; how long is the pole ?
10. There is an orchard, in which 3 sevenths of the
trees bear cherries, and 2 sevenths bear peaches,
and 10 trees bear plums; how many trees are there
in the orchard ? and how many of each sort ?
11. There is a school, in which 2 ninths of the
boys learn arithmetic, 3 ninths learn grammar, 1
ninth learn geography, 1 ninth learn geometry, and
12 learn to write ; how many are there in the school ?
and how many attending to each study ?
12. A man sold a watch for 63 dollars, which was
7 fifths of what it cost him ; how much did he gain
by the bargain ?
SECT. 6.] ARITHMETIC. 77
Miscellaneous Examples.
1. IF 1 yard of cloth cost 4 dollars, what will 5
yards cost?
2. A man bought 3 pounds of raisins, at 7 cents
a pound, and 16 oranges, at 4 cents apiece, and 1
pound of candles for 16 cents ; what did they all
come to?
3. A hoy had 37 apples; he kept five himself,
and divided the rest equally among four compan-
ions; how many did he give them apiece?
4. Two men are 40 miles apart, and both travel-
ling the same way ; the hindermost man gains upon
the other 5 miles each day ; in how many days will
he overtake him ?
5. Two men are travelling the same way; one
travels at the rate of 38 miles a day ; the other, who
is behind the former, travels 44 miles ; how much
does he gain of the first each day? and in how
many days would he gain 60 miles ?
6. A fox is 80 rods before a greyhound, and is
running at the rate of 27 rods in a minute ; the
greyhound is following at the rate of 31 rods in a
minute ; in how many minutes will the greyhound
overtake the fox ?
7. If two yards of cloth cost 6 dollars, what would
4 yards cost? What would 12 yards cost?
8. If 8 sheep cost 24 dollars, what would 3 cost?
9. If 4 tons of hay will keep 3 horses through the
winter, how many tons will keep 30 horses the same
time ?
10. If a man spends 8 shillings a day, how many
dollars will he spend in a week ?
11. Bought 10 pieces of cloth, each containing 5
yards, for 100 dollars ; what was it a piece ? and
..how much a yard ?
78 ARITHMETIC. [PART 1.
12. If the wages of 12 weeks come to 60 dol-
lars, what is that a month ? and how much for 5
weeks ?
13. If 7 horses eat 14 bushels of oats in 1 week,
how many bushels would 15 horses eat in the same
time ?
14. If 3 horses eat 8 bushels of oats in 2 weeks,
how long would it take them to eat 40 bushels ?
15. If 1 horse eat 1 bushel of oats in 3 days, in
how many days would 4 horses eat 36 bushels?
16. If 2 'men spend 12 dollars in I week, how
many dollars would 3 men, at the same rate, spend
in 5 weeks?
17. If a staff 3 feet long cast a shadow of 2 feet
at 12 o'clock, what is the length of a pole that casts
a shadow IS foot at the same time of day ?
18. If 47 gallons of water, in 1 hour, run into a
cistern containing 10S gallons, and by a pipe 38
gallons run out in an hour, how much remains in the
cistern in an hour 1 and in how many hours will the
cistern be filled ?
19. If 4 men can do a piece of work in 8 days,
how many men would it take to do the same work
in 4 days?
20. If 6 men can do a piece of work in 9 days,
in how many days would 2 men do it?
21. If 8 men can do a piece of work in 5 days,
in how many days would they do a piece of work 4
times as large?
22. If 7 men can do a piece of work in 3 days,
how many men would it take to do a piece of work
5 times as large in the same time ?
23. If 8 men can do a piece of work in 4 days,
in how many days would 2 men do a piece of work
one half as large ?
24. A man bought a cask of wine containing 63
gallons, 3 sevenths of which leaked out ; and he
SECT. 7.] ARITHMETIC. 79
sold the remainder for 36 dollars; how much per
gallon did he sell it for ?
25. If a cask of wine cost 48 dollars, what is 5
eighths of it worth 1
26. A man bought 7 oranges for 6 cents apiece,
and sold them all for 54 cents ; how much did he
gain by the bargain ?
27. A man bought 8 yards of cloth for 56 dol-
lars, and sold it again for 9 dollars a yard ; how
much did he gain by the bargain ?
28. A man bought 8 barrels of flour for 40 dol-
lars ; how much must he sell it at per barrel to gain
16 dollars ?
29. A man bought five firkins of butter at 7 dol-
lars a firkin ; how must he sell it per firkin to gain
10 dollars ?
30. A man gave 35 cents for his breakfast, which
was 5 eighths of what he gave for his dinner ; what
did he give for his dinner?
31. A ship's crew of 6 men have provision for 3
months ; how many months would it last 1 man ?
32. A ship's crew have provision sufficient to last
1 man 27 months ; how long would it last 9 men ?
33. A ship's crew have provision sufficient to last
3 men ten months ; how long would it last 5 men ?
34. A man built 40 rods of wall in a certain time,
another man can build 9 rods while the first builds
5 ; how much would he build in the same time ?
SECTION VII.
If the combinations in this section should be found
too difficult, they may be omitted until reviewing the
book.
A.. 1. A MAN, being asked the age of his eldest
80 ARITHMETIC. [PART!
son, answered, that his youngest son was six years
old, and that 2 thirds of the youngest son's age was
just 1 fifth of the eldest son's age. Required the
age of the eldest son.
2. 2 thirds of 6 is 1 fifth of what number ?
3. A man, being asked how many sheep he had,
said that he had them in two pastures ; in one pas-
ture he had eight ; and that 3 fourths of these was
just 1 third of what he had in the other. How
many were there in the other ?
4. 3 fourths of 8 is 1 third of what number ?
5. Two boys talking of their ages, one said he
was 9 years old. Well, said the other, 2 thirds of
your age is exactly 3 fourths of my age ; now, if you
will tell me how old I am, I will give you as many
apples as I am years old. What was his age ?
6. 2 thirds of 9 is 3 fourths of what number ?
7. Two boys counting their money, one said he
had ten cents. The other says, 4 fifths of your
money is exactly 2 sevenths of mine ; now, if you
will tell how many I have, I will give you 1 half of
them. How many had he ?
8. 4 fifths of 10 is 2 sevenths of what number ?
9. 5 sixths of 12 is 2 thirds of what number?
10. 6 sevenths of 14 is 4 ninths of what number?
11. 6 fifths of 15 is 2 thirds of what number?
12. 7 ninths of 18 is 2 fifths of what number ?
13. 4 fifths of 20 is 8 sevenths of what number ?
14. 8 sevenths of 21 is 6 tenths of what number?
15. 5 sixths of 24 is 10 sevenths of how many
times 5?
16. 3 sevenths of 28 is 2 eighths of how many
times 7?
17. 4 fifths of 30 is 6 sevenths of how many
times 8 ?
18. 6 eighths of 32 is 8 ninths of how many
times 5 ?
SECT. 7.] ARITHMETIC. 81
19. 4 ninths of 36 is 8 tenths of how many times
6?
20. 3 fourths of 40 is 5 sevenths of how many
times 8 ?
21. 6 ninths of 45 is 3 fifths of how many times
7?
22. 5 sixths of 48 is 10 sevenths of how many
times 3 ?
23. 4 sevenths of 63 is 6 fifths of how many
times 8 ?
24. 5 ninths of 72 is 4 sevenths of how many
times 9 ?
B. 1. 4 fifths of 15 is 6 tenths of how many
thirds of 21 ?
2. 4 thirds of IS is 8 ninths of how many sevenths
of 35?
3. 6 sevenths of 21 is 2 thirds of how many thirds
of 24?
4. 5 fourths of 24 is 10 sevenths of how many
fifths of 40 ?
5. 5 eighths of 32 is 2 fifths of how many fifths
of 35?
6. 4 sevenths of 63 is 6 eighths of how many
ninths of 45 ?
7. 3 sevenths of 56 is 4 ninths of how many
fourths of 28 ?
8. 3 eighths of 64 is 6 tenths of how many sixths
of 30?
9. 2 eighths of 72 is 3 tenths of how many fifths
of 40?
C. 1. Two times eleven are how many?
2. Two times twelve are how many ?
3. Two times thirteen are how many ?
4. Two times fourteen are how many ?
5. Two times fifteen are how many ?
82 ARITHMETIC. [PART 1.
6. Two times sixteen are how many ?
7. Two times seventeen are how many 1
8. Two times eighteen are how many ?
9. Two times nineteen are how many ?
10. Two times twenty are how many?
11. Three times eleven are how many ?
12. Three times twelve are how many ?
13. Three times thirteen are how many ?
14. Three times fourteen are how many ?
15. Three times fifteen are how many ?
16. Three times sixteen are how many?
17. Three times seventeen are how many ?
18. Three times eighteen are how many?
19. Three times nineteen are how many ?
20. Three times twenty are how many ?
21. Four times eleven are how many?
22. Four times twelve are how many ? '
23. Four times thirteen are how many ?
24. Four times fourteen are how many ?
25. Four times fifteen are how many ?
26. Four times sixteen are how many?
27. Four times seventeen are how many ?
28. Four times eighteen are how many ?
29. Four times nineteen are how many?
30. Four times twenty are how many ?
31. Five times eleven are how many 1
32. Five times twelve are how many ?
33. Five times thirteen are how many ?
34. Five times fourteen are how many ?
35. Five times fifteen are how many ?
36. Five times sixteen are how many ?
37. Five times seventeen are how many ?
38. Five times eighteen are how many ?
39. Five times nineteen are how many ?
40. Five times twenty are how many?
41. Six times eleven are how many?
42. Six times twelve are how many ?
SECT. 7.] ARITHMETIC.
43. Six times thirteen are how many 1
44. Six times fourteen are how many 1
45. Six times fifteen are how many ?
46. Six times sixteen are how many ?
47. Six times seventeen are how many ?
48. Six times eighteen are how many ?
49. Six times nineteen are how many ?
50. Six times twenty are how many ?
51. Seven times eleven are how many?
52. Seven times twelve are how many ?
53. Seven times thirteen are how many ?
54. Seven times fourteen are how many?
55. Seven times fifteen are how many ?
56. Seven times sixteen are how many ?
57. Seven times seventeen are how many ?
58. Seven times eighteen are how many ?
59. Seven times nineteen are how many 1
60. Seven times twenty are how many 1
61. Eight times eleven are how many ?
62. Eight times twelve are how many ?
63. Eight times thirteen are how many ?
64. Eight times fourteen are how many ?
65. Eight times fifteen are how many ?
66. Eight times sixteen are how many ?
67. Eight times seventeen are how many ?
68. Eight times eighteen are how many ?
69. Eight times nineteen are how many 1
70. Eight times twenty are how many ?
71. Nine times eleven are how many 1
72. Nine times twelve are how many?
73. Nine times thirteen are how many ?
74. Nine times fourteen are how many ?
75. Nine times fifteen are how many ?
76. Nine times sixteen are how many ?
77. Nine times seventeen are how many ?
78. Nine times eighteen are how many ?
79. Nine times nineteen are how many?
84 ARITHMETIC. [PART 1.
80. Nine times twenty are how many ?
81. Ten times eleven are how many 1
82. Ten times twelve are how many ?
83. Ten times thirteen are how many ?
84. Ten times fourteen are how many?
85. Ten times fifteen are how many ?
86. Ten times sixteen are how many ?
87. Ten times seventeen are how many ?
88. Ten times eighteen are how many ?
89. Ten times nineteen are how many ?
90. Ten times twenty are how many ?
SECTION VIII.
A. 1. IF you cut an apple into two equal parts,
what is one of those parts called?*
2. How many halves of an apple will make 4he
whole apple?
3. If you cut an apple into 3 equal paits, what is
1 of those parts called? What are 2 of the parts
called ?
4. How many thirds of an apple will make the
whole apple?
5. If you cut an apple into 4 equal parts, what is
1 of those parts called ? What are 2 of those parts
called ? What are three of them called ?
6. How many fourths of an apple make the
whole apple ?
7. If an apple be cut into 5 equal parts, what
is one of the parts called ? What are 2 of the parts
called ? What are 3 of the parts called ? What are
4 of the parts called ?
* See Sect. III. Art. B., remark before question 1 and 17.
SECT. 8.] ARITHMETIC. 85
8. How many fifths of an apple make the whole
apple ?
9. If an apple be cut into 6 equal parts, what is
1 of the* parts called? What are 2 of the parts
called? What 3? What 4? What 5?
10. How many sixths of an apple make the whole
apple ? ,
11. If an apple be cut into 7 equal parts, what is
1 of the parts called ? What are 2 of the parts
called? What 3? What 4? What 5? What 6?
Let the instructer ask the pupil the divisions of a
unit in this manner as far as the division into 10 parts.
It would be well to ask them further. Then let him
begin again, and suppose an orange instead of an
apple. After applying the division to several differ-
ent things, Plate II. may be explained and used.
It will often be found useful to refer the pupil to the
divisions of some sensible object. For the explana-
tion of Plate II. see the Key.
12. A man had a bushel of corn, and wished to
give 1 half of a bushel apiece to some laborers ;
how many could he give it to ?
13. How many halves are there in 1 ?
14. A man divided 2 barrels of flour among his
laborers, giving them 1 half of a barrel apiece;
how many men did he give it to ?
15. How many halves are there in 2 ?*
16. In 3 bushels of corn how many half bushels ?
17. How many halves are there in 3?
18. A boy divided 4 oranges among his compan-
ions, giving them 1 half of an orange apiece; how
many boys did he give them to ?
19. How many halves are there in 4 ?
* Be careful to make the pupil use the plate. He might answer the
questions without, but he will not understand their meaning so welK
86 ARITHMETIC. [PART 1.
20. A man, having some laborers, gave them 1
half a dollar apiece ; it took 3 dollars and 1 half a
dollar to pay them ; how many laborers were
there?
21. How many halves are there in 3 and 1 half?
22. How many halves are there in 5 ?
23. How many halves are there in 7 and 1 half?
24. How can you tell how many halves there are
in any number?
Answer. Since there are two halves in one, there
will be twice as many halves as there are whole
ones.
25. If you had 1 orange, and should divide it
among your companions, giving them 1 third apiece,
how many could you give it to ?
26. How many thirds are there in 1 ?
27. If you cut 2 oranges each into 3 pieces, how
many pieces would they make ?
28. If you cut 3 oranges into 3 pieces each, how
many pieces would they make ?
29. If you cut 4 apples each into 3 pieces, how
many pieces would they make ?
30. How many thirds are there in 2 ? In 3 ? In 4?
In 5?
31. How can you tell how many thirds there are
in any number?
Answer. Since there are 3 thirds in one, there will
be 3 times as many thirds as there are whole ones.
32. If you had 2 bushels and 1 third of a bushel
of corn to give to some poor persons, how many
could you give it to if you should give them 1 third
of a bushel apiece ?
33. How many thirds are there in 2 and 1 third ?
34. If a horse can eat 1 third of a bushel of oats
in 1 day, how many days would it take him to eat
3 bushels and 2 thirds of a bushel ?
35. How many thirds are there in 3 and 2 thirds ?
SECT. 8.] ARITHMETIC. 87
36. If 1 horse can eat 1 third of a bushel of oats
in a day, how many horses will it take to eat 5 bush-
els and 2 thirds of a bushel in the same time?
37. In 5 and 2 thirds how many thirds ?
38. In seven and 1 third hgw many thirds ?
39. If 1 horse will eat 1 fourth of a ton of hay in
1 month, how many horses will eat a ton in the same
time ? How many will eat 2 tons ? How many will
eat 3 tons ? 4 tons 1 5 tons ? 6 tons ?
40. How many fourths are there in 1? In 2? In
3? In 4? In 5? In 6?
41. How can you tell how many fourths there are
in any number ?
42. How many fourths are there in 4 and 1 fourth?
43. In 5 tons of hay and 3 fourths of a ton, how
many fourths of a ton 1
44. In 5 and 3 fourths how many fourths ?
45. In 7 and 1 fourth how many fourths ?
46. In 9 and 3 fourths how many fourths ?
47. If a horse eat 1 fifth of a ton of hay in a month,
how many horses will eat a ton in the same time ? 2
tons ? 3 tons ? i tons ? 5 tons ?
48. How many fifths are there in 1? In 2? In 3?
In 4? In 5? In 7? In 9?
49. How can you tell how many fifths there are
in any number ?
50. In 2 and 1 fifth how many fifths?
51. In 3 dollars and 2 fifths of a dollar how many
fifths of a dollar ?
52. In 3 and 2 fifths how many fifths ?
53. In 5 and 3 fifths how many fifths ?
54. In 6 and 4 fifths how many fifths ?
55. How many sixths are there in 1? In 2? In
3? In 4? In 5? In 7? In 8?
56. In 2 and 2 sixths how many sixths ?
57. In 3 and 4 sixths how many sixths ?
58. In 6 and 5 sixths how many sixths ?
88 ARITHMETIC. [PART 1.
59. How many sevenths are there In 1 ? In 2 ?
In 3? In 4? In 6? In 9? In 10?
60. In 3 and 1 seventh how many sevenths?
61. In 5 and 3 sevenths how many sevenths?
62. In 7 and 5 sevenths how many sevenths ?
63. How many eighths are there in 1 ? In 3? In
5? In8?
64. In 2 and 3 eighths how many eighths ?
65. In 3 and 5 eighths how many eighths ?
66. In 5 and 7 eighths how many eighths?
67. How many ninths are there in 1 ? In 2 ? In
7? In5?
68. In 2 and 2 ninths how many ninths ?
69. In 4 and 3 ninths how many ninths ?
70. In 6 and 4 ninths how many ninths ?
71. In 8 and 7 ninths how many ninths?
72. How many tenths in 1 ? In 2 ? In 5? In 8?
73. In 3 and 3 tenths how many tenths ?
74. In 4 and 7 tenths how many tenths?
75. In 8 and 9 tenths how many tenths?
76. In 7 and 4 tenths how many tenths ?
77. In 9 and 8 tenths how many tenths ?
78. In 7 and 4 sevenths how many sevenths ?
79. In 9 and 2 thirds how many thirds ?
80. In 10 and 3 fourths how many fourths?
81. In 8 and 4 fifths how many fifths ?
82. In 7 and 5 ninths how many ninths?
B. 1. If you give 4 men 1 half of a barrel of flour
apiece, how many barrels will it take ?
2. In 4 halves how many times 1 ?
3. If you give 3 boys 1 half of an orange apiece,
how many oranges will it take ?
4. In 3 halves how many times 1 ?
5. If you give five men 1 half of a dollar apiece,
how many dollars will it take ?
6. In 5 halves how many times 1 ?
SECT 8.] ARITHMETIC. 89
7. In 6 halves how many times 1 ?
8. In 7 halves how many times 1 ?
9. How can you tell how many whole ones there
are in any number of halves ?
10. A man divided some corn among 6 persons,
giving them 1 third of a bushel apiece ; how many
bushels did it take ?
11. In 6 thirds how many times 1 ?
12. In 5 thirds how many times 1 ?
13. A man gave eight paupers 1 third of a dollar
apiece ; how many dollars did it take ?
14. In 8 thirds how many times 1 ?
15. In 10 thirds how many times 1 ?
16. How can you tell how many whole ones there
are in any number of thirds ?
17. If a man spends 1 fourth of a dollar in one
day, how many dollars will he spend in 8 days?
How many in 7 days? How many in 11 days ?
18. In 8 fourths how many times 1 ?
19. In 7 fourths how many times 1 ?
20. In 11 fourths how many times 1 ?
21. In 13 fourths how many times 1 ?
22. In 18 fourths how many times 1 ?
23. How can you tell how many whole ones there
are in any number of fourths ?
24. If 1 fifth of a barrel of beer will last a family
1 day, how many barrels will last them 10 days?
How many 8 days? 11 days ? 17 days ?
25. In 10 fifths how many times 1 ?
26. In 8 fifths how many times 1 ?
27. In 1 1 fifths how many times 1 ?
28. In 17 fifths how many times 1?
29. In 18 sixths how many times 1 ?
30. In 23 fifths how many times 1 ?
31. In 21 sevenths how many times 1 ?
32. In 24 eighths how many times 1 ?
33. In 36 ninths how many times 1 ?
8*
90 ARITHMETIC. [!»ABT 1.
34. In 30 tenths how many times 1 ?
35. In 35 fourths how many times 1 ?
36. In 37 eighths how many times 1 ?
37. In 43 fifths how many times 1 ?
38. In 48 ninths how many times 1 ?
39. In 53 tenths how many times 1 ?
40. In 57 eighths how many times 1 ?
41. In 76 tenths how many times 1 ?
42. In 78 ninths how many times 1 1
SECTION IX.
A.. 1. IF a breakfast for 1 man cost 1 third of a
dollar, what would a breakfast for two men cost ?
2. How much is 2 times 1 third ?
3. If it take you I third of an hour to travel 1
mile, how long will it take you to travel 3 miles ?
4. How much is 3 times 1 third ?
5. If 1 man can eat 1 third of a pound of meat at
a meal, how much can 5 men eat?
6. How -much is 7 times 1 third?
7. If 1 man can eat 2 thirds of a pound of meat
for dinner, how many thirds of a pound would 3
men eat?
8. How much is 2 times 2 thirds ?
9. A man gave to 4 paupers 2 thirds of a dollar
apiece ; how many thirds of a dollar did he give
them ? How many dollars ?
10. 5 times 2 thirds are how many thirds ? How
many times 1 ?
11. If you give 3 men 1 fourth of a dollar apiece,
how many fourths of a dollar will it take ?
12. 3 times 1 fourth are how many fourths ?
13. If you give 3 men 3 fourths of a bushel of
SECT. 9.] ARITHMETIC. 91
corn apiece, how many fourths of a bushel will it
take ? How many bushels ?
14. 5 times 3 fourths are how many fourths ? How
many times 1?
15. If 1 horse eat 1 fifth of a bushel of oats in a
day, how much will 4 horses eat in the same time ?
16. 3 times 1 fifth are how many fifths?
17. If 1 man can earn 3 fifths of a dollar in a day,
how much can he earn in 4 days ?
18. 7 times 3 fifths are how many fifths? How
many times 1?
19. If a family consume 2 sevenths of a barrel of
flour in a week, how much would they consume in
5 weeks ?
20. 6 times 2 sevenths are how many sevenths?
How many times 1?
21. 5 times 3 eighths are how many eighths?
How many times 1 ?
22. How much is 6 times 3 fifths ?
23. How much is 7 times 5 sixths ?
24. How much is 5 times 4 ninths ?
25. How much is 6 times 8 ninths?
26. How much is 7 times 9 tenths ?
27. How much is 5 times 7 tenths ?
28. How much is 6 times 7 eighths ?
29. How much is 9 times 5 eighths ?
30. How much is 8 times 5 sevenths ?
31. How much is 7 times 5 sixths?
32. . How much is 8 times 7 fourths ?
33. How much is 7 times 4 fifths ?
34. How much is 5 times 3 eighths ?
IS. 1. If 1 bushel of wheat cost a dollar and 1
half, what will two bushels cost ?
2. How much is 2 times 1 and 1 half?*
* This is to be understood, 2 times 1 and 2 times 1 half, and to
be answered thus : 2 times 1 are 2; and 2 times 1 half are 2 halves
or 1, which, added to 2; makes 3,
92 ARITHMETIC. [PART 1.
3. If a barrel of cider cost 2 dollars and a half,
what will 3 barrels cost ?
4. How much is 4 times 2 and 1 half ?
5. If a barrel of beer cost 3 dollars and a half,
what will 2 barrels cost ?
6. How much is 5 times 3 and 1 half 1
7. How much is 6 times 3 and 1 half?
8. If a box of butter cost 2 dollars and 1 third of
a dollar, what will 3 boxes cost ?
9. How much is 4 times 2 and 1 third?
10. If you give to two persons 3 bushels and 1
third of a bushel of wheat apiece, how many bushels
will it take ?
11. How much is 5 times 3 and 1 third?
12. If you give to 4 persons each 2 oranges and 1
fourth of an orange, how many oranges will it take?
13. How much is 5 times 2 and 1 fourth ?
14. If it take 3 yards and 2 thirds of a yard of
cloth to make a suit of clothes, how many yards will
it take to make 2 suits ?
15. How much is 4 times 3 and 2 thirds ?
16. If a family consume 2 bushels and 2 thirds of
a bushel of malt in 1 month, how much will they
consume in 3 months ?
17. How much is 5 times 2 and 2 thirds ?
18. How much is 4 times 3 and 3 fourths ?
19. How much is 2 times 3 and I fourth?
20. How much is 3 times 3 and 3 fourths ?
21. How much is 3 times 5 and I fourth ?
22. If a horse eat 3 tons and 1 fifth of a ton of
hay in a year, how much will 2 horses eat in the
same time ?
23. How much is 4 times 3 and 1 fifth ?
24. If a man can travel 4 miles and 2 fifths of a
mile in one hour, how far will he travel in 3 hours?
25. How much is 5 times 4 and 2 fifths ?
26. How much is 3 times 5 and 3 fifths ?
SECT. 9.] ARITHMETIC. 93
27. How much is 4 times 6 and % fifths 1
28. How much is 3 times 2 and 1 sixth 1
29. How much is 2 times 5 and 3 sixths ?
30. If a yard of cloth cost 4 dollars and 5 sixths
of a dollar, what will 4 yards cost ?
31. How much is 7 times 4 and 5 sixths 1
32. How much is 2 times 3 and 3 sevenths ?
33. How much is 3 times 4 and 3 sevenths 1
34. If a bushel of wheat cost 7 shillings and 3
eighths, what will 5 bushels cost at that rate ?
35. How much is 3 times 4 and 5 eighths 1
36. How much is 4 times 3 and 7 eighths 1
37. A man bought 8 yards of cloth, at 9 dollars
and 3 tenths a yard ; how much did it come to ?
38. How much is 6 times 2 and 2 ninths ?
39. How much is 4 times 5 and 3 ninths 1
40. A man bought 10 barrels of cider at 3 dollars
and 5 sixths a barrel ; how much did it come to ?
41. How much is 3 times 7 and 2 tenths 1
42. What cost 8 barrels of beef, at 9 dollars and
3 sevenths a barrel?
C. 1. A boy wished to give 5 other boys 1 half
pint of chestnuts apiece ; how many pints would it
take?
2. A boy wished to give 3 other boys 3 fourths of
an orange apiece ; how many oranges would it take ?
3. A man gave to 10 persons 1 fifth of a bushel
of corn apiece ; how many bushels did it take ?
4. A man gave to 7 men 3 fourths of a gallon of
beer apiece ; how many gallons of beer did it take ?
5. If it take 1 yard and 1 fourth of a yard of cloth
to make a pair of pantaloons, how many yards would
it take to make 8 pairs ?
6. If a family consume 2 bushels and 3 fifths of a
bushel of grain in 1 week, how many bushels would
they consume in 4 weeks ?
94 ARITHMETIC. [PART 1.
7. If a horse eat 3 bushels and 4 sevenths of a
bushel of oats in 1 week, how many bushels would
he eat in 8 weeks 1
8. If a horse eat 5 loads and 3 eighths of a load of
hay in 1 year, how many loads would 6 horses eat ?
9. If a man travel 4 miles and 5 ninths in an
hour, how many miles would he travel in 8 hours ?
10. If, in an orchard of 10 trees, each tree bears
8 bushels and 3 sevenths, how many bushels will
the whole orchard bear 1
11. If a man can build 5 rods and 7 eighths of a
rod of wall in 1 day, how many rods can he build
in 8 days ?
12. If 3 men can build a piece of wall in 4 days
and 3 fifths of a day, how many days would it take
1 man to build it?
13. If 1 man can build 7 rods and 2 fifths ef wall
in a day, how many rods would 10 men build ?
14. If 1 man build 3 rods and 2 ninths of wall in
1 day, how many rods would 3 men build in 4 days ?
15. If it take 1 yard and 3 sevenths of a yard of
cloth to make 1 pair of pantaloons, and 2 yards
and 4 sevenths for a coat, how many yards would
it take to make 3 pairs of pantaloons and 3 coats?
SECTION X.
A. 1. IF a yard of cloth cost 3 dollars, what will
1 half of a yard cost ?
2. What is 1 half of 3?*
3. If a barrel of beer cost 5 dollars, what will 1
lialf of a barrel cost?
* See Sect. V. Art. A.
SECT. 10.] ARITHMETIC. 95
4. What is 1 half of 5?
5. If 2 barrels of cider cost 7 dollars, what is
that a barrel ?
6. What is 1 half of 7?
7. What is 1 half of 9?
8. What is 1 half of 11?
9. What is 1 half of 13?
10. What is 1 half of 15?
11. If you divide 1 bushel of wheat equally
among 3 persons, what part of a bushel will you
give them apiece?
12. If 3 yards of cloth cost 1 dollar, what part
of a dollar will 1 yard cost?
13. What is 1 third of 1?
14. How could you divide 2 oranges into 3 equal
parts? that is, how can you find 1 third of 2
oranges ? *
15. One third of 2 oranges will be the same as
how many thirds of one orange ?
16. If you divide 2 bushels of wheat equally
among 3 persons, what part of a bushel will you
give them apiece?
17. If 3 bushels of corn cost 2 dollars, what part
of a dollar will 1 bushel cost ?
NOTE. One third of two things is twice as much as
one third of one thing. One third of one is one third,
and consequently one third of two things is two thirds.
In the same manner, one third of four things is four
thirds of one thing. If four oranges be cut each into
three parts, and then one part of each be taken, it will
make four pieces, each of which is one third of one
orange. Hence one third of four oranges is four thirds
of one orange, that is, one whole one and one third.
18. If 3 bushels of wheat cost 4 dollars, how
much is that a bushel?
* Divide each orange into three parts, and then take one part from
each.
96 ARITHMETIC. [PART 1.
19. What is one third of 2 ? Of 4 ?
20. If 3 gallons of wine cost 5 dollars, what is
that a gallon ?
21. What is 1 third of 5? Of 71 Of 8? Of 10?
Of 11?
22. If a bushel of apples be divided equally
among 4 persons, what part of a bushel will they
have apiece? What would they have apiece, if 2
bushels were divided among them ? What, if 3 bush-
els? What, if 5 bushels? What, if 6 bushels ?
23. What is 1 fourth of 1? Of 2? Of 3? Of 5?
Of 6? Of 7? Of 9? Of 10?
24. If a bushel of malt will serve 5 persons 1
month, how much will serve 1 person the same time ?
25. If 2 barrels of cider will serve 5 persons 1
month, how much will serve 1 person the same
time?
26. If 3 barrels of flour be divided among 5 men,
how much will each have? If 4 barrels were divided,
what would each have? What, if 6 barrels were
divided ? What, if 7 barrels were divided ?
27. What is 1 fifth of 1? Of 2? Of 3? Of 4?
Of 6? Of 7?
28. What is 1 sixth of 1 ? Of 2? Of 3? Of 4?
Of 5? Of 7? Of 8? Of 9? Of 10?
29. What is 1 seventh of 1? Of 2? Of 3? Of
4? Of 5? Of 6?
30. What is 1 eighth of 1? Of 2? Of 3? Of 4?
Of 5? Of 6? Of 7? Of 8? Of 9? Of 10?
31. What is 1 ninth of 1? Of 2 ? Of 3? Of 4?
Of5? Of6? Of7? Of8? Of9? OflO? Ofll?
32. What is 1 tenth of 1? Of 2? Of 3? Of 4?
Of 5? Of 6? Of 7? Of 8? Of 9? OflO? Of
11? Of 12? Of 13?
33. If 3 yards of cloth cost 2 dollars, what will
1 yard cost? What will 2 yards cost ?
34. If 1 bushel of wheat cost 2 dollars, what
SECT. 10.] ARITHMETIC. 97
will 1 third of a bushel cost? What will 2 thirds of
a bushel cost?
35. What is 1 third of 2 ?
36. What is 2 thirds of 2 ?
37. If a load of wood cost 5 dollars, what will 1
third of a load cost ? What will 2 thirds of a load
cost?
38. What is 1 third of 5 ?
39. What is 2 thirds of 5 ?
40. What is 2 thirds of 7 ?
41. If 4 bushels of salt cost 3 dollars, what will 1
bushel cost ? What will 3 bushels cost ?
42. What is 1 fourth of 3?
43. What is 3 fourths of 3 ?
44. If a barrel of cider cost 2 dollars, what will
1 fifth of a barrel cost? What will 2 fifths of a
barrel cost ?
45. What is 1 fifth of 2?
46. What is 2 fifths of 2?
47. What is one fifth of 7 ?
48. What is 3 fifths of 7?
49. If 7 gallons of gin cost 5 dollars, what will 1
gallon cost ? What will 4 gallons cost ?
50. What is 1 seventh of 5 ?
51. What is 4 sevenths of 5 1
52. What is 1 sixth of 4?
53. What is 5 sixths of 4 ?
54. If you divide 7 dollars among 8 men, what
part of a dollar will you give them apiece ? What
would three of them have ?
55. What is 1 eighth of 7 ?
56. What is 3 eighths of 7?
57. What is 1 ninth of 10?
58. What is 7 ninths of 10?
59. What is 1 tenth of 14 ?
60. What is 4 tenths of 14 ?
61. If 5 yards of cloth cost 17 dollars, what is
9
98 ARITHMETIC. [PART!.
that a yard ? What would 3 yards cost ? What would
8 yards cost ?
<IW. What is 3 fifths of 17?
63. What is 8 fifths of 17?
<>4. What is 5 sevenths of 20 ?
65. What is 2 ninths of 22 ?
66. What is 3 eighths of 27?
67. What is 2 thirds of 28 ?
68. What is 3 fourths of 31 ?
69. If 5 loads of hay cost 47 dollars, what is that
a load ? What will 2 loads cost ? What will 7 loads
cost ? What will 12 loads cost ?
70. What is 2 fifths of 47?
71. What is 7 fifths of 47?
72. What is 12 fifths of 47 ?
73. What is 4 sevenths of 48?
74. What is 4 ninths of 50 ?
75. What is 2 sevenths of 58?
76. What is 3 eighths of 61 ?
77. What is 4 tenths of 73 ?
78. What is 8 ninths of 65 ?
79. What is 9 tenths of 78 ?
80. What is 7 eighths of 70 ?
B. 1. If you divide 7 apples equally between
2 boys, how many would you give them apiece ?
2. How can you divide 5 oranges equally among
3 persons ?
3. Divide 3 bushels of corn equally among 5 men,
how much would you give them apiece ?
4. A boy had 7 pears, and gave away 3 fifths ol
them ; how many did he give away ? and how many
had he left ?
5. If 2 yards of cloth cost 3 dollars, what is that
a yard ?
6. If 4 yards of cloth cost 2 dollars, what is that
a yard ?
SECT. 11.] ARITHMETIC. 99
[Let the answers be given in dollars and cents, or
in shillings. 1 dollar is 100 cents.]
7. If 5 bushels of corn cost 7 dollars, what is that
a bushel ?
8. If a man receive 8 dollars for 6 days' work,
what is that per day ?
9. If 3 bushels of wheat cost 8 dollars, what will
2 bushels cost ?
10. A man had 39 dollars, and gave away 3
sevenths of it ; how much did he give away ?
11. If 4 yards of cloth cost 10 dollars, what will
3 yards cost ?
12. If 3 barrels of cider cost 8 dollars, what will
10 barrels cost ?
13. If 7 pounds of flour cost 40 cents, what will
10 pounds cost ?
14. If 4 firkins of butter cost 26 dollars, what
will 7 firkins cost ?
15. If 3 men can do a piece of work in 7 days,
how long will it take 1 man to do it? How long
would it take 4 men ?
16. If 2 cocks will empty a cistern in 3 hours, in
how long a time would 1 empty it 1 In how long a
time would 7 cocks empty it ?
SECTION XI.
A. 1. BOUGHT 1 half of a yard of cloth for 1
dollar and 1 half; what was that a yard?
2. 1 and 1 half is the half of what number ?*
3. If 1 third of a yard of cloth cost 1 dollar and 3
fourths of a dollar, how much does a yard cost ?
* See Sect. VI. Art. A., and Sect. IX Art. E^
100 ARITHMETIC. [PART 1.
4. 2 and 3 fourths is I third of what number ?
5. If 1 half of a barrel of beer cost 2 dollars
and 1 fourth of a dollar, how much will a barrel
cost?
6. 2 and 3 fourths is one half of what number ?
7. If 1 fourth of a box of lemons cost 3 dollars
and 2 thirds of a dollar, what will a box cost?
8. 2 and 2 thirds is 1 fourth of what number?
9. 3 and 1 half is 1 fourth of what number ?
10. If 1 third of a barrel of pork cost 4 dollars
and 1 third of a dollar, what will 1 barrel cost?
11. 4 and 2 thirds is 1 third of what number?
12. If 1 sixth of a barrel of fish cost 2 dollars and
1 fifth of a dollar, what will a barrel cost?
13. 3 and 2 fifths is 1 sixth of what number?
14. If* 1 fifth of a barrel of salmon cost 3 dollars
and 2 sevenths of a dollar, what is that a barrel ?
15. 4 and 3 sevenths is 1 fifth of what number ?
16. If a man can travel 4 miles and 3 sevenths of
a mile in 1 ninth of a day, how far will he travel in
a whole day ?
17. 5 and 3 sevenths is 1 ninth of what number?
18. 2 and 4 fifths is 1 seventh of what number ?
19. 6 and 3 eighths is 1 eighth of what number ?
20. 7 and 5 sixths is 1 third of what number?
21. 8 and 3 sevenths is 1 fourth of what number?
22. 5 and 3 tenths is 1 seventh of what number ?
23. 8 and 5 ninths is 1 fifth of what number ?
24. 9 and 4 fifths is 1 eighth of what number ?
25. 6 and 4 tenths is 1 tenth of what number ?
26. 7 and 4 ninths is 1 fifth of what number?
27. 8 and 7 tenths is 1 seventh of what number ?
B. 1. If 2 thirds of a barrel of beef cost 3 dollars,
what does 1 third of a barrel cost ?
2. 3 is 2 times what number ? *
* See Sect. VI., Art. B.
SECT. 11.] ARITHMETIC. 101
Ans. 3 is 2 times the half of 3; but 1 half of 3 is
3 halves, or 1 and 1 half;* therefore 3 is 2 times 1
and 1 half.
3. If 3 fifths of a yard of cloth cost 4 dollars, what
will 1 fifth of a yard cost ?
4. 5 is 3 times what number ?
5. If 3 sevenths of a barrel of pork cost 2 dollars,
what will 1 seventh ,of a barrel cost ?
6. 2 is 3 times what number ?
7. If 4 thirds of a bunch of shingles cost 5 dol-
lars, what does 1 third of a bunch cost ?
8. 7 is four times what number ?
9. 2 is 4 times what number ?
10. A man bought 4 ninths of a barrel of flour for
three dollars; what would be the price of 1 ninth
of a barrel at the same rate ?
11. 5 is 4 times what number?
12. A man bought 5 eighths of a hundred weight
of sugar for 6 dollars ; what would 1 eighth of a
hundred weight cost at the same rate ?
13. 7 is 5 times what number ?
14. 3 is 5 times what number?
15. 8 is 5 times what number ?
16. 9 is 4 times what number ?
17. 11 is 6 times what number ?
18. 13 is 7 times what number?
19. 14 is 8 times what number?
29. 17 is 5 times what number ?
21. 18 is 8 times what number?
22. 17 is 9 times what number?
23. 15 is 10 times what number?
24. 20 is 9 times what number ?
25. 22 is 10 times what number ?
26. 24 is 7 times what number ?
27. If 2 thirds of a barrel of beef cost 3 dollars,
* See Sect. X., Art. A.
9*
102 ARITHMETIC. [PART 1.
what will 1 third of a barrel cost ? What will the
whole barrel cost?
28. * If 5 is 2 thirds of some number, what is 1
third of the same number? 2 and 1 half is 1 third of
what number ? Then 5 is two thirds of what number?
29. If 3 fourths of a barrel of flour cost 5 dollars,
what will 1 fourth of a barrel cost ? What will the
whole barrel cost?
30. If 8 is 3 fourths of some number, what is 1
fourth of the same number ? 2 and 2 thirds is 1 fourth
of what number ? Then 8 is 3 fourths of what number ?
31. A man bought 2 sevenths of a barrel of oil for
5 dollars ; how much will 1 seventh cost at the same
rate ? how much would a barrel cost ?
32. If 9 is 2 sevenths of some number, what is 1 sev-^
enth of the same number ? 4 and 1 half is 1 seventh o£'
what number ? Then 9 is 2 sevenths of what number ?
33. Bought 5 eighths of a chaldron of coal for 7
dollars ; what is the price of 1 eighth, at the same
rate ? What is the price of the whole chaldron ?
34. If 12 is 5 eighths of some number, what is 1
eighth of the same number? 2 and 2 fifths is 1
eighth of what number? Then 12 is 5 eighths of
what number ?
35. If a man can do 7 ninths of a piece of work
in 4 days, how long would it take him to do 1 ninth
of it ? How long would it take him to do the whole ?
36. If 3 is 7 ninths of some number, what is 1
ninth of the same number ? 3 sevenths is 1 ninth of
what number ? Then 3 is 7 ninths of what number ?
37. If 3 fifths of a chaldron of coal cost 8 dollars,
what is the whole chaldron worth ?
38. 7 is 3 fifths of what number?
39. A man bought 6 sevenths of a cask of raisins
for 5 dollars ; what was the whole cask worth ?
40. 8 is 6 sevenths of what number ?
* See Sect. VI., Art. B.; Examples 8 and following.
SECT. 11.] ARITHMETIC. 103
41. A man had 4 sixths of a week's board for 3
dollars ; how much is that for a whole week ?
42. 3 is 5 sixths of what number ?
43. 9 is 5 eighths of what number ?
44. 10 is 7 thirds of what number?
45. 1 1 is 6 fifths of what number ?
46. 12 is 7 fifths of what number?
47. 15 is 2 sevenths of what number ?
48. 17 is 3 tenths of what number?
49. A man bought 5 eighths of a cask of wine for
19 dollars; what would the whole cask cost?
50. 19 is 3 eighths of what number?
51. 21 is 4 fifths of what number ?
52. Bought 5 ninths of a ton of logwood for 23
dollars ; what would a ton cost at that rate ?
J53. 23 is 7 ninths of what number ?
54. 21 is 3 tenths of what number ?
55. 29 is 4 sevenths of what number ?
56. 31 is 5 eighths of what number ?
57. 33 is 6 sevenths of what number ?
58. 38 is 4 ninths of what number ?
59. A man bought 1 barrel and 1 seventh of a
barrel of wine for 41 dollars; what was that a bar-
rel?
NOTE. 1 barrel and I seventh of a barrel is the
same as 8 sevenths of a barrel If 8 sevenths of a
barrel cost 41 dollars, what docs a barrel cost 1
60. 35 is 8 sevenths of what number ?
61. A man bought 1 ton and 4 fifths of a ton (that
is, 9 fifths of a ton) of fustic for 43 dollars ; what
was that a ton ?
62. 52 is 9 fifths of what number ?
63. Bought 1 ton and 2 sevenths of a ton of log*
wood for 48 dollars ; what was that a ton ?
64. 67 is 9 sevenths of what number ?
65. 53 is 7 tenths of what number ?
66. 58 is 9 eighths of what number ?
104 ARITHMETIC. [PART 1.
67. 61 is 10 ninths of what number ?
68. Bought 2 barrels and 1 fifth of a barrel of gin
(that is, 11 fifths of a barrel) for 65 dollars; what
was that a barrel ?
69. 65 is 9 fifths of what number ?
70. 71 is 8 elevenths of what number?
C* 1. A boy gave away 2 apples and 1 half,
which was 1 fourth of all he had ; how many had
he?
2. A man gave away 3 dollars, which was 2 fifths
of all the money he had ; how much had he ?
3. A man sold a cow for fifteen dollars, which
was 4 fifths of what she cost him ; how much did
he lose by the bargain ?
4. A man sold a piece of cloth for 37 dollars,
which was 9 eighths of what it cost him; how
much did he gain by the bargain ?
5. There is a pole 3 fifths under water, and 7
feet out of the water ; how long is the pole ?
6. A man sold a piece of cloth for 47 dollars, by
which bargain he lost 2 ninths of what the cloth
cost him ; how much did it cost him, and how much
did he lose?
Miscellaneous Examples.
1. If a staff 5 feet long cast a shadow 4 feet at 12
o'clock, what is the length of a pole that casts a
shadow 67 feet at the same time ?
2. If 53 gallons of water, in 1 hour, run into a
cistern containing 97 gallons, and 44 gallons run out
in an hour, in what time will it be filled ?
3. A man bought a cask of wine containing 75
gallons ; 2 sevenths of it leaked out, and he sold the
remainder for 1 dollar a gallon ; how much did he
sell it for ?
SECT. 11.] ARITHMETIC. 105
4. A cask of wine cost 67 dollars; what is 5
eighths of it worth ?
5. A man bought 9 oranges for 6 cents and 2 sev-
enths apiece, and sold them for 67 cents ; what did
he gain by the bargain ?
6. A man bought 10 yards of broadcloth for 70
dollars; how must he sell it per yard in order to
gain 14 dollars 1
7. If, when the days are 12 hours long, a man
perform a journey in 3 days, how many hours is he
in performing it 1
8. If a man perform a journey in 36 hours, how
many days would he be in performing it, when the
days are 9 hours long ?
9. If, when the days are 11 hours long, a man can
perform a journey in 5 days, in how many hours will
he perform it 1 In how many days when the days are
9 hours long ?
10. What number added to 2 fifths of 33 will
make the number 17?
11. How many yards of cloth, that is 1 quarter of
a yard wide, will line 10 yards that is 3 quarters
wide?
12. 8 yards of cloth, that is 1 quarter wide, are
equal to how many yards that is 4 quarters wide ?
13. How many yards of cloth, that is 3 quarters
wide, are equal to 7 yards that is 5 quarters wide ?
14. How many yards of cloth, that is 6 quarters
wide, are equal to 37 that is 4 quarters wide ?
15. If a piece of cloth 5 quarters wide be worth
37 dollars, what is a piece of the same length, 3
quarters wide, worth?
16. If cloth 4 quarters wide is worth 8 dollars a
yard, what is 1 yard of the same kind of cloth, that
is 5 quarters wide, worth ?
106 ARITHMETIC. [PART 1.
SECTION XII.
PARTS of one are called fractions. Fractions may
be expressed by figures, as well as whole numbers.
It requires two numbers to express a fraction; one
to show into how many parts one is divided, and
the other to show how many of those parts are
used. For example, if we wish to express one half,
(which means that one is divided into two equal
parts, and that one part is used,) we must use the
figure 2 to express that one is divided into two equal
parts, and the figure 1 to show that one part is
used. And these must be written in such a man-
ner that we may always know what each of them
is intended to express.
One half is usually written thus, % ; one number
above a line, and the other below it. The number
below the line shows into how many parts one is
divided, and the number above the line shows how
many parts are used.
One third is written -J-
Two thirds f
One fourth £
Three fourths |
Two fifths |
Example. $ of an apple signifies that the apple is
to be cut into 7 equal parts, and that 3 parts are
to be used.
Let us apply an example to Plate II. | refers to
a square divided into 8 parts, and signifies that 5
parts are to be used.
We may observe, that, when one is divided into
3 parts, the parts are called thirds ; when one is
divided into 4 parts, the parts are called fourths, &c. ;
that is, the fraction takes its name from the number
of parts into which one is divided. The number
SECT. 12.] ARITHMETIC. 107
under the line is called the denominator, because
it gives name to the fraction ; and the number
above the line is called the numerator, because it
shows the number of parts used. Thus -£$, 10 is
the denominator, and 3 the numerator.
N. B. The pupil must be made familiar with
this mode of expressing fractions, and must be able
to apply it to any familiar objects, as apples, oran-
ges, &/c., and to the table, before he is allowed to
proceed any farther. Particular care must be taken
to make him understand what the denominator sig-
nifies, and what the numerator, as explained above.
The denominator should always be explained first.
The following examples are a recapitulation of
some of the foregoing sections, for the purpose of
showing the application of the above method of
writing fractions.
See Section VIII. A.
A. 1. In 2 how many times £ ? Ans. £.*
2. In 3 how many times £ ? Ans. f .
3. In 2 how many times ^ 1 Ans. f .
4. In 4 how many times -J ?
5. In 6 how many times £ 1
6. In 7 how many times £ ?
7. In 8 how many times -£ ?
8. In 2£f how many times ^ ?
9. In 3£ how many times £ 1
10. Reduce 4-1 to an improper fraction.^
11. Reduce 3f to an improper fraction.
12. Reduce 5f to an improper fraction.
* When the numerator is larger than the denominator, the fraction
js called ati improper fraction.
t 2£ is read two and 1 half. It is called a mixed number.
\ That is, to find how many fifths there are in four and 1 iAh.
108 ARITHMETIC. [PART 1.
13. Reduce 6| to an improper fraction.
14. Reduce 8T3^ to an improper fraction.
15. Reduce 9^ to an improper fraction.
B. I. £ are how many times 1 ?
2. f are how many times 1 ?
3. \ are how many times 1 ?
4. £ are how many times 1 ?
5. f are how many times 1 ?
6. JT& are how many times 1 ?
7. Jjfi- are how many times 1 ?
8. 2f- are how many times 1 ?
9. z/- are how many times 1 ?
10. £J are how many times 1 ?
See Section IX.
A. 1. How much is 3 times
2. How much is 4 times ^ ?
3. How much is 3 times f ?
4. How much is 4 times ^ ?
5. How much is 5 times f ?
6. How much is 6 times f ?
7. How much is 8 times
8. How much is 9 times
9. How much is 10 times ^?
10. How much is 9 times f ?
B. 1. How much is 3 times
2. How much is 4 times 3| ?
3. How much is 5 times 6|^ ?
4. How much is 6 times 4 j ?
5. How much is 7 times 5f ?
6. How much is 8 times 6f ?
7. How much is 4 times lO
SECT. 12.] ARITHMETIC. 109
8. How much is 9 times 7 1 ?
9. How much is 8 times 9f ?
10. How much is 10 times 7-| ?
See Sections V. & X.
1. What is £ of 6 ? 2. What is J of 6?
3. What is | of 8 ? 4. What is | of 9 ?
5. What is f of 9? 6. What is | of 10?
7. What is f of 14 ? 8. What is | of 5 ?
9. What is f of 5 ? 10. What is £ of 7 ?
11. What is f of 7 ? 12. What is f of 35 ?
13. What is -i of 17? 14. What is f of 26?
15. What is I of 27 ? 16. What is | of 37 ?
17. What is f of 47 ? 18. What is f of 42 ?
19. What is f of 65 ? 20. What is f of 75 ?
See Sections VI. & XI.
A. 1. 2 is £ of what number?
2. 4 is £ of what number ?
3. 8 is £ of what number ?
4. Ij- is ± of what number ?
5. 2f is -J- of what number ?
6. 4f is ^ of what number ?
7. 6|- is -f of what number ?
8. 74 is % of what number ?
9. 8| is £ of what number?
10. 9T^j- is T\j of what number ?
B. 1. 4 is f of what number?
2. 6 is f of what number ?
3. 8 is ^ of what number ?
10
110 ARITHMETIC. [PART 1.
4. 12 is ^ of \fhat number ?
5. 15 is f of what number ?
6. 18 is f- of what number ?
7. 20 Is f of what number ?
8. 24 is f of what number ?
9. 28 is J of what number ?
10. 30 is T&J of what number ?
11. 3 is § of what number ?
12. 4 is £ of what number ?
13. 5 is f of what number ?
14. 8 is g of what number ?
15. 9 is ^ of what number ?
16. 17 is f of what number ?
17. 25 is | of what number?
18. 38 is f of what number ?
19. 43 is | of what number ?
20. 54 is -y*- of what number ?
Miscellaneous Examples.
1. A man sold 8 yards of cloth for 3f dollars a
yard ; what did it come to?
2. A man sold a horse for 76 dollars, which was
•§• of what it cost him ; how much did it cost him ?
3. A man sold £ of a gallon of wine for 40 cents ;
what was that a gallon ?
4. If it will take If yards of cloth to make a coat,
how many yards will it take to make 7 coats ?
5. If 1 horse consume 3^ bushels of oats in 2
days, how much would 2 horses consume in 5 days ?
6. If, when the days are 9£ hours long, a man per-
form a journey in 10 days, in how many days would
he perform it when the days are 12 hours long?
7. A man sold 8 yards of cloth for 7-| dollars a
yard, and received 8 firkins of butter at 6f dollars
a firkin ; how much was then due to him ?
8. Two men are 38 miles apart, and are travel-
SECT. 13.1 ARITHMETIC. Ill
ling towards each other, one at the rate of 3 miles an
hour, the other 2 miles ; how much do they approach
each other in an hour ? How much in 2 hours ? In
how many hours will they meet ? At what distance
from each place from which they set out ?
SECTION XIII.
A. 1. If you give ± of an orange to one boy, and
£ to another, how much more do you give the first
than the second]
2. ± of an orange is how many £ of an orange?
3. If you give £ of an orange to one boy, and £
to another, how many % would you give away 1
How many £ would you have left ?
4. $ and £ are how many £?
5. A man gave to one laborer ^ of a bushel of
wheat, and f to another ; how many % of a bushel
did he give to both ? How many bushels ?
6. J- and | are how many £ ? How many times 1 ?
7. A man gave £ of a barrel of flour to one man,
and f of a barrel to another ; to which did he give
the most ?
8. £ is how many £?
9. A man bought -£ of a bushel of wheat at one
time, and f of a bushel at another ; at which time
did he buy the most?
10. £ is how many ^ ?
11. A man bought | of a yard of cloth at one
time, and £ of a yard at another ; at which time
did he buy the most ?
12. f are how many £?
13. A man wished to give'£ of a bushel of wheat
112 ARITHMETIC. [PART 1.
to one man, and $ of a bushel to another ; but he
could not tell how to divide it. Another man, stand-
ing by, advised him to divide the whole bushel into
six equal parts first, and then take £ of them for
one, and -J of them for the other. How many parts
did he give to each 1 How many to both ? How
many had he left?
14. £ is how many £ ? £ is how many £ ? £ and
J are how many £?
15. A man, paying some money to his laborers,
gave each man £ of a dollar, and each boy £ of a
dollar ; how much more did he give to a man than
to a boy?
16. What is the difference between £ and £?
17. If a man can earn £ of a dollar in a day, and
a boy £ of a dollar, how much does the man earn
more than the boy?
18. What is the difference between £ and £?
19. A boy, distributing some nuts among his com-
panions, gave £ of a quart to one, and £ of a quart
to another; how much more did he give to one
than to the other ?
NOTE. Change them to sixths.
20. What is the difference between -J and £?
21. A man, having two bushels of grain to dis-
tribute among his laborers, wished to give £ of a
bushel to one, and f of a bushel to another, and the
rest to a third ; but was at a loss to tell how to divide
it ; at last he concluded to divide each bushel into
six equal parts, or sixths, and then to distribute
those parts. How many sixths did he give to each ?
22. § is how many £?
23. A man had a horse, and a cow, and a sheep.
The horse would eat § of a load of hay in the win-
ter, the cow J-, and the sheep ^. How many £ of a
load would each eat ? How many £ would they all
eat ? How many loads ?
SECT. 13.] ARITHMETIC. 113
24. A boy, having a quart of nuts, wished to di-
vide them so as to give one companion £, another
J. and a third £ of them ; but, in order to make a
proper division, he first divided the whole into
eight equal parts, and then he was able to divide
them as he wished. How many eighths did he
give to each! How many eighths had he left for
himself?
25. £ is how many £? £ is how many £? £ and
£ and £ are how many £?
26. A man gave f of a barrel of flour to one
man, and f of a barrel to another; to which did he
give the most ? How much ?
27. Which is the larger, £ or |? How much the
larger ?
28. A boy, having a pound of almonds, said he
intended to give £ of them to his sister, and £ to
his brother, and the rest to his mamma. His mam-
ma, smiling, said she did not think he could divide
them so. O yes, I can, said he ; I will first divide
them into twelve equal parts, and then I can divide
them well enough. Pray how many twelfths did
he give to each ?
29. 4- is how many y1^ 1 £ is how many T^- ? £
and £ are how many yL- ?
30. Mr. Goodman, having a pound of raisins, said
he would give Sarah £, and Mary £, and James £
of them ; and he told Charles he should have the
rest, if he could tell how to divide them. Well, said
Charles, I would first divide the whole into twelve
equal parts, and then I could take £, and £, and £
of them. How many twelfths would each have?
31. •£ and £ and £ are how many y1^?
32. George bought a pine-apple, and said he
would give £ of it to his papa, and f to his mamma,
and y3^ to his brother James, if he could divide it.
James took it, and cut it into twenty equal pieces,
JO*
114 ARITHMETIC. [PART 1.
and then distributed them as George had desired.
How many twentieths did he give to each ?
33. £ is how many -fa? -J- is how many 2^? f is
how many fa ? fa is how many -fal
34. £ is how many -
35. £ is how many
36. £ is how many
37. £ is how many
38. f are how many £ ?
39. § are how many
40. £ is how many T
41. £ are how many
42. £ are how many
43. | are how many
44. ^ are how many -
45. |- are how many
46. f are how many
47. y3^ are how many ^j ?
48. Reduce £ to sixths and £ to sixths*
49. f and § are how many £?
50. Reduce £ and £ to eighths.
51. £ and £ are how many £?
52. \ and |- are how many J?
53. f and £ are how many £ ?
54. |^ and \ are how many £?
55. |- and f are how many £ ?
56. £ and f are how many T\j ?
67. £ and £ and £ are how many £ ?
58. £ and £ and -J^ are how many -fa 1
59. § and ^ are how many -j-1^ ?
60. § and £ and £ are how many -fa ?
6L f and -fa and £ are how many -fa ?
€2. £ and | and f and £ and ^- are how many fa t
63. £ and f are how many TJ¥ ?
64. § and -J are how many y1^ ?
65. | and f are how many T^ ?
€6. f less J are how many £ 7
SECT. 13.] ARITHMETIC. 115
67. f and £ less T5^, are how many T^ ?
68. £ less f are how many rJF ?
69. f less f- are how many ^ ?
70. | less f are how many ^ ?
71. £, and £ , and £, and T5^, less |, are how many
&<
72. £, and £, and f , and TV, and ^, less J, are
how many ^ ?
73. | and f are how many -^1
74. f and ^ are how many -Jg- ?
75. |- and f are how many ^?
When the denominators in two or more frac-
tions are the same, the fractions are said to have a
common denominator. Thus f and | have a com-
mon denominator. We have seen that, when two
of more fractions have a common denominator, they
may be added and subtracted as well as whole
numbers. We add or subtract the numerators, and
write their sum or difference over the common
denominator. The first part of the process in the
above examples was to reduce them to a common
denominator.
76. Reduce f and f to a common denominator.
NOTE. They may be reduced to twelfths.
If it cannot be immediately seen what number
must be the common denominator, it may be found
by multiplying all the denominators together; for
that will always produce a number divisible by all
the denominators.
77. Reduce f and -| to a common denominator.
78. Reduce f and J and | to a common denom-
inator.
79. Reduce ^ and ^ to a common denominator.
8Q. Reduce ^ and f to a common denominator.
116 ARITHMETIC. [PART 1.
81. Reduce £ and f and £ to a common denomi-
nator.
82. Add together f and f .
83. Add together f and ^.
84. Add together f and ^g-.
85. Add together £ and £ and -&.
86. Subtract % from £.
87. Subtract -fa from f .
88. Subtract | from §.
89. Subtract } from £.
B* 1. Mr. F. said he would give £ of a pine-ap-
ple to Fanny, and f to George, and the rest to the
one that could tell how to divide it, and how much
there would be left. But neither of them could tell ;
so he kept it himself. Could you have told, if you
had been there ? How would you divide it ? How
much would be left ?
2. A man sold l£ bushels of wheat to one man,
4J bushels to another ; how many bushels did he
sell to both ?
3. A man bought 6J bushels of wheat at one
time, and 2£ at another ; how much did he buy in
the whole ?
4. A man bought 7f yards of one kind of cloth,
and 6J yards of another kind ; how many yards in
the whole ?
5. A man bought f of a barrel of beer at one
time, 2^ barrels at another, and 6f at another ; how
much did he buy in the whole ?
6. A man bought one sheep for 4§ dollars, and
another for 5^ dollars ; how much did he give for
both?
7. There is a pole standing, so that ^ of it is in
the mud, and f of it in the water, and the rest out
of the water; how much of it is out of the
water?
SECT. 13.] ARITHMETIC. 117
8. A man, having undertaken to do a piece of
work, did $ of it the first day, -J of it the second
day, and £ of it the third day ; how much of it did
he do in three days ?
9. A man having a piece of work to do, hired
two men and a boy to do it. The first man could
do | of the work in a day, and the other £ of it,
and the boy ^ of it ; how much of it would they all
do in a day ?
C. It will be seen, by looking on Plate III. thatf
is the same as £, and that f is the same as £, and
that |j- is the same as f ; f- , f , can therefore be re-
duced to £, and | to §. This is called reducing
fractions to their lowest terms.
1. Reduce f to its lowest terms.* Ans. f .
2. Reduce T5^ to its lowest terms.
3. Reduce f to its lowest terms.
4. Reduce -fa to its lowest terms.
5. Reduce ^J to its lowest terms.
6. Reduce T5T to its lowest terms.
7. Reduce -f^ to its lowest terms.
8. Reduce ^f to its lowest terms.
9. Reduce -J^- to its lowest terms.
10. Reduce ^ to its lowest terms.
11. Reduce f J to its lowest terms.
12. Reduce §i to its lowest terms.
13. Reduce £|. to its lowest terms.
14. Reduce f J to its lowest terms.
Note. It will be seen by the above section, that,
if both the numerator and denominator be multi-
plied by the same number, the value of the fraction
will not be altered ; or, if they can both be divided
by the same number without a remainder, the frac-
tion will not be altered.
* If this article should be found too difficult for the pupil, he may
omit it till after the next section.
118 ARITHMETIC. [PART 1.
SECTION XIV.
A. 1. A BOY, having £ of an orange, gave away £
of that ; what part of the whole orange did he give
away?
2. What is £ of J?
3. If you cut an apple into three pieces, and then
cut each of those pieces into two pieces, how many
pieces will the whole apple be cut into ? What part
of the whole apple will one of the pieces be ?
4. What is £ of £ ?
5. A boy had £ of a pine-apple, and cut that half
into three pieces, in order to give away -J of it.
What part of the whole apple did he give away ?
6. What is £ of £ 1
7. If an orange be cut into 4 parts, and then
each of the parts be cut in two, how many pieces
will the whole be cut into?
8. What is £ of £?
9. A man, having £ a barrel of flour, sold £ of
that • how much did he sell ?
10. What is ± of £?
11. If an orange be cut into 4 equal parts, and
each of those parts be cut into 3 equal parts, how
many parts will the whole orange be cut into?
12. What is £ of £ ?
13. A boy, having ^ of a quart of chestnuts,
gave away £ of what he had. What part of the
whole quart did he give away ?
14. What is £ of £?
15. What is | off 1
16. A man, owning -J of a ship, sold -J of his share ;
what part of the ship did he sell, and what part did
he then own ?
17. What is £ of |?
18. What is £ of £?
SECT. 14.] ARITHMETIC. 119
19. What is £ of |?
20. What is £ of £?
21. What is ^ of |?
22. What is £ of £ ?
23. What is £ of | ?
24. What is £ of £ ?
25. What is | of |?
26. What is -i of |?
27. What is ^ of £?
28. What is | of £ ?
29. A boy, having f of an orange, (that is, 2
pieces,) gave his sister £ of what he had; how
many thirds did he give her ?
30. What is £ of §?
31. A boy, having f of a pine-apple, said he
would give one half of what he had to his sister, if
she could tell how to divide it. His sister says, You
have got j-, or three pieces ; if you cut them all in
two, you can give me J- of them. But £ of £ is £ ;
therefore I shall have f of the whole pine-apple.
32. What is £ of £?
33. A man, owning j- of a share in the Boston
Bank, sold -J- of his part; what part of a share .did
he sell ?
34. What is £ of f?
35. A man, owning f of a ship, sold £ of his
share; what part of the whole ship did he sell!
What part had he left?
36. What is £ of £ ?
37. What is £ of |?
38. What is j off?
39. What is | of f ?
40. What is | of J^?
41. What is£of f ?
42. A man, owning f of a share in the Boston
Bank, sold £ of his part; what part of a whole
share did he sell ?
120 ARITHMETIC. [PART 1.
43. What is £ of £ ?
44. What is J of f ?
45. A boy, having % of a water-melon, wished to
divide his part equally between his sister, his bro-
ther, and himself, but was at a loss to know how to
do it ; but his sister advised him to cut each of the
fifths into 3 equal parts. How many pieces did
each have ? and what part of the whole melon was
each piece ?
46. What is £ of £ ?
47. What is £ of f?
48. What is | of | ?
49. What is | of | ?
50. What is TV of |?
51. What is 4 of J?
52. What is f of £?
53. What is | of |?
54. What is J of f?
55. What is £ of f ?
56. What is § of | ?
57. What is | of | ?
58. What is f of f?
59. What is £ of f?
60. What is | of f?
61. What is} of f ?
62. What is $ of £ ?
63. What is | of f 1
64. What is T^ of f ?
65. Whatis-^of f?
66. What is f of -3% ?
67. What is I of f ?
68. What is f of f ?
69. If a yard of cloth cost 2£ dollars, what will
£ of a yard cost ?
' 70. What is £ of 2J?
71. A boy had 2£ oranges, and wished to give £
of them to his sister, and £ to his brother ; but he
SECT. 14.] ARITHMETIC. 121
did not know how to divide them equally. His bro-
ther told him to cut the whole into halves, and
then cut each of the halves into 3 pieces. What
part of a whole orange did each have ?
72. What is | of 2i?
73. A man bought 4 bushels of corn for 3§ dol-
lars; what part of a dollar did 1 bushel cost?
Change the 3f to thirds, and then find ^ of ^ as
above.
74. What is £ of 5f?
75. If 5 bushels of wheat cost 7f dollars, what
is that a bushel ?
76. What is £ of 7f?
77. A man bought 6 gallons of brandy for 8f dol-
lars ; what was that a gallon ?
78. What is £ of 8£ ?
79. A man bought 7 gallons of wine for 8f dol-
lars ; how much was that a gallon ?
80. What is | of 8f?
81. A man bought 10 pieces of nankin for 6|
.^dollars; how much was it a piece?
82. What is TV of 6f?
83. If 9 bushels of rye cost 7f dollars, what is
that a bushel ?
84. What is j. of 7§ 1
85. What is i of 5f ?
86. What is £ of 8|?
87. What is £ of 6^ ?
88. What is | of 9f ?
89. A man bought 7 yards of cloth for 18f dol-
lars; what was that a yard? What would 3 yards
cost at that rate ?
90. What is | of 18f ? What is f of 18| ?
91. A man bought 5 barrels of cider for 27f dol-
lars ; what was it a barrel ? What would 7 barrels
cost at that rate ?
92. What is i of 27f ? What is £ of 27f 1
122 ARITHMETIC. [PART 1.
93. If 6 barrels of flour cost 38f dollars, what
would 10 barrels cost at that rate ?
94. What is Y-of 38£?
B. 1. A man bought a piece of cloth for 42f dol-
lars, and was obliged to sell it for £ of what it cost
him ; how much did he lose ?
2. A man bought a quantity of flour for 53^ dol-
lars, and sold it for f of what it cost him ; how
much did he gain ?
3. If 7 men can do a piece of work in 4f days,
how long will it take 1 man to do it? How long
will it take 3 men to do it?
4. If 4 men can do a piece of work in 9f days,
how long would it take to do it if 7 men were em-
ployed ?
5. There is a pole standing so that ^ of it is in
the water, and £ as much in the mud ; how much
is in the mud ?
6. If a man can travel 13$- miles in 3 hours, how
many miles will he travel in 8 hours?
7. If 5 horses will eat 26|- loads of hay in a year,
what will 8 horses eat in the same time?
8. If 4 cocks will empty a cistern in 6£ hours,
how long will it take 7 cocks of the same size to
empty it ?
SECTION XV.
A. 1. A BOY, having 2 oranges, wished to give £ of
an orange apiece to his playmates ; how many could
he give them to ? If he had given § of an orange
apiece, how many could he have given them to ?
2. How many times ^ are there in 2 ? How many
times f are there in 2 ?
SECT. 15.] ARITHMETIC. 193
3. A man, having 3 bushels of corn, distributed it
among some poor persons, giving them £ of a bush-
el each; to how many did he give it ?
NOTE. Find first how many he would have given it
to, if he had given % of a bushel to each.
4. In 3 are how many times £ 1 How many times £ ?
5. If f of a barrel of flour will last a family one
month, how long will 4 barrels last the same family ?
How long will 6 barrels last? How long will 10
barrels last?
6. How many times is f- contained in 4? How
many times in 6 ? How many times in 10 ?
7. If £ of a bushel of wheat will last a family
one week, how many weeks will 6| bushels last the
same family ?
8. How many times is j- contained in 6£?
9. There is a cistern having a cock which will
fill it in -| of an hour ; how many times would the
cock fill the cistern in 3f hours ?
10. How many times is f contained in 3f ?
11. How much cloth, at l£ dollars (that is, f dol-
lars) a yard, can be bought for 4 dollars ?
12. How many times is 1^- or f contained in 4 ?
13. A man distributed 8£ bushels of wheat among
some poor persons, giving 1 ± bushels to each ; how
many did he give it to 1
14. How many times is 1 £ contained in 8^- ?
15. If a soldier is allowed 1£ pounds (that" is, § of
a pound) of meat in a day, to how many soldiers
would 6f pounds be allowed ?
16. How many times is 1-J- contained in 6f ?
17. If If tons of hay will keep a horse through
the winter, how many horses will 10 tons keep ?
18. How many times is If contained in 10 ?
19. At 2£ dollars a box, how many boxes of rai-
sins can be bought for 10 dollars?
20. How many times is 2£ contained in 10?
134 ARITHMETIC. [PART 1.
21. At 1$ dollars a pound, how many pounds of
indigo can be bought for 9f dollars?
22. How many times is If contained in 9f ?
23. At If dollars a barrel, how many barrels of
raisins can be bought for 9^ dollars ?
24. How many times is If contained in 9£?
25. At J of a dollar a piece, how many pieces of
nankin can be bought for 8 J dollars ?
26. How many times is J contained in 8$ ?
27. At £ of a dollar a pound, how many pounds
of tea can be bought for 7f dollars ?
28. How many times is % contained in 7f ?
29. How many times is 3£ contained in 7f ?
30. How many times is 5£ contained in 17 ?
31. How many times is 4£ contained in 9f ?
32. How many times is 3^ contained in 12f ?
11. 1. At^of a dollar a pound, how many pounds
of meat can be bought for £ of a dollar ?
NOTE. Change £ to tenths.
2. How many times is -fa contained in £?
3. A man, having £ of a barrel of flour, distributed
it among some poor persons, giving them £ of a
barrel apiece ; how many did he give it to?
NOTE. Change both fractions to twelfths; that is,
reduce them to a common denominator.
4. How many times is £ contained in f ?
5. If a pound of almonds cost ^ of a dollar, how
many pounds can be bought for f of a dollar?
NOTE. Reduce the fractions to a common denomi-
nator.
6. How many times is ^ contained in § ?
7. If a piece of nankin cost f of a dollar, how
many pieces can be bought for 4f dollars ; that is,
Y- dollars ?
8. How many times is § contained in 4 j^ ?
9. If a bushel of barley cost f of a dollar, how
PART 1.] ARITHMETIC. 125
many bushels can be bought for £ of a dollar ? How
many for If dollar?
10. How many times is & contained in £ ? How
many times in If 1
11. How many times is £ contained in f ?
12. How many times is f- contained in ^1
TABLES OF COINS, WEIGHTS, AND MEASURES.
Table I. — FEDERAL MONEY.
10 mills make 1 cent.
10 cents " 1 dime.
10 dimes " 1 dollar.
10 dollars " 1 eagle.
Table II. — STERLING MONEY.
4 farthings, q.
12 pence
20 shillings
6 shillings
28 shillings
make
«
t(
u
ft
1 penny.
1 shilling.
1 pound.
1 dollar.
1 guinea.
s.
£.
Table III.— TROY WEIGHT.
24 grains, gr. make 1 pennyweight dwt.
20 pennyweights " 1 ounce. oz.
12 ounces " 1 pound. Ib.
Table IV. — AVOIRDUPOIS WEIGHT.
16 drams, dr. make 1 ounce. oz.
16 ounces " 1 pound. Ib,
S8 pounds qr.
126
ARITHMETIC.
[PART 1,
4 quarters make 1
20 hundred weight " 1
hundred weight,
ton.
cwt.
T.
2£ inches
4 nails
4 quarters
3 quarters
5 quarters
6 quarters
Table V. — CLOTH MEASURE.
make
1 nail. nl.
1 quarter of a yard. qr.
1 yard. y.
1 ell Flemish.
1 ell English.
1 aune, or ell French.
4 gills
2 pints
4 quarts
3l£ gallons
63 gallons
2 hogsheads
2 pipes
Table FJ.— WINE MEASURE.
make
1 pint.
1 quart.
1 gallon.
1 barrel.
1 hogshead.
1 pipe.
1 tun.
Table VII. — DRY MEASURE.
2 pints
8 quarts
4 pecks
make
1 quart.
1 peck.
1 bushel.
Table F///.— MEASURE OF TIME.
60 seconds, sec. make 1 minute.
60 minutes " 1 hour.
24 hours " 1 day.
7 days " 1 week.
4 weeks " 1 month.
13 months 1 day and \
6 hours, or 365 V " 1 year,
days and 6 hours .;
pt.
qt.
gal.
bar.
hd.
qt.
pk.
bu.
mm.
h.
d.
w.
m.
PART 1.] ARITHMETIC. 127
For convenience of reckoning, it is usual in calen-
dars to call the year 365 days for 3 successive years,
and every fourth year 366, (for in 4 years the six
hours' overplus amount to a day,) which is called
bissextile or leap year. This day is added to Feb-
ruary.
The common year is divided into twelve months,
which are sometimes called calendar months, be-
cause they are the months used in calendars.
The names of the months, and the number of
days in each, are as follows : —
Names. Number of days.
I ( 1. January 31
•J ( 2. February 28, in leap year 29.
ts. ( 3. March 31
f { 4. April 30
m ( 5. May 31
6. June 30
7. July 31
8. August 31
9. September ..... 30
10. October 31
•< I 11. November , 30
If 12.
M
December . . . . .31
Miscellaneous Examples.
1 . In 2 pounds how many ounces ?
2. In 8 yards how many quarters ?
3. In 3 quarters of a yard how many nails?
4. t3g. of a dollar is how many cents ?
5. How many farthings is | of a penny ?
6. How many pence is f of a shilling ?
7. f of a yard is how many quarters and nails?
138 ARITHMETIC. [PART 1.
8. In £ «£. how many shillings?
9. How much is £• of a shilling ?
10. How much is f of a bushel of wheat?
11. How much would $ of a barrel of wine cost,
at one dollar a gallon ?
12. How much would £ cwt. of sugar cost, at 8
cents a pound ?
13. How much is f of a day ?
14. How much is £ of a day ?
15. How much is f of a week ?
16. How much is f of an hour ?
17. How much would $ of a hogshead of wine
cost, at 2 dollars a gallon ?
18. If a man spend 28 dollars in a month, what
is that a week ? How much a day ?
19. If a man spend 16 dollars a week, what is
that a day ?
20. If a man buy 4 bushels of grain for 5 dollars,
how much is that a bushel ?
21. If wine is 2 dollars a gallon, how much is
that a pint ?
22. If you give 5 cents a gill for wine, what is
that a pint? What is it a quart? What is it a gal-
lon ?
23. If wine is worth 20 cents a pint, what is that
a gill ? What is it a quart ? What is it a gallon ?
24. If a yard of cloth is worth 7 dollars, what
are 2f yards worth ?
25. If a man earn 11 dollars a week, what is
that a day ? What for 3 days ? What for 4J days ?
26. If a man earn 2f dollars in a day, what will
he earn in a week?
27. What is $ of a hogshead of wine ?
28. 1 farthing is what part of a penny ?
29. 2 farthings is what part of a penny ?
30. 3 farthings is what part of a penny ?
81. 1 penny is what part of a shilling ?
PART 1.] ARITHMETIC. 129
32. 2 pence is what part of a shilling ?
33. 3 pence is what part of a shilling?
34. 5 pence is what part of a shilling ?
35. 6 pence is what part of a shilling ?
36. 7 pence is what part of a shilling?
37. 8 pence is what part of a shilling ?
38. 9 pence is what part of a shilling ?
39. 10 pence is what part of a shilling ?
40. 11 pence is what part of a shilling?
41. 1 shilling is what part of a pound ?
42. 2 shillings is what part of a pound ?
43. 3 shillings is what part of a pound ?
44. 4 shillings is what part of a pound 1
45. 5 shillings is what part of a pound ?
46. What part of a pound is 6 shillings? 7 shil-
lings? 8 shillings? 9 shillings ? 10 shillings? 11
shillings ? 12 shillings ? 13 shillings ? 14 shillings ?
15 shillings ? 16 shillings ? 17 shillings ? 18 shil-
lings ? 19 shillings ?
47. How many farthings are there in a shilling?
48. One farthing is what part of a shilling ?
49. 2 farthings is what part of a shilling? 3 far-
things? 4 farthings? 5 farthings? 6 farthings ? 7
farthings ? 8 farthings ? 9 farthings ? 10 farthings ?
50. How many pence are there in a pound ?
51. One penny is what part of a pound ?
52. What part of a pound is two pence ? 3 pence ?
4 pence ? 5 pence ? 6 pence ? 7 pence ? 8 pence ?
11 pence ? 15 pence ? 27 pence ? 35 pence?
53. How many pence are there in 1 shilling and
6 pence?
54. In 2 shillings and 4 pence, how many pence ?
55. In 4 shillings and 5 pence, how many pence ?
56. In 5 shillings and 8 pence, how many pence ?
57. In 9 shillings and 11 pence, how many pence ?
58. What part of l£ is 2s. 6d. ?
59. 3s. 5d. is what part of 1^?
130 ARITHMETIC. [PART 1.
NOTE. Reduce the whole to pence.
60. 7s. 8d. is what part of 1 £1
61. What is the price of 2 yards of cloth, at 3s.
4d. a yard ?
62. What will 8 yards of cloth cost, at 2s. 8d. a
yard?
63. What will 4 bushels of wheat cost, at 5s. 9d.
a bushel ?
64. What must you give for 4 barrels of cider, at
2£ dollars a barrel ?
65. If 3 bushels of wheat be divided between two
men, how much would they have apiece ?
66. If 4 bushels of corn be divided among 5 men,
how much would they have apiece ?
67. If 3 bushels of corn be divided among 7 men,
how much would they have apiece ?
68. How many nails are there in 1 yard ?
69. How many nails are there in 4 yards ?
70. How many nails are there in 5 yards and 2
nails ?
71. In 7 yards and 3 quarters, how many quarters ?
72. In 4 yards, 2 quarters, and 3 nails, how .many
nails ?
73. 1 nail is what part of a quarter ?
74. 3 nails is what part of a quarter ?
75. 1 nail is what part of a yard ?
76. What part of 1 yard is 3 nails ? 5 nails ? 7
nails ? 10 nails 1 15 nails I
77. In 8 quarters of a yard how many yards 1
78. In 12 quarters of a yard how many yards I
79. In 10 quarters of a yard how many yards ?
80. In 15 quarters of a yard how many yards ?
81. In 12 nails how many quarters of a yard?
82. In 16 nails how many quarters of a yard ?
How many yards?
83. In 24 nails how many quarters of a yard ?
How many yards ?
PART 1.] ARITHMETIC. 131
84. In 35 quarters of a yard how many yards ?
85. In 45 nails how many yards ?
86. In 63 nails how many yards?
87. At 2 cents a nail, what would 4 yards of cloth
cost?
88. At 2f dollars for 1 quarter of a yard, what
would 2 yards cost ?
89. 1 oz. is what part of a Ib. ?
90. What part of a Ib. is 2 oz. ? 3 oz. ? 4 oz. ? 5
oz.? 7oz.? 10 oz.? 15 oz.?
91. What part of a qr. of 1 cwt. is 1 Ib. ? 2 Ibs. ?
31bs.? 4 Ibs.? 7 Ibs.? 9 Ibs. ? J41bs. ? 18 Ibs. ?
23 Ibs. ?
92. At 3 cents for 1 oz. what would 1 Ib. cost ?
93. At 3 cents for 2 oz. what would 1 Ib. cost?
94. At 3 cents for 8 oz. what would 1 Ib. cost?
95. At 5 cents for 10 oz. what would 1 Ib. cost?
96. At 8 shillings for 4 Ibs. what would 10 Ibs.
cost?
97. If a man consume 1 Ib. and 3 oz. of meat in
a day, how much would he consume in a week ?
98. If a man spend 2f dollars in a day, how
much would he spend in a week ?
99. If a man travel 3f miles in an hour, how far
would he travel in 3 hours ? How far in 7 hours ?
How far in 12 hours ?
100. If 2 men start from the same place, and trav-
el in opposite directions, one at the rate of 3f miles
in an hour, and the other 4-i miles, how far will
they be apart at the end of 1 hour ? How far at the
end of 2 hours ? How far at the end of 3 hours ?
How far at the end of 7 hours ?
101. Two men start from the same place, and
travel the same way, one at the rate of 4-f miles in
an hour, the other at the rate of 4f miles in an hour ;
how far will they be apart at the end of 1 hour ?
How far in 2 hours ? How far in 5 hours ? How far in
132 ARITHMETIC. [PART 1.
10 hours? How far in 3 days, if they travel 10
hours in a day ?
102. How many yards of cloth, at 5 dollars a
yard, must be given for 8 barrels of flour, at 7 dol-
lars per barrel ?
103. What part of a month is 1 week ? 2 weeks ?
3 weeks ?
104. What part of a year is 1 month 1 2 months ?
3 months ? 4 months 1 5 months ? 6 months ? 7
months? 8 months? 9 months? 10 months? 11
months ?
105. What part of 1 month is 1 day ? 2 days T
3 days? 7 days? 8 days? 11 days? 15 days? 18
days ? 20 days? 24 days ? 37 days ?
106. If 5 bushels of oats will keep 7 horses
through the winter, how many bushels will it take
to keep 12 horses the same time ?
107. If you give 7 men 2-J bushels of corn apiece,
how many bushels would it take for the whole ?
108. A man, failing in trade, was able to pay his:
creditors only 4 shillings on a dollar ; how much
would he pay on 2 dollars ? How much on 3 dollars ?
How much on 7 dollars? How much on 10 dollars?
109. A man, failing in trade, is able to pay only 9
shillings on a pound ; how much would he pay on
a debt of 2 pounds ? How much on 3 pounds ? How
much on 12 pounds ?
110. A man, failing in trade, is able to pay only 4
shillings and 7 pence on a dollar ; how much would
he pay on a debt of 7 dollars?
111. If 6 dollars' worth of provisions will serve 3
men 5 days, how many days will it serve 1 man ?
How many days will it serve 2 men? How many
days will it serve 8 men ?
112. If 10 dollars' worth of provision will serve 7
men 4 days, how many days will it serve 9 men?
113. If 12 dollars' worth of provisions will serve 7
PART 1.] ARITHMETIC. 133
men 3 days, how many men would it serve 1 day?
How many 2 days ? How many 8 days?
114. If 11 dollars' worth of provision will serve 6
men 8 days, how many men will it serve 5 days ?
115. If 8 dollars' worth of provisions will serve 7
men 5 days, how many days would 16 dollars'
worth of provision last 4 men ?
116. If 1 peck of wheat afford 12 ten-penny
loaves, how many penny loaves may be obtained
from it ? How many two-penny loaves ? How many
three-penny loaves? How many seven-penny loaves ?
117. If 1 peck of wheat afford 11 eight-penny
loaves, how many ten-penny loaves will it afford?
118. A man, having hired some men and some
boys, agreed to give each man 3 shillings, and each
boy 2 shillings ; how much would it take to pay a
man and a boy ? How much 2 men and 2 boys ?
How much 7 men and 7 boys?
119. A man, having 18 shillings to pay among
his laborers, would give to every man 2 shillings, and
to every boy 1 shilling; the number of men and
boys was equal ; how many were there of each ?
120. A gentleman, having 50 shillings to pay
among his laborers, would give to every man 8
pence, and to every boy 4 pence ; the number of
men and boys was equal ; how many were there of
each ?
121. Two men bought a bushel of corn ; one gave
1 shilling, the other 2 shillings ; what part of the
whole did each pay ? What part of the corn must
each have ?
122. Two men bought a barrel of flour for 8
dollars; one gave 3 dollars, the other 5 dollars;
what part did each pay ? and what part must each
have?
123. Three men, A, B, and C, hired a garden ;
A paid 6 dollars, B 5 dollars, and C 9 dollars ; how
12
134 ARITHMETIC. [PART 1.
much did they all pay ? What part of the whole
did each pay? They sold the produce for forty
dollars ; what part of it must each have ? What
did each one's share amount to?
124. Three men bought a lottery ticket for 10
dollars; the first gave 3 dollars; the second 5 dol-
lars, and the third 2 dollars. They drew a prize of
120 dollars. What was each man's share ?
125. Two men hired a pasture for 58 dollars;
one put in 7 horses, and the other 3 horses ; what
ought each to pay ?
126. Three men commenced trade together; they
put in money in the following proportion ; the first,
3 dollars, as often as the second put in 4, and as
often as the third put in 5 ; they gained 87 dollars.
What was each man's share of the gain ?
127. Two men hired a pasture for 32 dollars ;
the first put in 3 sheep for 4 months ; the second
put in 4 sheep for five months ; how much ought
each to pay ?
Note. Three sheep for four months is the same
as 12 sheep for 1 month ; 4 sheep for 5 months is
the same as 20 sheep for 1 month. This question
is therefore the same as if one man put in 12 sheep,
and the other 20 sheep.
128. Two men, A and B, traded in company ;
A put in 1 dollar for 4 months, and B 2 dollars for
3 months, and they gained ninety cents ; how many
cents must each have ?
129. Three men, A, B, and C, traded in compa-
ny, and put in money in the following proportions ;
A put in 4 dollars as often as B put in 3, and as
often as C put in 2 ; A's money, was in 2 months,
B's 3 months, and C's 4 months, and they gained
100 dollars ; what was each one's share ?
130. Two men, A and B, traded in company ;
A put in 2 dollars as often as B put in 3 ; A's money
PART 1.] ARITHMETIC. 135
was employed 7 months, and B's money 5 months ;
they gained 58 dollars; what was each man's share
of the gain ?
1-M. Three men, A, B, and C, traded in company,
and put in money in the following proportions ; A
put in % dollars as often as B put in 4, and as often
as C put in 6 ; B's money was in twice as long as
C's, and A's two times as long as B's ; they gained
88 dollars ; what was each one's share of the gain ?
NOTE. Interest is a reward or premium allowed by
a debtor to a creditor for the use of money. The
usual interest for 1 year, and that which is establish-
ed by law in this country, is 6 cents on a dollar, 6
dollars on a hundred dollars, or 6 pounds on a hun-
dred pounds ; or, in fine, T^ of the sum, whatever be
the denomination. It is called 6 per cent., that is, 6
on the hundred, because it is always reckoned by the
hundred. So 3 per cent., 4 per cent., fyc. signify
Ttf^j Ttftf' 4*c-> or so much on a hundred.
132. The interest of 1 dollar being 6 cents for 1
year, what is the interest of 7 dollars for the same
time ? What is the interest of 10 dollars ? Of 15
dollars? Of 20 dollars? Of 30 dollars? Of 50 dol-
lars? Of 75 dollars? Of 100 dollars? Of 118 dollars?
133. If the interest of 1 dollar is 6 cents for 1
year, what would it be for 2 years ? What would be
the interest of 8 dollars for two years ? Of 17 dol-
lars? Of 43 dollars?
134. If the interest of 100 dollars is 6 dollars for
a year, what would be the interest of 50 dollars for
trie same time? Of 2 hundred? Of 3 hundred ? Of
4 hundred ? Of 1 hundred and 50 ? Of 2 hundred
and 50?
135. If the interest of 100 dollars is six dollars
for 1 year, what would be the interest of it for 6
months? For 3 months? For 4 months? For 8
136 ARITHMETIC. [PART 1.
months? For 9 months? For 1 month? For 2
months ? For 5 months ? For 7 months ? For 10
months? For 11 months?
136. What is the interest of 100 and 32 dollars
for 2 years, at 6 per cent. ?
137. What is the interest of 300 dollars for 1
year and 6 months, at 6 per cent.?
138. What is the interest of 1 dollar for 6 months,
at 6 per cent. ? What for 2 months 1 What for 1
month ? What for 3 months ? 4 months ? 5 months ?
7 months? 9 months? 11 months?
139. What is the interest of 57 dollars for 1 year
and 7 months, at 6 per cent. ?
140. What is the interest of 200 and 67 dollars
for 1 year and 4 months, at 6 per cent. ?
141. If the interest of 1 year is 6 per cent., what
would be the per cent, for 2 years ? For 3 years ?
For 6 months ? For 2 months ? For 1 month ? For
4 months ? For 5 months ? For 7 months ? For 8
months ? For 9 months ?
142. If the interest of 2 months, or 60 days, is 1
per cent., what would be the per cent, for 20 days?
What for 40 days? What for 15 days? What for
45 days? What for 12 days? What for 10 days?
What for 5 days ?
143. What is the interest of 100 and 37 dollars
for 2 years 3 months and 20 days ?
144. A can do a piece of work in 2 days ; how
much of it can he do in 1 day ?
145. B can do a piece of work in 4 days ; how
much of it can he do in 1 day ?
146. If A can do ^ of a piece of work in 1 day,
and B can do £ of it in 1 day, how much would
both do in a day ? How long would it take them
both together to do the whole ?
147. If 1 man can do a piece of work in 2 days,
and another in 3 days, how much of it would each
PART 1.] ARITHMETIC. 137
do in a day? How much would both together do?
How long would it take them both to do the whole ?
148. A cistern has 2 cocks ; the first will fill it in
3 hours, the second in 6 hours ; how much of it
would each fill in an hour? How much would both
together fill ? How long would it take them both to
fill it ?
149. A man and his wife found by experience,
that, when they were both together, a bushel of meal
would last them only 2 weeks ; but when the man
was gone, it would last his wife 5 weeks. How much
of it did both together consume in 1 week ? What
part did the woman alone consume in 1 week?
What part did the man alone consume in one week?
How long would it last the man alone ?
150. If 1 man could build a piece of wall in 5
days, and another man could do it in 7 days, how
much of it would each do in 1 day 1 How many
days would it take them both do it ?
151 . A cistern has 3 cocks ; the first would fill it
in 3 hours, the second in 6 hours, the third in 4
hours ; what part of the whole would each fill in 1
hour ? and how long would it take them all to fill it,
if they were all running at once?
15*2. A and B together can build a boat in 8 days,
and with the assistance of C they can do it in 5
days; how much of it can A and B build in 1 day?
How much of it can A, B, and C, build in 1 day ?
How much of it can C build alone in 1 day ? How
long would it take C to build it alone ?
153. Suppose I would line 8 yards of broadcloth
that is 1£ yards wide, with shalloon that is £ of a
yard wide ; how many yards of the shalloon will
line 1 yard of the broadcloth ? How many yards will
line the whole ?
154. If 7 yards of cloth cost 13 dollars, what will
10 yards cost?
12*
138 ARITHMETIC. [PART 1.
155. If the wages of 25 weeks come to 75 dollars,
what will be the wages of 7 weeks?
156. If 8 tons of hay will keep 7 horses three
months, how much will keep 12 horses the same
time?
157. If a staff 4 feet long cast a shadow 6 feet
long, what is the length of a pole that casts a shadow
58 feet at the same time of day 1
158. If a stick 8 feet long cast a shadow 2 feet in
length, what is the height of a tree which casts a
shadow 42 feet at the same time of day?
159. At 6 dollars per week, how many months'
board can I have for 100 dollars ?
160. A ship has sailed 24 miles in 4 hours ; how
long will it take her to sail 150 miles at the same
rate?
161. 30 men can perform a piece of work in 20
days ; how many men will it take to perform the
same work in 8 days ?
162. 17 men can perform a piece of work in 25
days ; in how many days would 5 men perform the
same work?
163. A hare has 76 rods the start of a greyhound,
but the greyhound runs 15 rods to ten of the hare ;
how many rods must the greyhound run to overtake
the hare ? %
164. A garrison has provision for 8 months, at
the rate of 15 ounces per day ; how much must be
allowed per day, in order that the provision may last
11 months?
165. If 8 men can build a wall 15 rods in length
in 10 days, how many men will it take to build a
wall 45 rods in length in 5 days ?
166. If a quarter of wheat affords 60 ten-penny
loaves, how many eight-penny loaves may be ob-
tained from it?
167. Said Harry to Dick, My purse and money
PART 1.] ARITHMETIC. 139
together are worth 16 dollars, but the money is
worth 7 times as much as the purse ; how much
money was there in the purse? and what is the
value of the purse ?
168. A man, being asked the price of his horse,
answered, that his horse and saddle together were
worth 100 dollars, but the horse was worth 9 times
as much as the saddle. What was each worth ?
169. A man, having a horse, a cow, and a sheep,
was asked what was the value of each. He answer-
ed, that the cow was worth twice as much as the
sheep, and the horse 3 times as much as the sheep,
and that all together were worth 60 dollars. What
was the value of each 1
170. A man bought an apple, an orange, and a
melon, for 21 cents; for the orange he gave twice
as much as for the apple, and for the melon he gave
twice as much as for the orange. How much did
he give for each ?
171. If 80 dollars' worth of provision will serve
20 men 24 days, how many days will 100 dollars'
worth of provisions serve 30 men 1
172. There is a pole £ and % under water, and
10 feet out ; how long is the pole ?
173. In an orchard of fruit trees, £ of them bear
apples, j- of them bear plums, -J- of them pears, 7
of them peaches, and 3 of them cherries; how
many trees are there in the whole, and how many
of each sort ?
*" 174. A farmer, being asked how many sheep he
had, answered, that he had them in 4 pastures ; in
the first he had ^ of his flock ; in the second £ ; in
the third £ ; and in the fourth 15 ; how many sheep
had he ?
175. A man, driving his geese to market, was
m.et by another, who said, Good morrow, master,
with your hundred geese ; says he, I have not a
140 ARITHMETIC. [PART 1.
hundred ; but if I had half as many more as I now
have, and two geese and a half, I should have a
hundred ; how many had he ?
176. What number is that, to which if its half be
added, the sum will be 60 ?
177. What number is that, to which if its third
be added, the sum will be 48?
178. What number is that, to which if its fifth
be added, the sum will be 54 ?
179. What number is that, to which if its half
and its third be added, the sum will be 55?
180. A man, being asked his age, answered, that,
if its half and its third were added to it, the sum
would be 77 ; what was his age ?
181. What number is that, which being increased
by its half, its fourth, and eighteen more, will be
doubled ?
182. A boy, being asked his age, answered, that,
if $ and £ of his age, and 20 more, were added to
his age, the sum would be 3 times his age. What
was his age ?
183. A man, being asked how many sheep he had,
answered, that, if he had as many more, £ as many
more, and 2£ sheep, he should have 100. How
many had he?
ARITHMETIC.
FAKT II.
KEY.
THE Key contains an explanation of the plates,
and the manner of using them. The manner of
solving the examples in each section is particularly
explained. All the most difficult of the practical ex-
amples are solved in such a manner, as to show the
principles by which they are performed. Care has
been taken to select examples for solution, that will
explain those which are not solved. Many remarks,
with regard to the manner of illustrating the prin-
ciples to the pupils, are inserted in their proper
places.
Instructers who may never have attended to frac-
tions, need not be afraid to undertake to teach this
book. The author flatters himself that the princi-
ples are so illustrated, and the processes are made so
simple, that any one, who shall undertake to teach
it, will find himself familiar with fractions before he
is aware of it, although he knew nothing of them
before ; and that every one will acquire a facility in
solving questions which he never before possessed.
The reasoning used in performing these small
examples is precisely the same as that used upon
large ones. And when any one finds a difficulty in
solving a question, he will remove it much sooner,
142 KEY [PART 2.
and much more effectually, by taking a very small
example of the same kind, and observing how he
does it, than by recuVring to a rule.
The practical examples at the commencement of
each section and article, are generally such as to
show the pupil what the combination is, and how he
is to perform it. This will teach the pupil gradual-
ly to reason upon abstract numbers. In each com-
bination, there are a few abstract examples without
practical ones, to exercise the learner in the com-
binations, after he knows what these combinations
are. It would be an excellent exercise for the pu-
pil to put these into a practical form when he is re-
citing. For instance, when the question is, How
many are 5 and 3? let him make a question in this
way : If an orange cost 5 cents, and an apple 3 cents,
what would they both come to? This may be done
in all cases.
The examples are often so arranged, that several
depend on each other, so that the preceding ex-
plains the following one. Sometimes, also, in the
same example, there are several questions asked,
so as to lead the pupil gradually from the simple to
the more difficult. It would be well for the pupil
to acquire the habit of doing this for himself, when
difficult questions occur.
The plates should be used for young pupils, but
they are not necessary for the older ones. The
plates for fractions, however, will frequently be use-
ful to these. The first plate need not be used much,
after the pupil is familiar with the multiplication
table.
The book may be used in classes, where itds con-
venient. The pupil may answer the questions with
the book before him "or not, as the instructor thinks
proper. A very useful mode of recitation is, for
the iristructer to read the example to the whole
PART 2.] KEY. 143
class, and then, allowing sufficient time for them to
perform the question, call upon some one to answer
it. In this manner every pupil will be obliged to
perform the example, because they do not know
who is to answer it. In this way it will be best for
them to answer without the book.
It will often be well to let the elder pupils hear
the younger. This will be a useful exercise for
them, and an assistance to the instructer.
Explanation of Plate I.
This plate, viewed horizontally, presents ten rows
of rectangles, and in each row ten rectangles.
In the first row, each rectangle contains one mark,
each mark representing unity or one. In the sec-
ond row, each rectangle contains two marks ; in the
third, three marks, &c.
The purpose of this plate is, first, to represent
unity either as a unit, or as making a part of a sum
of units: secondly, to represent a collection of
units, either as forming a unit itself, or as making a
part of another collection of units ; and thus to com-
pare unity and each collection of units with another
collection, in order to ascertain their ratios.
All the examples as far as the eighth section can
be solved by this plate. The manner of using it is
explained in the Key for each section in its proper
place.
The pupil, if very young, should first be taught
to count the units, and to name the different assem-
blages of units, in the following manner : — r
The instructer, showing him the first row, which
contains ten units insulated, requests the pupil to
put his finger on the first, and say, one ,* then on the
second, and say, and one are two ; and on the third, and
say, and one are three ; and so on. to ten : then^ com-
144 KEY. [PART 2.
mencing the row again, let him continue and say,
ten and one are eleven, &c.
After adding them, let him begin with ten, and
say, ten less one are nine, nine less one are eight, &/c.
Then, taking larger numbers, as twenty or thirty,
let him subtract them in the same manner.
Next, let him name the different assemblages, as
twos, threes, &,c. Afterwards, let him count the
number of units in each row.
Note. The sections, articles, and examples, are
referred to by the same marks which distinguish
them in Part I.
SECTION I.
A. THIS section contains addition and subtraction.
The first examples may be solved by means of beans,
peas, &/c., or by Plate I. The former method is
preferable, if the pupil be very young, not only for
the examples in the first part of this section, but for
the first examples in all the sections.
The pupil will probably solve the first examples
without any instruction.
Examples in addition and subtraction may be
solved by Plate I. as follows : —
How many are 5 and 3 ?* Select a rectangle con-»
taming 5 marks, and another containing 3 marks,
and ascertain the number of marks in both.
How many are 8 and 6 ? Select a rectangle contain-
* Figures are used in the Key, because the instructer is supposed to
be acquainted with them. They are not used in the first part of tf ^
book, because the pupil would not understand them so well as he v
the words.
SECT. 1.] KEY. 145
ing 8 marks, and another containing 6 marks, and
count them together.
How many are 17 and 5 ? Keeping 17 in the
mind, select a rectangle containing 5 marks, and
add them thus : — 17 and 1 are 18, and 1 are 19, and
1 are 20, and 1 are 21, and 1 are 22.
If you take 4 from 9, how many will remain ?
Select a rectangle containing 9 marks, and take
away four of them.
18 less 5 are how many ? Keeping 18 in mind,
select a rectangle containing 5, and take them away
1 at a time.
In this manner all the examples in this section
may be solved.
B & C. The articles B and C contain the com-
mon addition table as far as the first 10 numbers. In
the first the numbers are placed in order, and in
the second out of order.
The pupil should study these until he can find the
answers readily, and then he should commit the
answers to memory.
D. In this article, the numbers are larger than in
the preceding, and, in some instances, three or more
numbers are added together. In the abstract ex-
amples, the numbers from one to ten are to be added
to the numbers from ten to twenty.
4E. This article contains subtraction.
_j_ A
F. This article is intended to make the pupil
familiar with adding the nine first numbers to all
others. The pupil should study it until he, can an-
swer the questions very readily.
€r. In this article, all the preceding are combined
13
145 KEY. [PART 2.
together, and the numbers from 1 to 10 are added
to all numbers from 20 to 100; and subtracted in
the same manner.
18. f>7 and <> arc 03, and 3 are 06, and 5 are 71,
^and 2 are 73, less 8 are 65.
H. This article contains practical questions which
show the application of all the preceding articles.
0. 37 less 5 are 32, less 8 are 24, less 6 (which
he kept himself) are 18 ; consequently he gave 18
to the third boy.
SECTION II.
THIS section contains multiplication. The pupil
will see no difference between this and addition. .
It is best that he should not at first, though it may
be well to'explain it to him after a while.
A. This article contains practical questions, which
the pupil will readily answer.
I. Three yards will cost 3 times as much as 1
yard.
N. B. Be careful to make the pupil give a similar
reason for multiplication, both in this article, and
elsewhere.
This question is solved on the plate thus : in the
second row, count 3 rectangles, and find their sum,
2 and 2 are 4, and 2 are 0.
II. A man will travel 4 times as far in 4 hours as
he will in 1 hour. In the third row, count 4 times
3, and ascertain their sum.
15. There are 4 times as many feet iu 4 yards as
in 1 yard, or 4 times 3 feet.
SECT. 2.] KEY. 147
B. This article contains the common multiplica-
tion table, as far as the product of the first ten
numbers. The pupil should find the answers once
or twice through, until he can find them readily,
and then let him commit them to memory.
^43. 6 times 3. In the third row count 6 times
3, and then ascertain their sum. 3 and 3 are 6, &,c.
59. 7 times 9. In the ninth row count 7 times
9, or 7 rectangles, and ascertain their sum. 9 and
9 are 18, &c.
C. This article is the same as the preceding, ex-
cept in this the numbers are out of their natural
order.
I>. In this article multiplication is applied to prac-
tical examples. They are of the same kind as
those in article A of this section.
• 12. There are 8 times as many squares in 8 rows
as in 1 row. 8 times 8 are 64.
13. There are 6 times as many farthings in 6
pence as in 1 penny. 6 times 4 are 24.
17. 12 times 4 are 48.
Note. When a number is taken more than 10
times, as in the above example, after taking it 10
times on the plate, begin at the beginning of the
row again, and take enough to make up the number.
23. There are 3 times as many pints in 3 quarts
as in 1 quart. 3 times 2 are 6. And in 6 pints
there are 6 times 4 gills, or 24 gills.
28. In 3 gallons there are 12 quarts, and in 12
quarts there are 24 pints.
31. In 2 gallons are 8 quarts, in 8 quarts 16 pints,
in 16 pints 64 gills. 16 times 4 are 64.
35. In 1 gallon are 32 gills ; and 32 times 2
148 KEY. [PART 2.
cents are 64 cents. Or, 1 pint will cost 8 cents,
and there are 8 pints in a gallon. 8 times 8 are (>4.
38. They will be 2 miles apart in one hour, 4 miles
in 2 hours, &c.
SECTION III.
A. THIS section contains division. The pupil
will scarcely distinguish it from multiplication. It
is not important that he should at first.
Though the pupil will be able to answer these
questions by the multiplication table, if he has com-
mitted it to memory thoroughly ; yet it will be bet-
ter to use the plate for some time.
9. As many times as 3 dollars are contained in
15 dollars, so many yards of cloth may be bought
for 15 dollars. On Plate I., in the third row, count
fifteen, and see how many times 3 it makes. It is
performed very nearly like multiplication.
B. In this article the pupil obtains the first ideas
of fractions, and learns the most important of the
terms which are applied to fractions.* The pupil
has already been accustomed to look upon a collec-
tion of units, as forming a number, or as being itself
a part of another number. He knows, therefore,
that one is a part of every number, and that every
number is a part of every number larger than itself.
As every number may have a variety of parts, it is
necessary to give names to the different parts, in order
to distinguish them from each other. The parts
* As soon as the terms applied to fractions are fully comprehended,
the operations on them are as simple as those on whole numbers.
SECT. 3.] KEY. 149
receive their names, according to the number of
parts which any number is divided into. If the
number is divided into two equal parts, the parts are
called halves; if it is divided into three equal parts,
they are called thirds ; if into four parts, fourths,
&/c. ; and, having divided a number into parts, we
can take as many of the parts as we choose. If a
number be divided into five equal parts, and three
of the parts be taken, the fraction is called three
fifths of the number. The name shows at once into
how many parts the number is to be divided, and
how many parts are taken.
The examples in this book are so arranged, that
the names will usually show the pupil how the ope-
ration is to be performed. In this section, although
the pupil is taught to divide numbers into various
parts, he is not taught to notice any fractions, ex-
cept those where the numbers are divided into their
simple units, which is the most simple kind.
It will be best to use beans, pebbles, &,c. first ;
and then Plate I.
4. Show the pupil one of the rectangles in the
second row, and explain to him that one is 1 half
of 2.
7. In the second row count 3 units ; it will take
all the marks in the first, and 1 in the second rec-
tangle. Consequently it is 1 time 2, and 1 half of
another 2.
15. In the second row count 9. It will take all
the marks in the four first rectangles, and 1 in the
fifth. Therefore 9 is 4 times 2 and one half of
another 2.
18. Show the pupil a rectangle in the third row,
and ask him the question, and explain to him that 1
is 1 third of 3.
20. Since 1 is 1 third of 3, 2 must be 2 thirds of 3.
34. In the third row count 11. It will take 3
13*
150 KEY. [PART 2.
rectangles and 2 marks in the fourth. Therefore
11 is 3 times 3, and 2 thirds of another 3.
Proceed in the same manner with the other di-
visions.
This being one of the most useful combinations,
and one but very little understood by most people,
especially when applied to large numbers, the pu-
pil must be made perfectly familiar with it. Ask
questions like those in the book for large numbers,
and also some like the following : What part of 7
is 18 ? The answer will be If-.
C. The first ten figures are here explained.
They are used as an abridged method of writing
numbers, and not with any reference to their use in
calculating.
This article is only a continuation of the last.
All the numbers from 1 to 100 are introduced into
the two articles, and are divided by all the numbers
from 1 to 10 ; except that some of the largest are
not divided by some of the smallest.
2. The pupil answers first, how many times 2 is
contained in 12, then how many times 3.
45. 63 are how many times 5? In the fifth row
count 63. It will take 12 rectangles and 3 marks
in the 13th. It will be necessary to count once
across the plate, and begin again, and take 2 rec-
tangles and a part of the third. 63 is 12 times 5
and 3 fifths of another 5.
D. These examples, which are similar to those
in article A of this section, are solved in the same
manner.
5. It would take as many hours as 3 miles are
contained in 10 miles. 3 hours and £ of an hour.
20. They cost as many cents as there are 3 ap-
ples in 30 apples ; that is, 10 cents.
SECT. 4.] KEY. 151
21. 12 dollars a month : and 12 dollars a month
is 3 dollars a week ; that is, 18 shillings a week,
which is 3 shillings a day.
26. The whole loss was 35 dollars, which was 7
dollars apiece.
SECTION IV.
A. THIS article contains multiplication simply.
It is repeating a number a certain number of times
and a part of another time.
14. 6 times 5 are 30, and -| of 5 are 3, which
added to 30 make 33. On the plate in the fifth
row, take 6 rectangles and 3 marks in the seventh,
and ascertain their sum.
B. In this article the pupil is taught to change a
certain number of twos into threes, threes into fives,
&c. This article combines all the preceding oper-
ations.
24. 4 cords of wood will cost 28 dollars, and f of
a cord will cost 2 dollars, which makes 30 dollars.
30 dollars will buy 3 hundred weight of sugar and
f of another hundred weight.
29. 7 times 8 are 56, and f of 8 are 5, which
added to 56 make 61 ; 61 are 6 times 9, and £ of 9.
C. 1. 4 bushels of apples, at 3 shillings a bushel,
come to 12 shillings ; and 12 shillings are 2 dollars.
2. The 2 lemons come to 8 cents, and 8 cents
will buy 4 apples, at 2 cents apiece.
This is usually called Barter. The general prin-
ciple is to find what the article will come to, whose
price and quantity are given, and then to find how
much of the other article that money will buy.
152 KEY. [PART 2.
6. If 2 apples cost 4 cents, 1 will cost 2 cents,
and 4 will cost 8 cents. Or 4 apples will cost 2
times as much as 2 apples.
22. Find how many times 2 pears are contained
in 20 pears, which is 10 times. 10 times 3 cents
are 30 cents. Or, first find what 20 pears would
come to, at 3 cents apiece ; and since it is 2 for 3
cents, instead of 1 for 3 cents, the price will be
half as much.
23. See how many times you can have 5 cents
in 30 cents, and you can buy so many times 3 eggs.
30 is 6 times 5, and 6 times 3 are 18. 18 eggs.
24. 10 dollars a week, and 40 dollars a month.
25. 5 dollars are 30 shillings, which is 10 shil-
lings a day.
26. 5 dollars apiece.
SECTION V.
IN this section the principle of fractions is applied
to larger numbers, but such as are divisible into the
parts proposed to be taken. The pupil who is fa-
miliar with what precedes, will easily understand
the examples in this section. They require nothing
but division and multiplication.
A. Let the pupil explain each example in the
following manner : — What is 1 sixth of 18 ? Ans. 3.
Why ? Because 6 times 3 are 18 ; therefore if you
divide 18 into 6 equal parts, one of the parts will
be 3.
To find this answer on the plate ; on the 6th row,
the pupil will find 3 times 6 make 18; this will
direct him to the third row, where he will find 6
SECT. 5.} KEY. 153
times 3 are 18. Consequently, he will see 18 divided
into 6 equal parts. It will be well to let the pupil
prove a large number of the examples on the plate.
The pupil will be very likely to say, 3 is the 6th
part of 18, because 3 times 6 are 18. Be careful to
make him say it the other way, viz. 6 times 3 are 18.
14. 1 third of 9 is 3 ; f is 2 times as much as £,
therefore f of 9 is 6.
19. 1 barrel will cost % part of 12 dollars ; 3 bar-
rels will cost | of 12 dollars. 7 barrels will cost J
of 12 dollars.
37. What is $ of 32 ? $ of 32 is 4, f are 5 times
4, or 20.
B. 11. | of 20 is 4 ; £ are 7 times 4, or 28 ; and
28 is 4 times 6, and ^ of 6.
C. 3. 1 half of 10 is 5, f of 10 are 4 ; 5 and 4
are 9. He gave away nine, and had 1 left.
4. 1 yard will cost £ of what 3 yards cost. £ of
6 dollars is 2 dollars.
5. 2 yards will cost 1 half of what 4 cost ; or 6
dollars.
6. 3 apples will cost ^ of what 9 cost ; or 6 cents.
7. 2 is § of 3 ; therefore 2 oranges will cost f of
what 3 cost, f of 18 cents are 12 cents.
8. | of 25 are 20. The 10 apples cost 20 cents,
which was 2 cents apiece.
11. f of 42 are 12, and 6 times 12 are 72. 72
dollars.
13. 3 is f of 4. J of 12 dollars are 9 dollars.
Or, 4 yards at 12 dollars is 3 dollars a yard, and 9
dollars for 3 yards.
14. Solved like the 13th. Ans. 15 cents.
15. Since 1 is $ of 3, 7 is J of 3. £ of 15 cents
are 35 cents. Or, 3 oranges at 15 cents is 5 cents
apiece : 7 times 5 are 35 cents.
154 KEY. [PART 2.
Note. In questions of this kind, it is generally the
simplest way to find what 1 article will cost ; then it
may easily be told how much any number will cost.
19. 4 men would do it in 1 half the time that 2
would do it. Or, you may say, If 2 men would do
it in 6 days, 1 man would do it in 12 days, and 4
men in £ of that time, or three days.
SECTION VI.
A. 4. 2 halves of any number make the whole
number. Therefore 2 is 1 half of 2 times 2, or 4.
It is £ of 4 times 2, or 8.
Let the pupil answer these questions in the fol-
lowing manner : — 4 is -J- of 3 times 4 ; 3 times 4 are
12. 5 is ij- of 7 times 5 ; 7 times 5 are 35.
B. 2. 4 is 2 times 2.
4. 6 is 2 times 3.
16. 2 thirds of any number is twice as much as $
of the same number. If 4 is § of some number,
then 1 half of 4, or 2, is £ of that number : 2 is £ of
6 ; therefore 4 is f of 6.
20. If 6 is f of a number, £ of 6, or 2, is | of the
same number : 2 is £ of 8 ; therefore 6 is f of 8.
23. It is evident that $ of a pound will cost only
•J of what £ will cost. If f cost 6 cents, -f will cost
2 cents, and the whole pound 14 cents.
26. It will probably be perceived, by this time,
that, ^ of a number being given, it is necessary to
find ^-, and then the number is easily found ; 4 be-
ing f , 2 is |, and 2 is | of 14.
45. 24 being f , £ of 24 or 3 will be | ; 3 is £ of 27.
SECT. 6.] KEY. 155
C. 6: 20 being f, 5 is |, and 5 is | of 35 ; and 35
is 5 times 6, and % of 6.
I>. 4. 18 is 3 times 6, and 6 is £ of 4 times 6, or
24. Ans. 24 dollars.
6. 54 is | of 48 ; 12 yards at 48 dollars is 4 dol-
lars a yard. He gained 6 dollars.
7. 10 feet is f of 15 feet.
8. If f are under water, there must be -J out of
the water. 4 is £ of 12.
9. If f are under water, there must be f out of
the water. 6 is f of 10.
10. | and f are -f . fy bear cherries and peaches ;
consequently, the 10 which bear plums must be the
other f ; 10 is f of 35. 10 bear peaches, and 15
bear cherries.
11. f, and f, and £, and £, are £; therefore 12
must be the other f of the whole. The whole
number is 54.
Miscellaneous Examples.
6. The ^greyhound gains upon the fox 4 rods in
a minute. It will take him 20 minutes to gain 80
rods.
8. f of 24. Or you may say, 1 sheep would cost
3 dollars, and 3 sheep 9 dollars.
9. 30 horses will eat 10 times as much as 3
horses.
11. 10 dollars apiece, and 2 dollars a yard.
12. 5 dollars for 1 week, 20 dollars for a month,
and 25 dollars for 5 weeks.
14. It would take them 5 times as long to eat 40
bushels, as it would to eat 8 bushels.
15. 4 horses would eat 4 bushels in 3 days, and
it would take them 9 times as long to eat 36 bush-
els. Ans. 27 days.
156 KEY. [PART 2.
16. If 2 men spend 12 dollars in 1 week, 1 man
will spend 6 dollars in 1 week, and 30 dollars in 5
weeks, and 3 men would spend 3 times as much, or
90 dollars.
17. The shadow of the staff is § of the length
of the staff; therefore the shadow of the pole must
be f the length of the pole. 18 feet is \ of 27
feet.
20. It would take 2 men 3 times as long to do it
as it would 6 men.
23. 8 men would do a piece of work 1 half as
large in 2 days, and it would take 2 men 4 times as
long to do it, or 8 days.
28. He must sell it for 56 dollars in order to gain
16 dollars. 56 dollars is 7 dollars per barrel.
29. It cost him 35 dollars, and he must sell it for
45 to gain 10 dollars ; 45 dollars is 9 dollars a firkin.
30. Ans. 56 cents.— See Section VI.
33. If it would last 3 men 10 months, it would
last 1 man 30 months, and 5 men 6 montjis.
34. There are 8 times 5 in 40 ; and since the
other would build as many times 9, as the first does
5, he would build 8 times 9, or 72 rods.
SECTION VII.
A. 13. £of 20is4,f are 16; 16 being f, 2 isj;
2 is | of 14, and 16 is f of 14.
16. f of 28 are 12 ; 12 is 2 times 6, and 6 is £ of
48, (12 is f of 48) and 48 is 6 times 7 and f of 7.
B. 1. ± of 15 are 12; 12 is 6 times 2; 2 is ^
of 20 (12 is ft of 20) ; £ of 21 is 7 ; 20 is 2 times
7 and f. of 7.
SECT. 8.] KEY. 157
2. | of 18 are 24; 24 is f of 27; | of 35 is 5;
27 is 5 times 5 and f of 5.
C. This article contains the multiplication table,
in which the numbers from 10 to 20 are multiplied
by the ten first numbers.
SECTION VIII.
Explanation of Plate II.
PLATE I., which has been used in the preceding
sections, presents each unit as a simple object, and
undivided. Plate II. presents the units as divisible
objects, the different fractions of which form parts,
and sums of parts, of unity.
This plate is divided into ten rows of equal
squares, and each row into ten squares.
The first row is composed of ten empty squares,
which are to be represented to the pupil as entire
units. The second row presents ten squares, each
divided into two equal parts by a vertical line ; each
of these parts, of course, represents one half. In
the third row, each square is divided into three
equal parts, by two vertical lines, each part repre-
senting one third, &c., to the tenth row, which is di-
vided into ten equal parts, each part representing
one tenth of unity.
N. B. In Plates II. and III., the spaces, and not
the marks, are to be counted.
Be careful to make the pupil understand, 1st,
that each square on the plate is to be considered as
14
158 KEY. [PART 2.
an entire unit, or whole one. 2d, explain the divi-
sions into two, three, four, &,c. parts. 3d, teach him
to name the different parts. Make him observe thv-it
the name shows into how many parts one is divided,
and how many parts are taken, in the same manner
as it does when applied to larger numbers. 4, for
example, shows that one thing is to be divided into
7 equal parts, and 4 of those parts are to be taken.
4th, make the pupil compare the different parts to-
gether, and observe which is the largest. Ask him
such questions as the following : Which are the
smaller, halves or thirds? Ans. Thirds. Why?
Because the more parts a thing is divided into, the
smaller the parts must be.
A. 15. On Plate II., count two squares in the
second row, and then ascertain the number of spaces
or halves in them. There are 4 halves.
21. In the 2d row, take 3 squares and 1 space in
the 4th square; then count the spaces. Ans. 7
halves.
37. In the third row, take 5 squares, and 2 spaces
in the 6th ; then count the spaces or thirds. Ans.
17 thirds.
54. In the 5th row, take 6 squares, and 4 spaces
in the 7th square ; then count the spaces or fifths.
Ans. 34 fifths.
H. 2. This operation is the reverse of the last.
In the 2d row, count 4 spaces or halves, and see how
many squares or whole ones it takes. It will take 2.
38. In the 9th row, count 48 spaces or 9ths, and
see how many squares or whole ones it takes. It
will take 5 squares and 3 spaces in the 6th. Ans.
5 whole ones and £.
SECT. 9.] KEY. 159
SECTION IX.
A. 2. £ signifies that 1 thing is divided into 3
equal parts, and 1 part taken. Therefore 2 times
1 third is 2 parts, or f .
6. 7 times £ is J, or 2£.
10. On the plate in the 3d row, 5 times f are J^0-,
which takes 3 squares and 1 space. Ans. 3£.
24. In the 9th row, take 4 spaces or 9ths, and
repeat them 5 times, which will make %°-, and will
take 2 squares and 2 spaces. Ans. 2f .
15. 4. 4 times 2 are 8, and 4 times 1 half are 4
halves, or 2, which added to 8 make 10.
18. 4 times 3 are 12, and 4 times f are *£-, or
three whole ones, which added to 12 make 15.
32. 2 times 3 are 6, and 2 times ^ are f , which
added to 6 make 6^.
40. 10 barrels of cider at 3 dollars and | a bar-
rel ; 10 barrels at 3 dollars would be 30 dollars,
then 10 times £ is -5^, or 8 and f of a dollar. Ans.
38f dollars.
C. 2. J to each would be 3 times f , or f , which
are 2£ oranges.
3. J£. or 2 bushels.
4. 7 times j- are -^L, or 5£ gallons.
5. 8 yards and f or 2 yards, that is, 10 yards.
6. 4 times 2 are 8, and 4 times -f are -^, or 2f ,
which added to 8 make 10f bushels.
12. It would take 1 man 3 times as long as it
would 3 men. Ans. 13f days.
14. 3 men would build 3 times as much as 1 man ;
and in 4 days they would build 4 times as much as
in 1 day. ^.715. 38f rods.
15. Ans. 12 yards.
360 KEY. [PART 2.
SECTION X.
A. 21. £of 1 is £. £ of 2 is 2 times as much, or
f . i of 4 is f, or If i of 5 is f , or If. £ of 6
is f, or 2. | of 7 is J, or 2f
27. £ of 1 is f | of 2 is f . £ of 3 is f . £ of
7 is £, or If
This manner of reasoning may be applied to any
number. To find J of 38 : it is Sf-, for | of 38 is 38
times as much as j- of 1, and ^ of 1 is | ; consequent-
ly | of 38 is ^8., and $p. is 5f
40. To find f of a number, £ must be found first,
and then f will be 2 times as much. -J of 7 is J,
and 2 times J are 4^-, or 4f .
74. £ of 50 is ^°-, or 5| ; f is 4 times as much ;
4 times 5 are 20, 4 times % are ^, or 2f , which
added to 20 make 22f .
NOTE. The manner employed in example 40th is best for small
numbers, and that in the 74th for large numbers.
B. 2. Am. If apiece.
3. £ of 3 is f ; f of a bushel apiece.
4. f of 7 is 4^ ; he gave away 4|, and kept 2|.
6. 1 half dollar a yard, or 50 cents.
7. i of 7 is I, or If ; f of a dollar is f of 100
cents, which is 40 cents. Ans. 1 dollar and 40 cents
a bushel.
8. | of 8 is If. f of 100 is 33f . Ans. 1 dollar
and 33f cents ; or it is 1 dollar and 2 shillings.
9. If 3 bushels cost 8 dollars, 1 bushel will cost
2 dollars and f, and 2 bushels will cost 5£ dollars.
Ans. 5 dollars and 2 shillings, or 33f cents.
13. If 7 pounds cost 40 cents, 1 will cost 5|- cents ;
10 pounds will cost 57| cents.
16. 1 cock would empty it in 6 hours, and 7 cocks
SECT. 11.] KEY. 161
would empty it in | of 6 hours, or f. of 1 hour,
which is f- of 60 minutes ; ^ of 60 minutes is
minutes.
SECTION XL
A, 2. 2 halves of a number make the number ;
consequently 1 and 1 half is the half of 2 times 1
and 1 half, which is 3.
15. 4f. is | of 5 times 4 and f , which is
17. 4$ is | of 9 times 4f , which is 39f
IB. 4. 5 is 3 times % of 5, which is -J, or If.
30. If 8 is | of some number, $ of S is £ of the
same number. % of 8 is 2§ ; 2f is £ of 4 times 2f ,
which is lOf ; therefore 8 is £ of lOf.
40. If8isf iofSisf; *of8isf; fis|of^,
or 9f ; therefore 8 is f of 9f.
52. If |^ of a ton cost 23 dollars, £ of a ton must
be -£ of 23, that is, 4f dollars, and the whole would
cost 9 times as much, that is, 41f .
69. £ of 65 is 7f : 7f is | of 5 times 7f , which is
38£. 65 is f of 36f
C. 4. 37 is f of 32f , which taken from 37 leaves
4£. Ans. 4^- dollars.
5. 7 feet must be f of the whole pole.
6. If he lost f , he must have sold it for £ of what
it cost. 47 is £ of 60f . Ans. 60 dollars and 42f
cents.
Miscellaneous Examples.
1. The shadow of the staff is f of the length of
the staff; therefore the shadow of the pole is % of
the length of the pole. 67 is £ of 83f . Ans. 83f
feet.
2. 9 gallons remain in the cistern in 1 hour. It
will be filled in 10 hours and £ ; £ of 60 minutes
14*
163 KEY. [PART 2.
are 46 minutes and f ; f of 60 seconds are 40 sec-
onds. Ans. 10 hours, 46 minutes, 40 seconds.
10. Find f of 33, and subtract it from 17. Ans. 3£.
11. It will take 3 times 10 yards.
13. 5 is £ of 3 ; it will take f as much. Or, 7
yards, 5 quarters wide, are equal to 35 yards 1 quar-
ter wide, which is equal to llf yards that is 3
quarters wide.
15. £ of 37 dollars.
16. £ as much.
SECTION XII.
THE examples in this section are performed in
precisely the same manner as those in the sections
to which they refer. All the difficulty consists in
comprehending that fractions expressed in figures
signify the same thing as when expressed in words.
Make the pupil express them in words, and all the
difficulty will vanish. Let particular attention be
paid to the explanation of fractions given in the
section.
VIII. A* 6. In 7 how many £ ? expressed in
words, is, In 7 how many sixths ? Ans. *•£-.
14. Reduce Sy3^ to an improper fraction ; that is,
In 8 and 3 tenths, how many tenths ? Ans. f§.
B. 8. %f- are how many times 1 ? that is, In 23
sevenths how many whole ones? Ans. 3^'
IX. B. 3. How much is 5 times 6f ? that is,
How much is 5 times 6 and 4 sevenths ? Ans. 32f .
V. &, X. 15. What is f of 27? that is, What is
5 eighths of 27? Ans. 16|.
VI. & XI. A. 8. 7f is £ of what number? that
is, 7 and 6 sevenths is 1 eighth of what number ?
Ans. 62f .
SECT. Ifc] KEY. 163
B. 4. 12 is f of what number ? that is, 12 is 3
sevenths of what number ? Ans. 28.
12. 4 is £ of what number ? that is, 4 is 3 fifths of
what number ? Ans. 6f .
Explanation of Plate III.
Plate III. is intended to represent fractions of
unity, divided into other fractions ; it is, therefore,
an extension of Plate II. It differs from it only
in this, that, besides the vertical divisions, the
squares are divided horizontally, so as to cut the
fractions of the square into fractions of fractions.
The horizontal lines are dotted, but they are to be
considered as lines.
This plate, like the preceding, is divided into
ten rows of squares, each row containing ten equal
squares. In the first row, the first square is undi-
vided ; the 9 following squares are divided by hori-
zontal lines into from two to ten equal parts. In all
the other squares, the vertical divisions are the same
as in Plate II. ; and, besides this, each row is divided
horizontally in the same manner as the first row.
By means of this double division, the second row
presents a series of fractions, from halves to twen-
tieths. The 3d row presents a series from thirds
to thirtieths, and so on to the tenth row, which
presents a series from tenths to hundredths.
The 2d row, besides presenting halves, fourths,
sixths, eighths, &c., shows also halves of halves,
thirds of halves, fourths of halves, &c., and shows
their ratios with unity'.
The 3d row, besides thirds, sixths, ninths, &/c.,
shows halves of thirds, thirds of thirds, &c., and
their ratios with unity. The other rows present
analogous divisions.
164 KEY. [PART 2.
SECTION XIII.
THE operations in this section are the reducing
of fractions to a common denominator, and the ad-
dition and subtraction of fractions. The examples
will generally show what is to be done, and how it
is to be done. Plate III. will be found very useful
in explaining the operations, by exhibiting the di-
visions to the eye.
1. The first example may be illustrated by the
second square in the second row. This square is
divided into halves by a vertical line, and then into
fourths by the horizontal line. It will be readily
seen that £ makes ,2 fourths, and that the first had
twice as much as the second. The plate will not
be so necessary for the practical questions as for
the abstract. In the second example, therefore, it
will be more useful than in the first.
4. It will readily be seen on the second square of
the second row, that £ and £ are £.
8. It will be seen in the third square of the sec-
ond row, that £ makes f .
10 and 12. In the second square of the third row,
it will be found, that £ makes f ; and that f make £.
25. la the fourth square of the second row, it
will be seen that 1 half is f ; and in the second
square of the fourth row, i is f ; both together make
f , and £ makes £ .
27. In the second square of the fourth row, J is
the same as f .
33. In the fifth square of the fourth row, it will
be seen that £ (made by the vertical division) con-
tains ^y ; and in the fourth square of the fifth row,
•J contains ^j, and -| contain •/§ ; and in the second
square of the tenth row, y1^ contains -£$.
When these questions are performed in the mind,
the pupil will explain them as follows. He will
SECT. 13.] KEY. 165
probably do it without assistance. Twenty twen-
tieths make one whole one. £ of 20 is 5, and |- of
20 is 8, and ^ of 20 is 2 ; therefore £ is ^, f is
/^, and yiy is -£$. All the examples should be ex-
plained in the same manner.
45. In the 8th row, the 7th square is divided ver-
tically into 8 parts, and horizontally into 7 parts ;
the square, therefore, is divided into 56 parts ; 3 of
the vertical divisions, or f , contain f £.
51. 1 half is |, and £ is f , which added together
make f .
61. f is ?8<y, -£$ is-/u> i ls -fv> which added togeth-
er make J-f .
67. f is T82-, f is T92-, which added together make
ij ; from -ij take -f^, and there remains -ff- , or 1.
82. It will be easily perceived that these exam-
ples do not differ from those in the first part of the
section, except in the language used. They must
be reduced to a common denominator, and then
they may be added and subtracted as easily as
whole numbers, f is |£, and f is T\, and both to-
gether make J-f , or ly^.
, 86. £ is f , and £ is |. If f be taken from f there
B. This article contains only a practical applica-
tion of the preceding.
3. This example and some of the following con-
tain mixed numbers, but they are quite as easy as
the others. The whole numbers may be added
separately, and the fractions reduced to a common
denominator, and then added as in other cases, and
afterwards joined to the whole numbers. 6 and 2
are 8 ; 1 half and £ are f , making in the whole 8f
bushels.
5. 6 and 2 are 8 ; f and £ and f are £J or 1£J,
which joined with 8 make 9£J.
166 KEY. [PART ±
C. It is difficult to find examples which will aptly
illustrate this operation. It can be done more con-
veniently by the instructer. Whenever a fraction
occurs, which may be reduced to lower terms, if it
be suggested to the pupil, he will readily perceive
it and do it. This may be done in almost any part
of the book, but more especially after studying the
13th section. Perhaps it would be as well to omit
this article the first time the pupil goes through the
book, and, after he has seen the use of the operation,
to let him study it. It may be illustrated on Plate
III. in the following manner :
8. £f . Find all the squares which are divided
into 24 parts. There are 4 squares which are di-
vided into 24 parts, viz. the; 8th in the 3d row, the
3d in the 8th row, the 6th in the 4th row, and the
4th in the 6th row. Then see if exactly 18 can be
found in one or more of the vertical divisions. In
the 6th square of the 4th row, there are exactly 18
divisions in three vertical divisions ; but those 3 ver-
tical divisions are £ of the whole square, because it
is divided into fourths vertically ; therefore £f are
equal to £ .
13. £| Find the squares which are divided into
56 parts ; they are the 8th in the seventh row, and
the 7th in the 8th row ; see if in either of them,
one or more of the vertical divisions contain exactly
42 parts. In the 7th of the 8th row, 6 vertical di-
visions contain exactly 42 : these divisions are f of
the square, for it is divided vertically into 8 parts.
But f may be still reduced to f , as may be seen by
looking on the 3d square of the 4th row; therefore
f | is equal to £.
SECT. 14.] KEY. 167
SECTION XIV.
A. THIS section contains the division of fractions
by whole numbers, and the multiplication of one frac-
tion by another. Though these operations sometimes
appear to be division, and sometimes multiplication,
yet there is actually no difference in the operations.
The practical examples will generally show how
the operations are to be performed, but it will be
well to use the plate for young pupils.
1 and 2. In the second row, the 2d square is di-
vided vertically into halves, and each of the halves
is divided into halves by the horizontal line ; J of £
is therefore £ of the whole.
3 and 4. In the third row, the 2d square shows
that £ of -J is £.
16 and 17. In the 5th row, the 3d square shows
that £ of £ is T\ of the whole.
33. Since f of a share signify 3 parts of a share,
it is evident that -J of the three parts is 1 part, that
is, i-
39. ^ signify 9 pieces or parts, and it is evident
that | of 9 parts is 3 parts, that is, f .
43. We cannot take -J of 5 pieces ; therefore we
must take £ of -£, which is ^, and f is 5 times as
much as £ ; therefore ^ of f is T5ff. This may be
readily seen on the plate. In the sixth row, jthird
square, find -| by the vertical division; then, these
being divided each into three parts by the hori-
zontal division, and -J of each being taken, you will
have y\.
5*2. In the 4th row, the 3d square shows that -J of
£ is TV, and § must be twice as much, or fa.
56. In the fifth row, the 3d square shows that -J-
of |. is T2^ ; but | must be twice as much as ^ ; there-
fore § of f are ^.
168 KEY. [PART
78. 8f is^,|
79. 8| is Sf-, | of | is ^, consequently J of
is ££, or 1«.
86. We may say, £ of 8 f is 2, and 2| over ; then
2J. is 32, and £ of ^ is £f ; hence $ of 8$ is 2ff
90. | of 18f is 2§£, and £ is 3 times as much, or
7ft-
B. 4. It would take 1 man 4 times 9f , or 37^
days ; and 7 men would do it in ^ of that time,- that
is, in 5^f days.
SECTION XV.
A. THIS section contains the division of whole
numbers by fractions, and fractions by fractions.
1. Since there are §• in 2, it is evident that he
could give them to 6 boys if he gave them £ apiece,
but if he gave them § apiece, he could give them to
only one half as many, or 3 boys.
5. If ^ of a barrel would last them one month, it
is evident that 4 barrels would last 20 months ; but
since it takes f of a barrel, it will last them but one
half as long, or 10 months.
7. 6f is -2¥7-. If £ of a bushel would last a week,
6f bushels would last 27 weeks ; but since it takes
J, it will last only £ of the time, or 9 weeks.
13. If he had given % of a bushel apiece, he
might have given it to 17 persons : but since he gave
3 halves apiece, he could give it to only ^ of that
number, that is, to 5 persons, and he would have 1
bushel left, which would be § of enough for another.
23. 9f is £T<s and If is -y1-- If it had been only
-f of a dollar a barrel, he might have bought 66
barrels for 9f dollars; but since it was 4^ a bar-
SECT. 15.] KEY. 169
rel, he could buy only -^ of that number, that is, 6
barrels.
25 and 26. Ans. 9f
31. 4i is -2^, and 9f is A£. Now, | is contained
in 4^. 48 times, and -^ is contained only ^ part as
many times, consequently only 2^r or 2f .
B. 1. J is !%•; consequently 5 pounds can be
bought for % of a dollar.
3. £• is T92-, and £ is -f^. If he had given only T^.
apiece, he could have given it to 9 persons; but
since he gave -fg, he could give it to only 1 half as
many, or 4£ persons.
5. | is ^ and 2. is ^.4. If a pound had cost ^
of a dollar, 14 pounds could be bought for ££ of a
dollar ; but since it costs ^-, only ^ as many can be
bought ; that is, 4f pounds.
9. 2. is £6.^ and 1-| is |^. If a bushel had cost
^y of a dollar, 65 bushels might have been bought ;
but since it cost £g, only TJF part as much could be
bought ; that is, 4T^ bushels.
12. f is ^F, and f is ^f ; T^ is contained in j-% 15
times, but -fa is contained only £ as many times ;
that is, 3f times.
Miscellaneous Examples.
5. f of a penny is f of 4 farthings. Ans. 2f
farthings.
6. f of 12 pence. Ans. 10 pence.
7. f of 4 quarters is 2 quarters and f of a quar-
ter ; f of a quarter is f of 4 nails, which is 1|^ nails.
Ans. 2 quarters, 1^ nails.
13. f of 24 hours is 15 hours.
14. |^ of 24 hours is 14 hours and f of an hour ;
f of 60 minutes is 24 minutes. Ans. 14 hours, 24
minutes.
15
170 KEY. [PART 2.
28. There being 4 farthings in a penny, 1 farthing
is £ part of a penny.
30. 3 farthings is £ of a penny.
31. 1 penny is -^ of a shilling, because there are
12 pence in a shilling.
34. 5 pence is -j^- of a shilling.
41. 1 shilling is -^ of a pound.
43. 3 shillings is ^ of a pound.
48. 1 farthing is ^ of one shilling
49. 2 farthings is ^2F, or ?V of a shilling. 5 far-
things is ¥%- of a shilling.
51. 1 penny is ^fa of 1 pound. 7 pence is ^J^
Ofcfl.
59. 3s. 5d. is 41 pence, which is £fc of £1.
75. 1 nail is ^ of a yard. 5 nails is -fa of a yard.
89. 1 oz. is -j^.of 1 Ib. 15 oz. is -}-£ of 1 Ib.
91. 1 Ib. is ^ of 1 quarter. 9 Ibs. is ^ of 1
quarter.
100. At the end of 1 hour, they would be 7 and £
miles apart ; in 7 hours, 7 times 7f , which is 54g
miles.
121. This is the principle of fellowship ; 3 shil-
lings were paid ; one paid £, the other f .
122. One paid f , the other f .
123. 20 dollars were paid in the whole ; one paid
^5-, another ^ and the third •£>.
126. 3 and 4 and 5 are 12. The first put in T^ ;
the second -£% ; the third T5^.
129. 4 dollars for 2 months is the same as 8 dol-
lars for 1 month ; 3 dollars for 3 months is the same
as 9 dollars for 1 month ; and 2 dollars for 4 months
is the same as 8 dollars for 1 month. The question
is the same as if A had put in 8 dollars, B 9 dollars,
and C 8 dollars. A must have ^, B ^\, and C
^, of 100 dollars.
131. A's money was in 4 times as long as C's.
It is the same as if A had put in 8 dollars for the
PART 2.] KEY. 171
same time, and B 8 dollars for the same time. A
must have ^ B 4-, and C ^, of 88 dollars.
The examples "127, 128, 129, 130, and 131, are
double or compound fellowship.
139. The interest of 50 dollars for 1 year and 6
months is 4 dollars and 50 cents, and for 1 month it
is 25 cents. The interest of 7 dollars for 18 months
(a dollar is £ of a cent a month) is 63 cents. The
whole amounts to 5 dollars and 38 cents.
140. The interest of 200 dollars for 1| year is
16 dollars. The interest of 67 dollars is 67 cents
for every 2 months ; for 16 months it will be 8 times
67 cents, which are 5 dollars and 36 cents. The
whole interest is 21 dollars and 36 cents.
143. The interest of 100 dollars for 2£ years,
is 13 dollars and 50 cents. The interest of 100
dollars for 60 days would be 1 dollar ; the interest
for 20 days will be % of a dollar, or 33£ cents.
The interest of 1 dollar for 2£ years is 13£ cents ;
for 10 dollars the interest would be 1 dollar and
35 cents, and for 30 dollars, 4 dollars and 5 cents.
The interest of 7 dollars for 2£ years is 7 times
13^ cents, or 94 j- cents. The interest of 37 dollars
for 60 days would be 37 cents, and for 20 days £ of
37 cents, or 12£ cents. The whole interest is 18
dollars and 95£ cents.
146. They would both together do f of the work
in 1 day, and it would take them £ of a day to do
the other £. Ans. l£ day.
147. -| would be done in 1 day, and it would take
£ of a day to do the other £. Ans. l-i day.
H(9- They both together consume £ of a bushel
hi a week, but the woman alone consumes only £ of
a bushel in a week. That is, they both together
consume -f$ in a week, but the woman alone only
^ ; consequently, the man alone would consume -fa ;
and a bushel would last him 3| weeks.
173 KEY. [PARTS.
152. A and B can build ^ of it in 1 day ; A, B,
and C, can build | of it in 1 day ; the difference be-
tween £ and £ is ft ; therefore C can build ^ of
it in 1 day ; and it would take him 13£ days to
build it alone.
164. Find how much they might eat in a day, in
order to make it last 1 month, and then it will be
easy to find how much they may eat in a day, to
make it last 11 months.
167. The money is 7 parts of the whole, and the
purse one part ; consequently the money is £, and
the purse £, of 16.
170. He gave one part for the apple, 2 parts for
the orange, and 4 parts for the melon. These make
7 parts. The apple 3 cents, the orange 6 cents, and
the melon 12 cents.
175. If to a number half of itself be added, the
sum is f of that number ; hence subtract 2£ from
100, and the remainder is f of the number of geese
that he had.
180. This must be reduced to 6ths. 1 half is g,
and i is §, and the number itself is £ . If, therefore,
to the whole number its half and its third be added,
the sum will be Jgi ; hence, 77 is -y- of the number.
181. £ is f ; therefore if to a number £ and £ of
itself be added, the whole number will be J; but
when 18 more is added to J, the first number is
doubled ; that is, the number is f of the first num-
ber ; therefore 18 is £ of the number.
VA 04342
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