;
IMi
w, BLAKEMAN TAYLOR & Co
BIW YOEK and CHICAGO.
Vi -
Education Library
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*
9
-A
PROGRESSIVE
[INTELLECTUAL ARITHMETIC;
ON THE INDUCTIVE PLAN.
BEING
A SEQUEL TO THE PROGRESSIVE PRIMARY ARITHMETIC, CONTAINING
MANY ORIGINAL FORMS OF ANALYSIS APPLICABLE TO A
GREAT VARIETY OF PRACTICAL QUESTIONS,
AND DESIGNED FOR
THE MORE ADVANCED CLASSES
IN
COMMON SCHOOLS AND ACADEMIES.
EDITED BY
DANIEL W. FISH, A.M.
IVISON, BLAKEMAN, TAYLOR & CO.,
PUBLISHERS,
NEW YORK' AND CHICAGO.
1878.
Q
ROBINSON'S
M a t h e matic al S erie s .
Graded to the wants of Primary, Intermediate, Grammar,
Normal, and High Schools, Academies, and Colleges,
Progressive Table Book.
Progressive Primary Arithmetic.
Progressive Intellectual Arithmetic.
Rudiment* of Written Arithmetic.
JUNIOK-CLASS ARITHMKT1C, Oral and Written. NEW.
Progressive Practical Arithmetic.
Key to Practical Arithmetic.
Progressive Higher Arithmetic.
Key to Higher Arithmetic.
New Klemeutary Algebra.
Key to New Momentary Algebra.
New University Algebra.
Key to New University Algebra.
New Geometry and Trigonometry. In one vol.
Geometry, Plane and Solid. In separate vol.
Trigonometry, Plane and Spherical. In separate vol.
New Analytical Geometry and Conic Sectio
New Surveying and Navigation. ^
New Differential and Integral Calculus.
University Astronomy Descriptive and Phys'cal.
Key to Geometry and Trigonometry, Analytical Geometry and Conic Sec-
tions, Surveying and Navigation.
Entered, according to Act of Congress, in the year 1858, and again in the year 1863, by
DANIEL W. FISH, A.M.,
In tne Clerk's Office of the District Court of the United States, for the Northern
District of New York.
''
Q A i *
RSI
PREFACE.
THE importance, and the practical benefit to be derived from the
study of Intellectual Arithmetic, not only as a preparation for busi-
ness life, but as a means of developing and strengthening the
thinking and reasoning powers, and of thorough mental culture,
can not be over-estimated. Not only is it a necessary- study for
young pupils, but indispensable to the more advanced student, as
a preparation for the prompt and accurate business man. And it
is believed that, as a general rule, the most critical, correct, and
ready students of mathematics are those who have been most
thoroughly drilled in intellectual arithmetic.
This work has been prepared more especially for advanced classes,
and is designed for those who have first been well taught in the
primary book, and for such as are pursuing the study of written
arithmetic, or algebra, and who have never been thoroughly exer-
cised in this branch of study.
Only a few of the many points of difference between this and
other similar works, and which, it is believed, renders this superior
to them, will be referred to.
It is more complete, comprehensive, and progressive In its char-
acter. The arrangement and classification are more strictly sys-
tematic, and in accordance with the natural order of mathematical
science. The development of principles, and their applications,
are shown by a more numerous selection, and greater variety of
appropriate examples, progressively arranged, commencing with
those that are simple and easy, and advancing to those more com-
plex and difficult.
At intervals, and especially in the closing sections of each chap-
ter, examples are given containing such a combination of principles,
and forms of analysis, as to require a knowledge of almost every
principle previously taught, thus affording the pupil a thorough
review, as well as requiring him to classify his knowledge of what
he has been over.
One of the most important, and, it is thought, one of the most
original and useful features of this work, is the full, concise, and
uniform system of ANALYSIS it contains, the result of long expe-
rience in the school-room.
Particular attention is invited to the subjects of Fractions, Per-
centage, and Interest ; their treatment is peculiar, and adapted to
obviate many of the difficulties, and greatly abbreviate most of the
operations in them.
M289985
4 PREFACE.
The chapter of Miscellaneous Examples will afford a valuable
and thorough drill to the advanced student of arithmetic or alge-
bra. They contain a great variety of principles, and while they
may be considered difficult, yet the full analysis given of every
principle, and the selection of numbers so adapted to the conditions
of the question as to produce results free from large and difficult
fractions, will render a mental solution of them comparatively easy.
In conclusion, we may be allowed to express the belie'f, that
in this work the thorough teacher will find a desideratum long
sought in this department of science, the means of mental dis-
cipline and development such as has been furnished by uo
similar treatise.
THE AUTHOR.
SUGGESTIONS TO TEACHERS.
Pupils of nearly the same degree of advancement should be
classed together. Regular exercises should be assigned to the
class, and sufficient time allowed them to thoroughly examine
their lesson before being called upon to recite.
The use of the book at the time of recitation should be strictly
prohibited, except, perhaps, in some of the more difficult lessons
in the latter part of the work.
Each example should be read but once, slowly and distinctly,
the pupils called upon promiscuously, who should arise, stand
erect, repeat the example, and then give the analysis. This will
secure close attention.
Every question should be clearly and thoroughly analyzed, and
the pupil required to adhere strictly to the forms of solution given,
unless better ones can be substituted ; and in no case should he
be allowed to omit the conclusion, commencing with " Therefore."
The class should be encouraged to detect and correct errors in
/statement or analysis, to criticise and make proper inquiries, all
of which should be signaled by the uplifted hand.
It is suggested that the class be occasionally exercised upon
"Ringing the Changes," as explained in the Appendix, and which
may be applied to a great number and variety of examples. It
will not only afford a valuable drill, but a pleasant and enlivening
exercise.
INTELLECTUAL ARITHMETIC.
CHAPTER I.
ADDITION.
1. 1. JAMES had 1 cent, and his father gave him
1 more ; how many had he then ?
2. If a slate pencil cost 2 cents, and a steel pen
1 cent, how many cents will both cost ?
3. George spent 2 cents for candy, and had 2 cents
left ; how many cents had he at first ?
4. A farmer sold a calf for 3 dollars, and a pig
for 2 dollars ; how many dollars did he receive for
both ?
5. Mary gave 2 cents for some tape, and 5 cents
for a thimble ; how many cents did she give for both ?
6. Martin gave John 3 apples, and kept 4 for him-
self; how many apples had he at first ?
7. There are 4 books on one desk, and 2 books
on another ; how many books on both desks ?
8. If a lemon cost 3 cents, and an orange 5 cents,
how many cents do both cost ?
9. In a Certain class there are 5 girls and 4 boys ;
how many pupils in the class ?
10. Samuel had 3 marbles, and his brother gave
him B more ; how many did he then have ?
11. Bought a barrel of apples for 2 dollars, and a
cord of wood for 4 dollars ; how many dollars did
both cost *f
1* ()
INTELLECTUAL ARITHMETIC. [2.
1. !j^4im^ JL ^re how many ?
2. 1 jjucU JU acejiow many ?
3. 2 and '1 -are how many ?
4. 2 and :) arejiow many:'
5. 2 and i^arejiow many?//*
6. 2 and ,"> are how many ?
7. -J aint ~J]ljirjiow many ?
8. 2 ]an^ < J..^areJy>w many ?
9. 2. 'and JLajj*ow many?
10. 2 ami 9-. jM-'-C. how many ?
11. 3__and 3 .qre how many ?
12.
13. gu3ir^I r ow many ?
14. 3 ^njLjS-^ are how many ?
15. 3^and X ___ are how many?
16. 3 ancr~8 _ are how many ?
17. 3 and are how many ?
18. 3 and 10 are how many?
19. 4 anjL-2 are how many ?
20. 4 and j^are how many ?
21. r aiiEIJ: are how many ?
22. 4_ and L are how many ?
23. 4__and _6 are how many ?
24. 4 ajiJ^n nrp how many?
25. 4 ^Hji j^ are how many ?
26. 4^jind 9 are how many ?
27. 4L3!ii^r^are how many?
28. &^nd^L- are how many ?
29. joT^ni^^ _ are how many?
30. v/jT and t 3^ are how many ?
31. ^ and _jL are how many?
32. ^^ jiudr 5 __ are how many ?
33. _5^ ^and 6 are how many ?
34. JL ^and ^^1-are how many ?
35. ^^JBa.jS^are how many ?
36. ^C^nji_^2---a?e how many ?
3.] INTELLECTUAL ARITHMETIC. 7
3. 1. A man bought a calf for 7 dollars, and
sold it for 3 dollars more than he gave for it ; how
much did he sell it for ?
2. Bought a barrel of cider for 2 dollars, and 20
bushels of apples for 8 dollars ; how many dollars
did the whole cost ?
3. If a coat cost 10 dollars, and a pair of panta-
loons 5 dollars, what will be the cost of both ?
4. A merchant bought 4 boxes of black tea, and
7 boxes of green tea ; how many boxes did he buy
of both kinds ?
5. A farmer sold 5 sheep to one man, and 9 to
another ; how many did he sell to both ?
6. Ralph walked 4 miles and rode 10 miles ; how
many miles did he go ?
7. Philip answered 8 questions in geography, and
Oliver 5 ; how many questions did both answer ?
8. If a quart of chestnuts cost 10 cents, and a
quart of walnuts 4 cents, what will be the cost pf
one quart of both ?
9.
6
and
3
are
how
many
?
10.
6
and
4
are
how
many
?
11.
6
and
5
are
how
many
?
12.
G
and
6
are
how
many
?
13.
6
and
7
are
how
many
?
14.
6
and
8
are
how
many
?
15.
6
and
9
are
how
many
?
16.
6
and
10
are
how
many
?
17.
7
and
4
are
how
many
?
18.
7
and
5
are
how
many
?
19.
7
and
6
are
how
many
?
20.
7
and
7
are
how
many
?
21.
7
and
8
are
how
many
?
22.
7
and
9
are
how
many
?
23.
rr
1
and
10
are
how
many
?
24.
8
and
3
arc
how
many
?
INTELLECTUAL ARITHMETIC. [4.
25. 8 and 4 are how many ?
26. 8 and 5 are how many ?
27. 8 and 6 are how many ?
28. 8 and 7 are how many ?
29. 8 and 8 are how many ?
30. 8 and 9 are how many ?
31. 8 and 10 are how many ?
32. 9 and 4 are how many ?
33. 9 and 5 are how many ?
34. 9 and 6 are how many?
35. 9 and 7 are how many ?
36. 9 and 8 are how many ?
37. 9 and 9 are how many ?
88. 9 and 10 are how many ?
39. 10 and 4 are how many ?
40. 10 and 5 are how many ?
41. 10 and 6 are how many ?
4. 1. Eli paid 9 shillings for a pair of skates,
and 6 shillings for a cap ; how many shillings did he
pay for both ?
ANALYSIS. Since Eli paid 9 shillings for a pair of
skates, and 6 shillings for a cap, he paid for both, the
sum of 9 shillings -and 6 shillings. The sum of 9 shil-
lings and 6 shillings is 15 shillings. Therefore, he paid
J 5 shillings for both.
2. A farmer sold some oats for 7 dollars, and a
ton of hay for 9 dollars ; how many dollars did ho
receive for both ?
3. Paid 8 cents for some raisins, and 6 cents for
some cloves ; how many cents did both cost ?
4. Nancy is 10 years old;, how old will she be
5 years from this time ?
5. A lady paid 7 dollars for a shawl, and 6 dol-
lars for a dress : what was the cost of both ?
S.j INTELLECTUAL ARITHMETIC. 9
6. There are 9 boys in one class, and 7 in anoth-
er ; how many in both classes ?
7. Margaret has 8 flower pots in one room, and 5
in another ; how many has she in the two rooms ?
8. If you read 10 pages to-day, and 7 pages to-
morrow, how many will you read in both days ?
9. If a firkin of butter is worth 9 dollars, and
two cords of wood are worth 8 dollars, what are both
worth ?
10. Myron has 6 young rabbits and 4 old ones ;
how many rabbits has he ?
11. There are 10 biids sitting on one limb, and
9 on another ; how many birds on both limbs ?
12. A boy, having some peaches given him, put 8
into his hat, and 7 into his pockets ; how many
peaches were given him ?
13. Edwin found 6 ripe pears under one tree, and
8 under another ; how many pears did he find under
both trees ?
14. In one yard are 5 cows, and in another 7 ;
liow many cows in both yards ?
15. If you work 8 examples in arithmetic to-day,
and 8 more to-morrow, how many will you work
in both days ?
16. Robert gave 8 nuts to Henry, and kept 9 ; how
many nuts had he at first ?
1. 7 and 4 are how many ?
2. 9 and 7 are how many ?
3. 6 and 7 are how many ?
4. 5 and 9 are how many ?
5. 8 and 7 are how many ?
6. 10 and 6 are how many ?
7. 9 and 6 are how many ?
8. 8 and 8 are how many ?
10 INTELLECTUAL ARITHMETIC. [5.
9. 6 and 8 are how many ?
10. 7 and 9 are how many ?
11. 8 and 10 are how many ?
12. 11 and 5 are how many ?
13. 13 and 7 are how many?
14. 14 and 6 are how many?
15. 12 and 4 are how many?
16. 15 and 8 are how many?
17. 9 and 12 are how many ?
18. 16 and 8 are how many?
19. 17 and 10 are how many?
V 20. 15 and 11 are how many ?
21. 13 and 12 are how many?
t
^22.
22. How many are 2 and 2 ? 12 and 2 ? 22 and
2 ? 32 and 2 ? 42 and 2 ? 52 and 2 ? 62 and 2 ?
72 and 2 ? 82 and 2 ? 92 and 2 ?
23. How many are 3 and 3 ? 13 and 3 ? 23 and
3 ? 33 and 3 ? 43 and 3 ? 53 and 3 ? 63 and 3 ?
73 and 3 ? 83 and 3 ? 93 and 3 ?
24. How many arc 4 and 4 ? 14 and 4? 24 and
4 ? 34 and 4 ? 44 and 4 ? 54 and 4 ? 64 and 4 ?
74 and 4? 84 and 4? 94 and 4?
25. How many are 5 and 5 ? 15 and 5 ? 25 and
5 ? 35 and 5 ? * 45 and 5 ? 55 and 5 ? 65 and 5 ?
75 and 5 ? 85 and 5 ? 95 and 5 ?
26. How many are 6 and 6 ? 16 and 6 ? 26 and
6 ? 36 and 6 ? 46 and 6 ? 56 and 6 ? 66 and 6 ?
76 and 6 ? 86 and 6 ? 96 and 6 ?
27. How many are 7 and 7 ? 17 and 7 ? 27 and
7? 37 and 7? 47 and 7 ? 57 and 7 ? 67 and 7?
77 and 7 ? 87 and 7 ? 97 and 7 ?
28. How many are 8 and 8 ? 18 and 8 ? 28 and
8 ? 38 and 8 ? " 48 and 8 ? 58 and 8 ? 68 and 8 ?
78 and 8 ? 88 and 8 ? 98 and 8 ?
29. How many are 9 and 9 ? 19 and 9 ? 29 and
O.] INTP:LLECTUAL ARITHMETIC. H
30. How many are 8 and 7 ? 18 and 7 ? 28 and
7 ? 38 and 7 ? 48 and 7 ? 58 and 7 ? 68 and 7 ?
78 and 7 ? 88 and 7 ? 98 and 7 ?
31. How many are 10 and 10 ? 20 and 10 ? 30
and 10 ? 40 and 10 ? 50 and 10 ? 60 and 10 ? 70
and 10 ? 80 and 10 ? 90 and 10 ?
32. How many are 4 and 8 ? 14 and 8 ? 24 and
8 ? 34 and 8 ? 44 and 8 ? 54 and 8 ? 64 and 8 ?
74 and 8 ? 84 and 8 ? 94 and 8 ?
33. How many are 6 and 5 ? 16 and 5 ? 26 and
5 ? 36 and 5 ? 46 and 5 ? 56 and 5 ? 66 and 5 ?
76 and 5 ? 86 and 5 ? 96 and 5 ?
34. How many are 11 and 9 ? 21 and 9 ? 31 and
9 ? 41 and 9 ? 51 arid 9 ? 61 and 9 ? 71 and 9 ?
81 and 9 ? 91 and 9 ?
35. How many are 9 and 7 ? 19 and 7 ? 29 and
7 ? 39 and 7 ? 49 and 7 ? 59 and 7 ? 69 and 7 ?
79 and 7 ? 89 and 7 ? 99 and 7 ?
36. How many are 4 and 8 ? 14 and 8 ? 24 and
8 ? 34 and 8 ? 44 and 8 ? 54 and 8 ? 64 and 8 ?
74 and 8 ? 84 and 8 ? 94 and 8 ?
37. How many are 3 and 10 ? 13 and 10 ? 23
and 10 ? 33 and 10 ? 43 and 10 ? 53 and 10 ?
63 and 10 ? 73 and 10 ? 83 and 10 ? 93 and 10 ?
6. Addition is the process of uniting several numbers
of the same kind into one equivalent number. The result
is called the Sam or Amount. The sign of addition is
written thus, +, and is called plus. When placed between
two numbers or quantities, it shows that they are to be
added together. Thus, 5 + 3 + 2 signifies that 5 and 3
and 2 are to be added together, and is read five, plus three,
plus two.
The sign of equality is written thus, :=. When placed
between two numbers or quantities, it signifies that the
latter is equal to the former.
12 INTELLECTUAL ARITHMETIC. [6.
1. 6 + 4 + 5 are how many?
2. 9 + 7 + 3 are how many?
3. 10 + 6 4- 4 are how many?
4. 13 -f- 8 + 6 are how many?
5. 12 + 10 + 9 are llow many?
6. 11 4" 6' 4" -7 are how many?
7. 8 -f 10 + 7 + 6 are now many?
8. 34 -jr ~ >} are uow man y ?
ANALYSIS. As many as the sum of 34 and 25 ; 34
is composed of 3 tens and 4 units, and 25 0/* 2 tens and 5
units ; 3 te?is and 2 fc>*s are 5 tos, or 50, arca 7 4 w
and 5 im/te are 9 units, which added to 50 makes
Therefore 34 + 25 are 59.
9. 46
10. 35
11. 40
31 are how many?
52 are how many ?
JLI. <v -t- 80 -f- 6 are how many ?
12. 19 4" 12 4~ 8 are how many ?
13. 60 4~ 72 4- 4 are how many?
14. 48 + 80 + 10 are how many ?
15. 13 4- 25 4- 7 are how many ?
16. 29 4- 24 4- 30 are how many?
17. 64 4- 40 4- 9 ai *e how many?
18. 29 + 17 + 12 are how many?
19. 7 4~ 37 4- 26 are how many ? ^
20. 14 4~ 39 4~ 4 are how many ?
21. 48 4~ 31 4~ 9 are h w many?
22. 56 4" 41 + 10 are how many ?
23. 75 4- 60 4- 20 are how many?
24. 16 4- 19 4~ 28 are how many?
25. 43 + 37 + 15 are how many?
26. 84 + 75 4- 20 are how many?
27. 68 4- 52 4- 45 are how many?
28. 35 + 77 4- 51 are how many ?
29. 80 + 95 + 12 are how many?
30. 42 4- ?& 4-17 + 9 are how many?
7.] INTELLECTUAL ARITHMETIC. 13
7. 1. A man gave 12 dollars for some corn, 8
dollars for a ton of hay, and 6 dollars for a barrel of
flour ; how much did he pay for all ?
ANALYSIS. He p^ v ^ the sum of 12 dollars, 8 dollars,
and dollars; 12 and 8 are 20, and 6 are 26 dollars.
Therefore, fyc.
2. Emily paid 12 shillings for a dress, 10 shillings
for a pair of shoes, and 6 shillings for a pair of
gloves ; what was the cost of the whole ?
3. A drover had 15 horses in one pasture, 13 in
another, and 9 in another ; how many horses had he ?
4. James has 10 cents, Henry has 7, and John
has 5 ; how many cents have the three boys ?
5. Bought a pound of candles for 14 cents, a
pound of coffee for 12 cents, and a bar of soap for
9 cents ; what was the cost of the whole ?
6. In a shop are 7 men, 9 boys, and 6 girls at
work ; how many persons at work in the shop ?
7. A lady sold 10 pounds of butter at one time,
12 pounds at another, and 8 pounds at another;
how many pounds did she sell in all ?
8. William bought an arithmetic for 15 cents,
a quire of paper for 12 cents, and a bottle of ink for
6 cents ; what was the cost of the whole ?
9. Peter, Dexter, and Irwin gave some money to
a beggar ; Peter gave him 10 cents, Dexter 11 cents,
and Irwin 13 cents ; how many cents did they all
give him ?
10. In a certain school there are 18 girls and 4
boys ; how many pupils in the school ?
11. A man bought a sleigh for 20 dollars, paid
10 dollars for repairing it, and 6 dollars for paint-
ing it ; what did it cost him ?
12. Morgan earned 25 cents one day, and 32 the
next ; how much did he earn in both days ?
14 INTELLECTUAL ARITHMETIC. [7.
13. A farmer sold a cow for 22 dollars, a ton of
hay for 11 dollars, and a calf for 7 dollars ; what
did he receive for all ?
14. A traveler walked 3C miles one day, 26
miles the next, and 21 miles the next ; how many
miles did he travel in the three days ?
15. One day Sarah wrote 18 lines, Maria 16, and
Carrie 15 ; how many lines did they all write ?
10. A man gave 60 dollars for a horse, 23 dollars
for a saddle, and 9 dollars for a bridle ; what was the
cost of the whole ?
17. Rollin received 36 new cents on his birthday,
34 on Christmas day, and 27 on New Year's day ;
how many cents did he receive ?
18. A man bought a piece of land for 56 dollars,
and paid 25 dollars for fencing it ; for how much
must he sell it to gain 15 dollars ?
19. Amasa, going into the orchard, found 12 ripe
apples under one tree, 15 under another, 11 under
another, and 9 under another; how. many apples did
he find under the four trees ?
20. A grocer bougfit some hams for 20 dollars,
some oats for 19 dollars, some fowls for 16 dollars,
and five barrels of apples for 10 dollars ; what
did he pay for all ?
21. A miller shipped by railroad 28 barrels of
flour at one time, 37 at another, and 40 at another ;
how many barrels did he ship ?
22. If Lucian gave 45 cents for a penknife, 15
cents for some paper, 12 cents for a folder, and 7
cents for some wafers, what did the whole cost him ?
23. A drover bought 26 sheep of one man, 30 of
another, 37 of another, and 40 of another ; how many
sheep did he purchase ?
24. Bought a horse for 75 dollars, and sold him
so as to gain 25 dollars ; how much did I receive for
him?
7.] INTELLECTUAL ARITHMETIC. 15
25. A lady went shopping, and bought a silk dress,
for 22 dollars, a muff for 16 dollars, a shawl for 14
dollars, and had 7 dollars left ; how much money
had she at first ?
26. Peleg bought a knife for 35 cents ; he gave
the knife and 12 cents for a sled, which he sold so as
to gain 10 cents on the cost; how much did he
receive for the sled ?
27. In a certain orchard are 31 apple trees, 27
peach trees, and 19 pear trees ; how many trees in
the orchard ?
28. A farmer raised 54 bushels of wheat, 66 bush-
els of oats, and 40 bushels of corn ; how many bush-
els of grain did he raise ?
29. If it be 62 feet from the ground to the belfry
of a church, and 76 feet from the belfry to the top
of the steeple, how high is the top of the steeple from
the ground ?
30. In March are 31 days, in April 30, in May 31 ;
how many days in those three months ?
31. If a farm contain 83 acres of cleared land
and 25 acres of wood land, how many acres are in
the whole farm ?
32. The distance from Boston to Worcester is 44
miles, from Worcester to Warren 29 miles, and from
Warren to Springfield 25 miles ; how many miles
from Boston to Springfield ?
33. Batavia is 37 miles west from Rochester, and
31 miles east from Buffalo by railroad ; what is the
distance from Rochester to Buffalo ?
34. A tailor bought three pieces of cloth, the first
containing 29 yards, the second 26 yards, and the
third 25 yards ; how many yards did the three pieces
contain ?
35. A mechanic sold a carriage for 15 dollars, and
a sleigh for 48 dollars ; what did he receive for both ?
16 INTELLECTUAL ARITHMETIC. [8.
36. A soldier was 33 years of age when ho en-
gaged in the battle of New Orleans ; what was his
age 44 years after ?
37. A farmer received 29 dollars for some pork,
18 dollars for some hay, 15 dollars for some oats, and
12 dollars for four cords of wood ; how many dollars
did he receive for the whole ?
38. Thomas hoed 13 rows of corn, Lyman 16 rows,
Cyrus 14 rows, and Warren 10 rows; how many
rows did they all hoe ?
39. A gentleman gave 80 dollars for a coat, 7
dollars for a vest, 11 dollars for a pair of pantaloons,
and 5 dollars for a hat ; what was the cost of all ?
40. Paid 60 dollars for a carriage, which, with 25
dollars, I gave for a horse ; paid 12 dollars for his
keeping, and then sold him for 20 dollars more than
he cost me ; for how much did I sell him ?
41. A man deposited 40 dollars in a bank, which
was 15 dollars less than his neighbor deposited at
the same time ; how much did both deposit ?
CHAPTER II.
SUBTRACTION.
8. 1. JAMES, having 5 cents, gave 2 cents for a
pencil ; how many cents had he left ?
ANALYSIS. Since James had 5 cents, and gave 2 cents
for a pencil, he had left the . difference between 5 cents
and 2 cents. The difference betitfeen 5 cents and 2 cents,
is 3 cents. Therefore, he had 3 cents left.
2. Mary had 3 peaches, and ate 1 of them ; how
many had she left ?
3. Ella had 6 pinks, and gave her sister 2 ; how
many had she left ?
8.J INTELLECTUAL ARITHMETIC. 17
4. On a rose bush were 7 roses, and Eliza picked
3 of them ; how many were left ?
5. A man bought 10 sheep and sold 5 of them ;
how many had he left ?
6. A bad boy robbed a bird's nest of 7 eggs, and
broke 4 of them ; how many remained unbroken ?
7. If there are 8 pigs in a pen, and five of them
get out, how many remain ?
8. If a merchant have 12 barrels of flour, and
lie sell 7 of them, how many barrels will he have
left?
9. Margaret is 11 years old, and Julia 4 years
younger ; what is Julia's age ?
10. Stephen had 9 marbles, and lost 3 of them ;
how many had he left ?
11. A man, earning 10 dollars a week, spends 6
dollars for provisions ; how many dollars has he
left?
12. Giles borrowed 12 dollars, and paid 7 dollars
of it ; how much of it remained unpaid ?
13. 5 less 3 are how many?
14. 9 less 2 are how many?
15. 7 less 4 are how many?
16. 11 less 3 are how many?
17. 8 less 2 are how many?
18. 6 less 4 are how many?
19. 10 less 6 are how many?
20. 7 less 5 are how many?
21. 14 less 8 are how many?
22. 12 less 7 are how many?
23. 9 less 8 are how many?
24. 15 less 9 are how many?
25. 20 less 6 are how many?
26. 23 less 10 are how many?
27. 18 less 4 are how many?
2
18 INTELLECTUAL ARITHMETIC. [9.
28. 11 from 14 leaves how many?
29. 9 from 12 leaves how many ?
30. 7 from 16 leaves how many?
81. 12 from 20 leaves how many?
32. 10 from 17 leaves how many?
33. 8 from 18 leaves how many ?
34. 6 from 15 leaves how many ?
35. 11 from 21 leaves how many ?
86. Take 5 from 12, and how many remain ?
37. Take 10 from 19, and how many remain ?
38. Take 7 from 11, and how many remain?
39. Take 9 from 13, and how many remain ?
40. Take 12 from 23, and how many remain ?
41. What is the difference between 14 and 6?
42. What is the difference between 20 and 8 ?
43. What is the difference between 19 and 15?
44. What is the difference between 11 and 22?
45. What is the difference between 17 and 6 ?
46. What is the difference between 25 and 9 ?
9. Subtraction is the process of finding the difference
between two numbers ; the result is called the Difference or
Remainder. The sign of subtraction is written thus, ,
and is read minus. Placed between two numbers, or quan-
tities, it shows that the one after it is to be taken from the
one before it. 9 5 shows that 5 is to be subtracted from
9, and is read nine minus jive.
1. How many are 10 6 ?
2. How many are 9 4 ?
3. How many are 12 7 ?
4. How many are 15 9?
5. How many are 20 8?
6. How many are 18 5 ?
7. How many are 17 3 ?
8. How many are 21 9?
ID.] INTELLECTUAL ARITHMETIC. 19
9. How many are 16 7 ?
10. How many are 19 9?
11. How many are 21 12?
12. How many are 17 10?
13. How many are 18 9?
14. How many are 20 11?
15. How many are 25 10?
16. How many are 32 12?
17. How many are 46 10?
18. How many are 57 34?
ANALYSIS. 57 is equal to 5 tens and 7 units, and 34 is
equal to 3 tens and 4 units ; 3 tens from 5 tens leave 2 tens,
or 20, and 4 units from 7 units leave 3 units, which added
to 20 make 23. Therefore, 57 34 = 23.
19. 39 27 are how many ?
20. 43 32 are how many ?
21. 29 17 are how many ?
22. 54 21 are how many ?
23. 67 45 are how many ?
24. 75 25 are how many ?
25. 89 74 are how many ?
26. 39 19 are how many ?
27. 41 30 are how many?
28. 96 81 are how many?
1O. 1. If you have 12 turkeys, and sell 9 of
them, how many will you have left?
2. A grocer had 10 boxes of lemons, and sold 6
boxes ; how many boxes had he left ?
3. There are 20 men and 8 boys at work in a
bookbindery ; how many more men than boys are
there ?
4. From a cistern holding 36 barrels of water, 12
barrels leaked out ; how many barrels remained ?
20 INTELLECTUAL ARITHMETIC. [1C.
5. A man, having 25 dollars due him, received a
ton of hay worth 11 dollars, and the remainder in
money ; how much money did he receive ?
6. There are 46 trees in an orchard ; 85 of them
are apple trees and the remainder peach trees ; how
many are peach trees ?
7. From a piece of broadcloth containing 27 yards,
15 yards were cut ; how many yards remained ?
8. A jeweler bought a watch for 60 dollars, and
sold it again for 75 dollars ; how much did he
gain ?
9. The whole number of pupils registered in a
certain school is 87, and but 65 are in attendance ;
how many are absent ?
10. Charles has 48 cents, and buys a slate for 16
cents ; how many cents has he left ?
11. There are 36 wild ducks in a flock, and a
huntsman fires at them and kills 13 ; how many are
left?
12. A farmer sold a cow for 22 dollars that cost
him 30 dollars ; how much did he lose by the bar-
gain ?
13. A grocer bought a quantity of sugar for 39
dollars, and sold it for 50 dollars ; how much did he
gain ?
14. In a school are 2? boys and 35 girls ; ho\*
many more girls than boys ?
15. The distance from Cincinnati to Miamisburg
by railroad is 49 miles, and to Dayton 60 miles;
what is the distance from Miamisburg to Dayton ?
16. In a warm day the thermometer indicated 85
degrees ; how many degrees was that above th^
freezing point, which is 32 degrees ?
17. A man died at the age of 77 years, having
been married 49 years ; what was his age when he
married ?
11.] INTELLECTUAL ARITHMETIC. 21
11. 1. 4 and 7 and 9, less 8, are how many ?
2. 9 and 12 and 6, less 7, are how many ?
3. 14 and 10 and 12, less 24, are how many ?
4. 20 and 16 and 5, less 14, are how many ?
5. 44 and 20 and 10, less 50, are how many ?
6. 27 and 15 and 12, less 30, are how many ?
7. How many are 9 + 12 + 15 25 ?
8. How many are 26 + 15 + 7 18 ?
9. How many are 40 + 10 + 8 20 ?
10. How much less than 64, is 25 + 20 ?
11. How much less than 56, is 28 + 16 ?
12. How much less than 100, is 46 + 34 ?
13. How many are 16 + 12 + 9 + 5 32
i 8?
14.' How many are 33 + 28 + 9 30 -f 15 ?
15. How many are 84 + 26 + 15 70 + 25 ?
16. A man, having 30 dollars, paid 15 dollars for
a coat, 5 dollars for a vest, and 4 dollars for a hat ;
how much had he left ?
ANALYSIS. He had left the difference between 30 dollars
and the sum of 15 -j- 5 -f- 4 dollars ; 15 dollars and 5
dollars are 20 dollars, and 4 dollars are 24 dollars, and 30
dollars less 24 dollars are 6 dollars. Therefore, fyc.
17. Ellen had 15 pi^ks, and she gave 6 to Mary
and 4 to Jane ; how many had she left ?
18. A woman, having 25 pounds of butter, sold 15
pounds at one time, and 10 pounds at another ; how
many pounds had she left ?
19. Amasa has 45 cents in three boxes; in the
first are 15 cents, in the second 19 cents ; how many
cents are in the third box ?
20. A boy found 8 apples under one tree. 10 under
another, and 6 under another ; he ate 3, gave away
7, and carried the remainder home; how many did
he take home ?
22 INTELLECTUAL ARITHMETIC. [11.
21. Martin, having 27 marbles, gave 12 to Albert,
and lost 5 ; how many had he left ?
22. Reuben had 16 cents ; Charles gave him 10,
Elisha gave him 9, and Henry gave him enough to
make his number 42 ; how many cents did Henry
give him ?
23. A merchant bought a hogshead of sugar for
50 dollars ; he paid 6 dollars for freight and customs,
and sold the whole for 75 dollars ; how much did he
gain ?
24. A man bought a watch for 40 dollars, a chain
for 15 dollars, and a key lor 3 dollars, and he sold
the whole for 50 dollars ; how much did he lose by
the bargain ?
25. Sarah bought a comb for 12 cents, some rib-
bon for 16 cents, a thimble for 10 cents, and some
thread for 6 cents ; how much had she left of fifty
cents, after paying for these articles ?
26. A drover bought 9 sheep of one man, 12 of
another, and 15 of another ; he afterwards sold 8
and butchered 5 ; how many had he left ?
27. From a piece of calico containing 26 yards
Jane bought a dress of 9 yards, and Josephine an-
other of 10 yards ; how many yards were left in the
piece ?
28. Richard, receiving 45 dollars for labor, paid
20 dollars for a cow, 7 dollars for a barrel of flour,
and 9 dollars for three cords of wood; how many
dollars had he left ?
29. Four men bought a horse for 80 dollars ; the
first gave 25 dollars, the second 20 dollars, and the
third 18 dollars ; how much did the fourth give ?
30. A boy had 12 marbles ; another boy gave him
10 more, another 9. and another enough more to
make his number 40 ; how many did the last boy
give him ?
11.] INTELLECTUAL ARITHMETIC. 23
31. A man traveled 22 miles one day, and 26
miles the second day, and on the third day he trav-
eled 30 miles on his return ; how many miles was he
from the place from which he first started ?
32. George sold two dozen eggs for 20 cents, one
Imshel of apples for 37 cents, and received a pair of
skates worth 75 cents ; how much did he still owe
for his skates ?
33. A man owed his grocer 18 dollars, his tailor
20 dollars, and a merchant 25 dollars ; he paid the
grocer 10 dollars, the tailor 12 dollars, and the
merchant 15 dollars ; how much did he still owe
them all ?
34. Bought a horse for 90 dollars, a cutter for 40
dollars, and a harness for 20 dollars, and then sold
the whole for 50 dollars more than the horse cost
me ; did I gain or lose by the bargain, and how
much ?
35. A lady bought a new bonnet for 7 dollars, a
dress for 12 dollars, a pair of shoes for 2 dollars, and
a parasol for 3 dollars ; she gave the merchant 3
ten dollar bills ; how many dollars must be re-
turned ?
36. A tailor bought a piece of cloth containing 31
yards, from which he sold 13 yards to one man, and
11 yards to another ; how many yards were left of
the piece ?
37. A farmer had 45 sheep in one lot, 37 in an-
other, and 30 in another ; from the first he sold
20, from the second 15, and from the third 12 ; how
many had he at first, and how many had he left?
38. A person, sitting down to play, counted his
money, and found that lie had 1 twenty dollar bill, 5
ten dollar bills, and 3 five dollar bills ; when he got
up from play, he had 7 ten dollar bills and 6 five dol-
lar bills ; did he gain or lose, and how much ?
24 INTELLECTUAL ARITHMETIC.
CHAPTER IIL
MULTIPLICATION.
2 times 1 are how many ? ; 2 times 7 are how many ?
2 times 2 are how many ?| 2 times 8 are how many?
2 times 3 are how many ? j 2 times 9 are how many ?
2 -times 4 are how many ?
2 times 5 are how many ?
2 times 6 are how many?
3 times 1 are how many?
3 times 2 are how many?
3 times 3 are how many ?
3 times 4 are how many?
3 times 5 are how many?
3 times 6 are how many?
2 times 10 are how many ?
2 times 11 are how many ?
2 times 12 are how many ?
3 times 7 are how many ?
3 times 8 are how many ?
3 times 9 are how many ?
3 times 10 are how many ?
3 times 11 are how many ?
3 times 12 are how many ?
4 times 1 are how many ?, 4 times 7 are how many ?
4 times 2 are how many ?! 4 times 8 are how many?
4 times 3 are how many ? 4 times 9 are how many ?
4 times 4 are how many ?; 4 times 10 are how many ?
4 times 5 are how many? 4 times 11 are how many ?
4 times 6 are how many ? 4 times 12 are how many?
1. How many are 2 times 6 ? 2 times 8 ? 2
times 7 ? 2 times 11 ? 2 times 9 ? 2 times 12 ?
2. How many are 3 times 5 ? 3 times 10 ? 3
times 9 ? 3 times 7 ? 3 times 6 ? 3 times 12 ?
3. How many are 4 times 4 ? 4 times 7 ? 4
times 9 ? 4 times 6 ? 4 times 10 ? 4 times 8 ?
4. At 9 dollars a ton, what will 3 tons of hay cost ?
ANALYSIS. Since one ton costs 9 dollars, 3 tons, which
are 3 times \ ton, will cost 3 times 9 dollars ; 3 times 9
dollars are 27 dollars. Therefore, at 9 dollars a ton,
3 tons of hay icill cost 27 dollars.
13.] INTELLECTUAL ARITHMETIC. 25
5. At 8 cents a quart, what will 4 quarts of blue-
berries cost ?
6. If you answer 11 questions at each recitation,
how many questions would you answer at 3 recita-
tions ? at 4 ?
7. What will 4 pairs of shoes cost at 12 shillings
a pair ?
8. If a ream of paper cost 3 dollars, what will 2
reams cost ? 3 reams ? 4 reams ?
9. Eight New York shillings make a dollar ; how
many shillings in 3 dollars ? in 4 dollars ? in 5
dollars ?
10. At 12 cents a yard, what will 3 yards of calico
cost ? 4 yards ?
13.
5 times 1 are how many ?| 5 times 7 are how many ?
5 times 2 are how many ? 5 times 8 are how many ?
5 times 3 are how many ? 5 times 9 are how many ?
5 times 4 are how many ?j5 times 10 are how many?
5 times 5 are how many ? j 5 times 11 are how many?
5 times 6 are how many ?|5 times 12 are how many ?
6 times 1 are how many ?| 6 times 7 are how many?
6 times 2 are how many ? 6 times 8 are how many ?
6 times 3 are how many ? 6 times 9 are how many ?
6 times 4 are how many ? 6 times 10 are how many?
6 times 5 are how many ? 6 times 11 are how many ?
6 times 6 are how many ? 6 times 12 are how many ?
7 times 7 are how many ?
7 times 1 are how many?
7 times 2 are how many ?
7 times 3 are how many ?
7 times 4 are how many ?
7 times 5 are how many ?
7 times 8 are how many ?
7 times 9 are how many ?
7 times 10 are how many?
7 times 11 are how many ?
./ I t/
7 times 6 are how many ? ! 7 times 12 are how many?
3
26 INTELLECTUAL ARITHMETIC. [14.
1. How many are 5 times 6 ? 5 times 8 ? 5 times
4 ? 5 times 3 ? 5 times 7 ? 5 times 12 ? 5 times 9 ?
5 times 11?
2. How many are 6 times 3 ? 6 times 10 ? 6 times
6 ? 6 times 9 ? 6 times 7 ? 6 times 11 ? 6 times 8 ?
6 times 12 ?
3. How many are 7 times 2 ? 7 times 7 ? 7 times
4 ? 7 times 9 ? 7 times 12 ? 7 times 10 ? 7 times 8 ?
7 times 11 ? 7 times 6 ?
4. If you recite 8 perfect lessons in 1 week, how
many would you recite in 5 weeks ? in 6 weeks ? in
7 weeks ?
5. What will G pounds of sugar cost at 10 cents a
pound ? at 9 cents ?
6. If you write 5 lines a day, how many lines will
you write in 4 days ? in 7 days ?
7. What cost 5 yards of ribbon, at 8 cents a yard ?
at 9 cents ?
8. What cost 7 barrels of apples at 2 dollars a
barrel ?
9. At 12 dollars a ton, what will 6 tons of hay
cost ? 7 tons ? 5 tons ?
10. If a boat sail 9 miles an hour, how far will
she sail in 4 hours ? in 6 hours ? in 7 hours ?
11. What cost 7 barrels of flour, at 11 dollars a
barrel ?
12. There are 7 days in 1 week ; how many days
in 5 weeks ? in 6 weeks ? in 9 weeks ?
14.
8 times 1 are how many ?|8 times 7 are how many?
8 rimes 2 are how many ?' 8 times 8 are how many?
8 times 3 are how many ?:8 times 9 are how many?
8 times 4 arc how many ? ! 8 times 10 are how many ?
8 times 5 are how many ? 8 times 11 are how many ?
f times 6 are how many ? 1 8 times 12 are how many ?
14.] INTELLECTUAL ARITHMETIC. 27
9 times 1 arc how many ?
9 times 2 are how many ?
9 times 3 are how many ?
9 times 4 are how many ?
9 times 5 are how many ?
9 times 6 are how many ?
9 times 7 are how many ?
9 times 8 are how many ?
9 times 9 are how many ?
9 times 10 are how many ?
9 times 11 are how many ?
9 times 12 are how many ?
10 times 1 are how many ? 1 10 times 7 are how many ?
10 times 2 are how many ? j 10 times 8 are how many ?
10 times 3 are how many ?
10 times 4 are how many ?
10 times 5 are how many ?
10 times 6 are how many ?
10 times 9 are how many ?
10 times 10 are how many ?
10 times 11 are how many ?
10 times 12 are how many ?
1. How many are 8 times 6 ? 8 times 4 ? 8 times
9 ? 8 times 5 ? 8 times 10 ? 8 times 7 ? 8 times 11 ?
8 times 8 ? 8 times 12 ?
2. How many are 9 times 3 ? 9 times 7 ? 9 times
4 ? 9 times 6 ? 9 times 8 ? 9 times 12 ? 9 times 9 ?
9 times 5 ? 9 times 11 ? 9 times 10 ?
3. How many are 10 times 10 ? 10 times 8 ? 10
times 4 ? 10 times 7 ? 10 times 11 ?
4. What will 8 barrels of flour cost, at 9 dollars a
barrel ?
5. What will 9 chairs cost, at 10 shillings apiece ?
6. If 5 boys can sit on one bench, how many can
sit on 7 benches ? on 8 benches ?
7. If Harvey can earn 10 dollars, in one month,
how many dollars can he earn in 8 months ? in 9
months ? in 7 months ?
8. If 8 men can do a piece of work in 7 days, how
many days will it take one man to do it ?
9. If two barrels of flour last 8 persons 3 months,
how long will they last one person ?
10. How many dollars will buy 10 tons of hay, at
12 dollars a ton?
28 INTELLECTUAL ARITHMETIC.
11. A farmer divided his farm into 9 fields, con-
taining 11 acres each ; how many acres in his
farm ?
12. If a man travel 6 miles an hour, how far will
he travel in 7 hours ? in 8 hours ? in 9 hours ? in 6
hours ?
13. When eggs are 11 cents a dozen, what will be
lie cost of 7 dozen ? of 9 dozen ? of 10 dozen ?
13.
11 times 1 are how many ?
11 times 2 are how many ?
11 times 3 are how many ?
11 times 4 are how many ?
11 times 5 are how many ?
11 times 7 are how many ?
11 times 8 are how many ?
11 times 9 are how many ?
11 times 10 are how many ?
11 times 11 are how many ?
v
11 times 6 are how many ? 11 times 12 are how many ?
12 times 1 are how many ? ( 12 times 7 are how many ?
12 times 2 are how many ? j 12 times 8 are how many ?
12 times 3 are how many ? 12 times 9 are how many ?
12 times 4 are how many ? 12 times 10 are how many ?
12 times 5 are how many ? 12 times 11 are how many ?
12 times 6 are how many ? 12 times 12 are how many ?
1. How many are 11 times 6 ? 11 times 10 ? 11
times 7 ? 11 times 12 ? 11 times 8 ? 11 times 11 ?
11 times 5 ?
2. How many are 12 times 10 ? 12 times 9 ? 12
times 7 ? 12 times 8 ? 12 times 6 ? 12 times 11 ?
12 times 5 ? 12 times 12 ?
3. What will be the cost of 11 bunches of grapes,
at 10 cents a bunch ?
4. What will be the cost of 12 barrels of pork, at
11 dollars a barrel ?
5. What will 11 yards of cloth cost, at 6 dollars a
yard ?
15.] INTELLECTUAL ARITHMETIC. 29
6. In an orchard are 12 rows of trees, and 11
trees in each row ; how many trees in the orchard ?
7. What will 11 turkeys cost at 6 dimes apiece ?
at 7 dimes ?
8. At 12 dollars a hundred, what will 7 hundred
oak posts cost ? 9 hundred ? 11 hundred ?
9. At 9 dollars a week, what will 5 weeks' board
cost ? 8 weeks' ? 11 weeks' ? 12 weeks' ?
10. If the fare for one person from Albany to Bos-
ton is 5 dollars, what will be the fare for 6 persons ?
for 9 persons ? for 12 persons ?
11. At 12 dollars a month, how many dollars can
a man earn in 4 months ? in 6 months ? in 9
mouths ? in 12 months ?
12. If it take 9 yards of calico for one dress, how
many yards will be required to make 4 dresses ? 7
dresses ?
13. If a farmer put his oats into 7 bins, each
containing 12 bushels, how many bushels has he ?
14. If I put 8 dollars in the savings bank every
month, how many dollars will I deposit in 7 months ?
in 8 months ? in 12 months ?
15. What will 11 dozen of eggs cost, at 9 cents
a dozen ? at 10 cents ? at 11 cents ?
16. If a quantity of provision will last 9 men 12
days, how long will the same provision last one man ?
17. When flour is 7 dollars a barrel, what must
be paid for 7 barrels ? for 8 barrels ? for 9 barrels ?
for 11 barrels ?
18. If 12 bushels of apples be picked from each
of 11 trees, how many bushels will be picked from
all?
1 G. Multiplication is the process of finding a number which shall
contain one of two given nu:nbers as many times as there are units in
the other j the result is called the Product.
30 INTELLECTUAL ARITHMETIC. [16.
The sign of multiplication is written thus, X, and, placed
between two numbers or quantities, shows that they are to
be multiplied together. 5 X 4 = 20, shows that 5 multi-
plied by 4 is equal to 20. The sign is commonly called
times. 5 X 4 is read 5 times 4.
1. 5x3 are how many ?
2. 7x4 are how many ?
3. 10x7 are how many?
4. 3x8 are how many V
5. 7x9 are how many ?
6. 12x6 are how many ?
7. 9x8 are how many ?
8. 8x11 are how many ?
9. 6x9 are how many ?
10. 5x7 are how many ?
11. 11 xlO are how many ?
12. 8x4 are how many?
13. 11 X 11 are how many ?
14. 12 x 12 are how many ?
15. What is the product of 7 X 6 ? of 9 X 5 ?
of 7 X 7 ? of 11 X 6 ?
16. What is the product of 4x3x2? of 5x
2 X 3? of 3 X 3 X9?
17. What will be the cost of 3 cows, at 23 dollars
apiece ?
ANALYSIS. Since 1 cow cost 23 dollars, 3 cows, which
are 3 times 1 cow, will cost 3 times 23 dollars. 23 is equal
to 2 tens and 3 units ; 3 times 3 units are 9 units, and 3
times 2 tens are 6 tens, or GO ; 60 and 9 are 69. Therefore,
if \ cow cost 23 dollars, 3 cows will cost 69 dollars.
18. At 31 cents a bushel, what will be the cost of
6 bushels of oats ?
19. What will be the cost of 7 bushels of apples,
at 25 cents a bushel ?
20. If a man travel 26 miles in one day, how far
can he travel in 10 days ?
21. Mary bought 6 yards of muslin at 18 cents a
yard ; what was the cost of the whole ?
22. Ephraim is 17 years old, and his father is 4
times as old ; how old is his father ?
16.] INTELLECTUAL ARITHMETIC. 31
23. At 50 dollars a share, what will 7 shares of
bank stock cost ?
24. What will 9 pounds of tea cost, at 56 cents a
pound ?
25. At 44 cents a day, what will 10 days' labor
amount to ?
26. At 3 dollars a yard, what will be the cost of a
piece of cloth containing 36 yards ?
27. What will be the cost of 12 pounds of coffee,
at 15 cents a pound ?
28. If the cars run 24 miles an hour, how far will
they run in 7 hours ?
29. In an orchard are 16 peach trees, and 6 times
as many apple trees ; how many apple trees in the ,
orchard ?
30. What will be the cost of 45 lemons, at 5 cents
apiece ?
31. If 11 men can do a piece of work in 13 days,
in what time will one man do the same work ?
32. If a man labor 12 months for 17 dollars a
month, how much will his wages amount to ?
33. At 3 dollars a barrel, what will be the cost of
28 barrels of potatoes ?
34. If in an orchard are 12 rows of trees, and
32 trees in each row, how many trees in the or-
chard ?
35. If a man can dig 28 bushels of potatoes iii ono
clay, how many bushels can he dig in 3 days ? in 4
days ? in 5 days ?
36. At 80 dollars apiece, what will be the cost of
3 horses ? of 4 horses ? of 6 horses ?
37. At 43 dollars an acre, what will 10 acres of
land cost ? 12 acres ?
38. If 8 horses eat 3 bushels of oats in one day,
how many bushels will they eat in 7 days ? in 9
days?
82 INTELLECTUAL ARITHMETIC. [17.
17. 1. What will be the cost of 6 tons of coal,
at 5 dollars a ton, and of 4 cords of wood at 3 dol-
lars a cord ?
ANALYSIS. They will cost the sum of the products of
X o dollars, and of 4 X 3 dollars ; 6 times 5 dollars are
30 dollars, and 4 times 3 dollars are 1 2 dollars ; 30 dollars
and 12 dollars are 42 dollars. Therefore, fyc.
2. A farmer sold 12 sheep at 4 dollars a head,
and 9 lambs at 2 dollars each ; how many dollars
did he receive for all ?
3. What will be the cost of 5 pounds of coffee at
15 cents a pound, and 14 pounds of rice at 6 cents a
pound ?
4. If two persons start from the same point, and
travel in opposite directions, one at the rate of 6
miles an hour, and the other 4 miles an hour, how
far apart will they be in 7 hours ? in 9 hours ? in 12
hours ?
5. If they travel in the same direction, how far
apart will they be in 5 hours ? in 7 hours ? in 24
hours ?
6. If I hire a man and his son to labor, the father
at 14 shillings a day, and the son at 8 shillings, how
much will be due them both in 6 days ? in 9 days ?
7. Asa has 16 marbles, and Omar 3 times as
many ; how many have both ?
8. A drover bought 28 sheep at 3 dollars a head,
and 5 cows at 26 dollars apiece ; what was the cost
of the whole ?
9. Justin bought 9 oranges at 4 cents apiece, 7
lemons at 3 cents apiece, and 10 load pencils at 2
cents apiece ; how much was the cost of the whole ?
10. Charles is twice as old as Byron, and Byron is
14 years old ; what is the sum of their ages ?
11. In a school-room 16 pupils can sit on each
18.] INTELLECTUAL ARITHMETIC. 33
of 5 seats, 8 on each of 3 seats, and 20 on the re-
mainder of the seats ; how many pupils can be seat-
ed in the room ?
12. A farmer has 30 sheep in each of 3 pastures,
35 in each of 2 pastures, and 40 in another pas-
ture ; how many sheep has he in all ?
13. If one boy earns 12 cents a day, another 15
cents a day, and another 20 cents a day, how much
can the 3 boys earn in 5 days ?
14. A man bought 9 yards of cloth for a suit of
clothes, at 5 dollars a yard ; he paid 7 dollars for
making the coat, 2 dollars for making the pantaloons,
and 1 dollar for making the vest ; what did his suit
cost him ?
18. 1. A farmer bought a horse for 85 dollars,
for which he gave 7 tons of hay at 9 dollars a ton,
and the remainder in money ; how much money did
he pay ?
ANALYSIS. He paid in money, the difference between
85 dollars and the product of 7 times 9 dollars ; 1 times 9
dollars are 63 dollars, and 85 dollars less 63 dollars are 22
dollars. Tfierefore, c.
2. John worked 5 days for 15 cents a day, and
Norman worked 4 days for 20 cents a day ; how
much more did Norman earn than John ?
3. A drover bought 35 sheep at 2 dollars a head,
and sold them for 90 dollars ; how much did he gain ?
4. A mechanic earned 32 dollars a month for
five months, and his apprentice 12 dollars a month
for the same time ; how much more did the one earn
than the other ?
5. If a man earn 90 cents a day, and pay 40 cents
a day for his board, how much will he save in 6 days ?
84 INTELLECTUAL ARITHMETIC. [18.
6. A farmer sold a grocer 7 pounds of butter at
20 cents a pound, and received in payment 12 pounds
of fish at 6 cents a pound ; how much was still due
the farmer ?
7< How much difference does Mr. Jones receive,
by exchanging 5 cows at 18 dollars a head, for 7
head of young cattle, at 9 dollars each ?
8. A merchant poured into a cask 14 quarts of
camphene 3 different times, and from the same cask
filled 2 cans, holding 8 quarts each, and 8 jugs, hold-
ing 6 quarts each ; how many quarts remained in
the cask ?
9. Nellie picked 6 quarts of blackberries, and
Laura picked 4 times as many, wanting 5 quarts ;
how many quarts did Laura pick ?
10. What is the difference in the cost of 30 yards
of cloth at 4 dimes a yard, and 25 yards at 3 dimes
a yard ?
11. A lady bought 6 yards of satin, at 2 dollars a
yard, 2 shawls at 9 dollars each, and some lace for 3
dollars ; she paid 4 ten dollar bills ; how much ought
she to receive back ?
12. A drover bought 50 sheep for 125 dollars ; he
sold 30 at 4 dollars a head, and the remainder at
3 dollars a head ; how many dollars did he gain by
the bargain ?
13. Three boys talking of their money, one said
he had 35 cents, another said he had twice as many,
and the third said he had as many as both the others,
wanting 10 cents ; how many cents had the last ?
14. Two men bought a horse for 75 dollars ; they
paid 2 dollars a week for keeping him, and at the
end of 12 weeks sold him for 90 dollars ; how much
did they lose by their bargain ?
15. A man, owing 100 dollars, gave a sleigh worth
40 dollars, 12 cords of wood, at 3 dollars a cord,
18.] INTELLECTUAL ARITHMETIC. 36
and the remainder in money ; how much money did
he pay ?
16. I bought a book-case for 28 dollars, a table
for 14 dollars, and 6 chairs at 2 dollars each ; I paid
for the table ; how much did I still owe ?
17. If a blacksmith earn 14 shillings a day, and a
joiner 11 shillings a day, what will the difference in
their earnings amount to in 12 days ?
18. Perry paid 3 cents for a lemon, Elisha twice
as much for a pineapple, and for a melon Albert paid
5 times as much, lacking 15 cents, as was paid for
both the lemon and the pineapple ; what would be
the cost of 3 melons at the same rate ?
19. At a public meeting are 35 gentlemen, and
3 times as many ladies lacking 30 ; how many ladies
are there, and how many ladies and gentlemen ?
20. Peter has 4 times 5 peaches, and Marcus has
3 times 6 ; how many will they both have left after
Peter gives away 7, and Marcus 5 ?
21. How many are 4 times 20, plus 3 times 10 2
22. How many are 7 times 15, plus 9 times 12 ?
23. How many are 12 times 11, minus 10 times 7 ?
24. How many are 10 times 11, less 15, plus 6
times 5 ?
25. How much less is 7 times 14, than 3 times 40 ?
26. How much less is 10 times 10, than 6 times
20 plus 25 ?
27. How much more is 8 times 16, than 7 times
13 minus 11 ?
28. How many are 9 times 12, plus 22 + 10 30 ?
29. How many are 15 X 12, + 20 X 5 ?
30. How many are 40 X 4, + 27 X 3 ?
31. How many are 16 X 9, 13 X 6 ?
32. How many are 10 + 12, X 6, 40 ?
33. How many are 14 X 8, + 18, + 12, 7 2
36 INTELLECTUAL ARITHMETIC. [19.
CHAPTER IV.
DIVISION.
19. 1. WHEN wheat is 2 dollars a bushel, how
many bushels can be bought for 12 dollars ?
ANALYSIS. Since 2 dollars will bin/ 1 bushel, 12 dol-
lars will buy as many bushels as 2 dollars, the price of \
bushel, is contained times in 12 dollars / 2 dollars is
contained in 12 dollars times. Therefore, at 2 dollars
a bushel, 6 bushels of wheat can be bought for 12 dollars.
2. How many peaches at 2 cents apiece, can bo
bought for 18 cents ?
3. If you can buy one lead pencil for 3 cents, how
many can you buy for 24 cents ?
4. For 16 dollars, how many cords of wood can
be bought at 4 dollars a cord ?
5. At 3 cents apiece, how many oranges can be
bought for 21 cents ?
6. In how many days can a man earn 20 dollars,
if he earn 2 dollars a day ?
7. Edward has 15 peaches, which he wishes to di-
vide equally among his 3 brothers ; how many must
he give to each ?
8. How many times 2 in 8 ? in 6 ? in 10 ? in 14 ?
in 16 ? in 20 ? in 24 ?
9. How many times 3 in 6 ? in 12 ? in 18 ? in
21 ? in 9 ? in 27 ? in 33 ? in 24 ? in 36 ?
10. How many times 4 in 12 ? in 8 ? in 16 ? in
20 ? in 36 ? in 24 ? in 40 ? in 32 ? in 48 ?
11. How many tons of coal, at 4 dollars a ton,
can be bought for 48 dollars ?
12. If one penholder can be bought for 4 cents,
how many can be bought for 20 cents ? for 32 cents ?
for 40 cents ?
INTELLECTUAL ARITHMETIC. 37
13. Paid 36 dollars for 3 gold chains ; what was
the cost of each ?
14. How many barrels of apples, at 3 dollars a
barrel, can be bought for 36 dollars ?
15. If you give 44 cents to 4 beggars, how many
do you give to each ?
16. If a man walk 3 miles an hour, how many
hours will it take him to walk 30 miles ?
17. Bought 4 barrels of flour for 28 dollars ; what
was the cost of one barrel ?
18. If I pay 27 cents for 3 pounds of sugar, how
much do I pay a pound ?
19. 15 are how many times 3 ? 5 ?
20. 86 are how many times 3 ? 6 ? 4 ?
21. 42 are how many times 6 ? 7 ? 3 ?
22. 24 are how many times 2 ? 6 ? 4 ? 8 ?
23. 16 are how many times 4 ? 2 ? 8 ?
24. 40 are how many times 8 ? 5 ? 4 ?
25. 72 are how many times 8 ? 6 ? 12 ?
26. 56 are how many times 7 ? 8 ? 4 ?
27. 48 are how many times 4 ? 6 ? 8 ? 12 ?
28. 32 are how many times 8 ? 2 ? 4 ?
29. 30 are how many times 5 ? 6 ? 3 ?
SO. 1. How many spools of thread, at 5 cents a
spool, can be bought for 40 cents ? for 50 cents ?
2. When lard is 7 cents a pound, how many
pounds can be bought for 56 cents ? for 63 cents ?
3. If a farmer divide 84 bushels of potatoes
equally among 7 laborers, how many bushels will
each receive ?
4. If an orchard contain 64 trees, and 8 trees in a
row, how many rows are there ?
5. If a man travel 72 miles ia 6 hours, how far
does he travel in one hour ?
88 INTELLECTUAL ARITHMETIC. [2O.
6. Levi paid 7 cents for his ball ; how many balls
at the same price could he buy for 28 cents ? for
56 cents ? for 63 cents ? for 84 cents ?
7. How many bins will be required to hold 72
bushels of- wheat, if each bin contain 8 bushels ?
8. At 5 dollars a week, how long will it take a
man to earn 45 dollars ? 50 dollars V
9. At 8 cents a pound, how many pounds of
sugar can be bought for 96 cents ?
10. If a man spend 6 cents a day for cigars, how
many days will 60 cents last him ?
11. A man bought some sheep for 48 dollars, at
4 dollars apiece ; how many sheep did he buy ?
12. If one man can do a job of work in 60 days,
in what time can 5 men do the same work ?
13. 45 are how many times 5 ? 9 ? 3 ?
14. 36 are how many times 4 ? 6 ? 9 ? 12 ?
15. 80 are how many times 10 ? 8 ? 4 ? 5 ?
16. 9l> are how many times 8 ? 6 ? 12 ? 4 ?
17. 44 are how many times 11 ? 4 ? 2 ?
18. 24 are how many times 2 ? 8 ? 4 ? 6 ? 12 ?
19. 60 are how many times 6 ? 5 ? 10 ? 12 ?
20. 63 are how many times 9 ? 3 ? 7 ?
21. 84 are how many times 7 ? 12 ? 4 ? 6 ?
22. 90 are how many times 9 ? 6 ? 10 ?
23. 108 are how many times 12 ? 9 ? 6 ?
24. 100 are how many times 10 ? 20 ? 5 ?
25. 99 are how many times 9 ? 11 ? 3 ?
26. 81 are how many times 9 ?
27. 66 are how many times 11 ? 6 ? 3 ?
28. 75 are how many times 15 ? 5 ? 25 ?
29. 88 are how many times 4 ? 11 ? 8 ?
30. 120 are how many times 12 ? 10 ? 8 ?
31. 140 are how many times 7 ? 10 ? 20 ?
82. 200 axe how many times 20 ? 40 ? 50 ?
31.] INTELLECTUAL VRITHMETIC. 39
31. 1. If 6 men receive 72 dollars for building
a barn, how many dollars will each man receive ?
NOTE. If the pupil is sufficiently acquainted with Fractions,
the following Analysis is often preferred.*
ANALYSIS. If 6 men receive 72 dollars, 1 man, which
is ^ of Q men, will receive \ of 72 dollars ^ of 72 dol-
lars is 12 dollars. Therefore if 6 men receive 72 dol-
lars for building a barn, each man will receive 12 dollars.
2. If 81 cents be paid for 9 dozen of eggs, what
will be the cost of 1 dozen ?
3. A mechanic sold a wagon for 77 dollars, and
took his pay in hay, at 11 dollars a ton ; how many
tons did he receive ?
4. How long will it take 12 men to perform a
piece of work that 1 man can do in 48 days ?
5. Paid 108 dollars for 9 thousand feet of lum-
ber ; what was the COSY of a thousand feet ?
6. How many barrels of flour can be bought for
56 dollars, at 7 dollars a barrel ? at 8 dollars ?
7. How many cows, at 12 dollars a head, can be
bought for 60 dollars? for 9n dollars? for 120 dollars?
8. When steak is 10 cents a pound, how many
pounds can be bought for 50 cents ? for 80 cents ?
for 100 cents ?
9. At 11 cents a pound, ho\v many pounds of
sugar can be bought for 88 cents ? for 99 cents ?
* When it is necessary to express a quantity less than a unit, we may
regard the unit as divided iiito some number of equal parts, and use
cue of these parts as a new unit of less value thau the unit divided.
Thus, if a yard be divided into two equal parts, each of the parts is
called one half; when into three equal parts, each of tue parts is called
one third wuen into four equal parts, each of the j>arts is called one
fourth ; when into Jive equal parts, each of the parts is called one-fifth ;
two of the parts, two fifths, &c. ; when into six equal parts, each of the
parts is called one sixth ; two of the parts, two sixths, three of the parts,
three sixths } &c.
40 INTELLECTUAL ARITHMETIC. [91.
10. A man planted an orchard of 132 trees, and
put 11 trees in a row ; how many rows in the orchard ?
11. A farmer sold a horse for 120 dollars; how
many cows at 12 dollars each would pay for the
horse ? How many young cattle at 8 dollars each ?
How many sheep at 5 dollars each ?
12. If 25 sheep cost 75 dollars, what will 1 sheep
cost?
13. If a boy read 7 pages a day, how long will it
take him to read 56 pages? to read 84 pages? 98
pages ?
14. At 15 cents a pound, how many pounds of
coffee can be bought for 45 cents ? for 60 cents ? for
90 cents ?
15. If 1 man can do a piece of work in 72 days,
in what time can 6 men do it ? 8 men ? 9 men ? 12
men?
16. If a certain quantity of provision will last 1
man 36 days, how many days will it last 4 men ? 9
men ? 12 men ?
17. A man earns 66 dollars in 11 weeks ; how
ttmfch does he earn in 1 week ? how much in 1 day ?
18. If a painter receive 100 dollars for painting 5
carriages, how much does he receive apiece ?
19. How many lots containing 12 acres each can
be sold from a farm of 120 acres ? how many lots of
20 acres each ?
20. If a steamboat run 11 miles an hour, how
long will she be in running 44 miles ? 88 miles ?
110 miles ?
21. How many men, at 8 dollars a month, can be
hired one month for 96 dollars? how many at 12
dollars ? how many at 16 dollars ?
22. If I travel 120 miles in 6 days, how many
miles do I travel in 1 day ?
INTELLECTUAL ARITHMETIC. 41
1. If 3 pounds of coffee cost 27 cents, what
will 6 pounds cost ?
ANALYSIS. If 3 pounds of coffee cost 27 cents, 1 pound,
which is 1 third of 3 pounds, will cost 1 third of 27 cents,
or 9 cents. If 1 pound cost 9 cercto, 6 pounds, which are
6 times 1 pound, will cost 6 ft'mes 9 cento, or 54 cents.
Therefore, if 3 pounds of coffee cost 27 cento, 6 pounds will
cost 54 cents.
Or thus : C pounds will cost 6 ta'wes 1 third of 27 ce??to; 1
<AiVc? o/ 27 cento is 9 cento, awe? 6 times 9 cento are 54 cents.
Therefore, fyc.
Or 6 pounds, which are 2 foVwes 3 pounds will c^st 2 times 27 cente, <fcc.
2. If 4 yards of broadcloth cost 20 dollars, what
will 9 yards cost ?
3. If 6 barrels of flour cost 54 dollars, what will
7 barrels cost ?
4. If 9 yards of ribbon cost 72 cents, what will
4 yards cost ?
5. If a man travel 40 miles in 8 hours, how far
will he travel in 6 hours ? in 9 hours ?
6. If 5 coats can bo cut from 20 yards of cloth,
how many yards would be required to cut 7 coats ?
' 7. If 9 weeks' board cost 45 dollars, what will 4
weeks' board cost ? 7 weeks' ? 12 weeks' ?
8. If a boy earn 60 cents in 5 days, how many
cents will he earn in 12 days ? in 20 days ?
9. What will be the cost of 10 cords of wood, if
6 cords cost 24 dollars ?
10. What will be the cost of 14 dozen of eggs, if
12 dozen cost 120 cents ?
11. If 18 apples are worth 6 peaches, how many
apples are 15 peaches worth ? 20 peaches ?
12. If a stage coach run 48 miles in 8 hours, in
how many hours will it run 60 miles ?
13. If 15 yards of cloth cost 75 dollars, what will
40 yards cost ?
4*
42 INTELLECTUAL ARITHMETIC.
14. How many melons can be bought for 100
/ents, at the rate of 3 for 60 cents ?
15. What will 42 pounds of beef cost, if 7 pounds
cost 56 cents ?
16. When apples are sold at the rate of 4 barrels
for 8 dollars, how many barrels must be given for 3
tons of coal at 6 dollars a ton ?
17. How many dozen of eggs, at 12 cents a dozen,
will pay for 15 pounds of sugar, at 8 cents a pound ?
18. How many gallons of molasses, at t,G cents a
gallon, will pay for 20 pounds of lard, worth 10 cents
a pound ?
19. If a farmer sell 5 firkins of butter, worth 12
dollars a firkin, for cloth worth 4 dollars a yard, how
many yards will he receive ?
20. If 5 barrels of cider are worth 15 dollars, how
many hundred weight of pork,worth 6 dollars a hun-
dred weight, would 8 barrels of cider buy ?
21. If 6 bushels of wheat are worth 12 dollars,
how many bushels of wheat must be given for 9 tons
of hay, worth 10 dollars a ton ?
22. At the rate of 3 for 4 cents, how many pears
can be bought for 24 cents ?
ANALYSIS. Since 4 cents will buy 3 pears, 24 cents will
buy as many times 3 pears as 4 cents is contained times
in 124 cents ; 4 cents are contained in 24 cents 6 times,
and 6 times 3 are 1 8 pears. Therefore, if 3 pears can be
bought for 4 cents, IS pears may be bought for 24 cents.
23. A man paid 72 cents for some oranges, at the
rate of 5 for 12 cents ; how many oranges did he buy?
24. How many pounds of nails may be bought for
60 cents, at the rate of 4 pounds for 20 cents ?
25. When eggs are sold at the rate of 9 for 11
cents, how many must be sold to receive 99 cents ?
26. At the rate of 16 miles in 4 hours, how many
miles would a man travel in 12 hours ?
INTELLECTUAL ARITHMETIC. 43
27. If 8 men can do a piece of work in 12 days,
how many days will it take 9 men to do the same ?
ANALYSIS. If 3 men can do the work in 12 days,
9 men can do it in ^ of 3 times 12 days / 3 times 12
days are 36 days, and ^ of 36 days are 4 days. There-
fore, <&c.
28. If 6 men can build a wall in 8 days, how
many men will be required to build it in 3 days ?
29. If 4 men can dig a ditch in 10 days, how
many days will it take 1 man to dig it ? 5 men ? 8
men ?
30. How long will it take 7 men to reap a field,
if it take 14 men 2 days to reap it ?
31. If 5 men can build a barn in 8 days, how
many men would be required to build it in 1 day ?
in 2 days ? in 4 days ?
32. A man bought some oranges for 44 cents, at
the rate of 5 for 11 cents, and divided them equally
among his 4 children ; how many did he give to
each ?
33. If 15 days' work will pay for 10 cords of
wood, at 3 dollars a cord, what is the price of 1
day's labor ?
34. A man engaged to labor 5 months for 80 dol-
lars, but by request continued 3 months longer, at
the same rate ; what amount will be due him for the
whole time ?
35. If you pay 60 cents for some lemons, at the
rate of 6 for 10 cents, and sell them at the rate of 9
for 20 cents, how many cents will you gain by the
operation ?
36. If two ships are 120 miles apart, and sail
directly toward each other, one at the rate of 9
miles an hour, and the other at the rate of 11 m^es
an hour, how many hours before they will meet ?
44 INTELLECTUAL ARITHMETIC. [23.
S3. Division \s the process of finding how many times
one number is contained in another ; and the result is called
the Quotient. The sign of division is written thus, ~, and,
placed between two numbers or quantities, shows that the
one on the left is to be divided by the one on the right.
24 ~ 8 = 3 shows that 24 divided by 8 is equal to 3.
1. How many times 9 are 6 times 6 ?
2. How many times 4 are 5 times 8 ?
3. How many times 8 are 6 times 12 ?
4. How many times 10 are 5 times 20 ?
5. How many times 15 are 3 times 25 ?
6. How many times 20 are 10 times 12 ?
7. How many times 16 are 8 times 8 ?
8. How many times 28 are 7 times 16 ?
9. How many times 25 are 5 times 30 ?
10. How many times 20 are 12 times 10 ?
11. 16 times 4 are how many times 8 ?
12. 9 times 12 are how many times 6 ?
13. 8 times 14 are how many times 16 ?
14. 4 times 42 are how many times 12 ?
15. 7 times 18 are how many times 9 ?
16. 14 times 10 are how many times 28 ?
17. 12 times 8 -j- 4 are how many times 10 ?
18. 6 times 15 -j- 20 are how many times 12 ? *
19. How many times 9 + 3 are 7 times 13 less 10 ?
20. How many times 18 4 are 8 times 12 + 2 ?
21. In 8 X 12, how many times 64 -=- 8 ?
22. In 6 X 20 + 12, how many times 55 -r- 5 ?
23. In 48 + 24 divided by 6, how many times 4 ?
24. How many times 12 7 are 9 times 8 ?
25. How many times 9 + 6 in 6 times 12 + 3 ?
26. How many times 27 -f- 9 + 7x3 are 8 X
11 + 20 ?
* Each sign affects only the number placed after it. When there is a
remainder in the result, let it simply be mentioned as such.
24.] INTELLECTUAL ARITHMETIC. 45
24. 1. How many vests, at 5 dollars each, will
pay for 7 weeks' board, at 6 dollars a week, and 18
dollars borrowed money ?
2. If a man receive 16 pounds of sugar in ex-
change for 20 pounds of cheese at 8 cents a pound,
what is the price of the sugar per pound ?
3. James paid 36 cents for some oranges at 3
cents apiece, and, after eating 5 of them, he gave
the remainder, at 4 cents apiece, for a knife ; how
much did the knife cost him ?
4. Omar bought 30 peaches at the rate of 2 for 3
cents, and, after eating 3 of them and giving 3 to
his brother, he sold the remainder at the rate of 3
for 7 cents ; did he gain or lose by the bargain, and
how much ?
5. A farmer sold 9 sheep at 12 shillings a head,
and 11 bushels of oats at 2 shillings a bushel ; how
many yards of cloth, at 10 shillings a yard, will pay
the debt ?
6. If I buy 40 turkeys at the rate of 5 for 3 dol-
lars, and sell them at the rate of 8 for 7 dollars, how
much will I gain by the operation ?
7. If a drover buy 24 lambs for 30 dollars, at how
much must he sell them a head to gain 18 dollars on
the cost ?
8. If 5 men ouy a mowing machine for 120 dol-
lars, and rent it 3 weeks for 15 dollars a week, and
then sell it for 100 dollars, what is each man's share
of the gain ?
9. If 5 barrels of flour are worth 30 dollars, how
many yards of cloth worth 3 dollars a yard will 2
barrels of the flour buy ?
10. Four men agree to build 120 rods 01 wall for
48 dollars, and to share equally ; but when the wall
is half built, 2 men quit, and the others finish it ;
how many dollars must each receive ?
46 INTELLECTUAL ARITHMETIC.
11. How many sheep can I buy for 75 dollars at
the rate of 3 for 7 dollars, and have 12 dollars left ?
12. A man, having 80 dollars, bought 9 cords of
wood at 4 dollars a cord, 3 barrels of flour at 8 dol-
lars a barrel, and the remainder he expended for
cloth at 4 dollars a yard ; how many yards of cloth
did he buy ?
13. If 4 bushels of wheat are worth 12 bushels of
corn, how many bushels of corn are equal in valuo
to 10 bushels of wheat ?
14. If 12 bushels of wheat make 3 barrels of flour,
how many bushels of wheat will make 7 barrels ?
15. A cistern holding 140 gallons has a pipe by
which 30 gallons will run into the cistern in one
hour, and another that will discharge 20 gallons in
an hour ; when both are running, in what time will'
the cistern be filled ?
16. If a quantity of provision serve 7 men 12
days, how long would it serve 4 men ?
17. A merchant bought 15 bushels of clover seed
for 75 dollars ; he wishes to keep 3 bushels for his
own use, and to sell the remainder so as to make 9
dollars on the first cost of the whole ; how much
must he ask a bushel ?
18. A man bought a span of horses for 80 dollars
each ; he spent 40 dollars in fitting them for market,
and then sold them for 800 dollars ; what was his
gain on each horse ?
19. A tailor bought 15 yards of one kind of cloth
for 60 dollars, and 25 yards of another kind for 75
dollars ; what was the difference in the price per
yard ?
20. A thief, having 36 miles the start of an officer,
travels at the rate of 6 miles an hour ; the officer
pursues at the rate of 9 miles an hour ; in how many
hours will he overtake the thief ?
84.] INTELLECTUAL ARITHMETIC. 47
21. Add 5 to 22, divide the sum by 9, multiply
the quotient by 12, subtract 6 from the product, and
add 10 to the remainder ; what will be the result ?
22. Divide 40 by 8, multiply the quotient by 9,
to the product add 11, from the sum take 6, and
what will remain ?
23. Multiply 20 by 4, subtract 40, add 10, divide
by 5, multiply by 7, and what is the result ?
24. Prom 36 subtract 6, divide the remainder by
5, multiply the quotient by 12, add 15 to the product,
subtract 77 from the sum, divide the remainder by
10, and what will be the quotient ?
25. Add 5 and 7 to 8, multiply the sum by 3,
from the product subtract 5, divide the remainder
by 11, multiply the quotient by 6, add 20 to the
product, and what will be the sum ?
26. Subtract 10 from 6 multiplied by 8, to the
remainder add 4, divide the sum by 6, multiply the
quotient by 11, subtract 7 from the product, divide
the remainder by 10, add 30 to the quotient, and
what will be the sum ?
27. Multiply 15 by 5, add 5, divide by 8, multiply
by 6, subtract 10, divide by 5, add 2, multiply by 11,
and what will be the product ?
28. 10 + 8-f-6x9 7x5 + 20 -MO + 8 =
how many ?
29. 8x9-7- 12 + 8x5 + 10 -Ml x8 + 20 -r
12 + 4 X 7 3 = how many ?
30. 25 5-7-2x9 + 10-7-20x12 + 15 5
-- 7 + 8 + 2 = how many ?
31. 36-7-9x12 + 8-7-7x5+10 25-7^5 +
4x9 11 -MO X 6 -T- 3 = how many ?
NOTE. The above examples will afford a thorough and profitable
mental drill to the pupil, and also familiarize the signs and terms used
in the preceding pages, and which must be well understood in the com-
mencement of the study of Written Arithmetic. The teacher can extend
the number of examples at pleasure.
48 INTELLECTUAL ARITHMETIC.
CHAPTER V.
REDUCTION.
25. TABLE OF UNITED STATES MONEY.
10 mills (m.) make 1 cent, c.
10 cents " 1 dime, d.
10 dimes " 1 dollar, 9.
10 dollars " 1 eagle, E.
NOTE. Dollars and cents are separated by a period (.) ; thus, $2.10, ig
read 2 dollars 10 cents.
A Simple Number contains but one name or denomina-
tion ; thus, 5, 84, 12 bushels, 65 dollars, are simple numbers.
A Compound Number contains two or more names or
denominations ; thus, 5 dollars 25 cents, 16 bushels 3 pecks,
3 days 10 hours, are compound numbers.
Reduction is the process of changing numbers from one
denomination to another, without altering their value ; thus,
changing pounds to ounces, ounces to pounds, dollars to
cents, &c., is reduction.
1. How many mills in 2 cents? in 3 cents?
2. How many cents in 1 dime ? in 3 dimes ?
3. How many dimes in 3 dollars? in 4 dollars?
4. How many dollars in 2 eagles? in 5 eagles? v
5. How many cents in 4 dimes and 7 cents?
6. How many dimes in 3 dollars and 6 dimes?
7. If Jam3S earned 12 dollars, and his father 3
eagles, how many dollars did they both earn ?
8. A man has 4 eagles, 4 five dollar bills, and 4
dimes ; how many dollars and cents has he ?
9. How many dimes are equal to 40^ cents? to 70
cents?
10. What is the difference in the cost of 8 yards
of cloth at 2 dimes a yard, and 6 yards at 30 cents a
yard?
26, 37.] INTELLECTUAL ARITHMETIC. 49
26. TABLE OF ENGLISH MONEY.
- 4 farthings (far.) make 1 penny, d.
12 pence " 1 shilling, s.
20 shillings 1 pound, jE.
NOTE. A sovereign (sov.) is equal in value to one pound.
1. How many farthings in 1 penny? in 3 pence?
in 6 pence ? in 9 pence ?
2. How many pence in 1 shilling ? in 4 shillings ?
in 5 shillings ? in 9 shillings ? in 10 shillings ?
3. How many pence in 8 farthings ? in 24 far-
things ? in 48 farthings ?
4. How many shillings in 24 pence ? in 84 pence ?
in 108 pence ?
5. How many pounds in 40 shillings ? in 120 shil-
lings ?
6. At 5 shillings a yard, how many pounds will
20 yards of carpeting cost ?
7. At 7 shillings a pair, how many pairs of shoes
can be bought for 2 sovereigns ?
27. TABLE OF TROY WEIGHT.
24 grains (gr.) make 1 pennyweight, pwt.
20 pennyweights " 1 ounce, oz,
12 ounces " 1 pound, Ib.
1. How many grains in 2 pennyweights ? in 4 ?
2. How many pennyweights in 3 ounces ? in 5 ?
3. How many pennyweights in 96 grains ? in 124
grains ?
4. In 120 pennyweights how many ounces ? in
200 pennyweights ?
5. What will a gold chain, weighing 15 penny-
weights cost, at 8 dimes a pennyweight ? how many
dollars and cents ?
4
60 INTELLECTUAL ARITHMETIC. [28.
28. TABLE OF AVOIRDUPOIS WEIGHT
16 drams (dr.) make 1 ounce, oz.
16 ounces " 1 pound, lb.
100 pounds " 1 hundred weight, cwt.
20 hundred weight " 1 ton. T.
1. How many drams in 3 ounces ? in 4 ounces ?
j;i 6 ounces ?
2. How many ounces in 32 drams ? in 48 drains ?
in 64 drams ?
3. How many ounces in 3 pounds ? in 5 Ibs. ? in
7 Ibs. ?
4. How many ounces in 2 Ibs. 6 oz. ? in 4 Ibs.
10 oz. ?
5. In 3 cwt. and 5 Ibs., how many pounds?
6. In 5 pounds, how many ounces V
7. How many pounds in 3 cwt. ? in 6 cwt. ? in
9 cwt. ?
8. In 16 tons, how many cwt. ? in 25 tons ?
9. In 1 ton and 5 cwt., how many cwt. ? how
many pounds ?
10. How many dollars will 3 cwt. of flour cost, if
25 pounds cost 5 dimes ?
11. What will 1 ton and 5 cwt. of hay cost, if 5
cwt. cost 3 dollars ?
12. What will 2 cwt. 10 Ibs. of beef cost at 6 cents
a pound ?
13. If 8 ounces of tea cost 4 dimes, what will 2
pounds cost ?
14. What will be the cost of 4 cwt. of sugar, at 7
cents a pound ?
15. Which will cost the more, 5 cwt. of fish at 6
cents a pound, or 3 tons of hay at 50 cents a cwt. ?
16. What will 1 ton of hay cost at 1 cent a
pound ?
INTELLECTUAL ARITHMETIC. 5l
29. TABLE OF LONG MEASURE.
12 inches (in.) make 1 foot, ft.
3 feet " 1 yard, yd.
5 yards, or 16 feet, " 1 rod, rd.
40 rods " 1 furlong, fur.
8 furlongs, or 320 rods, " 1 mile, mi.
3 miles " 1 league, lea.
69 miles " 1 degree, dcg. or .
360 degrees " 1 circle of the earth.
1. How many inches in 3 ft. ? in 5 ft. ? in 10 ft. ?
in 12 ft. ?
2. How many feet in 4 yards ? in 6 yards ? in 9
yards ?
3. How many yards in 15 feet ? in 24 feet ?
4. How many yards in 2 rods ? in 4 rods ?
5. How many furlongs in 5 miles ? in 7 miles and
4 furlongs ?
6. How many furlongs in 80 rods ? in 120 rods ?
in 160 rods ?
7. In 2 miles 4 fur. 20 rods, how many rods ?
8. How many leagues in 9 miles ? in 18 miles ? in
36 miles ?
9. How many inches in 3 yds. 1 ft. 10 in. ?
10. In 108 inches how many yards ? in 144
inches ?
11. If a man travel 5 miles an hour, how long
will it take him to travel 20 leagues ?
12. If a ship sail 20 leagues a day, how long will
it take her to sail from New York to Liverpool, it
being 3000 miles ?
13. In 100 inches how many yards, feet, and
inches ?
14. At 8 dimes a foot, how many dollars will 3 yds.
1 ft. of iron railing cost ?
15. In 2 rods 5 feet ? how many inches ?
r >" INTELLECTUAL ARITHMETIC. [3O, 31.
3O. TABLE OF CLOTH MEASURE.
2 inches (in.) make 1 nail, na.
4 nails 1 quarter of a yard, qr.
4 quarters 1 yard, yd.
1. How many nails in 3 qrs. ? in 5 qrs. ?
2. How many nails in 2 yds. 2 qrs. ?
3. How many quarters in 5 yds. ? in 6 yds. 3
qrs.?
4. How many yards in 16 qrs. ? in 56 qrs. ?
5. How many qrs. in 36 nails ? in 64 na. ?
6. How many qrs. in 5 yds. ? in 6 yds. ? in 7
yds. ?
7. In 4 yards, how many nails ?
8. In 6 yards, how many qrs. ?
9. In 123 nails, how many yards ?
31. TABLE OF LAND OR SQUARE MEASURE.
144 square inches (sq. in.) make 1 square foot, sq. ft.
9 square feet " 1 square yard, sq. yd.
30 square yards " 1 square rod, P.
40 square rods " 1 rood, R.
4 roods " 1 acre, A.
640 acres " 1 square mile, Sq. M.
1. How many square feet in 5 square yards ? in
10 square yards ? in 20 square yards ?
2. How many square yards in 36 sq. ft. ? in 72
sq. ft. ? in 108 sq. ft. ?
3. How many square rods in 3 roods ? in 5 roods ?
in 7 roods ?
4. How many acres in 20 roods ? in 80 roods ?
5. At 4 dollars a square rod, what would 2 roods
10 rods of land cost ?
32. 33.] INTELLECTUAL ARITHMETIC. 3
32. TABLE OP CUBIC MEASURE.
1728 cubic inches (cu. in.) make 1 cubic foot, cu. ft.
27 " feet " 1 cubic y ard, cu. yd.
24f " feet " 1 perch of stone,Pch.
16 " feet " 1 cord foot, c. ft.
8 cord feet or j u l * d f d
128 cubic feet \
1. How many cubic feet in 2 cubic yards? in 3
cubic yards ?
2. How many cubic feet in 2 perch of stone ? in
4 perch ? in 6 perch ?
3. How many cord feet in 3 cords of wood ? in 7
cords ? in 9 cords ?
4. How many cord feet in 120 cubic feet of wood ?
in 200 cubic feet ?
5. How many cords in 24 cord feet? in 56 cord
feet?
33. TABLE OF LIQUID MEASURE.
4 gills (gi.) make 1 pint, pt.
2 pints " 1 quart, qt.
4 quarts " 1 gallon, gal.
311- gallons " 1 barrel, bar.
2 barrels, or 63 gallons " 1 hogshead, hhd.
1. How many gills in 4 pints ? in 6 pints ? in 10
pints ?
2. How many pints in 3 quarts ? in 8 quarts ? in
1 gallon ? in 3 gallons ?
3. How many gills in 4 quarts ? in 6 quarts ? in
8 quarts ?
4. How many quarts in 3 gallons? how many
pints ? how many gills in 3 quarts ? in 1 gallon and
2 quarts ?
5*
54 INTELLECTUAL ARITHMETIC. [84.
5. How many gills in 2 gallons 2 qts. ?
6. How many quart cups could be filled from a
10 gallon keg of wine ? how many pint bottles ?
7. In 48 gills, how many pints ? quarts ?
8. How many gallons in 64 gills ? in 96 gills ?
9. How many^quarts in 56 pints ? gallons ?
10. In 10 gallons, how many quarts? pints?
gills?
11. What will be the cost of 6 gallons of burning
fluid, at 15 cents a quart ?
12. If a pint of molasses cost 7 cents, what will a
gallon cost ?
13. If one gallon of oil cost 90 cents, what will 3
gallons and 2 quarts cost ?
14. If you should draw 10 gallons and 8 quarts
from a hogshead of molasses, how much would re-
main ?
34. TABLE OF DRY MEASURE.
2 pints (pt.) make 1 quart, qt.
8 quarts " 1 peck, pk.
4 pecks " 1 bushel bu.
1. How many pints in 1 peck ? in 2 pks. 4 qts. ?
2. How many quarts in 1 bu. 2 pks. ?
3. In 2 bu. 1 pk. 6 qts. how many quarts ?
4. In 64 quarts how many pecks ? bushels ?
5. In 80 pints how many pecks ?
6. At 3 cents a pint, what will be the cost of one
peck of chestnuts ?
7. If one bushel of hickory nuts cost 160 cents,
what will be the cost of 6 quarts ?
8. Bought 2 pecks of cranberries for 96 cents, and
Bold them for 5 cents a pint; how much was the
gain ?
INTELLECTUAL ARITHMETIC.
55
35.
TABLE OF TIME.
60 seconds (sec.)
60 minutes
24 hours
7 days
12 calendar months
365 days
366 days
100 years
make 1 minute, m.
" 1 hour, hr.
" 1 day, d.
" 1 week, w.
" 1 year, yr.
" 1 common year, yr.
" 1 leap year, yr.
" 1 century, C.
The divisions of the year, the names of the months,
and the number in each, are shown in the following
table :
Winter.
( 1st month
2d
, January
February
has
44
31
28*
days.
u
(3d
44
March
44
31
u
Spring.
{4th
44
April
44
30
u
(5th
44
May
44
31
M
(6th
44
June
44
30
a
Summer.
{7th
44
July
44
31
a
(8th
44
August
44
31
u
(9th
44
September
44
30
a
Autumn.
{ 10th
44
October
44
31
a
( llth
44
November
44
30
u
Winter.
12th
44
December
44
31
u
The following lines will help to remember the
number of days in each month :
Thirty days hath September,
April, June, and November;
AIL the rest have thirty-one,
Save February, which alone
Hath twenty-eight; and one day more
We add to "it, one year in four.
* In leap-year, 29,
66
INTELLECTUAL ARITHMETIC.
[36.
1. How many seconds in 2 minutes ? in 5 minutes ?
2. How many seconds in 10 minutes ? in 20
minutes ?
3. How many seconds in 1 hour ? in 2 hours ?
4. How many minutes in 3 hours ? in 5 hours ?
5. How many hours in 2 days ? in 3 days ?
6. In 2 days and 12 hours how many hours ?
7. In 6 weeks how many days ? in 9 weeks ?
8. How many weeks in 49 days ? in 84 days ?
9. How many weeks and days in 75 days ? in 90
days?
10. How many hours in 3 days 10 hours ?
11. In 36 months how many years ? in 84 months?
12. How many days from January 1st to March
10th, inclusive ?
13. How many days from April 10th to the 15th
of June ?
36.
MISCELLANEOUS TABLE.
12 units
make 1 dozen.
12 dozen
" 1 gross.
12 gross
" 1 great gross.
20 units
" 1 score.
24 sheets of paper
" 1 quire.
20 quires
" 1 ream.
56 pounds .
" 1 bushel of corn.
60 pounds
" 1 bushel of wheat.
196 pounds
" 1 barrel of flour.
200 pounds
u ( 1 barrel of beef, p
1 or fish.
1. What cost 1 ream of paper, at 20 cents a quire ?
2. What cost 3 dozen lemons, at 3 cents apiece ?
3. What cost 2 gross of pens, at 5 cents a dozen ?
4. At 7 dollars a hundred pounds, what will 2
barrels of pork cost ?
37.] INTELLECTUAL ARITHMETIC. 57
CHAPTER VI.
FRACTIONS.
37. 1. WHAT is understood by one half, one
third, one fourth, one fifth, &c. ?
ANS. When a number or thing, as a unit or an
apple, is divided into two equal parts, fc each of the
parts is called one half; when into three equal parts,
each of the parts is called one third ; two of the parts
are called two thirds, &c.
2. How many halves in 1 ?
3. How many thirds in 1 ?
4. If a peach be divided into 2 equal parts, what
is each part called ?
5. If an orange be divided into 3 equal parts,
what are the parts called ?
6. How many halves in 4?
ANALYSIS. Since in 1 there are 2 halves, in 4 there are 4
times 2 halves, or 8 halves. Therefore there are 8 halves in 4.
7. How many halves in 5 ? in 6 ?
8. How many halves in 7 ? in 7 and 1 half?
9. How many thirds in 3 ? in 4 ? in 5 ?
10. How many thirds in 7 ? in 8 ? in 9 ?
11. How many thirds in 6 and 1 third? in 6 and
2 thirds ?
12. How many halves in 5 ? how many thirds ?
13. How many halves in 10 and 1 half?
14. How many thirds in 9 and 2 thirds ?
15. In 8 halves how many whole things ?
ANALYSIS. Since in 2 halves there is one ivhole thing,
in 8 halves there are as many whole things as 2, the num-
ber of halves in \, is contained times in 8, which is 4
times. Therefore, in 8 haloes there, are 4 whole things.
58 INTELLECTUAL ARITHMETIC. [37.
16. How many whole things in 12 halves ? in 16
halves ? in 24 halves ?
17. How many whole things in 9 thirds? in 18
thirds ?
18. How many whole things in 10 halves? in 12
thirds ? in 21 thirds ? in 20 halves ?
19. How many half bushels in 3 bushels and 1
half?
20. John gave 6 peaches to his 2 brothers ; what
part of them did each have ?
21. What is 1 half of 6 ?
ANALYSIS. It is that number which taken 2 times
loill make G ; 2 times 3 is G. Therefore 3 is 1 half
0/6.
22. What is 1 third of 9 ? of 18 ? of 27 ? of 36 ?
23. What is 1 half of 8 ? of 12 ? of 18 ? of 22 ?
24. How can you take 1 half of an orange ? 1 third ?
25. How can you take 2 thirds of an apple ?
26. Will|am had 30 new pennies, and John had
1 half as many ; how many had John ?
27. If a pound of candles cost 15 cents, what will
1 third of a pound cost ? 2 thirds ?
28. George is 12 years old, and Mary is 2 thirds
as old as he ; how old is Mary ?
29. If a barrel of flour cost 10 dollars, and a
barrel of cider 1 half as much, what is the cost of
the cider ?
30. If 3 tons of hay cost 27 dollars, what part of
27 dollars will 1 ton cost ? 2 tons ?
31. A boy, having 33 cents, spent 2 thirds of them ;
how many had he left ?
32. Edwin gave 2 peaches and 1 half to his sister,
and 1 peach and 1 half to his brother ; how many
peaches did he give to both ?
33. A farmer, having 44 sheep, sold 1 half of them ;
how many had he left ?
38.] INTELLECTUAL ARITHMETIC. 59
38. 1. What do you understand by 1 fourth?
3 fourths ?
2. What do you understand by 1 fifth ? 2 fifths ?
3 fifths? 4 fifths?
3. What do you understand by 1 sixth ? 3 sixths?
5 sixths ?
4. How can you take 1 fourth of any thing ? 1
sixth ?
5. How can you take 3 fourths of any thing ? 3
fifths ?
6. How many fifths in 1 ? in 4 ? in 6 ?
7. How many fourths in 1 ? in 2 and 3 fourths ?
8. How many sixths in 2 ? in 3 and 4 sixths ?
. 9. In 4 and 5 sixths how many sixths ?
10. In 8 and 3 fourths how many fourths ?
11. In 7 and 4 fifths how many fifths ?
12. Which is greater, 1 fourth or 1 fifth ? 1 fifth
or 1 sixth ?
13. How many times 1 in 8 fourths ? in 16 fourths ?
14. How many times 1 in 15 fifths ? in 25 fifths ?
15. How many times 1 in 24 sixths ? in 36 sixths?
16. What is 1 fourth of 12 ? of 20 ? of 28 ?
17. What is 1 sixth of 18 ? of 30 ? of 42 ?
18. What is 1 fourth of 32 ? of 40 ? of 48 ?
19. What is 1 fifth of 10 ? of 20 ? of 50 ?
20. What is 1 sixth of 36 ? 2 sixths ? 5 sixths ?
21. What is 1 fifth of 40 ? 3 fifths ? 4 fifths ?
22. What is 3 fourths of 16 ? of 24 ?
23. What is 2 fifths of 10 ? of 25 ?
24. What is 3 sixths of 30 ? 5 sixths of 42 ?
25. If a ton of hay cost 12 dollars, what will 1
fourth of a ton cost ? 2 fourths ? 3 fourths ?
26. If 5 bushels of apples cost 90 cents, what part
of 90 cents will 1 bushel cost ? 2 bushels ? 3 bushels ?
27. When coal is 6 dollars a ton, what part of a
ton can be bought for 1 dollar ? for 2 dollars ? for 5
dollars ?
GO INTELLECTUAL ARITHMETIC. [38,
28. Bought a piece of land for 45 dollars ; what
is 1 fifth of it worth ? 2 fifths ? 3 fifths ?
29. If you pay 30 dollars for one month's labor,
what do you pay for 1 fifth of a month ? for 1 sixth ?
t>r 5 sixths ? for 4 fifths V
30. A man divided 48 cents equally among 6
beggars ; what part of the whole did each beggar
receive ? what part did 2 receive ? 3 ? 4 ? 5 ?
31. If in a basket of 54 eggs 1 sixth are bad, how
many are good ?
32. If a man earn 15 dollars a week, and spend
2 fifths of it, how much does he save ?
33. A drover, having GO sheep, sold 1 fourth of
them at one time, and 1 fifth at another ; how many
had he left ?
34. Rollin had 18 marbles, and gave 2 sixths of
them to Martin ; how many did he give Martin, and
how many did he keep for himself?
35. If Harry bought 6 pounds of coffee for 1 dol-
lar, what part of 1 dollar did 1 pound cost ? 2 pounds ?
3 pounds ? 4 pounds ?
36. When tea is 5 shillings a pound, what part of
a pound can be bought for 1 shilling ? 2 shillings ?
3 shillings ? 4 shillings ?
87. If a barrel of pork cost 15 dollars, what will
2 thirds of it cost ? 3 fifths ?
38. A gentleman is 60 years old, and his wife is 4
fifths as old ; how old is she ?
39. Andrew is 18 years old, and his sister, Jane, is
5 sixths as old ; how old is Jane ?
40. Paid 36 dollars for 1 acre of land ; what is 1
sixth of it worth ? 3 sixths ? 1 fourth ? 3 fourths ?
41. If you divide a barrel of flour equally among
5 poor families, what part of the barrel will 1 family
receive ? 2 -families ? 3 families ? 4 families ? 5 fam-
ilies ?
39. J INTELLECTUAL ARITHMETIC. 61
39. 1, What do you understand by 1 seventh ?
3 sevenths ? 4 sevenths ? 5 sevenths ?
2. What is meant by 1 eighth? 5 eighths? 1
ninth ? 3 ninths ? 5 ninths ? 7 ninths ?
3. What is meant by 3 tenths? 9 tenths? 5
twelfths ? 11 twentieths ?
4. How many ninths in a unit, or whole thing ?
how many tenths ? how many elevenths ? how many
twelfths ? how many twentieths ?
5. How many times 1 in 14 sevenths ? in 32
eighths ? in 45 ninths ?
6. How many times 1 in 70 tenths ? in 44 elevenths ?
in 30 fifteenths ? in 80 twentieths ?
7. What is 1 seventh of 42 ? of 63 ? of 84 ?
8. What is 1 eighth of 24 ? of 56 ? of 64 ?
9. What is 1 ninth of 36 ? 3 ninths ? 7 ninths ?
10. What are 3 tenths of 50 ? of 60 ? of 90 ?
11. What is 1 twelfth of 72 ? 5 twelfths ? 7 twelfths ?
12. What is 1 twentieth of 60 ? 14 twentieths ?
13. How many times 1 in 19 eighths ? in 40 ninths ?
in 65 twelfths ?
14. How many times 1 in 28 sevenths ? in 56
tenths ? in 52 elevenths ?
15. In 3 and 5 sevenths how many sevenths ?
16. In 6 and 3 eighths how many eighths ?
17. In 8 and 7 ninths how many ninths ?
18. In 9" and 3 twelfths how many twelfths ?
19. In 7 and 9 tenths how many tenths ?
20. If 10 pounds of sugar cost 80 cents, what part
of 80 cents will 3 pounds cost ? 5 pounds ?
21. If a man earn 72 dollars in 1 month, how
much will he earn in 1 fourth of a month ? 5 sixths ?
4 ninths ? 7 eighths ? 5 twelfths ?
22. What is the difference between 7 eighths of
64 and 5 ninths of 81 ?
6
62 INTELLECTUAL ARITHMETIC. [39.
23. What will 5 eighths of a bushel of chestnuts
cost at 10 cents a quart ?
24. What will be the cost of 4 sevenths of 14
dozen of eggs at 11 cents a dozen ?
25. If James has 108 cents, and Henry has 7
twelfths as many and 20 cents more, how many
cents has Henry V
26. What will be the cost of 1 ninth of a hogs-
head of molasses at 12 cents a quart ?
27. Meiizo, having 48 dollars, gave 1 sixteenth of
his money for a hat, 3 eighths for a coat, and 3
twelfths for some books and a quarter's tuition ; how
many dollars had he left ?
28. A man bought a horse for 120 dollars; he
gave 7 tenths as much for a carriage as he paid for
the horse, and 3 sevenths as much for a harness as
he paid for the carriage ; how much did the whole
cost him ?
29. If a man put 200 dollars at interest, and
receive 3 twentieths of it annually for its use, how
many dollars does he receive ?
30. If a vessel sail 12 miles an hour, how many
times 9 miles will she sail in 3 eighths of a day ?
31. Abel has 49 marbles, and Alonzo has 6 sevenths
as many wanting 10 ; how many has Alonzo ?
32. A man, having 300 cedar posts, sold 2 fifths
of them to A, 1 sixth to B, 3 tenths to G, and 40
posts to D ; how many had he left ?
33. What will be the cost of a gold watch weigh-
ing 1 third of a pound at 1 dollar a pennyweight ?
34. What will be the cost of 3 eighths of 2 gallons
of wine at 40 cents a pint ?
35. A grocer bought 27 dozen of eggs for 240
cents, and sold 5 ninths of them for 12 cents a
dozen, and the remainder for 10 cents a dozen ; how
much did he gain by the operation ?
4O.] INTELLECTUAL ARITHMETIC. 65
4O. 1. 9 is 1 fourth of what number ?
ANALYSIS. 9 is 1 fourth of 4 times 9 ; 4 times 9 is
36. Therefore 9 is 1 fourth of 36.
2. 7 is 1 fifth of what number ?
3. 10 is 1 eighth of what number ?
4. 12 is 1 tenth of what number ?
5. 20 is 1 half of what number ?
6. 15 is 1 fourth of what number ?
7. 30 is 1 sixth of what number ?
8. 9 is 1 twelfth of what number ?
9. 5 is 1 twentieth of what number ?
10. 11 is 1 seventh of what number ?
11. 9 times 4, and 3 fourths of 4 are how many ?
12. 6 times 9, and 7 ninths of 9 are how many ?
13. 10 times 12,and 5 twelfths of 12 are how many ?
14. 14 times 2, and 1 half of 2 are how many ?
15. 8 times ll,and 7 elevenths of 11 are how many ?
16. 5 times 20, and 13 twentieths of 20 are how
many ?
17. 64 are how many times 8 ? 5 ? 12 ?
18. 47 are how many times 9 ? 10 ? 8 ? 6 ?
19. 81 are how many times 8 ? 9 ? 11 ? 12 ?
20. 35 are how many times 7 ? 6 ? 8 ? 9 ?
21. 75 are how many times 5 ? 6 ? 7 ? 9 ? 25 ?
22. 3 fourths of 16 are how many times 6 ?
ANALYSIS. 3 fourths of 16 are as many times 6 as 6
is contained times in 3 times \ of 16 ; 1 fourth of 16
is 4, and 3 times 4 is 12 ; 6 is contained in 12, 2 times.
Therefore 3 fourths oflQ are 2 times 6.
23. 6 tenths of 100 are how many times 5 ?
24. 5 twelfths of 72 are how many times 3 ? 4 ?
6? 8?
25. 7 ninths of 63 3 eighths of 24 are how
many times 4 ? 5 ? 6 ? 7 ? 8 ?
26. 11 twentieths of 80 + 7 fifteenths of 45 are
how many times 6 ? 7 ? 9 ? 12 ? 13 ?
64 INTELLECTUAL ARITHMETIC. [41.
41. 1. If 1 fourth of a yard of cloth cost 12
cents, what will 1 yard cost ?
ANALYSIS. Tf I fourth of a yard cost 12 cents, 4
fourths, or 1 yard, will cost 4 times 12 cents ; 4 times
12 cents are 48 cents. Therefore, if 1 fourth of a yard
cost 12 cents, 1 yard will cost 48 cents.
2. If 1 fifth of a pound of tea cost 10 cents, what
will 1 pound cost ?
3. If 1 third of a gallon of molasses cost 20 cents,
what will 1 gallon cost ?
4. 20 is 1 third of what numher ?
5. 10 is 1 fifth of what number ?
G. If 1 half of a cord of wood cost 2 dollars, what
will 1 cord cost ? what will 2 cords cost ?
7. A man gave 2 shillings for his dinner, which
was 1 twelfth of all the money he had ; how much
money had he ?
8. A drover sold some sheep, and gained by the
sale 15 dollars, which was 1 fourth of what the sheep
cost him : what was the cost of the sheep ?
9. If I sixth of a ton of hay cost 3 dollars, what
will 3 rixths cost ? 5 sixths ? 1 ton ?
10. If a man can walk 6 miles in 1 eighth of a
day, how far can he walk in 1 day ? in 2 days ? in 3
days ?
11. James had 40 plums, and divided 1 fourth of
them equally among 5 of his companions ; how many
did he give to each ?
12. 6 is 1 eighth of what number ?
13. 15 is 1 fifth of what number ?
14. 40 is 1 third of what number ?
15. 1 fourth of 40 is how many times 5 ? 7 ?
16. 1 sixth of 90 is how many times 4 ? 5 ? 6 ?
17. I have 16 dollars, which is 1 third of what I
paid for 12 sheep ; how much did I pay a head ?
INTELLECTUAL ARITHMETIC. 65
18. A man plowed 36 acres of his farm; 3
fourths of this was just equal to 1 third of the whole
number of acres in his farm ; how many acres in his
farm ?
19. A pole stands 12 feet in the water, which is 1
fifth of the length of the pole ; how long is the pole ?
20. Bought a watch and chain ; for the chain I
gave 15 dollars, which was 1 fourth of what I paid
for the watch ; what did they both cost me ?
21. If 1 twelfth of an acre of land cost 5 dollars,
what will 2 acres cost ?
42. 1. If 3 fourths of a barrel of flour cost 6
dollars, what will one barrel cost ?
ANALYSIS. If 3 fourths of a barrel of flour cost 6
dollars, 1 barrel will cost 4 times 1 third of 6 dollars ;
\ third of 6 dollars are 2 dollars, and 4 times 2 dollars
are 8 dollars. Therefore, <&c.
2. If 4 fifths of a barrel of ale cost 8 dollars,
what will 1 barrel cost?
3. If 5 ninths of a month's wages amount to 25
dollars, what will a month's wages amount to ? 2
months' ?
4. What will 1 barrel of flour cost, if 5 sevenths
of a barrel cost 10 dollars ?
5. Sold a cow for 25 dollars, which was 5 fourths
of what she cost me ; what did she cost me ?
6. Bought 4 barrels of vinegar at 6 dollars a
barrel, which was but 3 ninths of its real value ; what
was its real /alue per barrel ?
7. A speculator bought some pork for 90 dollars,
which was 9 elevenths of what he sold it for ; how
much did he gain ?
5
66 INTELLECTUAL ARITHMETIC.
8. A tailor bought two pieces of cloth, the shorter
piece containing 24 yards ; 5 eighths the numoer of
yards in the shorter piece is 20 yards less than the
number of yards in the longer piece ; how many
yards in both pieces ?
9. If 7 tenths of an acre of land yield 28 bush-
els of wheat, what will 1 acre yield ? what will 3
acres yield ?
10. 16 is 4 fifths of what number ?
ANALYSIS. 1C is 4 ffths of 5 times I fourth of 16 ;
1 fourth of 16 is 4, and 5 times 4 is 20. Therefore, 16
is 4 fifths o/20.
11. 25 is 5 eighths of what number ?
12. 14 is 7 twelfths of what number ?
13. 9 is 3 tenths of what number ?
14. 12 is 4 fifteenths of what number ?
15. 20 is 2 sevenths of what number ?
16. 18 is 6 elevenths of what number 2
17. 21 is 3 halves of what number ?
18. 72 is 9 eighths of what number ?
19. 40 is 10 elevenths of what number ?
20. 36 is 12 twentieths of what number ?
21. 42 is 7 fifths of what number ?
22. 5 sixths of 24 is 2 thirds of what number ?
ANALYSIS. 5 sixths of 24 is 2 thirds of 3 times I
half of 5 times 1 sixth of 24 ; 1 sixth of 24 is 4, and
5 times 4 is 20; 1 half of 20 is 10, and 3 times 10 is
30. Therefore, I sixths of 24 is 2 thirds of 30.
23. 3 fourths of 16 is 6 sevenths of what number ?
24. 4 fifths of 30 is 8 elevenths of what number ?
25. 6 fourths of 28 is 7 ninths of what number ?
26. 9 twelfths of 60 is 5 thirds of what number ?
27. 8 ninths of 18 is 1 fourth of what number ?
28. 11 fifteenths of 45 is 3 halves of what num.
her.
43.] INTELLECTUAL ARITHMETIC. 67
29. 6 tenths of 100 is 5 twelfths of what number ?
30. 3 twentieths of 80 is 2 sixteenths of what ;
number ?
31. 9 eighths of 64 is 8 fifths of what number ?
32. Of a certain farm, 36 acres are woodland, and
5 sixths of the woodland is 3 fifths of the number of
acres of cleared land ; how many acres of cleared
land, and how many acres does the farm contain ?
33. Mr. Smith gave 5 sevenths of all the money
lie had for a horse ; he then paid 12 dollars for a
saddle and bridle, which was 3 fourths of all tho
money he had left ; how much money had he at first ?
34. A person, being asked his age, answered that
if he were twice as old, 1 third of his age would be
20 years ; how old was he ?
43. A FRACTION is one or more of the equal parts of
a unit. Fractions are expressed by figures in the following
manner, viz. :
l, one naif. f , two thirds.
1, one third. f , three fourths,
^, one fourth. |, seven eighths.
1, one fifth. -j^, nine tenths.
1, one sixth. -J, eleven twelfths.
^, one seventh. J-jJ, thirteen twentieths.
|, one eighth. 4|, four and 1 seventh.
^, one tenth. 9f , nine and 5 eighths, &c.
1. The number below the short horizontal line is
called the denominator.
2. The number above the line is called the numer-
ator.
3. The denominator shows into how many parts
the whole thing is divided.
68 INTELLECTUAL ARITHMETIC. [43.
4. The numerator shows how many of these parts
are taken or used.
5. A proper fraction is one whose numerator is
less than the denominator ; its value is less than a
unit.
G. An impi'oper fraction is one whose numerator
is equal to, or greater, than the denominator ; its
value is equal to, or greater, than a unit.
7. A mixed number is a whole number joined with
a fraction.
8. What kind of a fraction is | ? What is the 4
called ? What does it show ?
9. In the fraction -f , what is the 3 called ? What
does it show ?
10. Into how many equal parts must a thing be
divided to produce the fraction | ? how many of
these parts must be taken ?
11. What kind of a fraction is | ? Why ?
12. What kind of a number is 6f ? Why ?
13. How would you obtain of any thing ?
14. How would you obtain | of any thing ?
15. How many times | in 6 ?
ANALYSIS. Since in 1 there are 4 times J, or J, in 6
there are 6 times f , or 2. Therefore there are 24 times
til 6.
16. How many times J in 4 ? in 8 ? in 9 ? in 12 ?
17. How many times in 3 ? in 5 ? in 7 ? in 8 ?
18. How many times j in 2 ? in 4 ? in 5 ? in 7 ?
19. How many times ^ in 5 ? in 6 ? in 7 ? in 9 ?
20. How many times ^ in 3 ? in 4 ? in 6 ? in 7 ?
21. How many times | in 3| ? in 4| ? in 7| ?
22. How many times | in 2$ ? in 5f ? in 6f ?
23. How many times -^ in 5^ ? in 7 T V ? in
44.] INTELLECTUAL ARITHMETIC. 69
24. In 12J how many thirds ?
25. In 7| how many fifths ?
26. In 10f how many eighths ?
27. In 9 ^ how many twelfths ?
28. In 15| how many fourths ?
29. Reduce 20 1 to an improper fraction.
30. Reduce 8| to an improper fraction.
31. Reduce 7-J-J to an improper fraction.
32. Reduce 8| to an improper fraction.
33. Reduce 25f to an improper fraction.
34. How do you reduce a whole number to a
fraction having a required denominator ?
35. How do you change a mixed number to an
improper fraction ?
44. 1. In ^ how many times 1 ?
ANALYSIS. Since | equal 1, ^ are as many times 1
as | are contained times in J ? 6 -; | are contained in -^ *
times. Therefore l f are 4 ta'mes 1, or 4.
2. In %- how many times 1 ?
3. In 42- how many times 1 ?
4. In -^ how many times 1 ?
5. In -L2A how many times 1 ?
6. In II how many times 1 ?
7. In -Mp how many times 1 ?
8. How many times 1 in f | ? i n |. ? in f| ?
9. How many times 1 in -^ ? in 4^ ? in ^ ?
10. How many times 1 in -^ ? in - 4 ^-? in \\ ?
11. How many times 1 in \^- ? in J^- ? in
12. How many times 1 in -i^ ? in ^L ? in
IS. How many times 1 in ty ? in ^-^ ? in
14. How many times 1 in $|- ? in ^ ? in
70 INTELLECTUAL ARITHMETIC. [4*5.
15. How many times 1 in - ? in f f ? in
16. How do you reduce an improper fraction to a
whole or mixed number ?
17. Reduce f to an equivalent whole number.
18. Reduce * to an equivalent mixed number.
19. Reduce & to an equivalent mixed number.
20. Reduce ^ to an equivalent mixed number.
21. Express tiie value of *ffi in a whole number.
22. Express the value of f-|-g- in a mixed number.
23. Express the value of -^ in a mixed number.
45. 1. In f of 1, how many sixths ?
ANALYSIS. Since in I there are g, in of 1, there
are 2 times \ of g ; \ of is f , and 2 times are }.
Therefore, <c.
2. In ^ of 1 how many tenths ?
3. In I of 1 how many twentieths ?
4. In | of 1 how many fortieths ?
5. In | of 1 how many tenths ?
6. In -j of 1 how many sixteenths ?
7. In f of 1 how many twenty-firsts ?
8. In | of 1 how many thirty-sixths ?
9. In | of 1 how many twelfths ?
10. In | of 1 how many twenty-fifths ?
11. In -^ of 1 how many sixtieths ?
12. In | of 1 how many eighteenths ?
13. In -Jjj- of 1 how many fortieths ?
14. In 2%- of 1 how many eightieths ?
15. l n i| of 1 how many thirtieths ?
16. In | of 1 how many eighths ?
17. In I of 1 how many fourteenths?
INTELLECTUAL ARITHMETIC. 71
18. In f of 1 how many twenty-sevenths ?
19. In | of 1 how many fifty-sixths ?
20. In T 4 r of 1 how many forty-fourths ?
21. In -1--J- of 1 how many thirty-seconds ?
22. In ^ of 1 how many eightieths ?
23. In If of 1 how many ninetieths ?
24. In J I of 1 how many one hundred twentieths ?
25. How many sixths in if ?
ANALYSIS. Since in 1 there are ||, in % of 1 are of
||, or -j3 ff ; and ^ is contained in ||, 5 times. Therefore
in -f | are 5 sixths.
26. How many eighths in A ?
27. How many fourths in
Oft TT/S\rr wan\r -flii-nrlc! in 2
i* 1 T 9 2"
-|| ?
in
l. Jiuw many louiiiis 111 -g-g- ; i.
28. How many thirds in f$ ? in
29. How many sevenths in J| ? in t
30. How many twelfths in |- ? in ^
31. How many ninths in |^ ? in |-g-
32. How many sixths in || ? in ^| ' 4
33. How many eighths in |--| ? in |~- ?
34. How many fourteenths in J| ? in \
35. How many fifths in f ? in || ?
36. How many sevenths in ||- ? in ||- 1
37. How many fourths in || ? in ^-f ?
38. How many sixths in |~- ? in ^f ?
39. How many halves in || ? in || ?
40. How many tenths in | ? in -|^- ?
41. How many fifteenths in ||- ? in f f
42. How many elevenths in || ? in J|
43. How many sixteenths in |f ? in -J-j]
44. How many twentieths in ^j- ? in 1
45. How many twenty-fourths in || ? i
72 INTELLECTUAL ARITHMETIC. [46.
46. 1. Reduce ^ to its lowest terms.
NOTB. A fraction is snid to be in its lowest terms when no number
greater than 1 will exactly divide both numerator and denominator.
2.
Reduce
A
to
its
lowest
terms.
3.
Reduce
M
to
its
lowest
terms.
4.
Reduce
to
its
lowest
terms.
5.
Reduce
if
to
its
lowest
terms.
6.
Reduce
lo
to
its
lowest
terms.
7.
Reduce
H
T2
to
its
lowest
terms.
8.
Reduce
tt
to
its
lowest
terms.
9.
Reduce
?!
to
its
lowest
terms.
10.
Reduce
f!
to
its
lowest
terms.
11.
Reduce
ft
to
its
lowest
terms.
12.
Reduce
to
its
lowest
terms.
13.
Reduce
9$
to
its
lowest
terms.
14.
Reduce
M
to
its
lowest
terms.
15.
Reduce
to
its
lowest
terms.
16.
Reduce
to
its
lowest
terms.
17.
Reduce
to
its
lowest
terms.
18.
Reduce
j^
to
its
lowest
terms.
19.
Reduce
M
to
its
lowest
terms.
20.
Reduce
4y
to
its
lowest
terms.
21.
Reduce
J^.
to
its
lowest
terms.
22.
Reduce
w
to
its
lowest
terms.
23. How do you reduce a fraction to its lowest
terms ?
HEM ARKS. 1. If the numerator be multiplied by any number, the
value of the fraction will be multiplied as many times as there are units
in the multiplier.
2. If the numerator be divided by any number, the value of the fraction
will be divided as many times as there are units in the divisor.
3. .If the denominator be multiplied by any number, the value of the
fraction will be divided as many times* as there are units in the multi-
plier.
47.] INTELLECTUAL ARITHMETIC. 73
Fractions are said to have a common denominator
when their denominators are alike. Any number that can
be divided by each of the denominators of the given frac-
tions may be taken for the common denominator.
1. Reduce ^ and |- to fractions having a common
denominator.
ANALYSIS. 12 is exactly divisible by 4 and 3, and may
therefore be taken for a common denominator. Since in 1
there are \%, in % of 1 there must be of -ff , or -fa, and in
% of I there must be f of |f , or ^. Therefore and
are equal to -fa and -f^.
2. Reduce J and | to fractions having a common
denominator.
3. Reduce | and | to a common denominator.
4. .Reduce J and | to a common denominator.
5. Reduce | and f to a common denominator.
6. Reduce f and ^ to a common denominator.
7. Reduce | and | to a common denominator.
8. Reduce |, ^, and to a common denominator.
9. Reduce f , ^, and -^ to a common denominator.
10. Reduce ^, -^ and ^ to a common denomi-
nator.
11. Reduce J, -fg and to a common denominator.
12. Reduce |, |, | and | to a common denomi-
nator.
13. Reduce ^, ^ and ^ to a common denomi-
nator.
14. Reduce 1^, 2^ and f to a common denomi-
nator.
4. If the denominator be divided by any number, the value of the frac-
tion will be multiplied as many times as there are units in the divisor.
5. If both numerator and denominator be multiplied by the same
number, the value of the fraction will not be changed.
6. If both numerator and denominator be divided by the same number
the value of the fraction will not be changed.
74 INTELLECTUAL ARITHMETIC. [48.
48. 1. James has f of a dollar, John | of a
dollar, and George | of a dollar ; how many fifths
of a dollar have they all ? how many dollars ?
ANALYSIS. They all have the sum of J + I + t> e( l ua ^
to -* 5 - dollars, equal to 2 dollars. Therefore, fyc.
2. Jane had $ of a dollar and her father gave
her ^ of a dollar more ; how much had she then ?
3. Austin buys a pair of skates for of a dollar,
a pair of mittens for f of a dollar, and a slate for
of a dollar ; what is the cost of all ?
4. A grocer sold dozen eggs to one man, of a
dozen to another, and | of a dozen to another ; how
many dozen did he sell to all ?
NOTE. Fractions having different denominators must be reduced to a
common denominator before adding or subtracting.
5. Mary paid -| of a dollar for some ribbon, | of
a dollar for a pair of gloves and of a dollar for a
pair of hose ; what was the cost of the whole ?
6. Sold | of an acre of land to one man, f to
another and -J? to another ; how much was sold
to all?
7. A laborer dug a rod of ditch the first hour,
| of a rod the seconcf, | the third and -^ the fourth ;
how many rods did he dig in 4 hours ?
8. A farmer sold ^ of his grain to one man, to
another, ana ^ to another ; what part of his grain
did he sell ?
9. Henry paid of a dollar for a sled, J a dollar
for a kite and of a dollar for a string ; what was
the cost of the whole ?
10. Emma paid 2^ dollars for a dress, If dollars
for a pair of shoes and ^ of a dollar for some ribbon ;
what was the cost of the whole ?
NOTE. Add the whole numbers and the fractions separately, and then
unite their sums.
48.] INTELLECTUAL ARITHMETIC. 75
11. Milton saved -| of a dollar one week, f of a
dollar the next, 1|- the next and 2^ the next ; how
much did he save in 4 weeks ?
12. Albert picked 2^ bushels of apples from one
tree, 3f from another and 8 from another ; how
many bushels did he pick from the 8 trees ?
18. If it take 4 J yards of cloth for a coat, 2- yards
for a pair of pantaloons, and |- of a yard for a vest,
how many yards of cloth will it take for the whole ?
14. If a man walk 12 1 miles in the forenoon and
10 i miles in the afternoon, how many miles does he
walk during the day ?
15. Harvey bought a sled for 62i cents and sold
it for 12^ cents more than he gave lor it ; how much
did he receive for it ?
16. Three men bought a horse. A paid 25^
dollars, B paid 30|- dollars and C paid 35| dollars";
what was the cost of the horse ?
17. If coal is worth 5A dollars a ton, and wood
is worth 3 T 6 2- dollars a cord, what will be the cost of
one ton of coal and two cords of wood ?
18. Find the sum of ^ and -^.
19. Find the sum of fa -f and fa
20. Find the sum of fa | and -f$.
21. Find the sum of ^, -^ and f .
22. Find the sum of fa | and .
23. Add together fa ^ and -f-%.
24. Add together -^, T ^ and ^.
25. Add together ^, fa -^ and fa
26. Add together -^ , |- and |.
27. Add together |, fa -fo and fa
28. Add together fa fa fa 1 and
29. What is the sum of 2^ and s| ?
30. What is the sum of 8| and
76 INTELLECTUAL ARITHMETIC. [49.
31. What is the sum of 14, 10 , and 5J ?
32. What is the sum of 7-^ + 15| + 10J ?
33. What is the sum of 25-| + 13| + 5^ ?
34. What is the sum of 6$ + 7 j + 4f ?
35. What is the sum of A + 11 + 14| +
36. What is the sum of 3^ + 6J + | +
37. What is the sum of 6 + 14& + 5^ +
38. What is the sum of 20 + 7^ T +
49. 1. Amos having f of a dollar, gave f of a
dollar to a beggar ; how much had he left ?
ANALYSIS. He liad left the difference between % and f
of a dollar ; = f of a dollar. Therefore, $c.
2. Norman had | of a bushel of peaches ; he sold
| of a bushel ; what part of a bushel had he left ?
3. Nancy paid | of a dollar for a geography and
| of a dollar for a philosophy ; for which did she pay
the more, and how much ?
4. Two men own a piece of property together;
one owns ^ of it ; how much does the other own ?
5. When rye is worth ^ of a dollar per bushel
and corn is worth | of a dollar, what is the differ-
ence in the price ?
6. A grocer sells ^ of a box of tea from a box
full ; what part of the whole box is left ?
7. Henry can run of a mile in the same time
that John can run |^ ; which runs the farther, and
how much ?
8. Parley gave of a dollar for a vest and f of a
dollar for a pair of shoes ; what was the difference
in the cost ?
9. A gentleman, owning a boat, sold -^ of it;
how much of it did he still own ?
49.] INTELLECTUAL ARITHMETIC. 77
10. If Robert earn 7f dollars a week and Her-
man earn 6^ dollars a week, how much more than
Herman does Robert earn ?
11. Arthur gathered 10 quarts of chestnuts and
sold 7^ quarts ; how many quarts had he left ?
12. From a piece of cloth containing 12| yard?,
5| yards were cut ; how many yards remained ?
13. If I put 15| dollars in the bank at one time,
how much must I afterwards put in to make the sum
20 dollars ?
14. From a hogshead of molasses, 9 f gallons were
drawn ; how many gallons remained ?
15. Edgar will be 14 years old 3 ^ years hence ;
how old is he now ?
16. From a jug of molasses containing 1 gallon,
of a quart was used ; how much was left ?
17. If I buy a book for f of a dollar, how much
change must be returned to me for a 3 dollar bill ?
18. A lady, having 18 1 dollars, bought a bonnet
for 5f dollars ; how much money had she left ?
19. What is the difference between I- and f ?
I O
20. What is the difference between ^ and I ?
21. What is the difference between A and | ?
o
22. What is the difference between 1J and f ?
23. What is the difference between 3f and 1| ?
24. What is the difference between 16f and 9J?
25. What is the difference between 20 and 12| ?
26. How many are |- | ?
27. How many are 4|- ^ ?
28. How many are 12 6 T 9 ? ?
29. How many are 36 J 24f ?
30. How many are 25}
31. How many are 16 9| ?
32. .How many are 28 17| ?
78 INTELLECTUAL ARITHMETIC. [5O.
5O. 1. Mr. Smith sold of his estate to A, of
it to B and | to C ; what part remained unsold ?
2. A gambler lost in play and of his money ;
what part had he left ?
3. A lady paid of her money for a bonnet, J for
a shawl and |- for a pair of gaiters ; what part had
she left ?
4. A laborer worked 3 days for 1 J, |, and 1| dol-
lars ; had he received 2 dollars a day, how much
more would he have received ?
5. From a piece of calico containing 32| yards,
two dress patterns were cut, one containing 10
yards and the other 12 1 yards ; how many yards
remained in the piece ?
6. I deposited in the bank at one time $20^, at
another $15| ; how much more must I deposit to
make the amount $50 ?
7. Paid $i)| for a barrel of flour, $5f for a ton of
coal, and gave in payment a 10 dollar bill and a 5
dollar bill ; how much change must be returned to
me?
8. A merchant bought 5 barrels of flour, at $15 a
barrel, and paid $45-^- in goods and the remainder
in cash ; how much cash did he pay ?
9. If you divide $80 among three men, giving the
first J of it, and the second $35 1, what will the third
receive ?
10. A farmer, having 120 bushels of oats to sell,
sold 15J bushels more than i of them at one time,
and 20 bushels less than J ot them at another time ;
how many bushels had he left ?
11. Asaph gave ^ of his money for a sled, ^ for a
cap, and had 39 cents left ; how much had he at first ?
12. Morgan is lOf years old, Myron is 9| years
old, and the sum of their ages is 5| years more than
Martin's age ; how old is Martin ?
5O.] INTELLECTUAL AEITHMETIC. 79
13. James is 9J years old, and Lewis is 5^ years
less than twice as old as James ; what is the differ-
ence in their ages ?
14. A man, having $64, bought 3 cows ; for the
first -cow he gave ^ of it, for the second | of it, and
the remainder for the third ; what was the cost of
the third cow ?
15. A grocer drew from a hogshead 20f gallons
of molasses at one time, and 25|- gallons at another ;
how many gallons remained ?
16. What number must be taken from 36 that the
remainder may be 7f- ?
17. From what number must 9i be taken to leave
12* ?
18. What number must be added to 16^ that
the sum may be 25 J ?
19. A person being asked his age, replied that if
4| years were added to 12| years, the sum would be
| his age ; what was his age ?
20. A person, undertaking a journey of 60 miles,
traveled ^ of the distance the first day, of the
remainder the second day, and f of the remainder
the third day ; how far from his place of destination
was he at the close of the third day ?
21. What is. the difference between 17| and 5|
+ 6?
22. What is the difference between 25^ and 11|
+ 7|?
23. What is the difference between 9 + 12| and
8f +41?
24. From 301 take 201 + 3^.
25. From 40 lOf take 7 + 5J.
26. From 281 take 211 _ 54..
27. From f + \ + f take
28. From 3 + | + f take
80 INTELLECTUAL ARITHMETIC. [51.
51. 1. What will 5 pounds of tea cost at ^ of a
dollar a pound ?
ANALYSIS. Since 1 pound cost J of a dollar, 5 pounds,
which are 5 times 1 pound, will cost 5 times , or *, of a
dollar, equal to $3 J. Therefore 5 pounds of tea, at of a
dollar a pound, will cost
2. What cost 4 yards of linen at f of a dollar a
yard?
3. When sugar is \ of a dollar a pound, what will
25 pounds cost ?
4. If I of a pound of butter last a family 1 day,
how many pounds will last them one week ?
5. What will be the cost of 10 peaches at J of a
cent apiece ?
6. If a man earn ^ of a dollar a day, how much
will he earn in 12 days ?
7. If 1 peck of pears cost of a dollar, what will
2 bushels cost ?
8. How many loaves of bread must you give to 8
beggars, if you give them | of a loaf apiece ?
9. At I of a dime a pound, what will 15 pounds
of nails cost ?
10. What will 9 yards of silk cost at f of a dollar
a yard ?
11. George had $ of a dollar, and William had 8
times as much ; how much had William ?
12. If 1 horse eat ^ of a ton of hay in a month,
how much will 10 horses eat ?
13. How many are 9 times | ? 12 times | ? 14
times ^ ? 20 times -fa ?
14. If 1 man can reap ^ of an acre of rye in a
day, how much can 11 men reap ?
15. How many barrels of flour will be given to 9
poor families, if each receive ^ of a barrel ?
16. How many are 12 times ^ ? 16 times | ?
09.] INTELLECTUAL ARITHMETIC. 81
17. Bought 6 bushels of corn at $| a bushel, and
had $4 left ; how much money had I at first ?
18. If 1 quart of molasses cost $^ 3 g, what will 4
gallons cost at the same rate ?
19. If a boy give each of his 4 companions f of a
quart of chestnuts, and have 4|- quarts left, how
many quarts had he at first ?
20. How do you multiply a fraction by a whole
number ?
1. If 1 box of raisins cost $2f , what will 5
boxes co^t ?
ANALYSIS. If 1 box of raisins cost $2 j, 5 boxes, which
are 5 times 1 box, will cost 5 times $2 ; 5 times f are J^-,
or $3; 5 ft'wes $2 are $10; $3f added to $10 are
$13|. Therefore, if 1 fo# #/* raisins cost $2f, 5 fo#es
will cost $13f.
2. At 12 J cents a pound, what will 9 pounds of
butter cost f
3. At $6f a barrel, what will 7 barrels of flour
cost?
4. How many bushels of grain will 15 bags hold,
if they hold 2-J- bushels apiece ?
5. What will be the cost of 12 pounds of rice, afc
6^ cents a pound ?
6. What will 9 dozen of eggs cost at 11 J cents a
dozen ?
7. What will 7 yards of cloth cost at $5 A a yard ?
8. If 1 barrel of fish cost $14f , what will 6 barrels
cost ?
9. If 1 horse eat 3^ tons of hay in 5 months, how
many tons will 20 horses eat in the same time ?
10. If 1 pint of wine cost $1|, what will 1 gallon
cost ?
6
82 INTELLECTUAL ARITHMETIC. [53.
11. If 5 men can do a piece of work in lOf days>
how long will it take 1 man to do the same ? *
12. If a grain of gold is worth 4* cents, what is 1
pennyweight worth V
13. A grocer sold 9 pounds of coffee at 12 cents
a pound, and 7 pounds of sugar at 8| cents a pound ;
1 ow much did he receive for both ?
14. If it take 34 yards of cloth for a coat, and 24
yards for a pair of pantaloons, how many yards will
be required to make 3 of each ?
53. 1. If 1 pound of cheese cost 10 cents, what
will 5|- pounds cost ?
ANALYSIS. Since 1 pound cost 10 cents, 5g pounds,
which are 5 times 1 pound, will cost 5 times 10 cents ;
5 times 10 cents are 50 cents, f of 10 cents are 3 tines % of
10 cents ; of 10 cents is If cents, and g of 10 cents are 3
times 1% cents, or 3 cents, which added to 50 cents make
53 J cents. Therefore, fyc.
2. If a horse travel 6 miles an hour, how far will
he travel in 7| hours ?
3. If a person's expenses be $12 a week, what
will they be for 9| weeks ?
4. What will 4| barrels of cider cost at $3 a barrel ?
5. What will lOf pounds of honey cost at 12 cents
a pound ?
6. What will 15f bushels of potatoes cost at 6
shillings a bushel ?
7. If a pint of wine is worth 40 cents, what are 3f
pints worth ?
8. If a man can build 4 rods of wall in a day, how
many rods can he build in 7| days ?
9. If a barrel of flour last a family 6 weeks, how
long will 4-J barrels last them ?
INTELLECTUAL ARITHMETIC. 83
10. If stage fare is 4 cents a mile, what will it
cost to ride 20f miles.
11. If a merchant pay 12 shillings for 3 caps, and
sell them for 6| shillings apiece, what is his gain ?
12. Bought 9 yards of silk at 12 shillings a
yard, and 12f pounds of tea at 7 shillings a pound ;
how much more did the silk cost than the tea ?
13. If I buy 8| tons of coal, at $5 a ton, how
much less than $50 will it cost ?
14. A woman took 6| dozen eggs to market,
which she sold for 12 cents a dozen ; she received in
payment 4| yards of calico at 10 cents a yard, 3^
yards of ribbon at 8 cents a yard ; how much was
still her due ?
15. If | of a barrel of flour last a family 1 month,
how many barrels will last 8 such families 6| months ?
16. What is the product of 4^ times 9 multiplied
by 3?
17. What is the product of 5 times 3| multi-
plied by 2 ?
18. Multiply 4|- times 10 by | of 15.
19. Multiply 3| times 6 by | of 24.
20. Multiply 4-f by f of 16.
21. Multiply 2 times 9 by 2^ times 4.
22. Multiply 5 times 3 by -f of 15.
23. What is the sum of 8 times 7 T % and 4|
times 12 ?
24. What is the sum of 6| times 11 and 12
times 5f ?
25. What is the difference between 4f times 10
and 3 times 9| ?
26. What is the difference between 7 times 8J and
101 times 5 ?
27. How many are 12| times 7 ? 9f times 10 ?
28. How many are 5f times 15 ? 3^ times 40 ?
84 INTELLECTUAL ARITHMETIC.
54. 1. A man, owning of a store, sold 4 of
his share ; what part of the whole store dia he
sell?
ANALYSIS. He sold \ of \ of the store. Since $ is
equal to J, | of \ is equal to \ of |, or . Therefore,
he sold ^ of the store.
2. James had | of a dollar, and gave -J- of it to a
beggar ; what part of a dollar did he give away ?
3. A boy, having J of a watermelon, gave away *
of what he had ; what part of the whole melon did
he give away ?
4. If a yard of crape cost | of a dollar, what will
J of a yard cost ?
5. If I own of an acre of land, and sell of it,
what part of an acre do 1 sell ?
6. If a bushel of apples be worth % of a dollar,
what is J of a bushel worth ?
7. John is as old as his father, and Henry is
as old as John ? what part of his father's age is
Henry's age ?
8. A merchant, having | of a hogshead of molas-
ses, sold of what he had ; what part of the whole
hogshead did he sell, and what part had he left ?
9. Homer bought a box of figs, and Robert \
as many ; what part of a box did both buy ?
10. C owned 1 of a steamboat, and sold | of his
snare ; what part of the whole did'he sell ?
11. Mr. Jones, owning of a cotton mill, sold J
of his share to B, and B sold l of his share to C ;
what part of the whole did each have after the di-
vision ?
12. A cistern, being full of water, sprang a leak,
and before it could be stopped J of the water ran
out, but ^ as much ran in at the same time ; what
part of the cistern was emptied ?
13. A poor man, having 1 of a barrel of flour.
54.] INTELLECTUAL ARITHMETIC. 85
gave I of- it to a poor neighbor, who gave ^ of hi?
share to a poor woman ; what part of a barrel had
each then ?
14. Andrew, having $|, gave | of it for a knife ;
what part of a dollar did he pay for his knife ?
ANALYSIS. He paid f of of a dollar. 3 fifths of
are 3 times 1 fifth of . of & ^, cmd f o/ are 3 times
& or ^. Therefore, $c.
15. If a yard of cloth is worth $|, what fs \ of a
yard worth ?
16. Harriet, having |- of a yard of silk, gave f of
it to her sister ; what part of a yard did she give
away, and what part of a yard had she left ?
17. A man, owning | of a farm, sold | of his share
to his brother ; what part of the farm did each own ?
18. Hannah picked -f$ of a pailful of strawberries,
and on her way home spilled | of them ; what part
of a pailful had she left ?
19. What will be the cost of f of a bushel of
beans at $f a bushel ?
20. Jacob, having -| of a pound of candy, gave \
of it to Mary, and \ of it to Jane ; what part of a
pound had he left ?
21. A keeper of a saloon bought a cask of ale ; \
of it leaked out, f of the remainder he sold, and
what was left he kept for his own use ; what part of
the cask did he keep ?
22. How do you multiply one fraction by another ?
23. What is i of 1? 29. What is \ of \ ?
24. What is |- of \ ? 30. What is \ of \ ?
25. What is ^ of I ? 31. What is l of 1 ?
26. What are 4 of \ ? 32. What is J of f ?
2T. What are f of f ? 33. What are J- of f ?
28. What are A of -- ? 34.
86 INTELLECTUAL ARITHMETIC. [55.
55. 1. At $8| a barrel, what will | of a barrel
of flour cost ?
ANALYSIS. Since 1 barrel cost $8|, | of a barrel, which
is 3 times , will cost 3 times of $8|. of $8 ?'s (2, and
i o/$| 2*5 $, which added to $2 w/r/to 2-J. 3 ft'ww $2
are $6, and 3 times -J are , wfo'c^ aeWeo 1 to $G make $6g.
Therefore, fyc.
Or, $8 are e^*aZ to \ 4 -, awe? %of* = ^, or $6f.
2. At $6 a bushel, what will f of a bushel of
clover seed cost ?
3. At $12f a ton, what will g- of a ton of hay cost ?
4. If a man travel 24} miles in 1 day, how far
will he travel in of a "day ?
5. What will | of a barrel of beef cost, at $15 1 a
barrel ?
6. What will be the cost of \ of a cord of wood,
if 1 cord cost $5-t ?
7. At lOi shillings a day, how much can a man
earn in | of a day ?
8. If a man can cut 3i acres of rye in 1 day, how
much can he cut in | of a day ?
9. If I pay $6 1 for ajar of butter, what is of it
worth ?
10. Bought a horse for $75J, and sold him for |
of what he cost ; what was the loss ?
11. What is | of 14? of 20| ? of 13|? of 181?
12. What are f of 9| ? of 12$ ? of 16* ? of 214 ?
13. What are | of 21f ? | of 32J ? f of 23 J ?
14. At $| a yard, what will 6 yards of flannel
cost ?
ANALYSIS. Since 1 yard costs $, 6 yards will cost G
times $|. 6 times $f are -^ or $4, arco 7 % of % is -%, or
$, which added to $4 makes $4 . Therefore, $c.
Or, 6 are ^wa/ to ^, awe? ^- o/ f are ^J, or $4J.
Therefore, $c.
15. At $| a bushel, what will 8|- bushels of peaches
cost?
INTELLECTUAL ARITHMETIC. 87
16. If a man hoe of an acre of corn in 1 day,
how many acres can he hoe in 5| days ?
17. If a vessel sail 12-| miles in 1 hour, how far
does she sail in | of an hour ?
18. If a man has 22J- bushels of clover seed, and
he sell I of it, how much has he left ?
19. What will J of of 12 gallons of oil cost, at
a gallon ?
20. At $| a rod, what will it cost to dig |- of f of
28 rods of ditch ?
21. What will 4* yards of cloth cost at $2| a
yard ?
NOTE. Reduce the mixed numbers to improper fractions, and then
proceed as in multiplying one fraction by another.
22. If you earn 8|- shillings in 1 day, how much
can you earn in 2| days ?
23. What cost 2| dozen of eggs at 10|- cents a
dozen ?
24. What will 3f yards of shalloon cost at $1^
a yard ?
25. If a man can do a job of work in 5| days, in
what time can he do a job If times as large ?
26. Clorinda is 7| years old, and Augusta is as
old ; how old is she V
27. A man paid $16f for a cow, and 2J times as
much for a colt ; how much did he pay for" both?
28. What is the product of 3| times 9-| ?
29. What is the product of 81 times 10| ?
30. What is the product of % of f of 16 multi-
plied by 5f ?
31. What is the value of f of | of 6f ?
32. What is the value of | of |- of 30 ?
33. What is the value of f of | of | ?
34. What is 9 T ^ times f ? 41 times - ?
35. What is 10^ times 2| ? 2^ times 25J ?
88 INTELLECTUAL ARITHMETIC. [456.
1. If 3 pounds of raisins cost f f-, what will
1 pound cost ?
ANALYSIS. If 3 pounds cost $f , 1 pound, which is $
of 3 pounds, will cost of $, or $f . Therefore, fyc.
2. If 4 slates cost $|, what will 1 slate cost ?
8. If 5 pounds of sugar cost $-|, what will 1 pound
ost?
4. If 3 oranges are worth | of a melon, what part
of the melon is 1 orange worth ?
NOTE. It will be remembered that a fraction may be divided by a
whole number either by dividing the numerator or multiplying the denom-
inator by it. (See pages 72 and 73, notes.)
5. If 6 pounds of coffee cost $|, what will 1 pound
cost ?
6. If 4 apples cost |- of a shilling, what will 1
apple cost ?
7. If 5 yards of sheeting cost f f , what will 1 yard
cost?
8. If 4 figs cost | of a dime, what will 1 fig cost ?
9. What will 1 yard of linen cost, if 4 yards
cost $| ?
10. Marcus had f of a melon, which he wished to
divide equally between his 2 sisters ; what part must
he give to each ?
11. Oliver, having 12 quarts of blueberries, sold |
of them for $ |- ; what was that a quart ?
12. If 6 persons agree to share equally | of a
bushel of grapes, what part of a bushel will each
have ?
13. If you divide f of a barrel of flour among 3
poor families, what part of a barrel will you give to
each ?
14. How many times is 7 contained in f ? 5 in
if? 8inf ? lOin-^?
15. What is the quotient of || divided by 5 ? by 7 ?
16. What is the quotient of f divided by 3 ? by 4 ?
57.] INTELLECTUAL ARITHMETIC. 89
57. 1. At $| a yard, how many yards of silk
can be bought for $6 ?
ANALYSIS. As many yards as $, the price of 1 yard,
is contained times in $6 ; $6 is equal to $- 2 /, and $| is con-
tained in $- 2 / 8 times. Therefore, at $J a yard, 8 yards of
silk can be bought for $6
2. If a boy can earn $f a day, how long will it
/take him to earn $5 ?
3. How long will it take a man to spend $10 for
cigars, if he spend $ J- a day ?
4. When potatoes are $f per bushel, how many
bushels can be bought for $8 ?
5. At $| a pair, how many pairs of shoes can be
bought for $7 ?
6. How much butter can be bought for $3, at f j-
a pound ?
7. How many times is 4 contained in 4 ? in 7 ?
in 8?
8. If the cars run f of a mile a minute, how long
will they be in running 25 miles ?
9. If a horse eat | of a bushel of oats a day, how
long will 9 bushels last him ?
10. If a man walk |- of a mile in J of an hour,
how long will it take him to walk 12 miles ?
11. How many bushels of oats, worth $| a bushel,
will pay for -| of a barrel of flour, worth $9 a barrel ?
12. A farmer sold a grocer 1 ton of hay for $12,
and received -|- the amount in sugar at $| a pound,
-J in money, and the remainder in molasses at $f a
gallon ; how many pounds of sugar, and how many
gallons of molasses, did he receive ?
13. If | of a barrel of trout cost $12, what will 1
barrel cost ?
14. If | of a ton of hay cost $7, what will 1 ton
cost ?
90 INTELLECTUAL ARITHMETIC. [58.
15. When potatoes are worth $*- a bushel, and
corn $| a bushel, how many bushels of potatoes are
equal in value to 16 bushels of corn ?
16. How many yards of ribbon, at $ o f a shilling
a yard, can be bought for 7 shillings ?
17. If 1 man consume of a pound of meat in a
day, how many men would 6 pounds supply ?
18. If f of a hogshead of molasses cost $36, what
will one hogshead cost ?
19. If | of an acre of land sell for $21, what will
an acre sell for at the same rate ?
20. How many pounds of tea, worth $^ a pound,
must be given for 9 bushels of apples, worth $J a
bushel ?
21. How many times is | contained in 5 ? in 7 ?
in 9 ? in 11 ? in 15 ?
22. How many times is | contained in 12 ? in 18 ?
in 20 ? in 15 ?
23. How many times is contained in | ? in |-| ?
in ft? in I?
24. How many times is % contained in of 16 ?
in | of 27 ?
25. How is a whole number divided by a fraction ?
58. 1. At $2 a bushel, how many bushels of
wheat can be bought for
ANALYSIS. As many bushels as $2, the price of I bushel,
is contained times in $11. 11 are equal to A, and 2 is
contained in Aj*> - 1 /-, or 5 times. Tfierefore, fyc.
Or, 2 is contained in 11 J, 5 times, and 1^ or ^over ; and
2 is contained in |, f times, which added to 5 makes 5|
times. Therefore, fyc.
2. If 6 pounds of coffee cost $lf , what is the cost
of 1 pound ?
59.] INTELLECTUAL ARITHMETIC. 91
3. If a man walk 18 1 miles in 4 hours, how many
miles does he walk in 1 hour ?
4. How many times will 16| gallons of cider fill a
vessel that holds 3 gallons ?
5. How many oranges, at 3 cents ap.,ce, caij. be
bought for I of of 60 cents ?
6. At $4 a yard, how many yards of cloth can be
bought for $21f ?
7. If a turkey cost 7 shillings, how many turkeys
can be bought for 38^- shillings ?
8. At 5 shillings a gallon, how many gallons of
molasses can be bought for 24|- shillings ?
9. If a day laborer earn $8f in 10 days, how much
does he earn in 1 day ?
10. If a locomotive run 4| miles in 6 minutes,
how far does she run in 1 minute ?
11. If 12 bushels of oats cost $4|, what part of a
dollar will 1 bushel cost ?
12. If 7 pounds of coffee cost 8| dimes, What
will 1 pound cost ?
13. Divide 8| by 5 ; lOf by 7 ; 15J by 15.
14. Divide 9f by 7; 18|- by 12; 20 by !
of 16.
15. Divide f of 21 by f of 10. f of 29 by
of 63.
59. 1. At $f a pound, how many pounds of
tea can be bought for $ f ?
ANALYSIS. As many pounds as $, the price of 1
pound, is contained times in $. f equal $< f e<?wa/ ,
and 8 twentieths are contained in 15 twentieths 1| times.
Therefore, at $| a pound, Impounds can be bought for $.
Or thus : ?s contained in 1,5 ft'mis, wc? m } q/' 1 iV i*
contained f o/ 5 ^'712^5, or -^ ^'TTZ^ ; f z's contained in |, ^
f times, and ^ of ^f- is *-, or 1 J.
92 INTELLECTUAL ARITHMETIC. [59.
2. How many pounds of honey, at 1} a pound,
can be bought for $ J- ?
3. In of an acre of land, how many building
lots of -^0- of an acre each ?
4. If a horse eat f of a bushel of oats in a day
in how many days will he eat -$ of a bushel ?
5. At | of a cent apiece, how many slate pencils
can be bought for f of a cent ?
6. If a piece of ribbon | of a yard long be cut
into pieces of a yard in length, how many pieces
will there be ?
7. A man, owning | of a coal mine, divided his
share equally among his sons, giving them -^ each ;
how many sons had he ?
8. Among how many children can you divide -| of
If melons, if you give - of of a melon to each '(
9. At $J a yard, how many yards of silk can be
bought for $f ?
10. At | of -J- of a dollar a yard, how many yards
of ribbon can be bought for $ ^ ?
11. At $jV a pint, how much wine can be bought
12. When peaches are worth $| a bushel, how
many can be bought for $| ?
13. If | of a yard of silk cost $f , what will 1 yard
cost?
ANALYSIS. It will cost 3 times of $, or of $|.
of | is f , and 3 times | are f , or $ 1 . f J7ierefore, if % of
a yard of silk cost $J, 1 yard will cost $1|.
14. What will 1 dozen eggs cost, if | of a dozen
cost f of a dime ?
15. If f of a bushel of corn are worth -V - of a
bushel of rye, how many bushels of corn is 1 bushel
of rye worth ?
16. Horace had $f , which was f of what he paid
for a cap ; what was the cost of his cap ?
INTELLECTUAL ARITHMETIC. 93
17. If f of a yard of cloth cost $^, what will 1
yard cost ?
18 James bought -f of a pineapple, which was f
of what Luther bought ; what part of a pineapple
did Luther buy ?
19. If | of a drum of figs cost $|, what will 1
drum cost ?
20. How many times can a bottle holding % of f
of a gallon be filled from a demijohn containing f of
If gallons ?
21. Divide | by f; ^ by |,
22. Divide J-f by f ; T 9 T by f .
23. Dividef by ; f by f .
24. Divide J of | by | ; f of f by | of f .
25. Divide f| by of f ; 2 9 o by f
26. Divide f| by ^; ^ b 7 iV
27. How many times are f- contained in -
28. How many times are | contained in
29. How many times are f contained in -| ?
30. How many times are f contained in -i|- ?
31. How many times are |- contained in | ?
32. How many times are f of f contained in
33. How many times are -| contained in f of 1| ?
84. How many times are | of f contained in of f ?
35. |^ are f of what number ?
ANALYSTS. ^ are | o/*4 fo*/??es -i q/" | ; -j of ^ are 2
4 times ^ are if. Therefore^ | ar^ | o/ ^|.
36. f are f of what number ?
37. f are f of what number ?
38. -^ are -| of what number ?
39. | are f of what number ?
40. are of what number ?
94 INTELLECTUAL ARITHMETIC. [6O.
CO. 1. At !-{?- dollars a yard, how many yards of
cloth can be bought for $11 ?
ANALYSIS. As many yards as $1, the price of 1 yard,
is contained times in $11. $1 are $y; $11 are $$-,
and -y is contained in ^, G times. Therefore, at $1| a
yard, G yards of cloth can be bought for 11.
2. If a turkey cost $1 , how many can be bought
for $9 ?
3. A grocer paid $4f for some onions, at the rate
of $f a bushel ; how many did he buy ?
4. How many times will 4-f gallons of campheno
fill a vessel that holds of of 1 gallon ?
5. If 4| yards of velvet ribbon cost $|, what will
1 yard cost ?
6. If 3^ dozen of eggs cost $^, what is the cost
of 1 dozen ?
7. How many yards of flannel worth $f a yard
can be bought for 6 turkeys worth $>f apiece ?
8. If 1 man can do a piece of work in 6| days, in
what time can 4 men do the same ?
9. If a man chop l cords of wood in a day, how
long will it take him to chop 10J cords ?
NOTE. Reduce the mixed numbers to improper fractions, and then
divide the same as you divide one traction by another.
10. If 41 baskets of peaches are worth $3f , what
is 1 basket worth ?
11. If 51 bushels of apples cost $1*, what is the
cost of 1 bushel ?
12. If a man spend $lf a month for tobacco, in
what time will he spend $101 for the same purpose ?
13. At 4^ shillings a gallon, how many gallons of
molasses can be bought for 9| shillings ?
14. If a stage run 24| miles in 3^ hours, how far
does it run in 1 hour ?
15. Mr. B. distributed 16^ bushels of corn equally
6O.] INTELLECTUAL ARITHMETIC. 95
among some poor persons, giving them 1-J- bushels
each ; among how many persons did he divide it ?
16. Bought 1 of 7-J cords of wood for * of $32 ;
what did 1 cord cost ?
17. A father divided 11 3 1 acres of land among his 3
sons ; to the first he gave f of it, and the remainder he
divided equally between the other 2 ; how many acrea
f\ir\ oarVi vonoivo ?
cuviueu equally
did each receive ?
id each receive ?
18. Divide 9J by 6 ; 10J by 7 ; 12| by 31.
19. Divide 10J- by f ; 14 by f ; 20| by |.
20. Divide 7J by 3| ; 15 by 2| ; 8| by 5f
21. Divide 16J by 3J ; 20 J- by 1^ ; 11$ by 4|
22.. Divide | of $ of 14 by f of 3-J-.
23. - are how many times 6 ?
24. T 8_ are now many times 4 ?
25. -|{r are how many times 15 ?
26. J^ are how many times 6^ ?
27. 12 are how many times f ?
28. 16 are how many times | ?
29. 25 are how many times | ?
30. 14 are how many times -^ ?
31. 14f are how many times | ?
32. 6| are how many times |- ?
33. 21^ are how many times 5 ?
34. 32| are how many times 7 ?
35. 17| are how many times 8 ?
36. 15 are how many times 7J ?
37. 18^ are how many times 2J ?
38. 22|- are how many times 3f ?
39. 4 T 9 7 are how many times 6| ?
40. 7f are how many times 4J ?
41. 8| are how many times 3| ?
96 INTELLECTUAL ARITHMETIC. [61.
61. 1. What is of 3 ?
ANALYSIS. That number which, taken 4 times, will
make 3. Since 1 fourth of 1 is ^ 1 fourth of 3 is 3
times | of 1, w | o/ 1. Therefore \ of 3 s f 0/ 1, or f;
2. What is i of 3 ? of 4 ? of 6 ? of 7 ?
3. What is -J- of 7 ? of 3 ? of 9 ? of 15 ?
4. What is of 2 ? of 4 ? of 5 ? of 11 ?
5. What is \ of 3 ? of 5 ? of 7 ? of 13 ?
6. What is i of 2 ? of 3 ? of 5 ? of 7 ?
7. What is of 3 ? of 5 ? of 9 ? of 14 ?
8. What is j of 4 ? of 6 ? of 12 ? of 18 ?
9. What is -i of 5 ? of 3 ? of 20 ? of 30 ?
10. What is T V of 7 ? of 9 ? of 16 ? of 47 ?
11. What is ^ of 4 ? of 7 ? of 11 ? of 63 ?
12. What is ^ of 2? of 7? of 11? of 34? of 75?
13. What is ^ of 13? of 9? of 17? of50? of 91?
14. If 5 bushels of wheat cost $9, what will 1
bushel cost :
ANALYSIS. If 5 bushels cost $9, 1 bushel^ which is ^
of 5 bushels, will cost I of $9 ; I of $9 i* $li. There-
fore, if 5 bushels cost $9, 1 bushel will cost $l.
15. If 6 turkeys are worth $5, what are they
worth apiece ?
16. If you divide 6 oranges equally among 7 boys,
what part of an orange will each receive ?
17. If a man travel 29 miles in 9 hours, how far
does he travel in 1 hour ?
18. If 1 man can build a barn in 20 days, in what
time can 3 men build it ?
19. If a horse eat 5 bushels of oats in 8 days,
what part of a bushel does he eat in 1 day ?
20. What will 1 hat cost, if 4 hats cost $17 ?
INTELLECTUAL ARITHMETIC. 97
62. 1. What are f of 3 ?
ANALYSIS.
of 1, and 3
| of 3 are 3
times | are f
times j
or 2|.
3;i*/
lerejore, *
\ of I
are 2|
2.
What
are
1
of
5?
of
4?
of
7?
3.
What
are
f
of
4?
of
5?
of
8?
4.
What
are
1
of
2?
of
3?
of
9?
5.
What
are
1-
of
3?
of
5?
of
10?
of
12?
6.
What
are
I 8 o"
of
4?
of
7?
of
15?
of
18?
7.
What
are
1
of
2?
of
3?
of
14?
of
20?
8.
What
are
1
of
5?
of
8?
of
19?
of
28?
9.
What
are
JL
11
of
7?
of
13?
of
21?
of
30?
10.
What
are
f
of
2?
of
6?
of
15?
of
24?
11.
What
are
A
of
7?
of
36?
of
50?
of
63?
12.
What
are
1<L
of
35?
of
60?
of
77?
of
91?
13.
What
are
"A"
of
10?
of
15?
of
120?
of
150?
14.
If 4 cords
of wood cost
$14,
what will 5
cords
cost ?
ANALYSTS. If 4 cords of wood cost $14, 5 cords
will cost 5 times \ o/$14. j o/$14 is $3^, and 5 times
$vM are $17-.|. Therefore 5 cords of wood will cost
$17J.
15. If 3 bushels of quinces cost $7, what will 2
bushels cost ?
16. If 9 bushels of apples cost $4, what will 7
bushels cost ?
17. At 6 cents for 2 oranges, how many oranges
could you buy for 21 cents ?
18. If 4 men can do a piece of work in 15 days,
in what time can 6 men do the same ?
19. If a quantity of provision will last 5 men 9
days, how long would the same last 11 men ?
7
98 INTELLECTUAL ARITHMETIC. [63.
63. 1. If 1 pound of butter cost 19 cents, what
will | of a pound cost ?
2. A barrel of beer sold for $8 ; what was | of it
worth ?
3. Sold a cow for $18, which was only of her
real value ; what was her real value ?
4. Bought a carriage for $65, and afterwards
sold it for | of what it cost me ; how much did I
gain ?
5. A farmer, having 80 sheep, sold -^ of them ;
how many had he left ?
6. If I give 4 pounds of butter, worth 12-|- cents a
pound, for 8 pounds of sugar, what is the sugar
worth a pound ?
7. After paying | of my money for a piece of
land, I had $48 left ; how much money had I at
first?
8. A man bought a watch and chain for $50,
which was of what the watch alone cost ; what was
the cost of each ?
9. If a man can cut 9 cords of wood in % of a
week, how many cords can he cut in 4 weeks ?
10. A man paid $35 for a book-case, f of the
cost of his book-case was ^ of what he paid for a
bureau, and % the cost of the bureau was ^ of what
he paid for a table ; what did he pay for the bureau,
and for the table ?
11. If 4 quarts of chestnuts cost 37 cents, what ,
will 1 bushel cost ?
12. If a man can walk 7 miles in 2 hours, how
far can he walk in 3 days, by walking 10 hours a
day?
13. If |- of a barrel of flour is worth $8, what are
12 barrels worth ?
14. If 3 pecks of grass-seed cost 14 shillings,
what will 4 quarts cost ?
63.] INTELLECTUAL ARITHMETIC. 99
15. If 3 yards of silk cost $19J, what will 7 yards
cost?
16. If a man earn $32|- in 4 weeks, how much will
he earn in 2 weeks ?
17. If $9 will buy 4f yards of cloth, how much
will $1 buy ?
18. A man, having 56| acres of land, sold | of it ;
how many acres did he sell ?
19. If 9 horses eat 5| bushels of oats in a day,
how many bushels will 4 horses eat in the same time ?
20. If 3 yards of flannel cost $|, what will 8
yards cost ?
21. If 10 bushels of wheat are worth 22| bushels
of corn, how many bushels of corn are 3 bushels of
wheat worth ?
22. If 14 acres of meadow land produce 32 1 tons
of hay, how many tons will 5 acres of the same land
produce ?
23. A cistern is filled by 3 pipes in 6-f hours ; how
many pipes of the same size will be requirecTlo fill it
in ^ of an hour ?
24. A boy sold 5 quarts of strawberries at the rate
of 3 quarts for 13 cents ; how much did he receive
for the whole ?
25. A tailor bought 3 pieces of cassimere, each
piece containing 6| yards, for $48-^ ; how much did
it cost a piece, and how much a yard ?
26. If a pole 8 feet long cast a shadow 6 feet, how
long is a pole that casts a shadow 12} feet at the
same time ?
27. How much less than $17 will 9 yards of satin
cost, if 2 yards cost $3^ ?
28. If 3 yards of velvet cost $5J, how much more
than $9 will 6 yards of it cost ?
29. If a horse travel 40|- miles in 6 hours, how
far will he travel in 4 hours ?
100 INTELLECTUAL ARITHMETIC. [64.
64. 1. If 5 pounds of chalk cost 28 -| cents,
what will | of a pound cost?
2. If 3 pairs of socks cost (j^, how many pairs
can be bought for $2^ ?
3. Lucas is f as old as his father, who is 60 years
of age ; what is Lucas's age ?
4. If of a box of raisins cost $2-^, what will
of a box cost ?
5. A boy sold some peaches for | cents, at the
rate of 3 for 4 cents ; how many did he sell ?
6. If 7 bushels of oats cost $lf , what will 12f
bushels cost ?
7. If 2 1 yards of shalloon cost $4, what will 5
yards cost ?
8. Bought 4J barrels of cider at $3| a barrel, and
paid in wood at" $3 a cord ; how many cords of wood
paid for the cider ?
9. John has $5|, and Gilbert has If times as
much ; how much has Gilbert ?
10. Three men incur an expense of $27|- ; how
much more than $7-| must each pay ?
11. A man, being asked his age, said that, were he
3 times as old, y 1 ^ of his age would be 8 years ; how
old was he ?
12. If a pound of tea is worth 3^ times as much
as a pound of coffee, what will 6| pounds of coffee
cost, when tea is worth 50 cents a pound ?
13. If 2f poiinds of cheese cost 20 cents, what
will 12 pounds cost ?
14. If 34- barrels of apples cost $51, what will |
barrel cost ?
15. A stone mason worked llf days, and after
paying his board and other expenses with f of his
earnings, had $20 left ; how much did he receive a
day?
64.] INTELLECTUAL ARITHMETIC. 101
16. If 7 bushels of potatoes cost $3|-, how many
bushels can be bought for $14| ?
17. If J of a quart of braiicly cost $f , how many
gallons will $12f buy ?
18. How many yards of silk can be bought for 80
shillings, if 3 shillings be paid for f of a yard ?
19. If | of a shilling will buy $ of a pound of
butter, how many pounds will 7 shillings buy ?
20. How many gallons of molasses can be bought
for $31, if | of a gallon cost $-}- ?
21. How many oats can be bought for 10 shillings,
if 9 bushels cost 24 shillings ?
22. If 5 dozen of eggs cost 4| shillings, how many
dozen can be bought for 7 shillings ?
23. If 8 yards of velvet ribbon cost $!-, how many
yards can be bought for $3^- ?
24. How many pints of alcohol will 3 shillings
buy, if of a pint cost ^ of a shilling ?
25. If a man eat If pounds of bread in a day, how
many days will 221 pounds last him ?
26. How many pounds of sugar worth 10 cents a
pound can be had for 6| pounds of cheese worth 7^
cents a pound ?
27. If a horse consume 8 \ bushels of oats in 3
weeks, how many bushels would he consume in 5
weeks ?
28. A man gave 6|- pounds of butter at 12 cents
a pound for | of a gallon of oil ; what was the oil
worth a gallon ?
29. When 4 pounds of tea can be bought for $2|,
how much can be bought for $ T ^- ?
30. A man, having $10, gave f of his money for
clover seed at $3J a bushel ; how much did he buy ?
31. If I of a bushel of rye cost $|, how many
bushels can be bought for $7 ?
102 INTELLECTUAL ARITHMETIC. [6*5.
63. 1. If a yard of cloth cost $|, what will
of a yard cost ?
2. If ^ of 2 pounds of tea cost $1|, what will |
of 1 pound cost?
3. What will -| of a barrel of apples cost, if | of
a barrel cost $f- ?
4. If | of a pound of butter is worth i of a pound
of fish, how many pounds of fish are l| pounds of
butter worth V
5. What will be the cost of 6 barrels of flour at
the rate of $2 f for every | of a barrel ?
6. At $1 a quart, what part of a bushel of clover
seed can be bought for $3^ ?
7. Since I was 25 years old, f of my entire age
has passed ; what is my age ?
8. A man, engaging in play, lost ^ of his money,
after which he gained $75, when he found that he
had $375 ; how much money did he lose ?
9. If 4 horses consume 2| bushels of oats in 2J
days, how many bushels will 6 horses consume in
the same time ?
10. If I of a bushel of barley be given for f of a
bushel of corn worth $f a bushel, what is the barley
worth a bushel ?
11. A person having a drove of turkeys, after sell-
ing 35, found that he had -| of his drove left ; how
many turkeys in his drove at first ?
12. What number is that to which if | of itself
be added, the number will be 55 ?
13. Wl;at number added to 3 times | of 27 will
make the number 40 ?
14. If 24 is |- of some number, what is ^ of the
same number ?
15. What number taken from 2^ times 12| will
leave 20f ?
65.] INTELLECTUAL ARITHMETIC. 103
16. If $40 exceed by $10, J of f of what I gained
in the sale of a house and lot, how much did I gain ?
17. If I of 4 tons of coal cost $5, what will | of
2 tons cost ?
18. A young lady, being asked her age, replied
that 5 times ^ of 30 years was 4| years more than 8
times her age ; what was her age ?
19. In an orchard, ^ the trees bear apples, 1 bear
peaches, -^ bear pears and 15 trees bear plums;
how many trees in the orchard, and how many of
each kind ?
20. How many yards of cloth that is | of a yard
wide are equal to 12 yards | of a yard wide ?
21. A, B, and C can do a piece of work in 5 days ;
B and C can do it in 8 days ; in what time can A do
it alone ?
22. A can build a piece of wall in 3 days, and B
can build it in 4 days ; in what time can both build
it together ?
23. If | the value of a carriage is equal to f the
value of a horse, and the difference in their values is
$25, what is the value of each ?
24. A tailor paid $73, which was -^ of all the
money he had, for cloth, at the rate of $3 a yard ;
how many yards could he have purchased with all
his money, at the same rate ?
25. A father divided a piece of land among his 3
sons ; to the first he gave 10J acres, to the second |
of the whole, and to the. third as much as to the
other two ; how many acres had each ?
26. A farmer bought 2 cows for $56, paying | as
much for one as for the other ; how much did each
cost him ?
27. A and B can do a piece of work in 14 days ;
A can do f as much as B ; in how many (lays can
each do it ?
104 INTELLECTUAL ARITHMETIC. [66.
66. 1. 16 is I of what number ?
ANALYSIS. 16 is | of 5 times \ of 16 ; j of 16 is 4,
and 5 fo'mes 4 are 20. Therefore, 16 2*5 i o/"20.
Or thus : Since 16 zs f of some number, J o/* ^e number,
which is of %, is of 16, or 4; ; and 4 zs o/ 5 fr'wes 4^
or 20. Therefore, $c.
2.
42
is
f
of
what
number ?
3.
25
is
*
of
what
number ?
4.
16
is
of
what
number ?
5.
32
is
t
of
what
number ?
6.
60
is
A
of
what
number ?
7.
14
is
I
of
what
number ?
8.
48
is
1
of
what
number ?
9.
75
is
1
of
what
number ?
10.
84
is
J|
of
what
number ?
11.
55
is
ii
of
what
number ?
12.
64
is
A
of
what
number ?
13.
13
is
I
of
what
number ?
14.
18
is
i
of
what
number ?
15.
28
is
1
of
what
number ?
16.
40
is
of
what
number ?
17.
54
is
A
of
what
number ?
18.
10
is
~15
of
what
number ?
19.
12*
is
%
of
what
number ?
20.
15f
is
f
of
what
number ?
21.
19 1
is
1*0
of
what
number ?
22.
24f
is
A
of
what
number ?
23.
16^
is
of
what
number ?
24.
J
is
1
of
what
number ?
25.
i
is
f
of
what
number ?
26.
A
is
!
of
what
number ?
67.] INTELLECTUAL ARITHMETIC. 105
67. 1. 24 is I of how many times 10 ? *
ANALYSIS. 24 is f of as many times 10 as 10 is con-
tained times in 5 times \ of 24 ; \ of 24 s 6, and 5
tfmes 6 s 30; 10 is contained in 30, 3 times. There-
fore 24 is ^ of 3 times 10.
2. 18 is I of how many times 6 ?
3. 25 is | of how many times 15 ?
4. 36 is -| of how many times 4 ?
5. 28 is y 7 g of how many times 8 ?
6. 40 is T 5 2 of how many times 16 ?
7. 75 is | of how many times 20 ?
S. 64 is -| of how many times 24 ?
9. 48 is if of how many times J of 30 ?
10. 35 .is | of how many times j of 28 ?
11. 42 is-^ of how many times ^ of 63 ?
12. 84 is ^ of how many times | of 35 ?
13. 27 is T 9 g of how many times of 72 ?
14. 36 is | of how many times J of 21 ?
15. 14| is | of how many times ^ of 20 ?
16. 16 1 is J of how many times ^ of 54 ?
17. 12f is ^ of how many times J of 40 ?
18. I is | of how many times ^ of J ?
19. O f | i s | O f how many times |- of 15 ?
20. James gave 56 cents for a pair of skates, which
was ^ of what he gave for his sled ; what did his
sled cost him ?
21. A farmer paid $23^ for a cow, which was |
of what he received for a horse ; what was the dif-
ference in the price of the cow and the horse ?
22. A man sold 4J cords of wood at $2|- a cord,
which was | of what he received for a ton of hay ;
how much did he receive for the hay ?
* Please turn to 67 in the Appendix, page 170.
100 INTELLECTUAL ARITHMETIC. [68,
23. Paid $ for a cravat, which was f of what I
paid for a vest ; what was the cost of the vest ?
24. A gentleman bought a sleigh for $50, which
was -| of '6 times what he paid for a harness ; what
did the harness cost him ?
25. A lady bought a fur cape for $90, which was
l- of what she paid for 12 yards of silk ; what was
the cost of the silk a yard ?
26. A merchant sold 5 barrels of flour for $324,
which was f of what he received for all he had left,
at $4 a barrel ; how many barrels in all did lie sell ?
68. 1. | of 20 is how many thirds of 24 ?
ANALYSIS. | of 20 is as many thirds of 24 as \ of
24 is contained times in 4 times of 20. ^ ^ of 20 is 4,
and 4 times 4 is 16 ; 8 is contained in 16, 2 times.
Therefore, $ of 20 is | of 24.
2. -| of 56 is how many fourths of 28 ?
3. | of 64 is how many fifths of 60 ?
4. I of 48 is how many sevenths of 28 ?
5. | of 108 is how many fourths of 32 ?
6. ^ of 90 is how many twelfths of 84 2
7. TJL of 160 is how many thirds of 36 ?
8. ^ of 100 is how many fifteenths of 30 ?
9. | of lOf is how many sixths of 12 ?
10. I of 37f is how many tenths of 40 ?
11. f of 42| is how many ninths of 72 ?
12. - of 78| is how many thirds of 9 ?
13. f of 28^ i s how many halves of 38 ?
14. 11 O f 42 is how many fourths of 44 ?
15. ^ of 96 is how many fifths of 90 ?
lt>. Xi of 40 is how many ninths of 72 ?
69.] INTELLECTUAL ARITHMETIC. 10?
69. 1. of 24 is I of what number ?
ANALYSIS. of 24 & | o/*3 ^'raes i o/ 5 ft'me* J of
24 ; I of 24 ?',9 4, *5 times 4 i's 20 ; i o/ 20 *a 10, and 3
tfmes 10 & 30. Therefore | o/ 24 is f o/ 30.
2. |- of 56 is T 7 Q- of what number ?
3. _4_ O f 88 is ^ of what number ?
4. f of 36 is T 3 5 - of what number ?
5. | of 27 is f of what number ?
6. _7_ O f go is T 4 3- of what number?
7. f of 52 is | of what number ?
8. I of 81 is T 9 o of what number ?
9. f of 63 is I of what number ?
10. | of 100 is If of what number ?
11. T Y g of 96 is y\ of what number ?
12. | of f of 64 is | of what number?
13. | of |- of 75 is {f of what number ?
14. | of | of 56 is ^ of f of what number ?
15. 4 of * of 72 is i of 4 of what number ?
-J O
16. I- of f of 36 is -I of A of what number ?
25 . u O ^j
17. | of 35 is -| of how many times 7 ?
ANALYSIS. | o/* 35 is ^ of as many times 7 as 7 e^
contained times in 8 times \ of 3 times % of 35 ; ^. q/'
3o & 5, cm<7 3 i5/mes 5 s 15 ; 4 o/ 15 fs 3, a/^/8 times 3 ?'s
24 ; 7 is contained in 24 3 1 fo'mes. Therefore, f o/ 35
7.
18. -^ of 54 is | of how many times 5 ?
19. f of 24 is -^ of how many times 12 ?
20. f of 75 is T 9 of how many times 8 ?
21. i of 56 is ^ of how many times 20 ?
22. f of 108 is f of how many times 9 ?
108 INTELLECTUAL ARITHMETIC. [7O.
of 121 are f of how many times 7 ?
of 33 are | of how many times 6?
of 34 are f of how many times 14 ?
| of 30 are ff of how many times 20 ?
|- of ^ of 63 are ^ of | of how many times 11 ?
I of I of 50 are | of f of how many times 8 t
7O. 1. f of 56 are | of 3 times what number ? *
ANALYSIS. $ of 56 are - q/* 3 rime* o/ 9 riwe*
o/ 4 rimes 4 o/ 50 ; j o/ 56 is 8, cwd 4 times 8 t* 32 ;
o/ 32 a* 4, and 9 times 4 -w 36 ; ^ o/36 ^ 12. Tftere-
fore, 4 <^/ 5 6 are f o/ 3 rimes 12.
2. | of 45 are f of 6 times what number ?
3. | of 64 are | of 9 times what number ?
4. |- of 63 are -^ of 4 times what number ?
5. of 36 are ^ of 5 times what number ?
6. ^ 4 T of 110 arc f of 8 times what number ?
7. -j of 80 are % of 11 times what number ?
8. | of 72 are ^ of 5 times what number ?
9. | of 81 are || of 10 times what number ?
10. I of 21 are | of 8 times what number ?
11. 4 of 14 are i of 3 times what number ?
o I
12. | of | of 70 are ^ of 4 times what number ?
13. A man, being asked his age, answered that if
he were 3 times as old, \ of his age would be. 24
years ; how old was he ?
14. A gentleman divided his farm between his 2
sons, giving the elder 10 acres more than | of the
whole, and the younger the remainder, which was 22
acres more than \ of the whole ; how many acres in
the farm ?
* See 70 in the Appendix, page 170.
7O.] INTELLECTUAL ARITHMETIC. 109
15. A coal merchant sold | of what coal he had
on hand for $90, at the rate of $6 a ton ; how many
tons had he ?
16. Bought a piano for $300, and f of the cost of
the instrument was | of what I received of 9 young
ladies for its use one year; how much did each
young lady pay for its use ?
17. The tools in a shop are w^orth $80 ; f of this
is f of of 2 times the value of the stock ; what is
the v^lue of the stock ?
18. A peddler, after selling $200 worth of his stock
in trade, finds that | of the remainder is equal to |
of 3 times the amount sold ; what amount of stock
had he at first ?
19. A merchant, after selling from a cask of wine
15 gallons more than | of the whole, found that the
number of gallons left was just 3 times the number
of gallons sold ; how many gallons did the cask con-
tain at first ?
20. If a store be worth $1000, and -f of the value
of the store be equal to -f of 2 J times the value of the
goods it contains, what is the value of the goods ?
21. Two boys, comparing their money, one said
he had 50 cents ; the other said, " -$ of your money
is just -f% of 6 times my money ; " how many cents
had the latter boy ?
22. The distance from Boston to Albany is 200
miles, and |- of this distance is | times J the distance
from Albany to Niagara Falls, lacking 5 miles ; wha*
is the distance from Boston to Niagara Falls ?
23. The distance from Chicago to Alton is 268
miles, and -| of this distance plus 15 miles, is 4^ times
| of the distance from Alton to St. Louis ; what is
the distance from Chicago to St. Louis ?
24. 40 is | of f of of 8 times what number ?
110 INTELLECTUAL ARITHMETIC.
71. 1. | of 35 is I of how many thirds of 18 ? *
ANALYSIS. Of as many thirds of 18 as of 18 is
contained times in 9 times i of 4 times 4 of 35 ; 4 ^/'
35 s 5, and 4 fa'mes 5 is 20 ; of 20 s 4, and 9 fames 4
*'s 36 ; \ of 18 is 6, twd 6 /s contained in 36, 6 fa'mes*
Therefore, $ of 35 z's f o/ 6 tfwVds o/ 18.
2. | of 42 is * of how many fifths of 35 ?
3. | of 96 is | of how many eighths of 24 ?
4. | of 180 is of how many tenths of 50 ?
5. |- of 84 is | of how many sixths of 48 ?
6. -fa of 100 is -^Q of how many ninths of 45 ?
7. ^ of 45 is ^.j f now many halves of 48 ?
8. ^ T of 77 is - of how many twelfths of 72 ?
9. -\l of 30 is f of how many thirds of 21 ?
10. -& of 28 is 2% of how many fourths of 24 ?
11. | of 18 is -/ T of how many twelfths of 60 ?
12. I of 80 is of how many fourths of 40 ?
13. | of 42 is T 9 T of how many eighths of 72 ?
14. ^ of 39 is 2 3 ^ of how many ninths of 180 ?
15. f of | of 18 is 4 times what part of 35 ?
ANALYSIS.. Such a part o/* 35, as 35 as contained
times in \ of 5 times ^ of 6 times 4 of 18 ; \ of 18 is
6, and 6 times 6, zs 36 ; ^ of 36 ^s 4, and 5 ft'mes 4 ^s
20 ; | of 20 i's 5, and 35 es contained in 5, g 5 ^ or ^
times. Therefore | o/* | o/ 1 18 is 4 times %> or %, of 35.
16. | of | of 40 is 3 times what part of 24 ?
17. | of f of 49 is 2 times what part of 28 ?
18. -i- of T 9 ^- of 100 is 9 times what part of 25 ?
19. f of I of 48 is 7 times what part of 35 ?
20. $ of | of 60 is 7 times what part of 64 ?
21. f of | of 80 is 3 times what part of 96 ?
* Observe the " changes " in the Appendix, 71 9 page 171.
7.] INTELLECTUAL ARITHMETIC. Ill
72. 1. I of 36 is 4 of how many times f of 42 ?*
ANALYSIS. Of as many times f of 42 as 2 ^'raes |
o/* 42 ?s contained times in 9 ft'mes | of 4 w?es ^ q/
36 ; -J o/" 36 is 12, >?<# 4 ^'mes 12 is 48 ; j of 48 w 12,
??r? 9 times 12 *s 108 ; 4 of 42 ?'.<? 6, and 2 tfmes 6 ^s
12, and 12 ?*s contained in 108, 9 ta'mes. Therefore \
.of 36 /* | of 9 tfwie* f o/ 42.
2. I of 64 is I of how many times | of 15 ?
3. | of 27 is | of how many times f of 56 ?
4. - of 81 is | of how many times \ of 28 ?
5. f of 50 is ||- of how many times | of 60 ?
6. | of 36 is ^ of how many times \ of 27?
7. f of 32 is ^ of how many times ^ of 45 ?
8. T ^ of 150 is | of how many times f of 40 ?
9. | of 108 is -$ of how many times f of 16| ?
10. -fy of 65 is ^ of how many times ^ of ^ of 50 ?
11. | of 60 is -| of how many times f of 25 ?
12. | of 72 is of how many times -|- of | of 70 ?
13. of J of 84 is | of how many times ^ of f
of 96?
14. A of f of 90 is of f of how many times f
of | of 20 ?
15. ^ of |^ of 36 is ^ of -| of how many times |
of f of 120 ?
16. A man pays $400 rent, and | of this is just f
of -|- of twice his annual income ; what is his annual
income ?
17. A boy being asked his age said that 18 years was
2 years less than f times | of his age ; how old was he ?
18. The distance from Baltimore to Washington
is 39 miles; if 5 miles be subtracted from of this
For " changes " see Appendix, 72, page 178.
112 INTELLECTUAL ARITHMETIC. [72.
distance, the remainder will be f times \ the distance
from Philadelphia to Baltimore ; what is the distance
from Philadelphia to Washington ?
19. Mr. 13., who is 64 years of age, is 2f times as
old as his eldest son, and his eldest son is 2 times
as old as his youngest brother ; what is the difference
in the ages of the two brothers ?
20. Oscar earned a certain number of pennies ;
Rufus earned * as many minus 4, and David earned
| as many as Rufus, plus 7 ; Oscar earned 50 pen-
nies ; how many did Rufus and David earn ?
21. A man put his money into 4 packages ; in the
first he put f , in the second ^, in the third -J, and in
the fourth $24 ; how much money had he ?
22. A sold | of his sheep on Monday ; Tuesday
he bought f as many as he had sold ; he then had 60
sheep ; how many had he at first ?
23. Mrs. D.'s shawl cost $30 ; | of the cost of the
shawl was f of 6 times the cost of her dress ; what
was the cost of her dress ?
24. B and C engaging in play, B had f as much
money as C ; C gained $16, which was | of 1|- times
as much as he commenced with ; how much did he
commence with, and how much had B left ?
25. A farmer, being asked how many sheep he
had, replied that he had just sold 150, and that -| the
number he had sold was 6 times | of what he had
left ; how many had he left ?
26. Mr. S. gave J of 1-^ times his ready money for
a buggy, -| of what was left for a harness, and had
$12 remaining ; what did he pay for the buggy ?
27. A paid $4800 for his farm ; f of this is 4J
times f of 2 times what it cost to build the house ;
what was the cost of the house ?
28. 1 of f of 80 is I- of I of how many times |
of 4 of 100 ?
73.] INTELLECTUAL ARITHMETIC. 113
CHAPTER VII.
RATIO AND PROPORTION.
73. 1. What part of 4 is 1 ? of 4 is 3 ? *
ANALYSIS. Since 1 is % of 4, 3 must be 3 times part
of 4, or of 4. Therefore 3 t* f of 4.
2. What part of 3 is 1 ? of 3 is 2 ?
3. What part of 7 is 1 ? of 5 is 3 ?
4. What part of 8 is 2 ? of 9 is 5 ?
5. What part of 10 is 7 ? of 13 is 9 ?
6. What part of 8 is 9 ? of 12 is 7 ?
7. What part of 16 is 11 ? of 20 is 17 ?
8. What part of 40 is 9 ? What is the ratio ?
9. What part of 30 is 27 ? What is the ratio ?
10. What part of is f ? What is the ratio ?
ANALYSIS, f = |f , and |f ; the ratio of fg- to
|-| is the same as that of their numerators, 10 to 12 : what
vart of 12 is 10 ? -JJ I- Therefore f is { of fa and th*
ratio is .
11. What part of is ? of f is J ?
12. What part of | is f ? of ^ is ?
13. What part of f is f ? of T \ is f ?
14. What part of ^ is f ? of |- is f ?
15. What part of ^ is f ? What is the ratio ?
16. What part of 5 is f ? What is the ratio ?
1J. What part of 7 is f ? of 3^ is 2 ?
18. What part of 4| is f ? of 7^ is 3J ?
19. What part, of f is 2 ? of 2| is If ?
20. What part of -^ is 5 ? of f is 6 ?
* The quotient arising from dividing one number by another is called
the ratio. Thus, what part of 4 is 1 ? It is 1 divided by 4, equal to ^, the
ratio. What part of 4 is 3 ? It is 3 divided by 4, equal to |, the ratio.
114 INTELLECTUAL ARITHMETIC. [74.
74. 1. What part of 25 dollars is 5 dollars ? is
10 dollars ?
2. What part of 16 pounds is 4 pounds ? is 7
pounds ? is 12 pounds ?
8. What part of 1 bushel is 1 peck ? is 2 pecks
and 4 quarts ? is 6 quarts ?
4. What part of 1 pound is 12 shillings ?
5. What part of 1 dollar is 20 cents ? is 5 dimes ?
6. What part of 1 cwt. are 15 pounds ?
7. What part of 1 cwt. are 6 Ibs. ? 9 Ibs. ?
8. What part of 1 ton is 14 cwt. ? 15 cwt. ? 18 cwt. ?
9. What part of 3 pounds is 10 oz. ? is 1 Ib. 6 oz. ?
10. What part of 1 foot is 3 in. ? 10 in. ? 11 in. ?
11. What part of 2 yds. is 1 foot ? 1 foot 10 inches ?
12. What part of 1 mile is 3 furlongs ? 5 furlongs
20 rods ?
13. What part of 1 mile is 140 rods ? 160 rods ?
14. What part of 1 gallon is 1 quart ? 1 pint ?
15. What part of 1 hhd. is 5 gallons ? 21 gallons ?
16. What part of 1 quart is 1 pint ? 2 gills ?
17. What part of 1 hour is 15 minutes ? 40 min-
utes ?
18. What part of 1 day is 3 hours ? 8 hours ? 16
hours ?
19. What part of 3 weeks is 2 days ? 5 days ? 9
days?
20. What part of 1 year is 3 weeks ? 7 weeks ?
21. What part of 5 pounds 6 ounces is 3 pounds
10 ounces ?
22. What part of 3 hhd. is 1 hhd. 22 gallons ? ,
23. What part of 1 mile 3 fur. is 4 fur. 20 rods ?
24. What part of 3 pecks is 6 quarts 1 pint ?
25. Two pounds is what part of 1 cwt. ? of 2
cwt. ?
26. Eight inches is what part of a foot ? of 2 feet ?
of 1 yard ? of 4 yards ?
74.] INTELLECTUAL ARITHMETIC. 115
27. Three quarts is what part of a gallon ? of 5
gallons ? of 10 gallons ? of 1 hhd. ?
28. What part of 9 miles is | of 8 miles ?
29. AVhat part of 4 acres is | of 3 acres ?
SO. A man, having $15, gave $3 lor a hat, and
$4 1- for a pair of boots ; what part of his money did
he use, and what part had he left ?
31. A farmer, having 160 bushels of wheat, sold
-i of | of it, floured f of | of it, and kept the re-
mainder for seed ; w r hat part of the whole did he use
for each purpose ?
32. If I have sold f of f of 120 sheep, what part
of the whole flock have I left ?
33. One man can walk 4f miles while another
walks 3f miles ; what part of the distance that the
first walks does the second walk ?
34. A gentleman paid | of $39| for a coat, and
$9-| for a pair of pantaloons ; how many times as
much did he pay for the coat as for the pantaloons ?
35. Paid $7f for a barrel of flour, and 1| times
as much for a ton of hay ; what was the cost of the
liay ?
36. A man bought a horse for $225, and a car-f
riage for ^ of | of this sum ; w r hat part of $125 did'
the carriage cost ?
37. If a vessel sail 40 miles in 5 hours, what part
of 144 miles will she sail in 9 hours ? how many
miles ?
38. A man gave to his 3 sons 360 acres of western
land ; to the first he gave f of it, to the second | of
3|- times as much, and the remainder he gave to the
third ; what part of the whole did each receive ? how
many acres ?
39. A merchant gained on a hogshead of sugar
f of f 46f , which was | of 2 times what it cost him ;
what did it cost him ? what part of the cost was the
gain ?
116 INTELLECTUAL ARITHMETIC. [75.
75. 1. If G cords of wood cost $15, what will
4 cords cost ?
ANALYSIS. Four cords are , or of 6 cords. Ij
6 cords of wood cost $15, 4 cords will cost of $15 ;
o/ $15 is $5, awrf f o/ $15 is 2 times $5, or $10.
Therefore, fyc.
Or, Me JM/M/ may wse the form of analysis used in section
on page 41.
2. If 9 pounds of sugar cost 75 cents, what will
12 pounds cost '.'
3. If 6 men can cut 45 cords of wood in 3 days,
how many cords can 8 men cut in the same time ?
4. If 3* yards of cloth cost $10 J, how much will
7 yards cost ?
5. If 4| pounds of butter cost GG cents, what will
2| pounds cost ?
6. If a mechanic earn 18J shillings in 1 days,
how much can he earn in a week ?
7. If $7 1 will buy 3-J cords of wood, how many
cords can be bought for JlO-J ?
8. If barrel of flour cost $4 J, what will f of a
barrel cost ?
ANALYSIS. What part of is f ? = g, and f =. f ;
Me ra^'o o/* | to |- is the same as of 3 to 4, nz |-. 7%/?, smee
4 barrel of flour cost $4, q/* a barrel will cost ^ of $4 ;
| o/ f = % 6 - = $6. Therefore, $c.
9. If a man can run G miles in ^ of an hour, how
far can he run in 3^ hours ?
10. How many yards of carpeting | of a yard
wide are equal to 12 yards f of a yard wide ?
11. If a staff 3 feet long cast a shadow 5 feet in
length, what would be the length of the shadow cast
by a pole 13 feet long ?
12. If the sawing of f of a cord of wood cost 6J
shillings, what would the sawing of 3| cords cost ? "
7t5.] INTELLECTUAL ARITHMETIC. 117
13. John earns 9 cents as often as James earns 15
cents ; when John has earned 63 cents, how much
has James earned ?
14. Two persons start from different points and
travel towards each other ; the first one travels 7
miles while the other travels 5 miles, and when they
meet the first one has traveled 70 miles ; how far
apart were they when they started ?
15. If 4 men can build a wall 60 feet long in 6
days, in what time can 3 men build a wall 90 feet in
length ?
ANALYSIS. In $ of -- of 6 days, which are 12 days.
Or thus : if it will take 4 men 6 days to build a wall 60 feet
long, it iv ill take 1 man 4 times 6 days, or 24 days, and 3
men can do it in of 24, or 8 days ; to build a wall 1 foot
long will require ^ of 8, or ^ days, and to build a wall
90 feet long will require 90 times as many days, or 90 times
e 8 g-, equal to 12 days. Therefore, 8?c.
16. If 5 men can do a piece of work in 10 days,
how many men will do a piece of work 4 times as
large in 20 days ?
17. If a man perform a journey in 8 days, by
traveling 9 hours a day, how many days will he re-
quire to perform the same journey, if he travel 1 %
hours a day ?
18. If 5 horses eat 1|- tons of hay in 1 month, lio"W
many tons will 6 horses eat in 2|- months ?
19. How many days will it take 5 men tc- earn
$11|, if 3 men earn $f in \ of a day ?
20. If 1 horse eat 1^ bushels of oats in 3 days, in
how many days will 4 horses eat 49 bushels ?
21. If a barrel of flour will supply a family of 4
persons 5 weeks, how much will supply a family of
3 persons 2^ weeks ?
22. If 3 "furnaces consume 121 tons O f coa l in 7
days, how long will 1H tons last 5 furnaces ?
118 INTELLECTUAL ARITHMETIC. [76.
76. CHAPTER VIII.
PERCENTAGE.
THE term?, percentage and per cent., signify by the hun-
dred, or hundredth*, that is, a certain number of parts of each
one hundred parts, of whatever denomination. Thus, by 5
per cent, is meant f) cents of every 100 cents, $5 of every
$100, 5 bushels of every 100 bushels, &c. Therefore, 5
per cent, equals 5 hundredth?, ( T ff,) or -2\s- 8 per cent., 8
liundredths, ( T o?) r & of the given number or quantity.
1. A man, having $40, lost 3 per cent, of it ; how
many dollars did he lose ?
ANALYSTS. Since he lost 3 per cent., or T ^, on $1,
on $40 he lost 40 times T jfo, or fgg- of $1, equal to $1-J.
Therefore, $c.
2. What is 5 per cent, of $25 ? of $30 ? of $45 ?
3. What is 7 per cent, of $15 ? ol $40 ? of $200 ?
4. What is 6 per cent, of $18 ? of $37 ? of $54 ?
5. What is 3 per cent, of 50 bushels of corn ?
G. What is 10 per cent, of 160 tons of plaster ?
7. What is 25 per cent, of 96 gallons of wine ?
8. What is 12 per cent, of 33i pounds of sugar ?
0. What is 15 per cent, of 110 barrels of flour ?
10. What is 8 per cent, of 28 yards of cloth ?
11. What is f per cent, of $45 ?
ANALYSIS. Since 1 per cent of $45 is T 4 ^-, of 1 p*r
cent, is of T Vu, or T 3 <&, equal to $-&. Therefore, $c.
12. What is f per cent, of $28 ? of $44 ? of $56 ?
13. What is per cent, of $63 ? of $120 ?
14. What is f per cent, of 30 barrels of cider ?
15. What is 3J per cent of 70 sheep ? of 35 cows ?
16. What is 2^ per cent, of 34 hundred weight?
77".] INTELLECTUAL ARITHMETIC. 119
17. A man bought 75 barrels of apples, and on
opening them, found 8 per cent, of them spoiled ;
how many barrels did he lose ?
18. If I have $200 deposited in the bank, and
draw out 15 per cent, of it, how much remains ?
19. A farmer, having 176 sheep, sold 75 per cent.
f them, and kept the remainder ; how many did he
keep?
20. A young man, having $104, lost 12|- per cent,
of it in gambling ; how many dollars dicl he lose,
and what per cent, of the whole had he left ?
77. 1. What fraction of a number is 8 per cent,
of it?
ANALYSIS. Since 1 per cent, is T <jg-, 8 per cent, is 8
times T ^, or T JJ, /% Therefore 8 per cent, equals /$
of any number or thing.
2. What fraction of a number is 6 per cent. ? 7
per cent. ? 10 per cent. ? 12|- per cent. ?
3. What fraction of a number is 6-J- per cent. ?
16| per cent. ? 20 per cent. ? 37| per cent. ? 40 per
cent. ?
4. If you have 8120, and lose 33^ per cent, of it,
what part of it will you lose ?
5. James, having 250 marbles, lost 40 per cent, of
them at play ; what part of the whole had he left,
and how many did he lose ?
6. What per cent, of a barrel of flour is -f of it ?
ANALYSIS. Since the whole of any number or thing is
equal to 100 per cent., f of one barrel of flour is equal to
f of 100 per cent., or 40 per cent. Therefore, fye.
7. If a man save -| of his income, what per cent.
does he spend ?
8. If a merchant invest f of his money in dry
goods, what per cent, does he invest ?
120 INTELLECTUAL ARITHMETIC. [78.
78. 1. A farmer, having 20 sheep, lost 5 of them
by disease ; what per cent, did he lose ?
ANALYSIS. Since 20 sheep are 100 per cent., or all that
he had, 5 sheep are $, or , of all he had ; and of 100
per cent, is 25 per cent. Therefore, fyc.
2. What per cent, of 21 is 7 ?
3. What per cent, of 50 is 5 ? 10 ? 20 ? 25 ?
4. What per cent, of $40 is $5 ? $8 ? $25 ? $30 ?
5. What per cent, of 18 pounds is 3 pounds ? 12
pounds ?
6. What per cent, of 42 miles is 6 miles ? 7 miles ?
7. What per cent, of 12* is 2*- ? 6-J ?
8. What per cent, of 30 is 12^? 15 ? 20 ?
9. What per cent, of 18 is 9 ? 12 ? 15 ?
10. 1J is what per cent, of 9 ? of 15 ? of 21 ?
11. 2|- is what per cent, of 7 ? of 12* ? of 10 ?
12. -J is what per cent, of 3j"? of 16| "? of 20 ?
13. * is what per cent, of 8 ? of 12 ? of 20 ?
14. If from a hogshead of molasses a grocer draw
42 gallons, what per cent, of the whole will remain ?
15. From a box of tea containing 60 pounds, 15
poimds were sold at one time, and 25 pounds at
another ; what per cent, of the wholo remained un-
sold ?
16. In a heap of potatoes containing 150 bushels,
3 bushels of every 5 are bad ; what per cent, are
bad, and how many bushels are good ?
17. A gambler, having $250, lost $80 at play;
what per cent, of his money did he lose, and what
per cent, had he left ?
18. James, having 62 cents, gave 37-J- for a book ;
what per cent, of his money had he left ?
19. A grocer, having a barrel of sugar containing
200 pounds, sold ^ of it at one time, and ^ of the
remainder at another time ; what per cent, remained
unsold ?
79.] INTELLECTUAL ARITHMETIC. 121
79. 1. A person collects $240, and receives
per cent, commission on what he collects ; what com>
pensation does he receive ?
ANALYSIS. Since the commission is 5 per cent., or y^^
equal to $, of the sum collected, he receives ^ of $240,
which is $12. Therefore, fyc.
2. What will be the expense of collecting a note
of $800, at 10 per cent, commission ?
3. What must be paid for collecting a tax of $300,
allowing 12| per cent, commission ?
4. An agent sells $1200 worth of merchandise, at
25 per cent, commission ; how much will he receive
for his services ?
5. An auctioneer sold goods to the amount of
$500 ; what will his commission amount to, at 2^-
per cent. ?
6. Paid an attorney 3|- per cent, to collect a debt
of $640 ; how much did he receive for his services ?
7. If a man fail in business, and can pay but 40
per cent, of his debts, how much will a creditor
receive on a debt of $175 ?
8. Bought $1000 worth of books, receiving a com-
mission of 3| per cent. ; how much did I receive for
my services ?
9. If a broker sell $600 worth of New York Cen-
tral Railroad stock, and charge | per cent, for selL
ing, how much will he receive ?
10. At | of 1 per cent., what will be the expense
of negotiating a bill of exchange of $625 ?
11. A broker in Chicago exchanged $1200 on the
Chemical Bank of New York,* at f per cent. ; what
amount of brokerage did he receive ?
12. A real estate agent sells a house and lot for
$2000, and charges 1J per cent, commission for sell-
ing ; how much does he receive ?
122 INTELLECTUAL ARITHMETIC. [8O.
8O. 1. A merchant sends his agent $780 to ex-
pend for merchandise, after deducting his commis-
sion of 4 per cent. ; what sum will he expend ?
ANALYSIS. The money to be expended is -f gg of itself,
and the commission T Jjj, or ^, of this sum; hence the
whole amount sent to the agent is |JJ $, or ff. Since $780
* 5 M' 2T ^ 2*5 / & 7 8^ or $^> ^ commission; and f g- /s
25 feme* 30, or $750, ^e sz/w to fo expended.
2. I gave a broker $810 to invest in bank stock,
after deducting his commission of 1| per cent. ; what
was his commission, and how much did he invest ?
3. A distiller sent Mr. B. $1550, with which to buy
corn, after deducting his commission of 3^ per cent. ;
how much did he expend, and how much did his
commission amount to ?
4. How many bushels of wheat, at $1 a bushel,
can an agent buy for $1230, and retain 2| per cent,
commission for his trouble ?
5. A farmer employed a man to thresh his wheat,
agreeing to give him 12 per cent, of all he threshed ;
how many bushels must he thresh that the farmer
niay retain 68 bushels?
ANALYSIS. Since the thresher received T \j 2 jy, or -/$ of all
he threshed, the farmer received the difference between -{%$,
or J| and 2 \, which is f f . Then 66 bushels is f of the
whole number of bushels threshed ; since 66 bushels is ,
2*5 * 5 sV f 66, or 3 bushels, and f | is 25 times 3 bushels,
or 75 bushels. Therefore, fyc.
6. What amount must be collected on a rate bill,
that the collector may retain his fee of 5 per cent.,
and pay over $228 ? *
7. What amount of accounts must an individual
collect in order to pay over $1100, and retain 8^ per
cent, for collecting ?
81.] INTELLECTUAL ARITHMETIC. 123
81. 1. What must be paid for insuring a house
and furniture for $1000, at 1J per cent, premium V *
ANALYSIS. Since the premium is 1^ per cent., or ^^,
equal to -- ff of the amount insured, the premium on $1000
will be J s of $1000, or $12. Therefore, $c.
2. What will be the premium for insuring 200
barrels of flour, valued at $1200, at 4 per cent. ?
3. What must I pay annually for an insurance of
$900 upon my life, at 2| per cent. ?
4. What must be paid for insuring a case of cloths,
valued at $600, at 1| per cent, premium ?
5. What is the annual premium of insurance, at
| per cent., 011 a building valued at $4000 ?
6. If a merchant has his stock of goods insured
for $2500, at | of 1 per cent., what is the premium ?
7. A man owns | of a boat load of grain, valued
at $1600, and insures his interest at 1| per cent ;
what premium does he pay ?
8. At 2 per cent., what amount of insurance can
I obtain for $18 premium ?
ANALYSIS. Since 2 per cent, premium is T -<j, or ^ of
the amount insured, $18, the given premium, will be equal
to -i$ of the required amount of insurance ; $18 is -$ (>f 50
times $18, equal to $900. Therefore, $c.
9. At 3 per cent., what amount of insurance can
be obtained on a house for $75 ?
10. At -| per cent., what amount of insurance can
be obtained on a boat load of flour, for $24 ?
11. What amount of insurance can be had for
$45, upon a car load of horses, at 44- per cent. ?
12. If the rate is If per cent., and the premium
paid is $91, what will be the amount of insurance
obtained on a store and its contents ?
* The sum paid for insurance is called the premium.
124 INTELLECTUAL ARITHMETIC.
82. 1. If a man buy a horse for $90, for how
much must he sell him to gain 20 per cent. ?
ANALYSIS. Since he gains 20 per cent., or -^ (7 , equal to
\ of what the horse cost him, he must sell him for & of what
he cost ; of S'JO is $18, and g- <//' $ ( JO is 6 tofeg $18, or
$108. Therefore, $c.
2. A sleigh was bought for $f;0, and sold for 8 per
cent, more than it cost ; what was it sold for ?
3. Bought a quantity of sugar for $84, and sold it
at an advance of 12 per cent ; what was it sold for?
4. A grocer bought some tea for $45, but finding
it damaged, lie was obliged to sell it at a loss of 10
per cent. ; what did he sell it for ?
5. Paid -SI 2 for 15 yards of cloth; what must it
be sold for a yard to gain 25 per cent. ?
G. A drover bought 60 sheep for $180, and sold
them at a loss of 20 per cent, on the cost ; what
did he receive per head for them ?
7. A grocer bought 440 lemons for $12, and after
throwing away 10 per cent, of them, lacking 4, as
worthless, he sold the remainder so as to gain 88A
per cent, on the cost of the whole ; what did he sell
them for apiece ?
8. Bought 6 barrels of flour for $40 ; for what
must it be sold a barrel to gain 5 per cent, on the
cost ?
9. If I buy eggs at 10 cents a dozen, how must
I sell them to gain 30 per cent. ?
10. A speculator bought a quantity of pork for
$500, and sold it at a loss of 7 per cent. ; with the
proceeds he bought another quantity, upon which he
gained 20 per cent. ; what was his gain on the whole ?
11. Bought hay at $14 dollars a ton, which was
12| per cent, less than the market price ; what was
the market price ?
83.] INTELLECTUAL ARITHMETIC. 125
83. 1. What is gained per cent, by selling clover
seed for $6, that cost $4 ?
ANALYSIS. The whole gain is equal to the difference
between $6 and $4, which is $2. Since $4 gain $2, or
of itself, equal to f^, the gain will be 50 per cent. There-
fore, fyc.
2. What per cent, is gained by selling pork for
$7 a hundred weight, that cost $6 ?
3. What will be gained by expending $500 for
sugar, when it can be bought for 8 cents, and sold
for 8|- cents a pound ?
4. "if I buy $300 worth of sheep, and sell them
for $375, what do I gain per cent. ?
5. Mr. C. bought a horse for $153, and sold him
for $128 ; what per cent, did he lose ?
6. A grocer bought eggs for 9 cents a dozen, and
sold them for 12 cents ; what per cent, did he gain ?
7. What per cent, is lost in buying corn at 50
cents a bushel, and selling it at 45 cents ?
8. If a house bring J of its value every 4 years in
rent, what is the gain per cent, each year ?
9. Bought a horse for 20 per cent, less than $150,
and sold him for 10 per cent more than $150 ; what
per cent, was gained ?
10. A peddler bought oranges for 18 cents a dozen,
and sold them for 2 cents each ; what per cent, did
he gain ?
11. A farmer refused to sell his barley for 75
cents a bushel, and was afterwards obliged to take
60 cents a bushel for it ; what per cent, did he lose ?
12. Bought cotton cloth at 1 cents a yard, and
sold it for 9 cents a yard ; what was the gain per
cent. ?
13. Sold damaged cloth so as to lose J of what it
cost ; what per cent, was lost ?
126 INTELLECTUAL ARITHMETIC. [84.
84. 1. A tailor sold a coat for $24, by which he
gained 20 per cent, on the first cost ; what was its
cost?
ANALYSIS. Since he gained 20 per cent., or -J of the
cost, $24 is f of the cost ; since $24 is f , of $24, or $4,
is $ f the cost, and $4 is | of 5 times $4, or $20. There-
fore, $c.
2. A hatter sold a hat for $3, which was
per cent, above cost ; what did it cost ?
3. A jeweler sold a watch for $65, thereby gain-
ing 30 per cent, on its cost ; what was its cost ?
4. A mechanic sold a carriage for $108, which
was 10 per cent, less than it cost ; how much would
he have received for it had he sold it so as to gain
20 per cent. ?
5. If 16| per cent, is gained by selling flour at
$7 a barrel, what did it cost ?
6. A merchant hold 2 barrels of sugar for $24
each ; on one barrel he gained 20 per cent, and on
the other he lost 20 per cent. ; did he gain or lose
by the bargain, and how much ?
7. A grocer bought coffee so that he could sell it
for 18 cents a pound, and make a profit of 33 J per
cent. ; what did it cost him ?
8. A man bought a horse and carriage ; he sold
them both at 8 per cent, above cost, receiving $81
for the horse, and $108 for the carriage ; what was
the cost of both ?
9. A fruit dealer sold pine-apples at 96 cents a
dozen, thereby losing 4 per cent. ; what did they cost
apiece ?
10. B sells a horse to C, and gains 12 J per cent. ;
C sells him to D for $118, and thereby gains 18 per
cent. ; how much did the horse cost B ?
85.] INTELLECTUAL ARITHMETIC. 121
85. 1. Sold wheat so as to gain 16 cents on fv
bushel, which was 12 J per cent, of what ?+, cost ; what
did it cost ?
2. A broker exchanged $150 at J per cent. ; how
much brokerage did he receive ?
3. A man, having a farm of 200 acres, sold 75
acres of it ; what per cent, of the whole remains
unsold ?
4. What per cent, does William gain by selling
apples at -| of the cost ?
5. If I buy stoves at $12 each, and sell them at
8J per cent, profit, what will I gain on 1 stove ?
6. If a broker buy $800 worth of bank stock for
me, and charge ^ per cent, brokerage, how much
will he receive ?
7. Sold butter so as to gain |- as much as it cost ;
what per cent, was gained ?
8. Mr. A. lends his money so as to receive a sum
equal to of it for its use ; what per cent, does he
receive ?
9. A brewer sends a person $520 with which to
purchase barley, after deducting his commission of 4
per cent. ; how much does he expend, and how much
does his commission amount to ?
10. The cost of building a town house, and paying-
collector's fees of 5 per cent., was $945 ; what was the
cost of the house ?
11. Hiram agrees to dig farmer F.'s potatoes for
8 per cent, of what should be put into the farmer^
cellar ; he digs 324 bushels ; how many bushels
should be put into the cellar, and how many bushels
should Hiram receive ?
12. Bought $200 worth of dry goods, and $300
worth of groceries ; on the dry goods I lost 20 per
cent., but on the groceries I gained 15 per cent. ; did
I gain or lose on my whole capital, and how much ?
128 INTELLECTUAL ARITHMETIC. [S3.
13. A merchant bought 40 bushels of grass seed,
at the rate of 4 bushels for $ 6, and sold it at the
rate of 5 bushels for $8 ; what was his whole gain,
and what his gain per cent. ?
14. A drover bought 20 sheep for $50, and sold
them so as to gain 20 per cent. ; how much did he
receive a head for them ?
15. A man bought 60 cords of wood for $200,
and sold it at a loss of 5 per cent. ; at what price
did he sell it per cord '.'
16. If I buy rice at 4 cents a pound, and sell it
at 5 cents, what is my gain per cent. ?
17. Sold a cow at auction for $18 that cost $20 ;
what was the loss per cent. ?
18. A boy bought peaches at the rate of 3 for 2
cents, and sold them at the rate of 9 for 7 cents;
what was his gain per cent. ?
19. A tailor sold two coats for $26 each ; on one
he gained 30 per cent., and on the other he lost
80 per cent.; did he. gain or lose on both, and how
much ?
20. A music dealer sold a piano-forte for $230,
which was 8 per cent, less than its value ; had he
<?old it for $300, what would have been his gain per
uent. ?
21. Two men buy a lot of pork together ; A fur-
nishes $200, and B $150 ; their profits are $70 ; what
is their gain per cent., and what is each one's share
of the gain ?
22. If a ton of hay is bought for -f of the market
price, and sold for 5 per cent, more than the market
price, what per cent, is gained ?
23. A person engaged in speculation sold out his
interest for $2100, which was 30 per cent, less than
twice as much money as he began with ; how much
did he begin with, and what was his gain?
86.] f INTELLECTUAL ARITHMETIC. 129
86. INTEREST is a sum paid for the use of borrowed
money, or on demands after they become due.
The PRINCIPAL is the money lent, or the sum for which
interest is paid.
The AMOUNT is the sum of the principal and interest.
The RATE PER CENT, is the sum paid for the use of
$100 or 100 cents for one year. Thus, if $6 be paid for
the use of $100 one year, the rate is 6 per cent, per
annum.
The interest on any sum for one year is always a certain
number of hundredths of the principal. Thus, the interest
of any sum for one year, at 5 per cent., is 5 hundredths
Ortfo) or 2\r of the principal ; at 8 per cent, it is 8 hun-
dredths ( T |hj) or ITS f tne principal.
Practically, 30 days are considered a month, and 12
months a year, in computing interest. When no time is
named, one year is always understood.
The rate of interest is established by law; and taking
more than such rates is termed usury.
1. What is the yearly interest of 880, at 5 per cent. ?
ANALYSIS. Since the interest is 5 per cent., or y^, equal
~* 5TT f th & principal, the interest of $80 is ^V of $80,
or $4. Therefore, $c.
Or thus : Since the interest of $1, or 100 cents, is 5 cents
for one year, the interest of $80 is 80 times 5 cents, or 400
cents, equal to $4.*
2. What is the yearly interest of $200, at 6 per
cent. ?
3. What is the yearly interest of $125, at 4 per
cent. ?
4. What is the annual interest of $60, at 7 per
cent. ?
5. What is the annual interest of $95, at 5 per
cent. ?
* The fractional method is considered preferable in mental exercises.
9
130 INTELLECTUAL ARITHMETIC. ' [87.
6. What is the animal interest of $26, at 10 per
cent. ?
7. What is the annual interest of 8228, at
per cent. ?
8. What is the annual interest of $96.32, at 6
per cent. ?
9. What is the annual interest of $144.60, at 8
per cent. ?
10. What is the annual interest of $400, at 15
per cent. ?
11. What is the annual interest of $500, at 7
per cent. ?
12. What is the annual interest of $120.80, at 7
per cent. ?
13. What is the annual interest of $1000, at 9
per cent. ?
14. What is the annual interest of $150, at 3 per
cent. ?
87. 1. What is the interest of $150 for 3 years,
at 8 per cent. ?
ANALYSIS. Since the interest for 1 year at 8 per cent,
is T f tf, equal to /$, of the principal, for 3 years it is 3
times /-%, or ^, of the principal. ^ of $150 is $6, and
2 \ of $150 is 6 times $6, or $36. Therefore, fyc.
Or thus : Since the interest of $150 for 1 year at 8 per
cent, is $12, (86,) for 3 years it is 3 times $12, or $36.
2. What is the interest of f 30 for 2 years, at 5
per cent. ?
3. What is the interest of $50 for 4 years, at 6
per cent. ?
4. What is the interest of $90 for 5 years, at 7
per cent. ?
5. What is the interest of $75 for 3 years, at 12
per cent. ?
88.] INTELLECTUAL ARITHMETIC. 131
6. What is the interest of $64.96 for 2 years, at
6J per cent. ?
7. A's note of $25 has been due 2 years ; what is
the interest, at 6 per cent. ?
8. Borrowed $300 for 3 years, at 7 per cent, ;
what will be the interest ?
9. What is the interest on a note of $46.50, for 5
years, at 10 per cent. ?
10. I owe two notes, one for $145, due in 2 years,
at 5 per cent., and the other for $200, due in 3 years,
at 6 per cent. ; what will be the sum of the interest
that I must pay ?
11. What is the interest of $225 for 2 years, at 7
per cent. ?
12. What is the interest of $50 for 8 years, at 6
per cent. ?
13. What is the interest of $400 for 2 years, at
5| per cent. ?
"14. Gave a note for $88.96, due in 3 years, at 6|
per cent ; what will be the interest ?
15. Gave a note due at date, for $45.50, but did
not pay it until the end of 5 years ; what was the
interest due, at 8 per cent. ?
88. 1. At 8 per cent., what part of $1 equals
the use of it for 3 years 9 months ?
ANALYSIS. Since there are 12 months in 1 year, 9
months is T 9 5 , or of 1 year, and 3 years 9 months are equal
to 3f, or i-f- years. Since the interest of $1 for 1 year at
8 per cent, is yg^, equnl to /% of the principal, for 3 years
9 months, it is 3$ times, or Ig., of ^, equal to -fa of the
principal, or $ T 3 I7 . Therefore, $c.
2. At 6 per cent., what part of $1 equals the use
of it for 2 years 8 months ?
132 INTELLECTUAL ARITHMETIC. [88.
3. At 5 per cent., what part of any sum equals
the use of it for 2 years 6 months ?
4. At 7 per cent., what part of any sum equals
the interest of it for 1 year 8 months ?
5. What part of any sum equals the interest of it
for 3 years 2 months, at 4 per c< i nt. '.'
6. What part of any sum equals the interest of it
for 5 years 4 months, at 9 per cent. ?
7. At 6J per cent., for 3 years 4 months, what
part of the principal equals the interest ?
8. At 8 per cent., for 5 years 5 months, what part
of the principal equals the interest ?
9. At 7i per cent., for 2 years 11 months, what
part of the principal equals the interest ?
10. At 6 per cent., for 2 years 1 month, what part
of the principal equals the interest ?
11. At 10 per cent., for 4 years 10 months, what
part of the principal equals the interest ?
12. At 8 per cent., for 3 years 3 months and 18
days, what part of the principal equals the interest ?
ANALYSIS. Since 30 days make a month, 18 days are
.18.^ or of a month ; and since 12 months make a year,
I month is T V of a year ; of 1 month is -i- of r ^,
or ^ of I year, and 3f months, equal to -*-, is 18 times -^,
or ^ of a year. Hence the interest of $l/or 3 T 3 ^, or f.g
years, at 8 per cent., is T ^\ of it. (88 5 Ex. 1.) There-
fore, fyc.
13. What part of the principal equals the inter-
est, for 2 years 5 months and 10 days, at 6 per cent. ?
14. What part of the principal equals the interest,
for 1 year 8 months and 25 days, at 7 per cent. ?
15. What part of the principal equals the interest,
for 2 years 3 months and 6 days, at 6 per cent. ?
16. What part of the principal equals the interest,
for 2 years 11 months and 6 days, at 5 per cent. ?
89.] INTELLECTUAL ARITHMETIC. 133
17. At 7 per cent., for 8 months 10 days, what
part of the principal will equal the interest ?
18. At 6 per cent., for 5 years 7 months and 15
days, what part of the principal will equal the
interest ?
19. At 8-J- per cent., for 3 years 5 months and
12 days, what part of the principal equals the in-
terest ?
20. At 7 per cent,, for 11 months 3 days, what
part of the principal will equal the interest ?
21. At 4 per cent., for 8 years 6 months and 15
days, what part of the principal will equal the in-
terest ?
89. 1. What is the interest of $40 for 3 years
4 months, at 6 per cent. ?
ANALYSIS. 3 years 4 months equals 3, or *g- years;
and the interest of any sum for *g- years, at 6 per cent.,
equals % of it. (88.) | of $40 is $8, the required in-
terest. Therefore, fyc.
Or thus : Since the interest of $1 for 1 year, at 6 per
cent., is 6 cents, for $40 it is 40 times 6 cents, or $2.40, and
for 3 years it is 3 J times $2.40, equal to $8.
2. What is the interest of $150 for 2 years 10
months, at 8 per cent. ?
3. What is the interest of $80.40 for 4 years 2
months, at 7 per cent. ?
4. What is the interest of $240 for 1 year 9
months, at 5 per cent. ?
5. What is the interest of $120 for 5 years 8
months, at 7-|- per cent. ?
6. What is the interest of $90 for 4 years 7
months, at 4 per cent. ?
7. What is the interest of $500 for 10 months, at
6 per cent. ?
134 INTELLECTUAL ARITHMETIC. [89.
8. What is the interest of $128 for 2 years 9
months, at 6^ per cent. ?
9. What is the interest of $125 for 1 year 4
months, at 7 per cent. ?
10. What is the interest of $300 for 8 months, at
6 per cent. ?
11. What is the interest of $50 for 5 years 7
months and 15 days, at 8 per cent. ?
ANALYSIS. The interest of any sum for 5 years 7 months
and 15 days, at 8 per cent., equals 2 9 o of **> (88 ? Ex. 12 ;)
of $50 is $2, and ^ of $50 is 9 times $2, equal to
Therefore, $c.
12. What is the interest of $75.60 for 4 years 5
months and 10 days, at 9 per cent. ?
13. What is the interest of $50 for 6 years 4
months and 24 days, at 5 per cent. ?
14. What is the interest of $300 for 4 years 7
months and 15 days, at 8 per cent. ?
15. What is the interest of $288 for 8 months 10
days, at 7- per cent. ?
16. What is the interest of $1200 for 20 days, at
6 per cent. ?
17. What is the interest of $40 for 9 months 5
days, at 4^- per cent. ?
18. What is the interest of $36.72 for 5 years 6
months and 20 days, at 7 per cent. ?
19. What is the interest of $120 for 2 years 1
month and 6 days, at 12 J per cent. ?
20. What is the interest of $24 for 11 months and
20 days, at 15 per cent. ?
21. What is the interest of $2000 for 18 days, at
5 per cent. ?
22. What is the interest of $1500 for 2 months
and 24 days, at 6 per cent. ?
9Oo] INTELLECTUAL AEITHMETIC. 135
90. 1. What will be the amount of $60 for 2
years 6 months, at 6 per cent. ?
ANALYSIS. Since the interest is equal to $ of the prin-
cipal, (88 5 ) and the amount is equal to the principal and
interest, it follows that f g-, the principal, added to -$, the
interest, equals f , the amount. -^ of $60 is $3, and f $
of $60 is 23 times $3, or $69, the amount. Therefore, fyc.
NOTE, "When more convenient, the interest may first be found, to
which add the principal, and the result will be the amount.
2. What is the amount of $36 for 4 years 2
months, at 7 per cent. ?
3. What will $120 amount to in 2 years 11
months, at 8 per cent. ?
4. What will $96.48 amount to in 3 years 4
months, at 6^ per cent. ?
5. What will $80.50 amount to in 6 years 8
months, at 4-J- per cent. ?
6. What will $20.25 amount to in 12 years, at 5
per cent. ?
7. What will $1500 amount to in 10 months, at
10 per cent. ?
8. What will $150 amount to in 6 months and 20
days, at 6 per cent. ?
9. What will $250 amount to in 9 months and 18
days, at 7 J per cent. ?
10. What is the amount of $480 for 1 year 10
months and 25 days, at 5 per cent. ?
11. What is the amount of $500 for 3 years 4
months and 24 days, at 6 per cent. ?
12. What is the amount of $240 for 2 years 7
months and 6 days, at 7 per cent. ?
13. What is the amount of $1000 for 5 years 1
month and 6 days, at 4 per cent. ?
14. What is the amount of $120 for 10 years 6
months and 20 days, at 3 per cent, ?
136 INTELLECTUAL ARITHMETIC. [91.
91. 1. What principal in 3 years 9 months, at
8 per cent., will give $90 interest ?
ANALYSIS. Since the interest for 3 years 9 months, at 8
per cent., is ^ of the principal, (88, j $90 must be -fa of
the required principal Since $90 is T V yV is $ of $90, or
$30, and $30 is -^ of 10 times $30, or $300, the principal.
Therefore, $c.
2. What principal, in 1 year 8 months, at 9 per
cent., will give $8t> interest?
3. What principal in 11 months, at 6 per cent.,
will give $22 interest ?
4. What principal in 7 months 15 days, at 8 per
cent., will give $40 interest?
5. A rents a store to B for the yearly rent of $300,
which is 12 per cent, of its cost ; what did it cost ?
6. A gentleman bought 6 horses, for which he paid
a sum of money which in 2 years 6 months, at 4 per
cent., would have given him $90 interest; what did
he pay apiece for the horses ?
7. A person deposits | of his money in the bank,
which brings him an animal income of $1500, at 10
per cent. ; the remaining | he invests in business
which yields him 20 per cent, profit ; what are his
entire capital and his yearly income ?
8. Two persons engaged in trade ; A furnished 4
of the capital, and B f ; at the end of 3 years and 4
months they found that they had made a clear profit
of $5000, which is 121 per cent, per annum on the
money invested ; how much capital did each furnish ?
9. A capitalist invested -f of his money in railroad
stock, which depreciated 5 per cent, in value ; the
remaining -f he invested in bank stock, which at the
end of 1 year had gained $1200, which was 12 per
cent, of the investment ; what was the whole amount
of his capital, and what was his entire loss or gain ?
92.] INTELLECTUAL ARITHMETIC. 137
92. 1. What principal will, in 3 years 4 months,
at 5 per cent, amount to $700.
ANALYSIS. Since the interest for 3 years 4 months, at 5
per cent, is ^ of the principal, (88,) the principal |, added
to the interest , equals the amount % ; hence $700 is % of
the principal Since $700 is , % is | of $700, or $100,
and $100 w o/ 6 times $100, or $600. * Therefore, $c.
2. What principal in 5 years, at 6 per cent., will
amount to $1300 ?
3. What principal in 1 year 6 months, at 8 per
cent., will amount to $280 ?
4. What principal in 4 years 2 months, at 3 per
cent., will amount to $630 ?
5. What principal in 10 years, at 7 per cent., will
amount to $340 ?
6. What principal in 9 months, at 10 per cent.,
will amount to $172 ?
7. What principal. in 18 months, at 4 per cent.,
will amount to $530 ?
8. The amount due on a bond and mortgage is
$2900, it having been on interest 7 years 6 months,
at 6 per cent. ; what was the principal ?
9. Two men bought 200 acres of land, and at the
end of 2 years sold it for $1500, which was an ad-
vance of 12|- per cent, per annum on its cost ; what
did it cost an acre ?
10. A man invested f of all he was worth in the
lumber trade, and at the end of 2 years 8 months
sold out his entire interest for $3100, which was a
yearly gain of 9 per cent, on the money invested ; how
much was he worth when he commenced trade ?
11. A's money is to B's as 2 to 3 ; if of A's
money be put at interest for 3 years 9 months, at
10 per cent., it will amount to $550 ; how much
money has each ?
138 INTELLECTUAL ARITHMETIC. [93.
93. 1. In what time, at 5 per cent., will $60
gain $15 interest ?
ANALYSIS. The gain, or interest, is , or % of the prin-
cipal, equal to 25 per cent, of the principal. (77 ? Ex. 7.)
Jf a given principal gain 5 per cent, in 1 year, it will re-
quire as many years to gain 25 per cent, as 5 per cent, is
contained times in 25 per cent., or 5 years. Therefore, c.
Or thus : Since $60 in 1 year, at 5 per cent., will gain
$3 interest, it will require as many years for $GO to gain
$15 interest, as $3 is contained times in $15, which is 5
times.
2. How long will it take $120 to gain $24 interest,
at 6 per cent. ?
3. In what time will $500 gain $35, at 7 per
cent. ?
4. In what time will $120 gain $40, at 10 pe*
cent. ?
5. In what time will $80 gain $6, at 3 per cent. ?
6. In what time will $300 gain $84, at 7 per
cent, ?
7. In what time will $160 gain $20, at 6 per
cent. ?
8. In what time will $49 gain $12.25, at 5 per
cent. ?
9. In what time will any given principal double
itself at 4 per cent. ?
ANALYSIS. Since, to double itself, any sum of money
must gain 100 per cent., at 1 per cent., it will require 100
years, at 4 per cent., ^ of 100 years, or 25 years. There-
fore, fyc.
10. In what time will a given principal double it-
self at 5 per cent. ?
11. In what time will a given principal double it-
self at 7 per cent, ?
94.] INTELLECTUAL ARITHMETIC. 139
12. In what time will a given principal double it-
self at 6 per cent. ?
13. In what time will a given principal double it-
self at 10 per cent. ?
14. -In what time will $500 double itself, at 12J
per cent. ?
15. In what time will $175 double itself at 20 per
cent. ?
16. In what time will a given principal treble it-
self at 5 per cent. ?
ANALYSIS. A given principal will treble itself at 1 per
cent, in 200 years ; at 5 per cent, in \ of 200 years, or 40
years.
17.* In what time will $100 treble itself at 6 per
cent. ? at 7 per cent. ? at 8 per cent. ? at 10 per
cent. ?
94. 1. A man loaned $75 for 4 years, and re-
ceived $18 interest ; what was the rate per cent. ?
ANALYSIS. Since the gain, or interest, for 4 years, is
|, or -/-$ of the principal, for 1 year it is J- of -$, or T {fo
of the principal, equal to G per cent. Therefore, fyc.
Or, thus : At 1 per cent., the interest of $1 for 4 years is
4 cents, and of $75 it is 75 times 4 cents, which are 300
cents, equal to $3. Since $3 interest is 1 per cent, of $75
for 4 years, $18 interest is as many per cent, as $3 is
contained times in $18, which is 6 times, equal to 6 per
cent.
2. At what per cent, will $200 gain $49 interest
in 3J years ?
3. At what per cent, will $60 gain $25 interest in
4^ years ?
4. At what per cent, will $90 gain $27 in 6
years ?
140 INTELLECTUAL ARITHMETIC. [94.
5. At what per cent, will $12 gain $5 in Ifr
years ?
6. At what per cent, will $400 gain $55 interest
in 2 years ?
7. At what per cent, will $150 gain $44 interest
in 8| years ?
8. At wiiat per cent, will $125 gain $37.50 interest
in 5 years ?
9. At what per cent, will $800 gain $75 interest
in 1 year and 3 months ?
10. At what per cent, will $180 gain $45 interest
in 8 years and 9 months ?
11. At what per cent, will $1250 gain $312.50
interest in 2 years 6 months ?
12. At what per cent, will $2500 gain $200 in-
terest in 1 year and 4 months ?
13. At what per cent, will a given principal double
itself in 5 years ?
ANALYSIS. Since a given principal must gain 100 per
cent, to double itself in 1 year, to double itself in 5 years will
require ^ of 100 per cent., or 20 per cent. Therefore, fyc.
14. At what per cent, will a given principal double
itself in 10 years ?
15. At what per cent, will a given principal double
itself in 12 years 6 months ?
16. At what per cent, will $200 double itself in 4
years ?
17. At what per cent, will $100 double itself in 8
years ?
18. At what per cent, will $150 double itself in
15 years ?
19. At what per cent, will $500 double itself in
16 years 8 months ?
20. At what per cent, will $175 double itself in
2 years 6 months ?
95.] INTELLECTUAL ARITHMETIC. 141
95. The present worth of a note payable at a future
time is such a sum as, put at interest at the given rate per
cent, until it becomes due, will amount to the face of the
note.
Discount is an allowance made for the payment of money
before it becomes due, and is obtained by subtracting the
present worth from the given sum or face of the note.
1. What is the present worth of $180 payable in
3 years 4 months at 6 per cent. ?
ANALYSIS. Since the interest for 3 years 4 months at
6 per cent, is of the principal, (88 5 ) the amount, $180,
is equal to the principal, -|, added to the interest, -J, equal to
f of the principal. Since $180 is f of the principal, ^ oj
$180, or $30, is i, and $30 is -i of 5 times $30, or $150.
Therefore, fyc.
NOTE. The discount is the difference between $180 and $150, which
is $30.
2. What is the present worth of a note for 836.90,
payable in 2 years 6 months, at 5 per cent. ?
3. What is the present value of a note for $54,
due in 5 years, and bearing interest at 7 per cent. ?
4. What is the present value of a debt of $124.40,
due in 4 years 2 months, at 8 per cent, interest ?
5. What is the discount of 2100, due in 1 year
3 months, at 4 per cent. ?
6. What is the discount on a note of $560, paya-
ble in 2 years, at 6 per cent, interest ?
7. What is the difference between the interest and
the discount of $130, due 10 months hence, at 10
per cent. ?
8. A man, having a span of horses for sale, offered
them for $480 cash in hand, or a note of $550 due
in 1 year 8 months without interest. The buyer
accepted the latter offer ? did the seller gain or lose
by his offer, and how much, allowing the money to
be worth 6 per cent. ?
142 INTELLECTUAL ARITHMETIC. [96.
96. 1. B sold a horse for | of 1| times the
cost ; what was the gain per cent. ?
2. A cow purchased for $28 in the spring was sold
for $21 in autumn ; what was the loss per cent. ?
8. An old lady bought 80 apples of John at the
lute of 2 for a cent, as many more of William at the
rate of 3 for a cent, and sold the whole at the rate
of 5 for 2 cents ; did she gain or lose, and at what
rate per cent. ?
4. Did she gain or lose on the purchase made of
John or William, and at what rate per cent, on the
amount of the purchase ?
5. A hogshead of molasses was bought for $20 ;
11^ per cent, of it leaked out ; how must the re-
mainder be sold per gallon to gain 40 per cent, on
the cost ?
6. A manufacturer sold 2 machines for $391 each ;
on one he gained 15 per cent., on the other he lost
15 per cent. ; did he gain or lose on the sale of both,
and how much ?
7. How much wheat must be taken to mill, that
6 bushels may be retained after riving toll of 6
pounds per bushel, or 10 per cent, for grinding ?
8. A man brought home 1500 feet of lumber after
giving 37|- per cent, of the whole for sawing ; how
much lumber did he have sawed ?
9. A farmer sold 2 horses for $150 apiece ; on one
he gained 25 per cent., on the other he lost 25 per
cent. ; did he gain or lose on both, and how much ?
10. For hulling barley, 36 per cent, is taken for
pay ; at this rate, how many bushels must be taken
that 32 bushels be returned after hulling ?
11. In a school district $57 was to be collected
for teacher's wages ; the collector was to have 5 per
cent, on all moneys collected ; how much must he
collect to pay his fee and the teacher ?
96.] INTELLECTUAL ARITHMETIC. 143
12. Hovi much plaster must A have ground, to
bring home 14 tons, after the miller has taken 12|-
per cent, of all he has ground ?
13. In how many years, at 5 per cent., will a
given principal amount to the same as it would at 7
per cent, in 3 years ?
14. At what rate per cent, in 12 years will a given
principal amount to the same as at 7 per cent, for 8
years ?
15. A farm rents for $500 a year, which is 6{ per
cent, on the purchase money, at $40 per acre ; how
many acres in the farm ?
16. Mr. Jones paid $44 interest on a note running
4 years 4 months and 24 days, at 5 per cent. ; what
was the face of the note ?
17. A's note of $200 was given for a carriage ; 5
years 7 months and 6 days afterwards, $256 was
given for the note ; what rate per cent, was paid ?
18. A note of $300 on interest at 6 per cent, was
given for a span of horses ; when the note was taken
up, $330 was paid for it ; how long did it run ?
19. B, on being asked how much he owed for his
farm, replied, " I pay 9 per cent, on the debt ; if I
paid but 7 per cent., it would be $60 in my favor
yearly.'' What did he owe ?
20. If Aaron had purchased his horses for 5 per
cent, less than he did, they would have cost him $190 ;
how much did he pay for them ?
21. If I had purchased my farm for 6 per cent,
less than I paid for it, it would have made a differ-
ence in the purchase money of $120 ; what did the
farm cost me ?
22. A speculator sold wheat for 90 cents a bushel,
and in so doing lost 10 per cent. ; what should he
have sold it for, to have gained 15 per cent. ?
23. A nurseiynian sold apple trees at $15 per
144 INTELLECTUAL ARITHMETIC. [96.
thousand, and cleared | of his receipts ; what per
cent, profit did he make ?
24. B pays $400 a year for the use of 100 acres
of land, giving 8 per cent, of its value ; what is it
worth per acre ?
25. I bought a stock of goods for $800, and paid
#15 for store rent, $85 for clerk hire, and sold them
at the end of 6 months ; what per cent, must be
made on the goods to cover expenses ?
26. In how many years will a man, paying inter-
est at 7 per cent, on a debt for land, pay the amount
of the debt in interest ?
27. A. bought a form for $3000 ; he realized $700
from the sale of wheat, $200 from corn and barley,
and $100 from fruit, and then sold the farm at a dis-
count of 25 per cent. ; did he gain or lose, and how
much ?
28. B bought a horse for $60, then sold him for
$100, and repurchased him for $80 ; what was the
gain per cent, on the original investment ?
29. A milkman sold milk for 4 cents a quart, by
which he cleared J of the receipts ; but milk becoming
scarce, he sold for 5 cents ; what per cent, did he
make at the latter price ?
30. A walked 5 per cent, of a journey of 300 ^
miles ; 20 per cent, of the remainder he rode in the
stage, and the rest of the way in the cars ; how far
did he ride in the cars ?
31. A mechanic sold a buggy for $54, and gained
20 per cent. ; he then sold another for $54, and losit
10 per cent. ; did he gain or lose, and how much ?
32. A dealer sold a melodeon for $190, and lost
5 per cent, on it ; he then sold another for the same
money, but made enough to balance the loss on the
first ; what per cent, did he make on the second ?
97.] INTELLECTUAL ARITHMETIC. 145
CHAPTER IX.
MISCELLANEOUS EXAMPLES.
97. 1. A gentleman, owning | of a vessel, sold
I of his share ; what part of the vessel does he still
own?
ANALYSIS. Since he sold of his share, he would have |
lefty and of ^ is -J. Therefore, fyc,
2. A man, owning ^ of a share in the Central Rail-
road, sold f of it ; what part of a share has he left ?
8. A man, having | of a barrel of flour, gave
of it to a poor neighbor ; what part of a barrel has
he left ?
4. Harry, having f of a dollar, gave | of it for a
knife ; what part of a dollar has he left ?
5. A man sold a watch for $18, which was | of
what it cost him ; how much did he lose ?
6. A grocer sold a quantity of cheese for $45,
which was | of what it cost him ; how much did he
gain ?
7. A threshing machine was sold for $120, at a
sacrifice of f of the cost ; what was the loss on it ?
8. Henry having lost -| of his money playing cards,
lent ^ of the remainder, when he found he had $22
left ; how much had he at first ?
9. A horse was sold for $80, which was of f of
twice the cost ; what was lost on the horse ?~
10. A man bought a wagon for $54, which was $
of twice as much as he sold it for ; what did he gain?
11. Homer had 15 marbles, which was f of 3
times as many as Henry had ; how many had Henry ?
12. Thomas gave 4 cents for a top, which was f
times J of what money he had ; how many marbles,
at 2 for a cent, could he buy with what he had left ?
10
146 INTELLECTUAL ARITHMETIC'. [97.
13. A farmer, after selling I of his flock of sheep,
had 20 left ; how many had lie at first ?
14. A pole was standing | of its length in the air,
| of the remainder in the water, and 3 feet in the
mud ; how many feet in the air ?
15. If from my age you take \ and f of my age,
the remainder will be 4 years ; how old am I ?
16. If to f of the cost of A's coat you add $12,
the sum will be $20 ; what was the cost of the coat V
17. A drover, being asked how many sheep he
had, replied, " If to J my number you add 80 sheep,
the sum will be 5 more than 3 times my number ; "
how many had he ?
18. John and James play at marbles ; John wins
16, which is f of twice as many as James now has ;
when they began, | of John's equaled | of James's ;
how many had each when they began ?
19. J- of Herbert's age, increased by 10 years,
equals 1| times his age ; how old is he ?
20. A cow. cost $40 ; f of this sum was -^ of 2f
times \ of the cost of a horse ; what was the cost of
the horse ?
21. A. was 25 years of age when he was married,
and his age was |- times | of his wife's age ; how old
was she ?
22. A post-office was robbed by a clerk, who took
-| of the money in the drawer ; when he was arrested,
he had made away with f of it, and had $10 left ;
how much was in the drawer at first ?
23. A's farm cost $1500 ; f of this is f of 4 times
the cost of the barn ; what did the barn cost ?
24. At the battle of Long Island, the Americans
lost 500 men ; | of this number was | of J^-Q of their
force ; how many men had they in the field ?
25. At Ridge field, 100 Americans were killed ; -f
of the slain was 7J- times y^ of the men on duty;
how many on duty ?
98.] INTELLECTUAL ARITHMETIC. 147
26. The American loss at the battle of Trenton,
in killed, frozen, and wounded, was 60 men ; | of
this number was 2| times T ^ of the number en-
gaged ; how many men in the engagement ?
27. At the assault on Quebec,, the Americans had
100 killed; f of this number was 6 more than If times
^ of the men sent; how many men were in the detach-
ment ?
28. The American loss in the battle of Port Royal
Island was 30 men ; 2^ times this number was 3|-
times yV of the number in the field ; how many in
the field ?
29. At Germantown, the American loss, in slain,
wounded, and prisoners, was 1200 ; f of this num-
ber was 8|- times T |Q- of | of the army, lacking 1000
men ; how many in the army ?
98. 1. Divide 28 into two numbers that' shall
be to each other as 5 to 2.
ANALYSIS. Since the numbers are to each other as 5 to
2, the given sum must be divided into 7 equal parts, and 5 of
those parts will equal one of the numbers, and 2 of them the
other ; the numbers will be % of 28 and of 28, equal to 20
and 8. Therefore, fyc.
2. Two men bought 40 bushels of potatoes ; one
paid $6, and the other $4 ; how many bushels should
each have ?
3. Two travelers, 49 miles apart, approach each
other ; one travels 4 miles an hour, the other 3 miles
an hour ; how many miles will each travel ?
4. A and B hired a horse and buggy for $25 ; A
used it 3 weeks, B 2 weeks ; what should each pay ?
5. Two men hired a pasture for $24 ; one put in
5 cows ? the other put in 3 cows ; what should each
pay?
148 INTELLECTUAL ARITHMETIC. [98.
6. Two men hired a field for $12 ; one put in 2
horses for 2 weeks, the other two horses for 4 weeks ;
what should each pay ?
7. John had 9 cents, and Jay had 7 ; they gave
them all for 32 figs ; how many hhould each of them
have ?
8. A and B enter into partnership ; A invests $10
for 5 months, B invests $5 for 8 months ; they gain
$45 ; what is each man's bhare of the gain ?
9. A, B, and C bought a horse ; A paid $5 as
often as B $4 and C $3 ; they gained $24 in selling
him ; what is each man's share of the gain ?
10. Three men agree to mow a field for $40 ; A
sends 2 men three days, B sends 3 men three days,
C sends 2 men two and a half days ; what should
each receive ?
11. A, B, and C traded in company; A put in
$30, B $50, and C such a sum that he took $10 of
the $42 gained during the year ; what was the gain
of A and B, and what C's stock ?
12. An ignorant partner, who furnished $500 of a
capital of $800, on which $240 had been gained,
took $145 for his share of the profits ; did he gain
or lose, and how much ?
13. Tw r o men paid $11 for the use of a pasture;
the first put in 5 calves, the second 4 colts ; what
should each pay, provided 2 colts consume as much
as 3 calves ?
14. A, B, and C hire a field for $42 ; A puts in 5
cows, B 3 cows, and C pays J of the rent ; how much
must A and B pay, and how many cows does C
put in ?
15. Three men paid $56 for a pasture ; the first
put in 9 oxen, the second 8 cows, and the third 10
calves ; what should each pay, if 4 cows eat as much
as 3 oxen, and 5 calves as much as 4 cows ?
99.] INTELLECTUAL ARITHMETIC. 149
16. Mary and Melissa wrote 60 lines in a copy
book ; Mary wrote as many lines as Melissa ; how
many lines did each write ?
17. A gardener gave 24 pears to some school
children, giving each girl 4 pears, and each boy 3 ;
the boys received as many pears as the girls ; how
many children were there ?
18. A father divided $96 among his two sons and
3 daughters, giving each son as much as each
daughter ; how much did each receive ?
99. 1. Three boys have 76 marbles; the second
has 2 more than the first, and the third has 6 more
than the second ; how many has each ?
2. James, Henry, and Joseph, together, have 72
apples ; Henry has 2 more than James, and Joseph
2 more than Henry ; if each had 110 more than
James, how many would all have ?
3. If eacli had as many as Joseph, how many
would all have ?
4. How many must Joseph give James that each
may have the same number ? How many has each ?
*5. A, B, and C built a warehouse for $600; B
paid $100 more than A, and C paid as much as A
and B ; how much did each pay ?
6. William bought a rifle, a watch, and a drum,
for $42 ; for the rifle he gave twice as much as for
the drum, and for the watch twice as much as for
the rifle ; what did he give for each ?
7. A span of horses, wagon, and harness cost
$225 ; the difference between the cost of the horses
and wagon was 1^ times the cost of the harness,
and the cost of the liorses was 4 times the cost of the
harness ; what was the cost of each ?
8. B traveled 114 miles in 3 days ; the first day
150 . INTELLECTUAL ARITHMETIC. [99.
he traveled | as far as on the last 2 days, and the last
clay as far as on the iirst two days ; how far did he
travel each day '!
9. A laborer in one week dug 5 rods more than
| the length of a ditch, and the next week he dug the
remaining 20 rods ; how long was the ditch ?
ANALYSIS. Since he dug 5 rods more than of the
ditch the Jirst week, the remainder, 20 rods, increased by 5
rods, must equal the ditch ; '20 rods -j- 5 rods =. 25 rods 9
which is of twice 25, or 50 rods. Therefore, fyc.
10. A farmer, owing a store debt, paid at 0110
time $10 more than . l , <>!' it, at another $20 more
than of the remainder, and gave a note of $20 for
the balance of the debt ; what was the amount of
the debt ?
11. Arthur gave 2 more than -*- of his pennies for
a top, 1 more than of the remainder for a whip,
and had 4 left ; how many pennies had he at first ?
12. A grocer, selling beer, drew -|- a gallon more
than -| the contents of a keg one day, the next day
he drew J a gallon more than |- of the remainder,
and the third day, when he went to draw, he found the
keg was empty ; how many gallons in the keg at first ?
13. A and B have $55 ; of A's money equals f
of B's ; how many dollars has each ?
ANALYSIS. Since % of As money equals of B's, 2
halves, or all of A's, will equal 2 times -f, or of It's ; then
A has 6, and S 5 equal parts of the money, and both hai9
11 parts ; T ] T of $55 is $5, or\ part; 6 times $5 are $30,
or As share ; and 5 times $5 are $25, or B's share. Ttiere-
fore, fyc.
14. If 34 apples he divided between John and
Mary, so that f of John's may equal | of Mary's,
how many will each have ?
1OO.] INTELLECTUAL ARITHMETIC. 151
15. A and B have together 83 sheep ; f of A's
flock, increased by 8 sheep, is equal to B's flock ; how
many sheep has each ?
16. If 10 cents be taken from of Mary's money,
the remainder will be equal to Susan's ; the girls,
together, have 39 cents ; how many cents has each ?
17. Homer has 3 marbles more than Silas, and -|
of Homer's marbles equal f of Silas's ; how many
has each ?
IOO. 1. One man will drink a gallon of beer in 3
days ; another will drink a gallon in 4 days ; how long
will a gallon last both of them ?
2. Julia can do the washing in 7 hours ; with
Jane's assistance she can do it in 4 hours ; in what
time can Jane do it ?
3. Two boys can pitch a quantity of hay in 5
hours ; the first can do it in 9 hours ; in what time
can the second do it ?
4. A cistern can be emptied by 2 pipes in 3|
hours, but 1 pipe will empty it in 7^ hours ; in how
many hours will the other empty it V
5. A can make a vest in | of a day, B in |, and
in of a day ; how many more vests can A and B
make in a day than C can ?
6. Susan can knit a pair of mittens in f of a
day, and Sarah can knit a pair in | of a day ; how
many pairs can both knit in a day ?
7. What part of a day must Sarah assist Susan
that Susan may complete 2 pairs ?
8. What part of a day must Susan assist Sarah
that Sarah may complete 3 pairs ?
9. Jason can hoe 10 rows of corn in | of an
hour, and Jesse can hoe them in f of an hour ; in
what time can both together hoe them ?
152 INTELLECTUAL ARITHMETIC. [1OO.
10. In what time can Jason hoe what remains
after Jesse has hoed | an hour ?
11. In what time can Jesse hoe what remains
after Jason has hoed ^ of an hour ?
12. How many rows can both hoe in 1 hour ?
13. A and B can clear a field in 15 days ; with
the help of C they can do it in 9 days ; in how many
days can C do it ?
14. A and B can dig a well in 6 days, A and C in
8 days, and B and C in 9 days ; in how fhany days
can each dig it, working alone ?
15. In how many days can they do it working
together ?
16. Patrick, Peter, and Philo can dig 40 rods of
ditch in 10 days ; Patrick and Peter can do it in 15
days ; Peter and Philo can do it in 13 J days ; in how
many days can Patrick and Philo dig 28 rods ?
17. How long will it take each man to dig 30 rods ?
18. Henry and Harlan can cut a field of corn in
6f days ; Harlan's day's work exceeds Henry's by ^
of Henry's ; in how many days can each cut the field ?
19. Three men can do a piece of work in 10 days ;
the first does f as much as the second, the third does
I as much as the first and second ; in how many
days will each do it ?
20. In how many days will the first and second
doit?
21. In how many days will the first and third do it ?
22. In how many days will the second and third
doit?
23. A, B, and C can do a piece of work in 8 days ;
B and C can do it in 12 days, and A and B in 10
days ; in how many days can each, working separately,
doit?
24. A cistern can be discharged by 3 pipes in 4
hours ; the first and second pipes will discharge it in
101.] INTELLECTUAL ARITHMETIC. 153
8 hours ; the first and third in 6 hours ; in what time
will each pipe discharge it ?
25. Three men can trench a field in 10 days ; A
and B can do it in 20 days ; B and C can do it in 15
days ; in how many days can A and C trench the
same field ?
26. A barrel of flour would last a brother, sister,
and servant 30 days ; when the brother was absent
it would last 60 days ; when the sister was absent it
would last 45 days ; how long would it last the sister
after all had used from it 20 days ?
1O1. 1. Benjamin, meeting some beggars, gave
each of them 5 cents ; had he given them 7 cents
apiece, it would have taken 8 cents more ; how many
beggars were there ?
ANALYSIS. Since 2 cents was the increase paid to I
beggar, and 8 cents the increase paid to all, there were as
many beggars as 2 is contained times in 8, which is 4 times.
TJierefore, fyc.
2. Edward gave 6 plums to each of his playmates,
and had 9 remaining ; had he given each of them 8,
he would have had but 1 left ; how many playmates
had he ?
3. Three times a certain number is 15 less than 6
times the same number ? what is the number ?
4. Susan, purchasing calico, found that if she
took that at 8 cents a yard she would have 11 cents
left ; but if she took that at 12 cents a yard she
would need 17 cents to pay for it ; how many yards
did she want ?
5. 4 times a number is 15 less than 6|- times the
same number ; what is the number ?
6. John, being asked how many marbles he had,
replied, " f of them exceed i of them by 4." How
many had he ?
154 INTELLECTUAL ARITHMETIC. [1<H.
7. 5J times a number is 21 greater than 3| times
the same number ; what is the number ?
8. If 24 chestnuts be divided among some boys,
giving 4 apiece to J of them, and 2 apiece to the
others, how many boys are tlu'iv ':
ANALYSIS. Since one boy received 4 chestnuts as often
as another received 2 chestnut s> 2 boys received j3 chestnuts',
there were as many times 2 boys as 6 is contained times
in 24, which is 4 times, and 4 times 2 boys are 8 boys.
Therefore, fyc.
9. Mary gave her chickens 26 grains of corn ; f
of them had 5 grains each, the others but 3 each ;
how many chickens had she ?
NOTE. The above gives the following question :
10. 5 times f of a certain number, plus 3 times J
of the same number, is 26 ; what is the number ?
11. A class received 29 merit marks; ^ of them
received 5 apiece, |- of the remainder 3 apiece, f of
the remainder 2 apiece, and the rest 1 apiece ; how
many scholars in the class ?
12. 26 melons were stolen, of the party getting
4 each, the rest 3 apiece, except 2, who obtained but
2 apiece ; how many were there in the party ?
ANALYSIS. Had the last 2 obtained 3 each, there would
have been 28 melons ; then 4 times of a number, plus 3
times of the same number, or J, icould equal 28. Since
28 is of a number, $ of 28, or 4, is , and 4 i s of 2
times 4, or 8. Therefore, fyc.
13. 4 times |- of a number, plus 3J times the
same number, is equal to 28 plus 5 ; what is the
number ?
14. Henry said that 4 years more than J of his
age, increased by 5 times ^ of it, and that sum by 4
times ^ of his age less 4, would make the sum 50
years ; how old was he ?
1O1.J INTELLECTUAL ARITHMETIC.. 155
15. A drover paid $5 a head for J of his flock, $4
a head for as many more, $3 a head for | of the
remainder, and $2 a head for the rest ; the flock cost
|115 ; how many sheep had he ?
16. A grocer received $10 for a lot of baskets ;
for J of them he had 6 dimes each, for -| of the re-
mainder plus 3, 4 dimes each, for J of the rest plus
2, 3 dimes each, and for the rest 2 dimes each ; how
many baskets did he sell ?
17. 5 times Henry's age is 3 more than 4 times |
of it increased by 6 times *- of it ; how old is he ?
ANALYSIS. 5 times Henry's age equals f, 4 times of
it, plus 6 times ^ of it, equals - 1 /-, and if- less * equals
of it, which, according to the conditions of the question, is
3 ; and 3 is ^ of 3 times 3, or 9. Therefore, fyc.
18. 6 times a number is 4 less than 7 times | of
it, plus 5 times ^ of the same number ; what is the
number ?
19. Bridget, wishing to get a certain number of
pounds of meat, found, if she took beef at 5 cents,
she would have 4 cents left ; if pork at 7 cents, she
had not money enough to pay for it ; and therefore
she took | of the quantity in beef, the rest in pork,
and paid for it ; how many pounds did she get ?
20. Pour times a number is 6 greater than 5 times
|- of the same number ; what is the number ?
21. A farmer, wishing to buy a flock of sheep,
found, if he paid $2 a head, he would have $60 left ;
if he paid $5 a head, he had not enough to pay for*
them ; if he took |- the number at each price, he
could just pay for them ; how many sheep would
his money buy ?
22. Twice f of a number equals the number plus
8 : what is the number ?
156 INTELLECTUAL ARITHMETIC. [1O3.
1. The sum of two numbers is 27 ; their
difference 5 ; what are tiie numbers ?
ANALYSIS. Since 27 is the sum of two numbers whose
difference is 5, 27 less o, or 22, would be twice the less num-
ber ; of 22 is 11, //<? /ess number, which increased by 5
equals 16, or Me greater number. Therefore, fyc.
2. James and John together have 19 peaches ;
John has 3 more than James ; how many has each ?
3. Hobart has 9 marbles more than Herbert, and
together they have 31 ; how many lias each ?
4. Two men are 37 miles apart, and travel
toward each other ; when they meet, one has trav-
eled 5| miles more than the other ; how far has
each traveled ?
5. Horace, finding he had 5 more apples than
Homer, gave him 3, when they together had 21 ;
how many had each then ?
6. Mary has twice as many berries as Martha, and
together they have 12 quarts ; how many quarts has
each ?
7. A farmer had 47 sheep in two fields ; in one
field there were 5 more than twice as many as in the
other ; how many were there in each field ?
8. There are 54 bushels of corn in two bins ; in
3ne bin there are 6 bushels less than half as many as
there are in the other ; how many bushels in each ?
9. A man bought a watch and chain for $96 ; the
watch cost $4 less than 3 times the cost of the chain ;
what was the cost of each ?
10. Hiram and Harvey built wall 11 days ; Hiram
received 2 dimes a day more than Harvey, and they
both received $25.30 ; what did each receive, and
how much per day ?
11. The sum of A's and B's ages is 60 years ; 6
years ago, A's age was J of B's ; what are their ages
now?
IO2.] INTELLECTUAL ARITHMETIC. 157
12. A horse and cow cost $124 ; the horse cost
$4 more than 3 times the cost of the cow ; what was
the cost of each ?
13. A farmer paid $24 for a colt and cow ; the
cost of the colt, multiplied by ^ the cost of the cow,
equaled the cost of the cow ; what did he pay for
each ?
14. A lady purchased a dish and cover for 24
dimes ; | of the cost of the dish, increased by the
difference between the cost of the dish and cover,
equals the cost of the cover ; what was the cost of
each ?
15. A grocer purchased 25 pounds of butter of
two women ; | of the number of pounds he took of
one, increased by the difference between the amounts
purchased of both, equals the number of pounds he
took of the other ; how many pounds did each sell ?
16. The sum of two numbers is 10 ; their differ-
ence is equal to ^ of tfre greater number ; what are
the numbers ?
17. A wagon cost $38 ; ^ of the cost of the wood
work exceeds the cost of ironing it by -l- of the dif-
ference between the expense of the two ; what was
the cost of each ?
18. Emily purchased some lace and some ribbon
for 30 cents ; ^ the cost of the lace, increased by ^
the difference between the cost of the lace and rib-
bon, equals the cost of the ribbon ; what did she pay
for each ?
19. Wilber paid 20 shillings for a pair of skates
and a knife ; twice the difference between the cost of
the two, plus J the cost of the knife, equals the cost
of the skates ; what did he pay for each ?
20. A horse and harness cost $34 ; | of the cost
of the horse is $1 more than the cost of the harness ;
what was the cost of each ?
158 INTELLECTUAL ARITHMETIC. [1O3.
1O3. 1. A grocer paid 812.60 for 30 bushels of
potatoes, giving 50 cents a bushel for good ones, and
20 cents a bushel for poor ones ; how many bushels
were good ?
ANALYSIS. If all had been good, he would have paid
$15, or $2.40 more than he did; each bushel of poor ones
made a difference of 30 cents; hence there were as -many
bushels of poor ones as 30 cents (the difference on 1 bushel)
is contained times in $2.40, (the difference on a//,) which is
8 times, equal to 8 bushels of poor ones ; the difference be-
tween 30 bushels and 8 bushels is 22 bushels, equal to the
good ones. Therefore, fyc.
2. A fanner bought 28 sheep for $76, paying $3
for old sheep, and $2 for yearlings ; how many old
sheep ?
3. A flour merchant paid $82 for 20 barrels
of flour, giving $4.50 for first quality, and $3.50
for second quality; how many barrels were first
quality ?
4. Henry sold his watch for $18, and by so doing
lost of what it cost ; what did it cost ?
5. A man, employing an equal number of men
and boys, paid each boy 5 dimes, and each man 7
dimes, and to them all he paid $7.20 ; how many
were there of both ?
6. An old lady sold an equal number of chickens
and turkeys for $4, receiving for chickens 3 dimes
each, and for turkeys 5 dimes each ; how many fowls
did she sell ?
7. A speculator bought an equal quantity of
wheat and beans, giving $1.50 per bushel for wheat,
and $1 for beans ; the wheat cost $7 more than the
beans ; how many bushels of beans did he buy ?
8. A man engaged to work 25 days, on condition
that he should receive $2 for every day's labor, and
1O3.] INTELLECTUAL ARITHMETIC. 159
pay $1 for board every day he was idle ; he received
at settlement $38 ; how many days was he idle ?
ANALYSIS. As many days as $3, (the difference made
by 1 idle day,} is contained times in $12, (the difference
made by all the idle days,) which is 4 times, llierefore he
was idle 4 days.
9. A man was engaged for 20 days to copy, re-
ceiving $2.50 for every day's labor, and forfeiting $1
for every day lie was idle ; he received $43 ; how
many days did he copy ?
10. Job, having some chestnuts, gave | of them
to John, who, after eating 6, gave | of what remained
to his sister, and still had 5 left ; how many had Job ':
ANALYSIS. If John gave to his sister, the 5 left was ;
5 is of 4 times 5, or 20, and 20, plus the 6 he ate, make,
26, which is f of Job's, fyc.
11. Mary gave ^ of her flowers to her playmates,
-| of the remainder to her teacher, and had 2 left ;
how many had she at first ?
12. A farmer sold | of 1| times his flock, and
had 12 sheep remaining ; how many sheep had he
at first ?
13. A man paid $5 more than f of his grocer's
bill ; soon after he paid -| of what remained unpaid,
when he found $3 was still due ; what was the
amount of his bill ?
14. John paid $10 more .than J of his year's
wages for clothes, $12 for a watch ; lie then lent f
of what he had left, and having paid $3 for a pair of
boots, had $5 left ; what were his wages ?
15. A laborer engaged to work 20 days, on con-
dition that he should have 12 dimes for a day's labor,
and pay 2 dimes for every idle day for board ; he
received as many dollars as he worked days ; how
many days was he idle ?
160 INTELLECTUAL ARITHMETIC. [1O4.
1O4. 1. A rope 28 feet in length was broken so
that f of the longer piece was equal to the shorter ;
what was the length of each piece ?
ANALYSIS. The longer piece is divided into 5 equal
parts, 2 of which equal the shorter ; the sum of the 2
pieces is 7 equal parts, one of which equals | of 28 feet,
or 4 feet ; 2 parts equal 2 times 4 feet, which are 8 feet,
the shorter piece ; 5 parts equal o times 4 feet, which are 20
feet, the longer piece. Therefore, fyc.
2. A tree 56 feet high was broken in a storm ; | of
the part standing was equal to the part broken off;
what was the length of each part ?
3. Henry and Horace have 45 marbles ; Horace
has | as many as Henry ; how many has each ?
4. A grocer, having opened a barrel containing
160 pounds of sugar, took out | as much as he left ;
how many pounds were left in the barrel ?
5. A traveler paid 63 cents for his supper and
fodging, paying f as much for his supper as for his
lodging ; what did he pay for his supper ?
6. A horse and wagon cost $170 ; the horse cost
1-J- times as much as the wagon ; what was the cost
of each ?
7. Henry traveled 140 miles in 2 days ; the first
day he traveled 1^ times as far as he did the second ;
how far did he travel each day ?
8. From Buffalo to Bergen is 50 miles, from Bergen
to Utica is 1| times as far as from Utica to Schenec-
tady, which is 280 miles from Buffalo ; how many
miles from Buffalo to Utica ?
9. David caught a trout 17 inches long ; the tail
was -| as long as the body, and the head was 3 inches
long ; how long was the tail ?
10. Divide 36 into 2 such parts that 1| times
one shall equal the other.
104*] INTELLECTUAL ARITHMETIC. 161
11. Homer and Harlan have 48 apples ; of
Homer's, plus 6, equal the number Harlan has;
how many has each ?
ANALYSIS. Harlan has 6 more than as many as Ho-
mer, which makes the sum 48, 6 greater than if he had but
% as many ; Homer's are divided into 4 parts, of which 3
parts plus 6 apples are equal to Harlan' s ; then 4 parts
plus 3 parts, plus G apples, equal 48, 7 parts equal 42,
and y of 42, or 6, equals 1 part. 3 parts plus 6 equal
24, the number Harlan has, and 4 parts equal 24, the num-
ber Homer has. Therefore, fyc.
12. Rochester is 52 miles from Geneva, via Vic-
tor ; i 5 y of the distance from Geneva to Victor, plus
4 miles, equals the distance from Victor to Roches-
ter ; how far is Victor from each place ?
13. From the ground to the top of a church steeple
is 146 feet ; | of the height of the steeple above the
church, plus 6 feet, is equal to the height of the church;
what is the height of the steeple above the church 2
14. A jar and cover weigh 18 pounds ; -| of the
weight of the cover, increased by 12 pounds, equals
the weight of the jar ; what is the weight of each ?
15. A coat and vest cost $19 ; ^ of the cost of the
coat, plus $3, equals the cost of the vest ; what is the
cost of each ?
16. Wilson has 15 oranges and lemons ; J of the
oranges equals ^ of the lemons ; how many "of eacli
has he ?
ANALYSIS. Since ^ of the oranges equals -J of the lemons,
f , or all the oranges, equal of the lemons ; we then have
the oranges and lemons divided into 5 equal parts, 2 parts
of which equal the oranges, and 3 the lemons, $c.
17. A father gave his 2 sons 50 cents; l of
George's money was equal to f of Abel's ; how many
cents had each ?
18. Andrew has 84 marbles ; -| of the gray ones
11
162 INTELLECTUAL ARITHMETIC. [1O4.
equal of the black ones ; how many of each kind
were there ?
19. The sum of two numbers is 62 ; | of one
equal | of the other ; what are the numbers ?
20. A purse and contents are valued at 46 shil-
lings ; | of the value of the purse is equal to | of
the value of what is in it ; what is the purse worth ?
21. A person, being asked the time of day, replied,
" The time past noon is equal to | of the time to
midnight." What was the hour ?
ANALYSIS. The time to midniyht is divided into 5 equal
parts, one of which equals the time past noon ; then 5 parts
pins 1 part equals 6 parts \and- 12 Itours, the time from
noon to midnight, must be ^ of the time it lacked of being
midnight ; since 12 is f , -, which is o/f, will be of 12
hours, which is 2 hours. Therefore the hour was 2 o'clock,
P.M.
22. What was the time, provided f- of the time
past midnight equaled the time to 10 o'clock, A. M.?
23. | of the time from this instant to 5 o'clock,
P. M., equals f of the time past midnight; what is
the time of day ?
24. Peter said to John, " If times my age is 1|
times your age, and the sum of our ages is 36 years."
What was the age of each ?
25. A bin whose capacity was 44 bushels was
partly filled with grain ; 2| times what was in it was
2| times what it would take to fill it ; how many
more bushels would fill it ?
26. A person, inquiring the distance to Cincinnati,
was told that 3|- times what it lacked of being 83
miles was 1-| times what it exceeded 39 miles ; how
far was he from Cincinnati ?
27. The number of miles that the distance from
Charleston to Columbia exceeds 100 miles equals 1J
times the distance it lacks of being 150 miles : how
far is it from Charleston to Columbia ?
1OO.] INTELLECTUAL ARITHMETIC. 163
1O> 1. Two pieces of cloth contain 38 yards;
| of the first piece equals -f of the second, plus 4
yards ; how many yards in each piece ?
ANALYSIS. If equal f -f- 4, ^ wz7/ e^waZ o/ 1 -j- 4,
wAicA is -i + 2, Jfo's zs of 3 times - + 2, or f -|- 6 ;
tf^ew we Aaz?e 1 piece equal to f ^e other plus 6 yards, or 1
pzece divided into 5 parts, and 3 of these parts plus 6 yards
equal the other piece ; hence 5 parts plus 3 j0ar& plus 6 yards
equal 38 yards, or 8 /?ars ano? 6 yards, equal 38 yards ; 38
yards minus 6 yards equal 32 yards, or 8 parts, fyc.
2. A lad inquiring his mother's age, his father
replied, u f of my age is 9 years more than |- of
your mother's, and the sum. of our ages is 72 years."
How old was his mother ?
8. Two men built 38 rods of wall ; | of what one
built, plus 4 rods, is equal to | of what the other
built ; how many rods did each build ?
4. Hiram and Richard sheared 67 sheep ; | of the
number Hiram sheared was 4 more than -| of the
number Richard sheared ; how many did each shear ?
5. What is the time, providing | of the time past
midnight, plus 1-^ hours, equals | of the time to
midnight ?
6. A, being asked his age, replied, " 1|- times what
I lack of being a hundred years old is 9 years more
than 1|- times what my ago exceeds 64." What was
his age ?
7. A rope was cut into 3 pieces; the first piece
was 5 feet long, the second was as long as the first
plus I of the third, and the third was as long as the
other two ; what was the length of the rope* ?
ANALYSIS. Since the third piece is as long as the other
two, it must be ^ of the rope ; the second piece, being as
long, is ^ ; and the 5 feet which the second exceeds J, with
the 5 feet of the first piece, make 10 feet, which must be the
other fourth ; 4 times 10 is 40. Therefore, fyc.
164 INTELLECTUAL ARITHMETIC.
8. The head of a fish is 7 inches long, the tail is
as long as the head and i of the body, and the body
is as long as the head and tail ; what is the length of
the fish ?
9. A piece of steel was broken into 3 pieces; the
first piece was j- of the whole, the second weighed 3
pounds, and the third weighed as much as the other
two ; what was the weight of the whole ?
10. Three men dig potatoes ; the first digs ^ of
the whole, lacking 2 bushels, the second digs 5 bush-
els, and the third digs as many as the other two ;
how many bushels did all dig ?
11. From Batavia to Avon is 24 miles ; this dis-
tance, plus the distance from Bath to Corning, and
12 miles more, equals the distance from Avon to
Bath ; from Bath to Corning is \- of the whole dis-
tance ; how far from Batavia to Corning ?
12. A farmer took money for stock, as follows :
$18 for swine, $6 more than of the whole for sheep,
and for cattle $7 less than f as much as for sheep
and swine ; how many dollars did he receive ?
13. A third and * a third of 12 is f of twice what
number ?
14. Stephen spent ^ of his money, and afterward
earned as much as he had spent, when he had $16.50
less than at first ; how much had he at first ?
15. If | of a ton of hay cost f of an eagle, how
many dollars will of a ton cost ?
16. A is 6 years more than -|- as old as B, C is |
as old as A, and B's age is equal to the sum of A's
and CV; what is the age of each ?
17. A owns 12 more than f as many acres as
B, C owns as many as A, and B owns 24 acres
than A and C ; how many acres has each.
1OO.] INTELLECTUAL ARITHMETIC. 1G5
1O6. 1. Malcom is 8 years old, and Martin 25 ;
in how many years will Malcom be -| as old as
Martin ?
ANALYSIS. 8 years since, Martin was 17 years of age;
in 17 years from that time, or, since Malcom is 8 years old,
in 17 less 8, or 9 years from this time, Malcom will be 17
years old, and Martin will be twice 17, or 34 years old.
Therefore, fyc.
2. A father is 22 years older than his son, and the
son's age is f of the father's ; how many years since
the son's age was J the father's ?
3. Helen is 9 years old, her sister is 22 ; in now
many years will Helen be - as old as her sister ?
4. Henry played at marbles with Charles, who
had 45 ; Henry won as many as he had to commence
with ; when they quit, each had the same number ;
how many had both ?
ANALYSIS. Since Henry won as many as he had, he left
off playing with twice as many as he began with ; and Charles,
now having the same number, must have had 3 times as
many as Henry when they commenced playiny. 45 is 3
times -J of 45, or 15, and 45 -\- 15 is 60. Therefore, fyc.
5. A farmer, having 60 sheep in one field, took as
many from them as he had in another field, and put
them with the others, when the flocks were equal ;
how many sheep had he ?
6. From a bin containing 52 bushels, 2 more than
as many bushels as were in another bin were taken
and put in the other bin, when both contained the
same quantity ; how many bushels in both bins ?
7. Uncle Simon's age, increased by the difference
between -| and % of his age, is 6 more than ^ times
his age ; how old is he ?
8. Henry earned 20 dollars in the spring ; in the
fall he earned as much as in the spring and as
10(3 INTELLECTUAL ARITHMETIC. [1O6.
much as in the summer, and in the summer as much
as in the spring and fall ; how much did he earn in
all?
9. A has 10 sheep, C has as many as A and | as
many as B, and B has as many as A and C ; how
many sheep have they all ?
ANALYSIS. Since B has as many as A and C he muit
have of the ivltole ; then 10 sheep, plus 10 -f~ % f i
phis , equals % -\- 20, or the whole number of sheep ; hence
20 equals the number, and 20 is \ of 6 times 20, or 120
sheep. Therefore, fyc.
10. A, owning 74 acres of land, sold B | as much
as B owned ; then B's farm was f as large as A's ;
how many acres in each farm ?
11. Mary has 10 more chickens than turkeys, yet
| of the turkeys equal \ the chickens ; how many
chickens has she ?
12. Hainan's coat cost $15 less than his whole
suit ; ^ of the price of the suit equaled | of the price
of the coat ; what did the suit cost ?
13. Daniel caught 14 pigeons more than his
brother, and 2f times his brother's equaled | of his ;
how many did each catch ?
14. One number is 31 times another ; if 15 be
added to each, one is just ^ of the other; what are
the numbers ?
15. The difference between the cost of a horse
and a buggy is $40 ; | of the cost of the buggy equal
-^5 of the cost of the horse ; what is the value of
each ?
16. Alice is l as old as her mother ; 5 years since
she was but ^ as old ; in how many years will she be
J as old ?
IT. Hobart, having 20 marbles more than Dwight,
plays with him ; Dwight w r ins |- as many as he had
1O7.] INTELLECTUAL ARITHMETIC. 16T
at first, when Hobart lias |- as many left as he com-
menced with ; how many had each at first ?
18. The difference between two numbers is 16 ;
if 4 be taken from the larger and added to the less,
2 1 times the larger will equal 3J times the smaller;
what are the numbers ?
19. Jason bought a watch, and had 820 remain-
ing ; he then gave 2 times the cost of the watch for
a rifle, and had -^ of his money left ; what did the
rifle cost ?
20. Find the ages of A, B, and C, by knowing
that C's age at A's birth was 5J times B's, and now
is equal to the sum of A's and B's ; also that if A
were now 3 years younger, or B 4 years older, A's
age would be equal to | of B's.
1O7. 1. Two boys are running in the same
direction ; one is 27 rods in advance of the other in
pursuit, who gains upon the foremost 3 rods in a
minute ; in how many minutes will he be overtaken ?
ANALYSIS. In as many minutes as 3, the number of rods
gained in one minute, is contained times in 27, the number
of rods to be gained.
2. A, after traveling 2 hours, at the rate of 5
miles an hour, was followed by B, at the rate of 7
miles an hour ; in how many hours would B over-
take A ?
3. A dog in pursuit of a hare which has 28 rods
the start, runs 9 rods while the hare runs 7 ; how far
will the dog run to catch the hare ?
4. Henry saves $5 while John saves $ 7 ; how
much will each have when the difference between
what each has saved is $30 ?
5. A and B started from the same point to run a
race ; A ran 84 rods and gave out, when ^ of the
168 INTELLECTUAL ARITHMETIC. [1O7,
distance B had run equaled the distance he was
ahead of A ; how far did B run ?
G. Three men bought a horse ; A paid $25, which
was ^ of what B and C paid ; B paid | as much a&
A and C ; what did B and C pay ?
7. The hour and minute hands of a watch aro
together at 12 o'clock ; when will they next be to-
gether ?
ANALYSIS. Since the minute hand passes the hour hand
11 times in 12 hours, if both are at 12, the minute hand will
pass the hour hand the jirst time in -J f of 1 2 hours, or in
l^Y hours, equal to 5^ minutes past 1 o'clock. There-
fore, $c.
8. A gentleman, being asked the time, replied,
" It is between 3 and 4 o'clock, and the hands are
directly opposite." What was the time ?
9. A fox is 40 leaps ahead of a hound, and takes
7 leaps to the hound's 4 ; but 3 of the hound's leaps
equal 6 of the fox's ; how many leaps will the fox
take before being caught ?
10. A wolf ran 80 rods to catch a sheep ; f of the
distance the sheep ran was equal to the distance be-
tween them when the chase commenced ; what was
the distance ?
11. The sum of f of A's and f of B's money, being
on interest 2 years 7 months and 6 days, at 5 per
cent., amounted to $2260 ; f of B's money was \\
times f of A's ; how much had each ?
12. B's fortune is 1J times A's ; the interest of \
of A's fortune and \ of B's for 5 years, at 6 per cent.,
is $600 ; what is the fortune of each ?
13. A drover paid $76 for calves and sheep, pay-
ing $3 apiece for calves and $2 for sheep ; he sold \
of his calves and f of his sheep for $23, and in so
doing lost 8 per cent, on their cost ; how many of
each did he purchase ?
APPENDIX.
THE author desires to call the attention of teachers to the numbered
Tariety of changes that may be wrought upon the examples of the Sec.
tions from which the following have been selected, as well as upon many
otheis of the same class, and which may appropriately be termed " Ring-
ing the Changes." It is not recommended that these changes be ap-
plied to ail the examples, but used as an occasional exercise, and limited
according to the time and convenience of the teacher.
It is recommended to teachers to exercise their classes in these
* changes," not so much for the novelty of them, as for the enlivening
effect produced upon the class, and especially for the peculiar facilities
they afford for thorough mental discipline ; and while, at first, to some
they may appear severe and too mathematical, they will very soon become
easy and comprehensible.
To aid beginners, the blackboard may be used to advantage ; and to
illustrate, take the example which has been written out in full in Section
71 5 page 171-3:
4- of 35 is I of how many thirds of 18 ?
which may be written on the board thus :
f x 35 = f X 7 X 18?
Call upon seme pupil to read the question according to thejfost form,
and to solve the question according to the form of analysis given ; and
the answer, if correct, placed over the 3, will fulfill the conditions of th
question; substitute this number, and erase another, thus :
which is read
4. of 35 is f of f of what number ?
Solve this as before, and so continue until all the numbers have been
erased, and replaced by solution.
AVhen the class have become familiar with the exercise, the use of the
blackboard should be dispensed with, and the pupil be required to " ring
the changes " by simply hearing the first form of the question read.
Each question, and its form of solution, should be given by the pupil,
but he may be allowed to give results in the solution that have been
previously determined by analysis, in order to shorten the work and avoid
monotonous repetition.
Members of the class should be called upon promiscuously, and en-
couraged to ask questions, and to detect errors either in the statement or
solution of questions. The teacher should read an example once, slowly
and distinctly, then call upon some one of the class who shall commence
operations, and continue to give questions, forms, solutions, and conclu-
sions, until signaled to sit, or another pupil detecting an error, raises
the hand, when some other one should be called upon, who should re-
sume where the last one left off; and so on.
A teacher who has not practiced a class upon these changes, cannot
conceive of the interest, close attention, and perfect enthusiasm that
may be created, as well as the thorough mental discipline attained, by this
exercise ; and it is firmly believed that, upon trial, it will commend itself
to the judgment and approval of every live teacher.
12 (169)
170 APPENDIX.
67. 1. 24 is of how many times 10 ?
24 is | of as many times 10 as 10 is contained times in
5 times : } of 24 ; J of 24 is 6, and 5 times 6 is 30; 10 is
contained in 30, 3 times. Therefore 24 is i of 3 times 10.
2. 24 is f of 3 times what number ?
24 is 1 of -3 times .'. of 5 times \ of 24 ; \ of 24 is 6,
and 5 times G is 30 ; ] of 30 is 10. Therefore 24 is /
of 3 times 10.
3. 24 is how many fifths of 3 times 10 ?
24 is as many fifths of 3 times 10 as of 3 times 10 is
contained limes in L>4. 3 times 10 is 30, and -J- of 30 is
0, and (') is contained in 24, 4 times. Therefore 24 is A
of 3 times 10.
4. 24 is 4 times what part of 3 times 10 ?
24 is 4 times such a part of 3 times 10 as 3 times 10
is contained times in -J of 24. 3 times 10 is 30. and \
of 24 is 6 ; 30 is contained in C, / or 1 times. There-
fore 24 is 4 times i, or |, of 3 times 10.
5. What number is f of 3 times 10 ?
4 times i of 3 times 10. 3 times 10 is 30, l of 30 is C,
and 4 times 6 is 24. Therefore 24 is 4 of 3 times 10.
7O. 1. 4 of 56 is -J- of 3 times what number ?
4 of 56 is | of 3 times -J- of 9 times of 4 times -\
of 56 ; i of 56 is 8, and 4 times 8 is 32 ; of 32 is 4,
and 9 times 4 is 36 ; -J of 36 is 12. Therefore, -$ of 56
is | of 3 times 12.
2. 4 of 56 is of how many times 12 ?
| of 56 is of as many times 12 as 12 is contained
times in 9 times J- of 4 times of 56 ; 4 f 56 is 8 ? an ^
4 times 8 is 32 ; i of 32 is 4, and 9 times 4 is 36 ; 12 is
contained in 36, 3 times. Therefore ^ of 56 is | of 3
times 12.
3. 4 of 56 is how many ninths of 3 times 12 ?
4 of 56 is as many ninths of 3 times 12 as ^ of 3 thru."?
APPENDIX. 171
12 is contained times in 4 times * of 50. ^ of 56 is 8,
and 4 of 56 is 4 times 8 ; 4 times 8 is 32 ; 3 times 12
is 36, ^ of 36 is 4, and 4 i-? contained in 32, 8 times.
Therefore | of 56 is f of 3 times 12.
4. 4 f 5(> is 8 times what part of 3 times 12 ?
Such a part of 3 times 12 as 3 times 12 is contained
times in | of 4 times % of 56. ^ f 56 is 8, and 4 times
8 is 32 ; of 32 is 4 ; 3 times 12 is 36, and 36 is con-
tained in 4, ^j or ^ times. Therefore ^ of 56 is 8 times
^, or |, of 3 times 12.
5. f of 3 times 12 is -f of what number ?
| of 3 times 12 is ^ of 7 times j of 8 times -g o. 3
times 12 ; of 36 is 4, and 8 times 4 is 32 ; | of 32 is
8, and 7 times 8 is 56. Therefore f of 3 times 12 is of 56.
6. f of 3 times 12 is how many sevenths of 56 ?
As many sevenths of 56 as % of 56 is contained times
in 8 times i of 3 times 12. 3 times 12 is 36, i of 36 is
4, and 8 times 4 is 32 ; ^ of 5G is 8, and 8 is contained
in 32, 4 times. Therefore f of 3 times 12 is 4 of 56.
7. | of 3 times 12 is 4 times what part of 56 ?
Such a part of 56 as 56 is contained times in \ of 8
times l of 3 times 12 ; 3 times 12 is 36 ; of 36 is 4,
and 8 times 4 is 32 ; \ of 32 is 8, and 56 is contained in
8, -% or \ times. Therefore | of 3 times 12 is 4 times
4, or 4, of 56.
71. 1. -f of 35 is f of how many thirds of 18 ?
Expressed by signs on the blackboard thus :
4x35 = X3 x!8?
Of as many thirds of 1 8 as ^ of 1 8 is contained times
in 9 times i of 4 times 4 of 35 ; i of 35 is 5, and 4 times
5 is 20 ; | of 20 is 4, and 9 times 4 is 36 ; | of 18 is 6,
and 6 is contained in 36, 6 times. Therefore -$ of 35 is
| of 6 thirds of 18.
2. 4 of 35 is f of f of what number ?
4 of 35 is 4 of TJ of 3 times A of 9 times ?- of 4 times
I J O U U
172 APPENDIX.
4 of 35 ; 4 of 35 is 5, and 4 times 5 is 20 ; of 20 is 4,
and 9 times 4 is 3G ; of 30 is 6, and 3 times 6 is 18.
Therefore } of 35 is $ of | of 18.
3. A of 35 is f of 6 times what part of 18 ?
fx35=f x*xl8'?
| of 35 is | of 6 times such a part of 18 as 18 is con-
tained times in J of 9 times -I of 4 times 4 of 35. 4 *
35 is 5, and 4 times 5 is 20 ; -!- of 20 is 4, and 9 times 4
is 36 ; J of 36 is 6, and 1 8 is comaim-d in 6, 7 fi g or -^ times.
Therefore $ of 35 is of 6 times ?, or , of 18.
4. A of 35 is how many ninths of f of 18 ?
4 x35 -gx g x 18?
;* of 35 is as many ninths of {j of 18 as -J of 6 times ^
of 18 is contained times in 4 times 4 f 35. 4 f 35 ^ s
5, and 4 times 5 is 20 ; J of 18 is 6, and 6 times 6 is 36 ;
i of 36 is 4, and 4 is contained in 20, 5 times. There-
fore 4 of 35 is of | of 18.
5. 4 of 35 is 5 times what part of f of 18 ?
f*35=*xf Xl8?
% of 35 is 5 times such a part of | of 18 as 6 times J
of 18 is contained times in of 4 times 4 of 35. 4 ^
35 is 5, and 4 times 5 is 20 ; 'i of 20 is 4 ; -J of 18 is 6,
and 6 times 6, is 36 ; 36 is contained in 4, ^ or times.
Therefore | of 35 is 5 times ^, or f, of | of 18.
6. f of f of 18 is 4 of what number ?
$ of | of 18 is 4 O f 7 times \ of 5 times of 6 times
^ of 18 ; I of 18 is 6, and 6 times 6 is 36 ; l of 36 is c ;
and 5 times 4 is 20 ; \ of 20 is 5, and 7 times 5 is 3&,
Therefore f of | of 18 is | of 35.
7. | of f of 18 is how many sevenths of 35 ?
x| x!8=: 7 x35 ?
f of | of 18 is as many sevenths of 35 as 4 of 35 is
contained times in 5 times ^ of 6 times \ of 18 ; \ of
18 is 6, and 6 times 6 is 36 ; ^ of 36 is 4, and 5 times 4
APPENDIX. 173
is 20 ; 4 of 35 is 5, and 5 is contained in 20, 4 times.
Therefore f of | of 18 is | of 35.
8. f of f of 18 is 4 times what part of 35 ?
x | x 18=A x 35?
| of | of 18 is 4 times such a part of 35 as 35 is con-
tained times in ] of 5 times i of 6 times j of 18. -J of
18 is 6, and 6 times 6 is 36 ; ^ of 36 is 4, and 5 times 4
is 20 ; -j of 20 is 5, and 35 is contained in 5, -/- or 4
times. Therefore f of 1 1 of 4 8 is 4 times 4, or $, of 35.
72. 1. | of 36 is A of how many times f of 42 ?
Of as many times f of 42 as 2 times 4 of 42 is con-
tained tinier in 9 times \ of 4 times ] of 36 ; -J f 36 is !?
and 4 times 32 is 48 ; j of 48 is 12, and 9 times 12 is
108 ; 4 of 42 is 6, and 2 times 6 is 12, and 12 is con-
tained in 108, 9 times. Therefore | Of 36 is | of 9 times
of 42.
2. f of 36 is f of 9 times f of what number ?
-| of 36 is 4 of 9 times f of 7 times of of 9 times |
of 4 times \ of 36 ; -J- of 36 is 12, and 4 times 12 is 48 ; \ of
48 is 12, and 9 times 12 is 108 ; lot* 108 is 12, ^ of 12 is 6,
and 7 times 6 is 42. Therefore | of 36 is | of 9 times $
of 42.
3. f of 36 is f of 9 times how many sevenths of 42 ?
| of 36 is ^ of 9 times as many sevenths of 42 as 4 of
42 is contained times in -J of 9 times | of 4 times J- of 36 ;
\ of 36 is 12, and 4 times 12 is 48 ; | of 48 is 12, 9 times
12 is 108 ; 1 of 108 is 12 ; 4 of 42 is 6, and 6 is con-
tained in 12, 2 times. Therefore - of 36 is 4 of 9 times
f of 42.
4. | of 36 is | of 9 times 2 times what part of 42 ?
| of 36 is | of 9 times 2 times such a part of 42 as 42 is
contained times in \ of \ of 9 times | of 4 times | of 36 ;
\ of 36 is 12, and 4 times 12 is 48 ; \ of 48 is 12, and 9
times 12 is 108 ; of 108 is 12, and -A of 12 is 6 ; 42 is
174 APPENDIX.
contained in 6, {\ 2 or I times. Therefore -j of 36 is of
9 tunes 2 times ,1, or r, of 42.
5. f of 36 is how many ninths of 9 times f of 42 ?
J of 3G is :is many ttint/ts of 9 times -ij of 42, ns -I of
9 times 2 times j of 42 is contained times in 4 times -*
of 36 ; * of 36 is 12, and 4 times 12 is 48 ; -\ of 42 is
6, and 2 times r> is 12 ; 9 times 12 is 108, and ^ of 108 is
12 ; 12 is contained in 48, 4 times. Therefore J of 30 is
$ of 9 times of 42.
6. | of 36 is 4 times what part of 9 times f of 42 ?
>j of 36 is 4 times such a part of 9 times ~ of 42, as
9 times 2 times 1 of 42 is contained times in | of 4 times .',
of 30 ; J of 36 is J2, and 4 times 12 is 48 ; : J of 48 is 12 ;
4 of 42 is 6, 2 times (> is 12, and 9 times 12 is 108;
108 is contained in 12, / r\ or .', times. Therefore -3 of
36 is 4 times, J or , of 9 times -5j of 42.
7. | of 9 times f of 42 is -J- of what number ?
4 of 9 times ~ of 42 is -jj of 3 times | of 4 times J rf 9
times 2 times ,} of 42. 4 of 42 is 6, 2 times 6 is 12, and
<) times 12 is 108 ; ^ of 108 is 12, and 4 times 12 is 48 ;
] of 48 is 12, and 3 times 12 is 36. Therefore of 9
times of 42 is :j of 36.
8. A of 9 times f of 42 is how many thirds of 36 ?
4 of 9 times f of 42 is as many thirds of 36, as { of
36 is contained times in 4 times J of 9 times 2 times ^
of 42; * of 42 is 6, 2 times 6 is 12, and 9 times 12 is
108 ; l of 108 is 12, and 4 times 12 is 48 ; | of 36 is 12,
and 12 is contained in 48, 4 times. Therefore | of 9
times f of 42 is -J of 36.
9. f of 9 times f of 42 is 4 times what part of 36 ?
^ of 9 times i of 42 is 4 times such part of 36 as 36 is
contained times in | of 4 times ^ of 9 times 2 times 4 of
42 ; 4 of 42 is 6, 2 times 6 is 12, and 9 times 12 is 108 ;
l of 108 is 12, and 4 times 12 is 48 ; \ of 48 is 12, and
36 is contained in 12, || or ^ times. Therefore ^ of 9
times of 42 is 4 times, ^ or |, of 36.
INTELLECTUAL ARITHMETIC.
175
MULTIPLICATION TABLE.
Once
1
2
3
4
5
Twice
are 2
are 4
are 6
are A 8
are 10
are 12
Jptimes
are 3
are 6
are 9
are 12
are 15
are 18
^tinies
are 4
are 8
are 12
are . 16
are 20
are 24
5 times
are 5
are 10
are 15
are 20
are 25
are 30
6 times
are 6
are 12
are 18
are 24
are 30
are 36
7 times
are 7
are 14
are 21
are 23
are 35
are 42
8 times
are 8
are 16
are 24
are 32 are 40
are 48
9 times
are 9
are 18 are 27
are 36 are 45
are 54
10 times
are 10
are 20
are 2-f
are 30
are 40 are 50
are 60
11 times
are 11
are 33
"are 3lT
are 44 are 55
are 66
12 times
are 1-2
are 24
are 48
are 60
are 72
Once
7
8
9
10
11
12
Twice
are 14
are 16
are 18
are 20
are 22
are 24
3 times
are 21
are 24
are 27
are 30
are 33
are 36
4 times
are 28
are 32
are 36
are 40
are 44
are 48
5 times
are 35
are 40
are 48
are 45
are 50
are 55
are CO
6 times
are 42
are 54
are 60
are 66
are 72
7 times
are 49
are 5o
are 63
are 70
are 77
are 84
8 times
are 56
are 64
are 72
are 80
are 88
are i)6
9 times
are 63
are 72
are 81
are 90
are 99
are 108
10 times
are 70
are 80
are 90
are 100
are 110
are 120
11 times
are 77
are 88
are 99
are 110
ai-e 121
are 132
.1:3 times
are 84
are 96
are 108
are 120
are 132
are 144
1
2
3
4
5
6
13 times
are 13
are 26
are 39
are 52
are 66
are 78
14 times
are 14
are 28
are 42
are 56
are 70
are 84
15 times
are 15
are 30
are 45
are 60
are 75
are 90
16 times
are 16
are 32
are 48
are 64
are 80
are 96
17 times
are 17
are 34
are 51
are 68 are 85
are 102
18 times
are 16 are 36
are 54
are 72 are 90 are 108
19 times
are 19 are 38
are 57
are 76 are 95 are 114
20 times
are 20 are 40 \ are 60 are 80 are 100 are 12J
21 times
are 21 are 42 i are 63
are 84 are 105 are Iv'J
22 times arc 22 are 44 are 66
are 88 are 110 are 132
23 times are 23 are 46
are 69
are 92 are 115 are 138
24 times are 24 ! are 48
are 72 are 96 1 are 120 are 144
25 times are 25 , are 50 , are 75 j are 100 j are 125 , are 150
176
INTELLECTUAL ARITHMETIC.
7
8
10
11
12
13 times
are 91
are 104
are 117
are 130
are 143
are 156
14 times
are 98
are 112
are 126
are 140
are 154
are 168
15 times
are 105
are 120
are 135
are 150
are 165
are 180
16 times
are 112
are 12d
are 144
are 160
are 176
are 192
17 times
are 119
are 136
are 153
are 17J
are 187
are 204
18 times
are 126
are 144
are 162
are 180
are 198
are 216
19 times
are 133
are 152
are 171
are 190
are 209
are 228
20 times
are 140
are 160
are 180
are 200
are 220
are 240
21 times
are 147
are 168
are 189
are 210
are 231
are 252
22 times
are 154
are 176
are 198
are 220
are 242
are 264
23 times
are 161
are 184
are 207
are 230
are 253
are 276
24 times
are 168
are 192
are 216
are 240
are 264
are 288
25 times
are 17o
are 200
are 225
are 250
are 275
are 300
13
14
15
16
17
18
13 times
are 169
are 182
are 195
are 208
are 22J
are 234
14 times
are 182
are 196
are 210
are 224
aie 238
are 252
15 times
are 195
are 210
are 225
are 240
are 255
are 270
16 times
are 208
are 224
are 240
are 256
are 272
are 288
17 times
are 221
are 238
are 255
are 272
are 289
are 306
18 times
are 234
are 252
are 270
are 288
are 306
are 324
19 times
are 247
are 266
are 285
are 304
are 323
are 342
20 times
are 260
are 280
are 300
are 320
are 340
are 360
21 times
are 273
are 294
are 315
are 336
are 357
are 378
22 times
are 286
are 308
are 330
are 352
are 374
are 396
23 times
are 299
are 322
are 345
are 368
are 391
are 414
24 times
are 312
are 336
are 360
are 384
are 408
are 432
25 tunes
are 325
are 350
are 375
are 400
are 425
are 450
19
20
21
22
23
24
13 times
are 247
are 260
are 273
are 286
are 299
are 312
14 times
are 266
are 280
are 294
are 308
are 322
are 336
15 times
are 285
are 30)
are 315
are 330
are 345
are 360
16 tunes
are 304
are 320
are 336
are 352
are 368
are 384
17 times
are 323
are 340
are 357
are 374
are 391
are 408
18 times
are 342
are 360
are 378
are 396
are 414
are 432
19 times
are 361
are 380
are 399
are 418
are 437
are 456
20 times
are 380
are 400
are 420
are 440
are 460
are 480
21 times
are 399
are 420
are 441
are 462
are 483
are 504
22 times
are 418
are 440
are 462
are 484
are 506
are 528
23 times
are 437
are 4
are 483
are 506
are 529
are 552
24 times
are 456
are 480
are 504
are 528
are 552
are 576
25 times
are 475
are 500
are 525
are 550
are 575
are 600
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