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same in 6 days ; in what time can they both do the work if
they labor together ?

30. If 6 men can do a piece of work in 10 days, how long
will it take 5 men to do it?

ANALYSIS. If G men can do a piece of work in 10 days, 1 man
will require 6 times as long, or 60 days to do the same work
Five men will require but one fifth as long as one man or 60^-5
=-12 days.



ANALYSIS 207

OPERATION.



10x6-^5=12 days.



6



Ans. | 12 days.



31. Three men together can perform a piece of work in 9
days. A alone can do it in 18 days, B in 27 days ; in what
time can C do it alone ?

32. A and B can build a wall on one side of a square
piece of ground in 3 days ; A and C in 4 days ; B and C in
6 days : what time will they require, working together, to
complete the wall enclosing the square ?

33. Three men hire a pasture, for which they pay 66 dol-
lars. The first puts in 2 horses 3 weeks ; the second 6 horses
for 2J weeks; the third 9 horses for 1J weeks: how much
ought eaeh to pay ?

ANALYSIS. The pasturage of 2 horses for 3 weeks, would he the
same as the pasturage of 1 horse 2 times 3 weeks, or 6 weeks ;
that of six horses 2^ weeks, the same as for 1 horse 6 times 2
weeks, or 15 weeks ; and that of 9 horses 1^ weeks, the same as
1 horse for 9 times H weeks, or 12 weeks. The three persons had
an equivalent for the pasturage of 1 horse for 6+15-f 12 -33 weeks ;
therefore, the first must pay ^j, the second i, and the third
41 of 66 dollars

OPERATION.

3 x2=6; then $66x T r =$12. 1st
21x6=15; " $66 x J$ =$30. 2d.
Ijx9 = 12; " $66 x if =$24. 3d.

34. Two persons, A and B, cuter into partnership, and gain
$175. A puts in 75 dollars for 4 months, and B puts in 100
dollars for 6 months : what is each one's share of the gain ?

35. Three men engage to build a house for 580 dollars.
The first one employed 4 hands, the second 5 hands, and the
third 7 hands. The first man's hands worked three times as
many days as the third, and the second man's hands twice as
many days as the third man's hands : how much must each
receive ?



208



ANALYSIS.



36. If 8 students spend $192 in 6 months, how much will
12 students spend in 20 months ?

ANALYSIS. Since 8 students spend $192, one student will spend
i of $192, in 6 months , in 1 month 1 student will spend -^ of
of $192- $4. Twelve students will spend, in 1 month, 12 times
as much as 1 student, and in 20 months they will spend 20 times
as much as in 1 month.



OPERATION.



24 2

-w i i n 20

XXTXyXY=$960.



48



20



$960. Ans.



31 If 6 men can build a wall 80 feet long, 6 feet wide,
and 4 feet high, in 15 days, in what time can 18 men build
one 240 feet long, 8 feet wide, and 6 feet high ?

ANALYSIS. Since it takes 6 men 15 days to build a wall, it
will take 1 man 6 times 15 days, or 90 days, to build the same
wall. To build a wall 1 foot long, will require - 8 \ r as long as to
build one 80 feet long ; to build one 1 foot wide, i as long as to
build one 4 feet wide ; and to build one 1 foot high, as long as
to build one 6 feet high, 18 men can build the same wall in ^
of the time that one man can build it : but to build one 240 feet
long, will take them 240 times as long as to build one 1 foot in
length ; to build one 8 feet wide, 8 limes as long as to build one
1 foot wide, and to build one C feet high, 6 times as long as to
build one 1 foot high.



OPERATION.



$ 2

15x0 1 1 1 1 &<0 $ $0

~~I X $0 X >I X X ;F$ X ~T x ;r x I ^

* *,!



15



Ans. i 30 days.

38 If 96/6s. of bread be sufficient to serve 5 men 12 days,
how many days will 57/6. serve 19 men?



ANALYSIS. 209

39. If a man travel 220 miles in 10 days, travelling 12
hours a day, in how many days will he travel 880 miles,
travelling 16 hours a day?

40. If a family of 12 persons consume a certain quantity
of provisions in 6 days, how long will the same provisions
last a family of 8 persons ?

41. If 9 men pay $135 for 5 weeks' board, how much
must 8 men pay for 4 weeks' board ?

42. If 10 bushels of wheat are equal to 40 bushels of
corn, and 28 bushels of corn to 56 pounds of butter, and 39
pounds of butter to 1 cord of wood ; how much wheat is 12
cords of wood worth ?

ANALYSIS. Since 10 bushels of wheat are worth 40 bushels of
corn, 1 bushel of corn is worth > of 10 bushels of wheat, or
i of a bushel ; 28 bushels are worth 28 times of a bushel of
wheat, or 7 bushels : since 28 bushels of corn, or 7 bushels of
wheat are*worth 56 pounds of butter, 1 pound of butter is worth
^g of 7=i of a bushel of wheat, and 39 pounds are worth 39
times as much as 1 pound, or 39*^=^ bushels of wheat; and
since 39 pounds of butter, or ^ bushels of wheat are worth 1 cord
of wood, 12 cords are worth 12 times as much, or 12x^=58
bushels.

OPERATION.

3

ro i n i 39 xt



V

rf A />> -|
V v i

2



39 o

n 3



117=5816^.



NOTE. Always commence analysing from the term which is
of the same name or kind as the required answer.

43. If 35 women can do as much work as 20 boys, and
16 boys can do as much as 7 men : how many women can
do the work of 18 men ?

44. If 36 shillings in New York, are equal to 27 shillings
in Massachusetts, and 24 shillings in Massachusetts are equal
to 30 shillings in Pennsylvania, and 45 shillings in Pennsyl-
vania are equal to 28 shillings in Georgia ; how many shil-
lings in Georgia are equal to 72 shillings in New York ?

14



210 PROMISCUOUS EXAMPLES



PROMISCUOUS EXAMPLES IN ANALYSIS.

1. How many sheep at 4 dollars a head must I give for 6
cows, worth 12 dollars apiece ?

2. If 7 yards of cloth cost $49, what will 16 yards cost ?

3. If 36 men can build a house in 16 days, how long will
it take 12 men to build it?

4. If 3 pounds of butter cost 7J shillings, what will 12
pounds cost ?

5. If 5 1 bushels of potatoes cost $2f, how much will 12 J
bushels cost ?

6. How many barrels of apples, worth 1 2 shillings a barrel,
will pay for 16 yards of cloth, worth 9s. Qd. a yard ?

7. If 31 J gallons of molasses are worth $9f , what are 5J
gallons worth ?

8. What is the value of 24| bushels of corn, at 5s. *ld. a
bushel, New York currency ?

9. How much rye, at 8s. Zd. per bushel, must be given
for 40 gallons of whisky, worth 2s. 9d. a gallon?

10. If it take 44 yards of carpeting, that is 1 J yards wide,
to cover a floor, how many yards of yards wide, will it
take to cover the same floor ?

11. If a piece of wall paper, 14 yards long and 1J feet
wide, will cover a certain piece of wall, how long must an-
other piece be, that is 2 feet wide, to cover the same wall ?

12. If 5 men spend $200 in 160 days, how long will $300
last 12 men at the same rate ?

13. If 1 acre of land cost of f of of $50, what will 3|
acres cost ?

14. Three carpenters can finish a house in 2 months ; two
of them can do it in 2J months : how long will it take the
third to do it alone ?

15. Three persons bought 2 barrels of flour for 15 dollars.
The first one ate from them 2 months, the second 3 months,
and the third 7 months : how much should each pay ?

16. What quantity of beer will serve 4 persons 18| days,
if 6 persons drink 7 gallons in 4 days ?



IN ANALYSIS. 211

17. If 9 persons use If pounds of tea in a month, how
much will 10 persons use in a year ?

18. If | of f of a gallon of wine cost f of a dollar, what
will 5 J gallons cost ?

19. How many yards of carpeting, 1| yards wide, will it
take to cover a floor that is 4f yards wide and 6 and three-
fifths yards long ?

20. Three persons bought a hogshead of sugar containing
413 pounds. The first paid $2J as often as the second paid
$3 J, and as often as the third paid $4 : what was each one's
share of the sugar ?

21. A, with the assistance of B, can build a wall 2 feet
wide, 3 feet high, and 30 feet long, in 4 days ; but with the
assistance of C, they can do it in 3 1 days : in how many days
can C do it alone ?

22. If two persons engage in a business, where one advances
$875, aritt the other $625, and they gain $300, what is each
one's share.

23. A person purchased f of a vessel, and divided it into 5
equal shares, and sold each of those shares for $1200 : what
was the value of the whole vessel ?

24. How many yards of paper, f of a yard wide, will be
sufficient to paper a room 10 yards square and 3 yards high ?

25. What will be the cost of 45#>s. of coffee, New Jersey
currency, if 9?6s. cost 27 shillings ?

26. What will be the cost of 3 barrels of sugar, each weigh-
ing %cwt. at 10c?. per pound, Illinois currency?

27. If 12 men reap 80 acres in 6 days, in how many days
will 25 men reap 200 acres ?

28. If 4 men are paid 24 dollars for 3 days' labor, how
many men may be employed 16 days for $96 ?

29. If $25 will supply a family with flour at $7.50 a bar-
rel for 2 months, how long would $45 last the same family
when flour is worth $6.75 per barrel ?

30. A wall to be built to the height of 27 feet, was raised
to the height of 9 feet by 1 2 men in 6 days : how many men
must be employed to finish the wall in 4 days at the same
rate of working ?



212 PROMISCUOUS EXAMPLES.

31. A, B and C, sent a drove of hogs to market, of which
A owned 105, B 75, and C 120. On the way 60 died :
how many must each lose ?

32. Three men, A, B and C, agree to do a piece of work,
for which they are to receive $315. A works 8 days, 10 J
hours a day ; B 9 j days, 8 hours a day ; and C, 4 days, 12
hours a day : what is each one's share ?

33. If 1 barrels of apples will pay for 5 cords of wood,
and 12 cords of wood for 4 tons of hay, how many barrels of
apples will pay for 9 tons of hay ?

34. Out of a cistern that is f full is drawn 140 gallons,
when it is found to be \ full : how much does it hold ?

35. If .7 of a gallon of wine cost $2.25, what will .25 of a
gallon cost ?

36. If it take 5.1 yards of cloth, 1.25 yards wide, to make a
gentleman's cloak, how much surge, f yards wide, will be
required to line it ?

37. A and B have the same income. A saves | of his
annually ; but B, by spending $200 a year more than A, at
the end of 5 years find himself $160 in debt : what is their
income ?

38. A father gave his younger son $420, which was | of
what he gave to his elder s.on ; and 3 times the elder son's
portion was \ the value of the father's estate : what was the
value of the estate ?

39. Divide $176.40 among 3 persons, so that the first shall
have twice as much as the second, and the third three times
as much as the first : what is each one's share ?

40. A gentleman having a purse of money, gave \ of it for
a span of horses ; of of the remainder for a carriage :
when he found that he had but $100 left : how much was in
his purse before any was taken out ?

41. A merchant tailor bought a number of pieces of cloth,
each containing 25^ yards, at the rate of 3 yards for 4 dol-
lars, and sold them at the rate of 5 yards for 13 dollars, and
gained by the operation 96 dollars : how many pieces did he
buy?



RATIO AND PROPORTION. 213



RATIO AND PROPORTION.

221. Two numbers having the same unit, may be com-
pared in two ways :

1st. By considering how much one is greater or less than
the other, which is shown by their difference ; and,

3d. By considering how many times one is contained in the
other, which is shown by their quotient.

In comparing two numbers, one with the other, by means
of their difference, the less is always taken from the greater.

In comparing two numbers, one with the other, by means
of their quotient, one of them must be regarded as a standard
which measures the other, and the quotient which arises by
dividing by the standard, is called the ratio.

222. Every ratio is derived from two numbers : the first
is called the antecedent, and the second the consequent: each
is called a term, and the two, taken together, are called a
couplet. The antecedent will be regarded as the standard.

If the numbers 3 and 12 be compared by their difference,
the result of the comparison will be 9 ; for, 12 exceeds 3 by 9.
If they are compared by means of their quotient, the result
will be 4 ; for, 3 is contained in 12, 4 tunes : that is,
3 measuring 12, gives 4.

223. The ratio of one number to another is expressed in
two ways :

1st. By a colon ; thus, 3 : 12 ; and is read, 3 is to 12 ; or,

o measuring 12.

12

2d. In a fractional form, as; or, 3 measuring 12.



231. In how many ways may two numbers, having the same unit, be
compared with each other ? If you compare by their difference, how do
you find it ? If you compare by the quotient, how do you regard one of
the numbers ? What is the ratio ?

222. From how many terms is a ratio derived ? What is the first
term called ? What is the second called ? Which is the standard ?

2~53. How may the ratio of two numbers be expressed ? How read ?



214 RATIO AND PROPORTION.

224. If two couplets have the same ratio, their terms are
said to be proportional : the couplets

3 : 12 and 1 : 4

have the same ratio 4 ; hence, the terms arc proportional,
and are written,

3 : 12 : : 1 : 4

by simply placing a double colon between the couplets. The
terms are read

3 is to 12 as 1 is to 4,
and taken together, they are called a proportion : hence,

A proportion is a comparison of the terms of two equal
ratios*

224. If two couplets have the same ratio, what is said of the terms ?
How are they written V How read ? What is a proportion ?

* Some authors, of high authority, make the consequent the stand-
ard and divide the antecedent by it to determine the ratio of the couplet.

The ratio 3 : 13 is the same as that of 1:4 by both methods ;
for, if the antecedent be made the standard, the ratio is 4 ; if the conse-
quent be made the standard, the ratio is one-fourth. The question is,
which method should be adopted V

The unit 1 is the number from which all other numbers are derived,
and by which they are measured.

The question is, how do we most readily apprehend and express the
relation between 1 and 4 ? Ask a child, and he will answer, "the dif-
ference is 3." But when you ask him, "how many 1's are there in
4V" he will answer, "4," using 1 as the standard.

Thus, we begin to teach by using the standard 1 : that is, by dividing
4byl.

Now, the relation between 3 and 13 is the same as that between 1
and 4; if then, we divide 4 by 1, we must also divide 13 by 3. Do we,
indeed, clearly apprehend the ratio of 3 to 12, until we have referred to
1 as a standard ? Is the mind satisfied until it has clearly perceived that
the ratio of 3 to 13 is the same as that of 1 to 4 ?

In the Rule of Three we always look for the result in the 4th term.
Now, if we wish to find the ratio of 3 to 13, by referring to 1 as a stand-
ard, we have

3 : 13 : : 1 : ratio,

which brings the result in the right place.

But if we define ratio to be the antecedent divided by the consequent,
we should have

3 : 12 : : ratio : 1,

which would bring the ratio, or required number, in the 3d place,



RATIO AND PROPORTION. 215

What are the ratios of the proportions,

3 : 9 : : 12 : 36?
2 : 10 : : 12 : 60?

4 : 2 : : 8 : 4?
9 : 1 : < 90 : 10?

225. The 1st and 4th ter-ms of a proportion are called the
extremes : the 2d and 3d terms, the means. Thus, in the pro-
portion,

3 : 12 : : 6 : 24

3 and 24 are the extremes, and 12 and 6 the means:

12 24

Since (Art. 224), Y^lp

we shall have, by reducing to a common denominator,
12x6_24x3
!Tx~6~ 6x3'

But since the fractions are equal, and have the same deno-
minators, their numerators must be equal, viz. ;

12x6=24x3; that is,

In any proportion, the product of the extremes is equal to
the product of the means.

Thus, in the proportions,

1 : 6 : : 2 : 12 ; we have 1 x 12= 6x2;
4 : 12 : : S : 24 ; " " 4x24 = 12x8.

220. Since, in any proportion, the product of the extremes
is equal to the product of the means, it follows that,

In all cases, the numerical value of a quantity is the number of times
which that quantity contains an assumed standard, called its unit of



If we would find that numerical value, in its right place, we must
say,

standard : quantity : : 1 : numerical value :
but if we take the other method, we have

quantity : standard : : numerical value : 1,
which brings the numerical value in the wrong place.



216 RATIO AND PROPORTION".

1st. If the product of the means be divided by one of the
extremes, the quotient will be the other extreme.

Thus, in the proportion

3 : 12 : : 6: 24, we have 3 x 24 = 12 x 6 ;

then, if 12, the product of the means, be divided by one of
the extremes, 3, the quotient will be the other extreme, 24 :
or, if the product be divided by 24, the quotient will be 3.

2d. If the product of the extreme? be divided by either of
the means, the quotient ivill be the other mean.

Thus, if 3 x 23=12 x 6 = 72 be divided by 12, the quotient
will be 6 or if it be divided by 6, the quotient will be 12.

EXAMPLES.

1. The first three terms of a proportion are 3, 9 and 12 :
what is the fourth term ?

2 The first three terms of a proportion are 4, 16 and 15 :
what is the 4th term ?

3. The first, second, and fourth terms of a proportion are
6, 12 and 24 : what is the third term ?

4. The second, third, and fourth terms of a proportion are
9, 6 and 24 : what is the first term ?

5. The first, second and fourth terms are 9, 18 and 48 :
what is the third term ?

227. Simple and Compound Eatio.

The ratio of two single numbers is called a Simple Eatio,
.and the proportion which arises from the equality of two such
ratios, a Simple Proportion.



225. Which are the extremes of a proportion ? Which the means ?
What is the product of the extremes equal to ?

226. If the product of the means be divided hy one of the extremes,
what will the quotient be ? If the product of the means be divided by
either extreme, what will the quotient be ?

227. What is a simple ratio ? What is the proportion called which
comes from the equality of two simple ratios? What is a compound

ratio ? What is a compound proportion ?



RATIO AND PROPORTION. 217

If the terms of one ratio be multiplied by the terms of an-
other, antecedent by antecedent and consequent by conse-
quent, the ratio of the products is called a Compound Ratio-
Thus,^if the two ratios

3 : 6 and 4 : 12

be multiplied together, we shall have the compound ratio
3x4 : 6x12, or 12 : 72 ;

In which the ratio is equal to the product of the simple
ratios.

A proportion formed from the equality of two compound
ratios, or from the equality of a compound ratio and a simple
ratio, is called a Compound Proportion.

228. What part one number is of another.

When the standard, or antecedent, is greater than the
number which it measures, the ratio is a proper fraction,
and is such a part of 1, as the number measured is of the
standard.

1. What part of 12 is 3 ? that is, what part of the stand-
ard 12, is 3 ?



12 : 3 : : 1 : I;
that is, the number measured is one-fourth of the standard.



2. What part of 9 is 2 ?

3. What part of 16 is 4?

4. What part of 100 is 20 ?

5. What part of 300 is 200 ?

6. What part of 36 is 144 ?



7. 3 is what part of 12 ?



8. 5 is what part of 20 ?

9. 8 is what part of 56 ?

10. 9 is what part of 8 ?

11. 12 is what part of 132 ?



NOTE. The standard is generally preceded by the word of, and
in comparing numbers, may be named second, as in examples 7,
8, 1), 10 and 11, but it must be always be used as a divisor, and
should be placed first in the statement.



238. When the standard is greater than the consequent, how may
the ratio be compared ? What part is 3 of 1 ? 5 of 1 ? What part is
4 of 2 ? 12 of 3 ? 7 of 5 ?



218



SINGLE RULE OF THREE.



SINGLE RULE OF THREE.

229. The Single Rule of Three is an application of the
principle of simple ratios. Three numbers are always given
aixl a fourth required. The ratio between two of the given
numbers is the same as that between the third and the required
number.



1. If 3 yards of cloth cost $12, what will 6 yards cost at the
same rate ?

NOTE. We shall denote the required term of tlie proportion by
the letter x.



STATEMENT.

yd. yd. $
3 : 6 : : 12

OPERATION.
12 o




: x



ANALYSIS. The condition, " at the same
rate," requires that the quantity 3 yards
must have the same ratio to the quantity 6
yards, as $12, the cost of 3 yards, to x dol-
lars, the cost of 12 yards.

Since the product of the two extremes is
equal to the product of the two means, (Art.
235), 3xz=Gxl2; and if 3x^=6x12, x
must be equal to this product divided by 3 : A^ C J-AQA
that is,

The 4th term is equal to the product of the second and third
terms divided by the first.

2. If 56 dollars will buy 14 yards of broadcloth, how many
yards, at the same rate, can be bought for 84 dollars ?



ANALYSIS. Fifty-six dollars, (being
the cost of 14 yards of cloth), has the
same ratio to $84, as 14 yards has to the
number of yards which $84 will buy

NOTE. When the vertical line is used,
the required term, (which is denoted by
a;), is written on the left



STATEMENT.

$ $ yd. yd.
56 : 84 : : 14 : x

OPERATION



21



229. What is the Single Rule of Three ? How many numbers are
fivcn ? How many required ? What ratio exists between two of the
given numbers ?



SINGLE RULE OF THREE. 219

230. Hence, we have the following

RULE I. Write the number which is of the same kind with
the answer for the third term, the number named in connection
with it for the first term, and the remaining number for the
second term.

II. Multiply the second and third terms together, and divide
the product by the first term : Or,

Multiply the third term by the ratio of the first and second.

NOTES. 1. If the first and second terms have different units,
they must be reduced to the same unit.

2. If the third term is a compound denominate number, it must
be reduced to its smallest unit.

3. The preparation of the terms, and writing them in their pro-
per places, is called the statement.

EXAMPLES.

1. If I can walk 84 miles in 3 days, how far can I walk in
11 days?

2. If 4 hats cost $12, what will be the cost of 55 hats at
the same rate ?

3. If 40 yards of cloth cost $170, what will 325 yards cost
at the same rate ?

4. If 240 sheep produce 660 pounds of wool, how many
pounds will be obtained from 1200 sheep?

5 If 2 gallons of molasses cost 65 cents, what will 3 hogs-
heads cost ?

6. If a man travels at the rate of 210 miles in 6 days, how
far will he travel in a year, supposing him not to travel on
Sundays ?

7. If 4 yards of cloth cost $13, what will be the cost of 3
pieces, each containing 25 yards ?

8. If 48 yards of cloth cost $67.25, what will 144 yards
cost at the same rate ?

9. If 3 common steps, or paces, are equal to 2 yards, how
many yards are there in 1 60 paces ?

10. If 750 men require 22500 rations of bread for a month,
how many rations will a garrison of 1200 men require ?

235. Give the rule for the statement. Give the rule for finding the
fourth term.



220 SINGLE RULE OF THREE.

11. A cistern containing 200 gallons is filled by a pipe
which discharges 3 gallons in 5 minutes ; but the cistern has
a leak which empties at the rate of 1 gallon in 5 minutes.
If the water begins to run in when the cistern is empty, how
long will it run before filling the cistern ?

12. If 14| yards of cloth cost $19*, how much will 19 J
yards cost ?

NOTE. First make the STATEMENT.

statement ; then change tlio yd. yd, $ $

mixed numbers to im- \\ : \C)1 . : IQi : %

proper fractions, after
which arrange the terms,
and cancel equal factors
according to previous in-

struction.



13. If - of a yard of cloth cost - of a dollar, what will
2 \ yards cost?

14. If y\ of a ship cost 273 2s. Qd., what will ^ of her
cost ?

15. If 1 T 4 T bushels of wheat cost $2*, how much will 60
bushels cost ?

16. If 4| yards of cloth cost $9.15, what will 13| yards
cost?

17. If a post 8 feet high cast a shadow 12 feet in length,
what must be the height of a tree that casts a shadow 122
feet in length, at the same time of day ?

18. If ^cwt. Iqr. of sugar cost $64.96, what will be the
cost of kcwt. 2qr. ?

19. A merchant failing in trade, pays 65 cents for every
dollar which he owes : he owes A $2750, and B $1975 :
how much does he pay each ?

20. If 6 sheep cost $15, and a lamb costs one-third as
much as a sheep, what will 27 lambs cost?

21. If 2/6s. of beef cost J of a dollar, what will 30/6*.
cost?

22. If 4-J- gallons of molasses cost $2f , how much is it per
quart ?

23. A man receives f of his income, and finds it equal to
$3724.16 : how much is his whole income ?



SINGLE RULE OF THREE. 221

24. If 4 barrels of flour cost $34 f, how much can be


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