54. A borrowed of B $600 for 3 years; how long ought A to lend B $800 to requite the favour? (2-3) How long ought he to lend him $900? (2) How long $500? (3-7-6) How long $1200? (1-6) A. 9 years, 4 mo. 6 days.

55. A gentleman bought 3 yards of broadcloth 1 yards wide; how many yards of flannel, which is only yd. wide, will line the same?

It is evident it will take more cloth which is only yd. wide, than if it were 11⁄2 yd. wide; hence 11⁄2 must be the middle term. A. 6 yds. Ratio, 2.

56. A regiment of soldiers, consisting of 800 men, are to be clothed, each suit containing 4 yds. of cloth, which is 14 yd. wide, and lined with flannel yd. wide; how many yards of flannel will be sufficient to line all the suits?

A. 8633 yds. 1 qr. 1} na.

FRACTIONS. 57. If of a barrel of flour cost of a dollar, what will of a barrel cost?

By analysis. It is plain, that if we knew the price of 1 barrel, of a barrel would cost as much. If of a barrel cost of a

8

dollar, §, or 1 barrel, will cost 8 times as much, that is,

OR, as is more than }, we may make the 2d, or multiplying

term, as in the foregoing examples, thus :—

bbls.bbls. $

금: 2F6. Then,

(Inverting § by ¶ XLVII.

$13, Ans.

64 X 1

64

6 X 5

30

= $13, Ans., as be

OR, multiplying by the ratio, thus; the ratio of 1 to 4 is ÷

} = 2 = 6, ratio ; then,

fore.

OR, which is obviously the same, having inverted the 1st, or dividing term, multiply all the fractions together; that is, ceed as in Division of Fractions, (¶ XLVII.) thus, 1×4×16 8 X 3X pro

120 =$17, Ans., as before.

64

The pupil may perform the following examples by either of the preceding methods, but the one by analysis is recommended, it being the best exercise for the mind.

58. If 3 lbs. of butter cost of a dollar, what cost ✈ lb. ?

3

59. If of a bushel of wheat cost of a dollar, what will

60. If 13 yds. of cloth cost $12, what will 1 yd. cost?

A. $1.

a pound, what will 40 pounds cost? A. $24. cost $225, what will 1 yd.

cost?

A. $2,823.

63. If of yd. cost $2, what is it a yard?

A. $5. 64. If 2 of off of $1 buy 20 apples, how many apples 3 will $5 buy? A. 4872 apples.

65. If & oz. of gold be worth $1,50, what is the cost of 1 oz.? A. $1,80.

66 If 16 yds. will make 8 coats, how many yards will it take for 1 coat? A. 264 yds.

67. If of of a gallon cost $, what will 5 gallons cost? A. $9.

68. If 6 yds. cost $53, what will 142 yds. cost? A. $1323. 69. If of cwt. of sugar cost $1, what will 40 cwt. cost? A. $824.

70. If 4 yd. of silk cost of $3. what is the price of 50 yds.? A. $314.

5

71. If 1 cwt. of flour cost $15, what will 12 cwt. cost? A. $1732.

75

72. It 3 yds. of cloth, that is 2 yds. wide, will make a cloak, how much cloth, that is only yd. wide, will make the samne garment?

The narrower the cloth, the more yards it will take; hence we make the greater the second term, thus; yl. : 24 yds. : : 3 yds.: 10 yds., Ans. 73. If I lend my friend $960 for of a year, how much ought he to lend me of a year to requite the favour?

He ought not to lend me so much as I lend him, because I am to keep the money longer than he; therefore, make the middle term. A. $8534. 74. If 12 men do a piece of work in 124 days, how many men will do the same in 6 days? A. 24 men. Ratio, 2.

75. A merchant, owning

$500; what was the whole vessel worth?

of a vessel, sells

; then, as

of his share for

of the vessel is $500, † is $250, and, or the whole vessel, is 5 × 250 $1250.

Or thus; of: 1: 500: $1250, Ans., as before.

76. If 14 lb. indigo cost $3,84, what will 49,2 lbs. cost at the same rate? A. $125,952.

77. If $293 buy 59 yds. of cloth, what will $60 buy? A. 120 yds.

78. How many yds. of cloth can I buy for $75, if 2673 yds. cost $373? A. 535 yds. Ratio, 2.

COMPOUND PROPORTION.

▼ LXXIV. 1. If 40 men, in 10 days, can reap 200 acres of grain, how many acres can 14 men reap in 24 days?

5

By analysis. If 40 men, in 10 duys, reap 200 acres, 1 man, in the same time, will reaps of 200 acres, that is, 5 acres in 10 days; and in 1 day, he will reap 1 of 5 acres = 1 = 1 an acre a day; then 14 men in 1 day will reap 14 times as much chich is, 14 X 7 acres; and in 24 days, 24 times 7 acres, =

165 acres, Ans.

Perform the following sums in the same manner.

;

2. If 4 men mow 96 acres in 12 days, how many acres can 8 men mow in 16 days?

First find how many acres 1 man will mow in 12 days; then, in 1 day. A. 256 acres.

3. If a family of 8 persons, in 24 months, spend $480, how much would 16 persons spend in 8 months? A. $320.

4. If a man travel 60 miles in 5 days, travelling 3 hours each day, how far will he travel in 10 days, travelling 9 hours each day?

डै of 60 12, and of 124 miles, the distance which he travels in 1 hour; then, 4 miles × 9 hours = = 36 × 10 days 360 miles, the Ans.

It will oftentimes be found convenient to make a statement,. as in Simple Proportion. Take the last example.-In solving this question, we found the answer, which is miles, depended on two circumstances; the number of days which the man travels, and the number of hours he travels each day.

Let us, in the first place, find how far he would go in 5 days, supposing he travelled the same number of hours each day. The question will then be :

If a man travel 60 miles in 5 days, how many miles will he travel in 10 days? This will give the following proportion, to which, and the next following proportion, the auswers, or fourth terms, are to be found by the Rule of Three; thus,

5 days 10 days: 60 miles :

miles ;

which gives for the fourth term, or answer, 120 miles. In the next place, we will consider the difference in hours; then the question will be,

If a man, by travelling 3 hours a day for a certain number of days, travel

120 miles, how many miles will he travel, in the same number of days, if ke travel 9 hours a day; which will give the following proportion:

3 hours: 9 hours: 120 miles : miles;

which gives for the fourth term, or answer, 350 miles.

in performing the foregoing examples, we, in the first.operation, multiplied 60 by 10, and divided the product by 5, making 120. In the noxt operation, we multiplied 120 by 9, and divided the product by 3, making 360, the answer. But, which is precisely the same thing, we may multiply the 60 by the product of the multipliers, and divide this result by the product of the divisors; by which process the two statements may be reduced to one; thus,

5 days 10 days

3 hours:

9 hours

In this example, the product of the multipliers, or second terms, is 9 X 10 = 99; and the product of the divisors, or first terms, is 3 × 5 = 15; then, 60 X 90 5400 15360 miles, the Ans., as before.

Note. It will be recollected, that the ratio of any two terms is the second divided by the first, expressed either as a fraction, or by its equal whole number.

OR, by comparing the different terms, we see that 60 miles has the same proportion to the fourth term, or answer, that 5 days has to 10 days, and that 3 hours has to 9 hours; hence we may abbreviate the process, as in Simple Proportion, by multiplying the third terms by the ratio of the other terms, thus:

=

The ratio of 5 to 10 is 2, and of 3 to 9 is 3. But multiplying 60 miles by the product of the ratios 2 and 3, that is, 6, is the same as multiplying 60 by them separately; then, 6 X 60 360 miles, Ans., as before.

Note. This method, in most cases, will shorten the process very materially, and in no case will it be any longer; for, when the ratios are fractions, multiplying the third term by them (according to the rule for the multiplication of fractions) will, in fact, be the same process as by the other method.

Q. From the preceding remarks, what does Compound Proportion, or Double Rule of Three, appear to be? A. It is finding the answer to such questions as would require two or more statements in Simple Proportion; or, in other words, it is when the relation of the quantity required, to the given quantity of the same kind, depends on several circumstances combined.

Q. The last question was solved by multiplying the third term by the product of the ratios of the other terms; what, then, may the product be called, which results from multiplying two or more ratios together? A. Compound Ratio.

From the preceding remarks we derive the following

RULE.

I. What number do you make the third term? A. That which is of the same kind or denomination with the answer.

II. How do you arrange all the remaining terms? A. Take any two which are of the same kind, and, if the answer ought

to be greater than the third terin, make the greater the second term, and the smaller the first; but, if not, make the less the second term, and the greater the first; then take any other two terms of the same kind, and arrange them in like manner, and so on till all the terms are used; that is, proceed according to the directions for stating in Simple Proportion.

III. How do you proceed next? A. Multiply the third term by the continued product of the second terms, and divide the result by the continued product of the first terms; the quotient will be the fourth term, or answer.

IV. How may the operation, in most cases, be materially shortened? A. By multiplying the third term by the continued product of the ratios of the other terms.

More Exercises for the Slate.

1. If 25 men, by working 10 hours a day, can dig a trench 36 feet long, 12 feet broad, and 6 feet deep, in 9 days, how many hours a day must 15 men work, in order to dig a trench 48 feet long, & feet broad, and 5 feet deep, in 12 days?

In this example, the second terms, 25 X 48 X 8 X 5 X 9,

432000, and

the first terms, 15 × 36 × 12 × 6 × 12, 466550. Then, the third term, 10 X 432000, = = 43200004665609h. 153 m., the fourth term, or Answer OR, by multiplying the third term by the ratios, thus: the ratio of 15 to 25 is 25 , of 36 to 48 is ‡, of 12 to 8 is, of

6 to 5 is, of 12 to 9 is, whose products, multiplied by the 5 X X 6000 2 X 5 X 3 X 10h. 3X3 X3 X 6 X 4

third term, are

Ans., as before..

648

9 h. 153 m.,

This method, it will be perceived, is much shorter than the former. But, had we selected terms whose ratios would be whole numbers, the process would have been shorter still, as is the case in the next question. The present example, however, may be rendered more simple by rejecting equal 5 X 4 X 2 X 5 X 3 X 10 h. 5 X terms, as in T XLI.; thus, the ratios 3 × 3 × 3 × 6 × 4

2 X 5 X 10 h. 500 —9h. 15† m., Ans., as before.

3 X 6

54

3 X

Let the pupil perform the following examples by the common rule of proportion first, then by multiplying by the ratio, and lastly by analysis.

2. If 5 men can build 10 rods of wall in 6 days, how many rods can 20 men build in 18 days?