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59. If of a bushel of wheat cost 1% of a dolla. what wil

I bushel cost? A. •

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

61. At $ 62. If

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

63. If of 64. If of

cost?

A. $ A. $2

A. $2,823

A. $5}.

yd. cost $2, what is it a yard? off of $1 buy 20 apples, how many apples will $5 buy? A. 487 apples.

65. If

4. $1,80.

oz. of gold be worth $1,50, what is the cost of 1 oz.?

66 If 16 yds. will make 8 coats, how many yards will it take for 1 coat? A. 27 yds.

67. Ifofof a gallon cost $8, what will 5 gallons cost! A. $9.

68. If 6 yds. cost $5, what will 14 yds. cost? A. $1338. 69. If of cwt. of sugar cost $ro, what will 40 cwt. cost? A. $824.

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

71. If 1 cwt. of flour cost $1%, what will z cwt. cost? A. $1792.

72. If 3 yds. of cloth, that is 24 yds. wide, will make a cloak, how much cloth, that is only yd. wide, will make the same garment?

The narrower the cloth, the more yards it will take; hence we make the greater the second term, thus; † yd. : 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 Fam to keep the money longer than he; therefore, make the middle term. A. $853.

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 of a vessel, sells

$500; what was the whole vessel worth?

of his share for

} of } == ; then, as 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 1 lb. indigo cost $3,84, what will 49,2 lbs. cost at the A. $125,952.

same rate?

77. If buy 59 yds. of cloth, what will $60 buy? 9. 120.

78. How many yds. of cloth can I buy for $75, if 267 yds cost $37? A. 5354 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?

By analysis. If 40 men, in 10 days, reap 200 acres,1 man, in the same time, will reap o of 200 acres, that is, 5 acres in 10 days; and in 1 day, he will reap of 5 acres == an acre a day; then 14 men in 1 day will reap 14 times as much, which is, 14 X 7 acres, 168 acres, Ans.

=

and in 24 days, 24 times 7 acres,

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?

of 60=12, and of 12=4 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 lays? This will give the following proportion, to which, and the next following proportion, the answers, or fourth terms, are to be found by the Rule of Three; thus,

miles;

5 days: 10 Inve: 60 miles : which gives for the fourth te r answer, 120 miles. In the next place. we will consider the differencen nours; then the question will be,

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

120 miles, how many miles will he travel, in the same number days, if he travel 9 hours a day; which will give the following proportion:3 hours: 9 hours: : 120 miles :

which gives for the fourth term, or answer, 360 miles.

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 next 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 multipkers, and divide this result by the product of the divisors; by which process the two statements may be reduced to one; thus, miles;

5 days 10 days

3 hours: 9 hours

:: 60 miles :

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In this example, the product of the multipliers, or second terms, is 9X 10: 90; and the product of the divisors, or first terms, is 3X5= 15; then, 60X96 =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 mate rially, 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 ne multiplica tion of fractions) will, in fact, be the same process as by the other method

Q. From the preceding remarks, what does Compound Pro portion, or Double Rule of Three, appear to be? A. It is find ing 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 is of the same kind or denominatio.

term? A. That which ith the answer.

11. How do you arrange all the remaining terms? 1. Take any two which are of the same kind, and, if the answer ought

than the third term, make the greater the second

to be greate smaller the first; but, if not, make the less the term, and second rm, and the greater the first; then take any other two tens 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 continu ed 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, 8 feet broad, and 5 feet deep, in 12 days?

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12 days : 9 days

In this example, the second terms, 25 X 48 X 8X5X9,; =432000, and the first terms, 15 × 36 X 12 X 6 X 12,466560. Then, the third term, 10 X 432000,= 43200004665609h. 15 m., the fourth term, or Answer OR, by multiplying the third term by the ratios, thus: the ratio of 15 to 25 is = §, of 36 to 48 is, of 12 to 8 is 3, of 6 to 5 is, of 12 to 9 is, whose products, multiplied by the third term, are 5X 4X2 5 X 3 X 10h h. = 9h 158 m., Ans., as before.

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=

6000

648

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 5X4X2X5 X 3 X 10 h. terms, as in T XLI.; thus, the ratios 3X3X 3X6X4 2 X 5 X 10 h. 3 X 6

=

500
54

--=9h 15§ m, Ans., as before.

=

5 X

3X

Let the pupil perform the following examples by the com mon 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 ild in 18 days?

Men 5 20

Days 8: 18}

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In this example, the ratios of 5 to 20, and of 6 to B, are and 4; then, 3 x 4 x 10 rods 120 rods, Ans.

=

The same by analysis. 1 man will build of 10 rods =

in 6 days, and in 1 day of 1= 18; that is, 1 man will build of a rod a day; then, 20 mcn × 18 days × ₫ =120 rods, Ans., as before.

3. If 4 men receive $24 for 6 days' work, how much will 8 men receive for 12 days work? A. $96.

4. If 4 men receive $24 for 6 days' work, how many men may be hired 12 days for $96? A. 8 men.

5. If 8 men, in 12 days, receive $96, how much will 4 men receive for 6 days' work? A. $24.

6. If 8 men receive $96 for 12 days' work, how long may 4 men be hired for $24? A. 6 days.

7. If 9 persons in a family spend $1512 in 1 year (or 12 ms.), how much will 3 of the same persons spend in 4 monus. A. $168.

8. If $2000 will support a garrison of 150 men 3 months, how long will $6000 support 4 times as many men? (The ratios are 3 and; then, 3 x x 3 mo..) A. 24 mo.

9. If $100 gain $6 in 1 year, what will $900 gain in 8 months? A. $36.

10. If $100 gain $6 in 1 year, in what time will $900 gain $36? A. 8 months."

11. If the transportation of 12 cwt. qrs., for 400 miles, cost$57,12, what will the transportation of 10 tons, for 75 miles, amount to? A. $168.

12. An usurer put out $150 at interest, and when it had been on interest 8 months, he received, for principal and interest, $160; at what rate per cent. per annum did he receive in

terest?

By cancelling the ratios and, the third term will be the answer. A. 10 per cent.

Questions on the foregoing.

1. What will 2 yds. of cloth cost, at 50 cents (or $4) a yard? What will 10 yds.? What will 100 yds.? What will 5 yds. ? What will 9 yds.?

2. At 25 cents (or $1) a yard, what will 4 yds. cost? What will 12 yds.? What will 40 yds.? What will 300 yds. ?

3. At $,331 (or $), what will 6 yds. cost? What will 9 yds. Will 24 yds.? Will 300 yds.? Will 7 yds.? Will 25. vds.?

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