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

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Or, as is more than 1, we may make the 2d, or multiply ing term, as in the foregoing examples, thus

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Or, multiplying by the ratio, thus; the ratio of} to is ÷

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Or, which is obviously the same, having inverted the 1st, or dividing term, multiply all the fractions together; that is, proceed as in Division of Fractions, (¶XLVII.) thus,

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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.? A. $16. 59. If of a bushel of wheat cost

bushel cost? A. $16.

62. If yd.

of a dollar, what will 1

A. $2,823.

60. If 1 yds. of cloth cost $12, what will 1 yd. cost? A. 61. At $ a pound, what will 40 pounds cost? A. $24. cost $22, what will 1 yd. cost? 63. If of yd. cost $2, what is it a yard? A. $51. 64. If of off of $1 buy 20 apples, how many apples will $5 buy? A. 487 apples.

€5. If oz. of gold be worth $1,50, what is the cost of 1 oz.?

A. $1,80.

66. If 167 yds. will make 8 coats, how many yards will it take

for 1 coat? A. 24 yds.

67. If of of a gallon cost $8, what will 5 gallons cost?

A. $95.

68. lf 6 yds. cost $53, what will 14 yds. cost?

A. $138.

69. If of cwt. of sugar cost $o, what will 40

cwt. cost?

A. $824.

17*

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

.1 $31.

71. If 1 cwt. of flour cost $1, what will

cwt. cost? A. $1732.

72. If 3 yds. of cloth, that is 2 yds. wide, will make a cloak, how much cleth, 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.: 21 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 favor?

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. $853.

74. If 12 men do a piece of work in 124 days, how many men will do the samne in 6 days? A. 24 mnen. 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 14 lb. indigo cost $3,84, what will 49,2 lbs. cost at the same rate? A. $125,952.

77. If $29 buy 59 yds. of cloth, what will $60 buy?

A 120 yds.

78. How many yds. of cloth can I buy for $75, if 2674 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?

By analysis. If 40 men, in 10 days, reap 200 acres, 1 man, in the same time, will reap

of 200

acres, that is, 5 acres, in

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of 5 acres an reap 14 times as much,

10 days; and in 1 day he will reap acre a day; then 14 men in 1 day will which is 14 X=7 acres, and in 24 days, 24 times 7 acres,= 168 acres, Ans.

Ferform the following sums in the same manner.

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

8 mn 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?

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of 60=12, and of 12=4 miles, the distance which he travels in 1 hour; then, 4 miles X 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 answers, 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 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 129. 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 multipliers, and divide this result by the product of the divisors; by which process the two statements may be reduced to one; thus,

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In this example, the product of the multipliers, or second terms, is 9 × 10 = 90; and the product of the divisors, or first terms, is 3x5= 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 102, 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 60360 miles, Ans., as before.

Note. This method, rost cases, will shorten the process very materially, and in no case will it be onger; 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 prod uct be called, which results from multiplying two or more ratios together?

A. Compound Ratio.

From the preceding remarks we derive the following

RULE.

Q. What number do you make the third term?

A. That which is of the same kind or denomination with the answer.

Q. 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 term, 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.

Q. 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.

Q. How may the operation, in most cases, be materially shortened? A. By multiplying the third term by the con. tinued 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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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, 466560. Then, the third term, 432000, ‚=4320000 ÷ 466560 = 9 h. 158 m., the fourth term, or Ans. Or, by multiplying the third term by the ratios, thus: the satio of 15 to 25 is, of 36 to 48 is, of 12 to 8 is, of C to 5 is, of 12 to 9 is, whose products, multiplied by the 5 X 4X2 X5 X 3 x 10 h. 6000 = h.—9 h. 15§. m.,

third term, are Ans., as before.

3 X

X3X6X4

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 terms, as 5 X 4X2X5 X 3 x 10 h. 3x3x3x6x4

in T XLI.; thus, the ratios

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5X2X5 X 10 h. 3X3X6

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 ods can 20 men build in 18 days?

Men 5 20

Days 6: 18)

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this example, the ratios of 5 to 20, and of 6 to 18, are 3 and then, 3 x 4 x 10 rods=120 rods, Ans.

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

=

40, for days, and in 1 day of 1048; that is, 1 man will do a rod a day; then, 20 men X 18 days X = 120 rods, Ans., as before.

3. If 4 en 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 may be hired 12 days 1 $96?

6

days' work, how many men A. 8 men.

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