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How many bushels at $1,202-50. How many at 80 cents?-78.
At 50 cents?-120. At 40 cents?-150. At 30 cents?-200.
A. 635 bushels

47. How much in length that is 6 inches in breadth will raake ■ square foot? (12 inches in length and 12 in breadth make 1 square foot; then, 6 inches in breadth will require more in length; that is, 6: 12: 12.)-24. How many 4 inches in breadth?-36. How many 8 inches in breadth ?-18. How many

16 inches in breadth ?-9. A. 87 inches.

48. If a man's income be $1750 a year, how much may be spend each day to lay up $400 a year? A. $3,70.

49. If 6 shillings make $1, New England currency, how much will 4 s. 6 d. make, in federal money?-,75. Will 2s. 6d. ?,413. Will 1 s. 6d. ?-25. Will 3 s. 9 d. ?-,62). A. $2,04†.

50. A merchant bought 26 pipes of wine on 6 months' credit, but, by paying ready money, he got it 3 cents a gallon cheaper; how much did he save by paying reaay money?-A. $98,28.

51. Bought 400 yards 2 qrs. of plaid for $406,80, but could sell it for no more than $300; what was my loss per ell French? A. $,40.

52. If 120 gallons of water, in 1 hour, fall into a cistern containing 600 gallons, from which, by 1 pipe, 20 gallons run out in 1 hour, and by another 50 gallons, in what time will the cistern be filled? A. 12 hours.

53. A merchant bought 40 pieces of broadcloth, each piece containing 45 yards, at the rate of $6 for 9 yards, and sold it again at the rate of $15 for 18 yards; how much did he make in trading? A. $300.

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

55. A gentleman bought 3 yards of broadcloth 14 yards wide; how many yards of flannel, which is only yd. wide, will line the same?

It is evident it will take inore cloth which is only yd. wide, than if it were 18 yd. wide; hence 1 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 43 yds. of cloth, which is 13 yd wide, and lined with flannel yd. wide; how many yards of fannel 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 dollas, 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 dollar, f, or 1 barrel, will cost 8 times as much, that is,

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Or, as is more than, we may make the 2d, or multiplyfng term, as in the foregoing examples, thus:

Bbls. Bbis.

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8

5X3 15
16 X 4 64
120

+ (Inverting by

¶ XLVII., then, 15X)=—=$13, Ans.

64

64

Or, multiplying by the ratio, thus; the ratio of to } is { + 6 x 5 30 16 16

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26, ratio; then,

= = = $11, Ans. as before.

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, (TXLVII.) 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? 4. $16.

of a dollar, what will 1

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. 62. If yd. cost $225, what will 1 yd. cost? A. $2,823. 63. If of yd. cost $2, what is it a yard? A. $51.

64. If off of

will $5 buy? A. 487

of $1 buy 20 apples, how many apples apples.

65. If oz. of gold be worth $1,50, what is the cost of 1 oz.? A. $180,

66. If 167 yds. will make 8 coats, how many yards will it tak for 1 coat? A. 274 yds.

67. If of of a gailon cost $, what will 5 gallons mestr

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68. 156 yds. cost $53, what will 14 yds. cost? A. $1338. 69. If of cwt. of sugar cost $, what will 40 cwt. cost?

A. $327.

70. 1 yd. of silk cost of $, what is the price of 50 yds.? lig yd. 17*

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

cwt. cost? A. $T7SZ.

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

The narrower the cloth, the more yards it will take; hence wo make the greater the second term, thus ; 2 yd. : 2} 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. $8531. 74. If 12 men do a piece of work in 121 days, how many met will do the saine 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 fo

of; then, as of the vessel is $500, is $250 =$1250.

and §, or the whole vessel, is 5 × 250

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 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?

=

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 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, and in 24 days, 24 times 7 acres,= 168 acres, Ans.

Perform the following sums in the same manner.

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

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

A. 256 acres.

1

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 3 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 hours a day; which will give the following proportion :

3 hours: 9 hours: 120 miles : miles;

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

5 days 10 days

3 hours: 9 hours

:: 60 miles : miles.

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 3 x 5 15; then, 60 X 90 5400+15=360 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 1, 6, is the same as multiplying 60 by them separately; then, € 60360 miles, Ans., as before.

Note. This method, in nost cases, will shorten the process very materially, d in no case will it be any 'onger; for, when the ratios are fractions, multi'ying the third term by them (according to the rule for the multiplication of iations) will, in fact, be the same process as by the other method.

Q. From the preceding remarks, what does Compound Proportion Double Rule of Three, appear to be?

A. It is finding the answer to such questions as would require two or more statements in Sim ple 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 act be called, which results from multiplying two or more ratios kugether?

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 denomi nation with the answer.

Q. How do you arrange all the remaining terms?

4. 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.

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