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Now, since the product of the extremes is equal to that of the means, times equals 6 times 12, or, according to the second arrangement, 12 times 6. Hence, if 12 times 6, or 72, be divided by 4, the first extreme, the quotient, 18, is evidently the other extreme, or the value of x.

196. 3. If 4 men can do a piece of work in 6 days, in how many days can 8 men do it?

By analyzing the example, we find that 4 men 6 days=1 man 24 days, and 1 man 24 days 8 men 3 days. 8 then is the answer. Moreover it is obvious, that if 4 men can do a piece of work in 6 days, twice_the_number of men will do it in half the time, or 3 days; and generally the greater the number of men, the less the time, and the reverse; and also, the longer the time, the less the number of men, and the reverse. In the above example, the ratio of the men, 4 to 82, but the ratio of the times, 6 to 3. Now, if we invert the first ratio, it becomes, 8 to 4; and we have two equal ratios, and consequently a proportion: i. e. 8:4:6:3, or 8:6::4 3. By the question, the proportion would stand, 8: 6:4: x; then 8x 4×6, and 23. Ans. Where more requires less or less requires that is, when one of the ratios is inverted, as explained in this article it is denominated inverse proportion; otherwise it is called direct propor

more,

tion.

I. Single Rule of Three.

197. When three terms of a proportion are given, the operation by which the fourth is found, is called the Single Rule of Three. All questions, which can be solved by the single rule of three, must contain three given numbers, two of which are of the same kind, and the other of the kind of the required answer; and from an examination of the preceding analysis, it will be seen that the given number, which is of the same kind as the answer, may always be one of the means in the proportion; and, since the proportion is not altered by changing the places of the means, (195) it may always be regarded as the first mean, or the middle one of the three given terms. Now if the conditions of the question require the answer to be greater than the given number of the same kind, or first mean, the other inean must obviously be greater than the first extreme; but if the answer be required to be less, the second mean must be less than the first extreme Hence we have the following general

RULE

198. Write down the given number, which is of the same kind as the answer, or number sought, for the second term. Consider whether the answer ought to be greater, or less, than this number; and if greater, write the greater of the other two given numbers for the third term, and the less for the first term; but if less, write the least of the other two given numbers for the third term, and the greater for the first. Multiply the second and third terms together, and divide the product by the first, the quotient will be the answer.

NOTE. Before stating the question, the first and third terms must be reduced to the same denomination, if they are not already so, and the middle term to the lowest denomination mentioned in it. The answer will be in the same denomination as the second term, and may be brought to a higher by reduction, if necessary.

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oz. d. 132: 66:: 396

66

2376

2376

132) 26136 (198d.
16s. 6d. Ans.

Here the several terms are reduced to the lowest denominations mentioned, before stating the ques

tion.

9. If 8 acres produce 176 bushels of wheat, what will 34 acres produce?

Ans. 748 bushels.

10. A borrowed of B 250 dollars for 7 months; afterwards B borrowed of A 300 dollars; how long must he keep it to balance the former favor? Ans. 5mo. 25d.

11. A goldsmith sold a tankard weighing 39oz. 15pwt., for £10 12s.; what was it per oz.? oz. pwt. £ s.

39 15:10 12:1 Ans. 5s. 4d.

12. If the interest of $100 for 1 year be 6 dolls., what will. be the interest of 336 dollars for the same time?

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100: 6:: 336 Ans. $20.16.

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$ mo.

6:12:: 100 Ans. 200 mo. 16, If $100 gain 6 dollars in 365 days, in how many days will a sum of money double at that rate, simple interest?

Ans. 6083 days.

17. A owes B £296 17s., but becoming a bankrupt, can pay only 78. 6d. on the pound; how much will B receive?

Ans. £111 63. 4d. 2qrs. 18. If 1 dozen of eggs cost 10 cents, what will 250 eggs cost? Ans. $2.187.

19, If a penny loaf weigh 9oz. when wheat is 6s, 3d. per bushel, what ought it to weigh when wheat is 8s. 24d. per

Lushel?

Ans. 6oz. 13drs.

20. How many yards of flannel 5qrs. wide, will line 20 yards of cloth 3qrs. wide?

Ans. 12 yds.

21. If a person at the equator be carried by the diurnal Imotion of the earth, 25000 miles in 24 hours, how far is he carried in a minute?

Ans. 1718 miles.

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366 365 :: 1 Ans. 23h. 56m. 4s. nearly.* 24. Bought 4 bales of cloth, cach containing 6 pieces, and each piece containing 27 yds at £16 4s. per piece; what a the value of the whole, and the price per yard?

Ans. £388 16s. and 12s. per yard.

25. If a hogshead of rum cost $75.60, how much water must be added to it to reduce the orice to $1 per gallon? Aus. 123 gal

26. If a board be 9 inches wide, how much in length will make a square foot?

Ans. 16in.

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35. A cistern containing 230 gallons, has 2 pipes; by one it receives 50 gallons per hour, and by the other discharges 35 gallons per hour; in what time will it be filled? Ans. 15h. 20m. 36. What will 39 weeks' board come to at $1.17 per week? Ans. $45,63.

37. If 40 rods in length and 4 in breadth make 1 acre, how many rods in breadth, that is 16 rods long will make 1 acre? Ans. 10 rods.

38. How many men must be employed to finish in 9 days, what 15 would do in 30 days? Ans. 50 men.

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39. The earth is 360° in circumference, and revolves on its axis in 24 hours; how far does a place move in one minute in lat. 44°, a degree in that latitude being about 50 miles? Ans. 124m. nearly. h. m. deg. m. 24×60: 360×50 :: 1.

m.

40. If the earth perform its diurnal revolution in 24 hours, in what time does a place on its surface move through one degree? Ans, 4 minutes. 360°: 24:: 1°

41. There is a cistern which has a pipe that will empty it in 6 hours; how many such pipes will be required to empty it in 20 minutes?

Ans. 18 pipes.

42. What is the value of 642 dollars against an estate which can pay only 69 cents on the dollar?

Ans. $442.98.

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2. Compound Proportion.

ANALYSIS.

199. 1. If a person can travel 96 miles in 4 days, when the days are 8 hours long, how far can he travel in 2 days, when the days are 12 hours long?

I. If a person can travel 96 miles in 4 days, he can travel (96—4—) 24 miles in 1 day, and if he can travel 24 in a day, which is 8 hours long, he can travel (248) 3 miles in 1 hour, and if he can travel 3 miles in an hour, he can travel, when the days are 12 hours long, (12x3=) 36 miles in 1 day, or (36×2) 72 miles in 2 days, which is the answer.

II. It must be evident that the distances travelled by a person going all the time at the same rate will be in proportion to the times in which they are travelled. In this case, 4 days, which are 8 hours long, are equal to (8X4) 32 hours, and 2d. 12 hours long equal (12×2) 24h. and hence we have this proportion, 32h.:: 96m.:: 24h.: x, or the distance travelled in the 2 days, which we find to be 72 miles as before.

III. It will be obvious, in the above question, that the distance travelled depends upon two circumstances, viz. the number of days and the length of the days. Now, supposing the days had all been of the same length, we should have had this proportion, viz. 4d.: 96m.:: 2d. x, or the distance travelled in 2 days; or, supposing the number of days had been the same in both cases, the proportion would stand, 8h.: 96m.: 12h. : x, or the distance travelled when the days are 12 hours long. Uniting these propor tions together, we have

4d.

8h.

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by which it appears that 96 is to be multiplied by 2 and 12, or (2×12=) 21, and divided by 4 and 8, or (4×8) 32, which is the same as the second method of solving the question.

200. 2. It 12 men can make 9 rods of fence in 6 days, when the days are 10 hours long, how many men will be required to make 18 rods of fence in 4 days, when the days are 8 hours long?

In this question, the number of days and their length being supposed to be the same in both cases, we should have this proportion, 9rds.: 12 men 18x, or the number of men required to build the 18 rods-supposing the number of rods to be the same in both cases, and the days to be of equal length, we should have this proportion, 4d.: 12 men: : 6d. :x, or the number required to build the fence in 4 days, and supposing the number of rods and also the number of days to be the same in both cases, we should have this proportion, 8 hours: 12 men: 10h. x, or the number required, when the days are 8 hours long. These three proportions combined, we have

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by which it appears that 9X4X8: 12:: 18×6×10: x, and multiplying the product of the third terms by the second, and dividing by the product of the first terms, we find the value of x to be 45 men, which is the answer.

DOUBLE RULE OF THREE.

201. A proportion which is formed by the combination of two, or more, simple proportions, as in the preceding examples,

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