sum; I A can pay 158. 4 d. in the pound, and B only 78. 63d. At the 4 same time, A has in his possession £1304. 17 more than B; what do the debts of each amount to? 39. 4 25 I 2 Answer, £3340.8.3For every 12 acres which one country contains, a second contains 56. The second country contains 17,300 square miles. How much 4 does the first contain? Again, for every 3 people in the first, there are 5 in the second; and there are in the first 27 people on every 20 acres. How many are there in each country? Answer, The number of square miles in the first is 3844, and its population 3,321,600; and the population of the second is 5,536,000. 2 If 42 yds. of cloth, 18 in. wide, cost £59. 14. 2, how much will 118 yds. cost, if the width be 1 yd.? I Answer, £332. 5. 2 If £9.3.6 last six weeks, how long will £100 last? 4 17 145 Answer, 65 weeks. 367 How much sugar, worth 93d. a pound, must be given for 2 cwt. of tea, worth 10d. an ounce? 4 35 39 Answer, 32 cwt. 3 qrs. 7 lbs. 243. Suppose the following question asked: How long will it take 15 men to do that which 45 men can finish in 10 days? It is evident that one man would take 45 × 10, or 450 days, to do the same thing, and that 15 men would do it in one-fifteenth part of the time which it employs one man, that is, in By this and similar 450 or 30 days. 15 reasoning the following questions can be solved. EXERCISES. If 15 oxen eat an acre of grass in 12 days, how long will it take 26 oxen to eat 14 acres? If 22 masons build a wall 5 feet high in 6 days, how long will it 6 take 43 masons to build 10 feet? Answer, 6 days. 43 244. The questions in the preceding article form part of a more general class of questions, whose solution is called the DOUBLE RULE OF THREE, but which might, with more correctness, be called the Rule of Five, since five quantities are given, and a sixth is to be found. The following is an example: If 5 men can make 30 yards of cloth in 3 days, how long will it take 4 men to make 68 yards? The first thing to be done is to find out, from the first part of the question, the time it will take one man to make one yard. Now, since one man, in three days, will do the fifth part of what 5 men can do, he will in 3 days 30 or 6 yards. He will, therefore, make one yard in 3 or in make 3×5 30 of a day. From this we are to find how long it will take 4 men to make 3 × 5 68 yards. Since one man makes a yard in of a day, he will make 30 5 3×5 days; and 4 men will 68 yards in 30 30×4 days, or in -If 5 men can make 30 yards in 3 days, how much can 6 men do in 12 days? Here we must first find the quantity one man can do in one day, which appears, on reasoning similar to that in the last 5×3 30 3×5 example, to be yards. Hence, 6 men, yards, and in 12 days will make From these examples the following rule may be drawn. Write the given quantities in two lines, keeping quantities of the same sort under one another, and those which are connected with each other in the same line. In the two examples above given, the quantities must be written thus: Draw a curve through the middle of each line, and the extremities of the other. There will be three quantities on one curve, and two on the other. Divide the product of the three by the product of the two, and the quotient is the answer to the question. If necessary, the quantities in each line must be reduced to more simple denominations (219), as was done in the common Rule of Three (238). EXERCISES. If horses can, in 2 days, plough 17 acres, how many acres will 93 horses plough in 47 Answer, 592ğ I 4 I days? 7 If 20 men, in 3 days, can dig 7 rectangular fields, the sides of each of which are 40 and 50 yards, how long will 37 men be in digging 53 fields, the sides of each of which are 90 and 125 yards? I 1252 Answer, 75 2451 20720 days. If the carriage of 60 cwt. through 20 miles, cost £14 10s. what weight ought to be carried 30 miles for £5.8 ? 9 Answer, 15 cwt. If £100 gain £5 in a year, how much will £850 gain in 3 years and 8 months? Answer, £155. 16. 8. SECTION III. INTEREST, ETC. 245. In the questions contained in this Section, almost the only process which will be employed, is the taking a fractional part of a sum of money, which has been done before in several cases. Suppose it required to take 7 parts out of 40 from £16, that is, to divide £16 16 into 40 equal parts, and take 7 of them. Each of these parts is £ 16 16×7 and 7 of them make × 7, or pounds (116). The process may 40 40 be written as in the accompanying example: £16 7 40)112(£2.168. 32 20 640 40 240 240 Suppose it required to take 13 parts out of a hundred from Let it be required to take 2- parts out of a hundred from £3 128. I 22 £3 128. × 2, £3 128. × 5 or (123) ; 100 200 The result, by the same rule, is so that taking 2 out of a hundred is the same as taking 5 parts out I 3 Take 5 parts out of 100 from £107 138. 43d. £56 38. 2d. is equally divided among 32 persons. does the share of 23 of them exceed that of the rest? 246. It is usual, in mercantile business, to mention the fraction which one sum is of another, by saying how many parts out of a hundred must be taken from the second in order to make the first. Thus, instead of saying that £16 128. is the half of £33 4s., it is said that the first is 50 per cent of the second. Thus, £5 is 2- per cent of I 2 I 2 200; because, if £200 be divided into 100 parts, 2 of those parts are £5. Also, £13 is 150 per cent of £8. 13. 4, since the first is the second, and half the second. Suppose it asked, How much per cent is 23 parts out of 56 of any sum? The question amounts to this: If he who has £56 gets £100 for them, how much will he who has 23 receive? 23 × 100 2300 This, by (238), is 56 I 41 per cent. 14 or 56 I or 41 Hence, 23 out of 56 is 14 16 X ICO 18 "" or 40 per cent. From which the method of reducing other fractions to the rate per cent is evident. Suppose it asked, How much per cent is £6. 12. 2 of £12. 3? Since the first contains 1586d., and the second 2916d., the first is 1586 out of 2916 parts of the second; that is, by the last rule, it is 158600 2916 1136 or 54 or £54.7.9 per cent, very nearly. The more expeditious 2916' I 2 way of doing this is to reduce the shillings, &c. to decimals of a pound. The rule in (221) will usually give the rate per cent to the nearest shilling, which is near enough for all practical purposes. For instance, in the last example, which is to find how much £6.608 is of £12·15, 6.608 × 100 is 660·8, which divided by 12·15 gives £54.38, or £54 • 7• Greater correctness may be had, if necessary, by using the table in (222). EXERCISES. How much per cent is 198- out of 233 parts ?-Ans. £85. 1. Goods which are bought for £193 12, are sold for £216. 13.4; how much per cent has been gained by them? Answer, A little less than £11. 18. 6. |