£ 8. d. 23 0 5 This method proceeds on the principle of reducing everything to its first element or lowest unit. It is very evident that if the price of a number of articles be given, we can find the price of one by dividing by that number; then having the price of 1, to find the price of 2, 3, 17, 20, 57, etc. 10x3=3C 230 4 2 68 10 20 17)212(12 3 £ S. d. etc., it is merely necessary to 17)690 12 6(40 12 6 Ans. multiply by 2, 3, 17, 20, 57, Hence in the above, to find the price of 1 sheep we divide by 17; then having the price of 1, in the third line we multiply by 30 to obtain the value of 30. We might have cancelled, but in the above example this is purposely avoided to show the mode of procedure when the sums present greater difficulties. this sum, if cancelling be resorted to, the work might stand thus : In Here 17 sheep are worth Then 1 sheep is worth 204 12 17)102(6 Therefore 30 sheep are worth 102 EXERCISES.-XVII. 1. It 4 books cost 10s. 6d., what will a dozen cost? 2. If 12 books cost £1, 11s. 6d., what will two cost? 3. Suppose 3 lbs. of sugar cost 10 d., what will 50 lbs. cost at the same rate? 4. Five pounds of sugar costing 1s. 3d., what will one hundredweight of the same cost? 5. Given that 12 lbs. of cheese cost 9s. 6d., what will 1 cwt. 2 qrs. 17 lbs. come to? 6. In a chest of tea there are 28 lbs.; supposing the price of one pound to be 2s. 3d., what is the value of the whole chest? 7. Allowing that three bottles of wine are worth 7s. 44d., find the price of seven dozen of the same quality. 8. Seek for the value of threescore sheep, when it is understood that three are worth £4, 2s. 6d. 9. If 8 yds. of cloth cost 24s., what price must I give for 29 yds.? 10. If 48 lbs. of tea cost £6, 8s., how much will 16 lbs. cost? 11. How much will 44 suits of clothes cost, if 12 suits cost £28, 6s. 9d.? 12. How much will 18 cwt. cost if 72 cwt. cost £57 ? 13. Bought 17 yds. for £11; how much will £33 buy? 14. How many yards can I buy for £20 if 49 yds. cost £56? 15. If 28 tons of coal cost £28, 15s. 2d.; how much will 62 tons cost? 16. If 14 tons of pig-iron cost £560, how much will 27 tons cost? 17. If 36 men can perform a piece of work in 105 da., in what time would 72 men perform it? 18. How long will 8 horses take to plough a field if 3 horses can do it in 8 da.? 19. A person travelling at the rate of 14 m. a day can perform a journey in 24 da.; in how many days would he perform it at the rate of 21 m. a day? 20. If 7 railway trucks weigh 12 tons 9 cwt. 3 qrs., how heavy are 18 such trucks? 21. Let us suppose that a joint of meat weighing 10 lbs. cost 6s. 14d., what is the price of a whole side of a bullock weighing 4 cwt. 1 qr. at the same rate? 22. A leg of mutton weighing 94 lbs. cost 6s. 4d.; find the value of six sheep, each weighing 1 cwt. 2 lbs., at the same price. 23. Forty books cost £7, 5s.; find the price of 139 copies of the same work. 24. Suppose that the price of a quarter and 4 bushels of wheat be 4 guineas; find the price of 1000 qrs. 25. A gentleman has on his estate 7341 trees; he values them at £5, 4s. 6d. for every three trees; find the value of the whole. 26. A man that can walk on the average 25 m. a day, performs a journey in 16 da.; in what time will another do it who can only walk 20 m. a day? 27. A piece of pork valued at 4s. 4d. weighed 8 lbs.; find the worth of the whole pig weighing 10 score lbs. at the same average price. 28. When 6 yds. cost 9s. 9d., how much will £10, 4s. 9d. buy? 29. How much cloth at 7s. a yard should be given in barter for 21 yds. at 7s. 6d.? 30. If the quartern loaf cost 8d. when wheat is selling at 64s. a quarter, what should it cost when wheat is at 56s. a quarter. 31. If the sixpenny loaf weighs 2 lbs. 3 oz. when wheat is selling at 64s. per qr., what should it weigh when wheat is selling at 56s.? 32. If 2 ac. 3 ro. of turnips be sold for £25, 6s., what should I pay for 41 ac. 1 ro. DOUBLE PROPORTION. Double or Compound Proportion consists of 5, 7, 9, 11, etc., terms. Either five terms are given to find a sixth, or seven given to find the eighth, etc. This is a very useful rule, so useful that it has been termed the Golden Rule. Every problem in Proportion consists of two parts-the supposition and the question. We must first learn to distinguish between the two. The supposition is that part of the problem in which the proposition is laid down; it begins generally with the words, if, suppose, etc. The question is that part of the problem where what is required is asked for; it begins with such words as what, how many, how much, how long, etc. The subject is best illustrated by examples. FIRST METHOD.—An innkeeper charged £28 for keeping 21 horses 14 days, when hay was selling at 8d. per stone; what should he charge for keeping 18 horses 42 days, when hay is selling at 9d. per stone? The supposition in this question is printed in italics. (1.) We ask in what terms is the answer to be? Ans. £; hence the £28 are placed in the third term. (2.) We go to the part of the question beginning “What should," etc., and take the 18 horses, and say, Will 18 horses cost more or less to keep than 21? Ans. Less; therefore less, or 18, is put for the middle term, and 21 for the first. (3.) Going on from the 18 we come to 42 days, and say, Will the keep of the same number of horses cost more for 42 days, or less, than for 14? Ans. More; hence put more, or 42, for the middle term, and 14 for the first. (4.) The next number is 9d. If the hay be 9d. per stone, will it cost more or less to bait the horses than if it be 8d. per stone? Ans. More. Then put more, or 9d., in the middle, and 8d. in the first term. Cancel as on the right hand, or place them thus and cancel : RULES.-(1.) Ask in what terms is the answer to be, and place whatever it may be in the third term, and be sure and write over it what it is: £, men, days, horses, interest, etc. (2.) Take each term in the question, and say, Will this number require more or less of the "answer" than the term in the supposition? If the reply be more, put the larger term in the middle; if less, the less; and the remaining term in the first place. (3.) The terms in the first places are the divisors, and the others the multipliers. Take another illustration: If £100 in 1 year gain £4, what will £459, 8s. 6d. gain in 4 years at the same rate per cent.? The supposition is printed in italics. d. £ (Interest.) As 100 459 8 6:4 1: 4 18 7 6 -459 8 63×4×4 25 =£73 10 2 Ans. (1.) What is the answer to be? Ans. Interest; hence £4 interest is written in the third term. (2.) The first quantity in the question is £459, so we say, Will £459 gain more interest or less interest than £100? Ans. More. Put more, or £459, 8s. 61⁄2d., in the second term, and £100 in the first term. (3.) The next quantity in the question is 4 years, so we say, In 4 years will the same money gain more interest or less than in 1 year? Ans. More; so put more, or 4, in the middle, and 1 in the first term. (4.) Cancel as before. SECOND METHOD.—First Principles-If 42 horses in 10 days eat 6 quarters of oats, how many quarters will 28 consume in 20 days? Here 42 horses in 10 days consume 6 quarters. 6 42 EXPLANATION OF THE ABOVE METHOD.-First, That which is given, or the "supposition," is written down as the first line, taking care that whatever is to form the answer comes at the end of the line. 6 4 × 10 Second, It is said that 1 horse in 1 day consumes which is obtained thus: We say, Will one horse consume more or less quarters than 42? It is evident that 1 consumes 42 times less than 42, and therefore we divide the 6 by 42. Next question is: In one day will they consume more or less quarters than in 10? Of course, in 10 days they consume 10 times as much as in 1 day, so we divide by 10 to find 1 day's consumption. consume 42 × 10 Third, In the lower lines is written: 28 horses in 20 days 6 x 28 x 20 which is found thus: We say, Will 28 horses consume more or less quarters than one; the obvious answer is more, and 12 times more; hence we multiply by the 12. Again, In 20 days will they consume more or less than in 1 day? Answer is as before, more, and 20 times more; hence we multiply by 20, and finish the sum by cancelling. In all cases the question is asked about the lower term, but in comparison with the number, immediately above that one, about which we are asking the question. Distinguishing carefully the supposition from the demand, some advocate drawing a line as below, and then placing the terms above and below the line, where the principle is exactly the same as already explained. If 4 boats rowed by 2 men each, carry 5 tons 12 miles in 4 hours, how many hours will 6 boats, rowed by 4 men each, take to carry 45 tons 8 miles? |