38. If a person pays half a guinea a week for his board, how long can he be boarded for £21? Ans. 40 weeks. 39. What is the value of a year's rent of 547 acres of land at the rate of 15s 6d the acre ? Ans. £423 18s 6d. 40. If a person drinks 20 bottles of wine per month, when it costs 2s per bottle, how much can he drink without increasing the expense when it costs 2s 6d per bottle ? Ans. 16 bottles. 41. A merchant bought 21 pieces of cloth, each containing 40 yards, for which he paid $1260; he sold the cloth at $1,75 per yard : did he make or lose by the bargain ? Ans. he gained $210. 42. A cistern containing 200 gallons is filled by a pipe which discharges 3 gallons in 5 minutes: but the cistern has a leak which empties 1 gallon in 5 minutes. Now if the water begins to run in, when the cistern is empty, how long will it be in filling? Ans. 8 hours 20 minutes. 43. What will be the cost of 895 feet of timber, at $6 per hundred feet? In this example, 100 OPERATION. feet of timber, is to the 100 895 Ans. given quantity 895 feet, :: 6 6 as $6, the cost of 100 100)5370 feet, is to $53,70, the 53,70 cost of 895 feet. If the timber had been sold at Ans. $53,70. the rate of $6 per thousand feet, the cost of 895 feet would have been $5,37, for we should have divided by 1000, instead of 100; that is, we should have removed the separating point in the product three places to the left. Hence, to find the cost of things sold by the 100, or 1000, we have the following RULE, Multiply the number of things by the price, and if the things be reckoned by the 100, cut off two places from the right, and if reckoned by the 1000, cut off three, and the figures to the left will be the answer in the same denomination as the given price. 44. What will be the cost of 1350 feet of boards at $11 per hundred ? Ans. $148,50. 45. What will be the cost of 36578 bricks at $6,50 per thousand ? Ans. $237,75 +. 46. What will be the cost of 6359 feet of boards at $9,25 per 100 feet ? Ans. $ ti 47. What will be the cost of 13918 feet of timber at *14,37 per thousand ? Ans. $200,00+. 48. What will 18759 oranges cost at $5,50 per hundred ? Ans. *1031,74+. 49. What is the cost of 6559 feet of round timber at $9,25 per 100 feet? Ans. * t. 50. What is the cost of 37032 feet of square timber at $85,72 per thousand feet? Ans. $3174,38+. Q. How do you find the cost of things sold by the hundred? How do you find the cost of things sold by the thousand ? QUESTIONS INVOLVING FRACTIONS. EXAMPLES 21 1. If of a yard of cloth cost $3,20, what will 21 yards cost ? We state the ques OPERATION. tion exactly as in : 21 :: 3,20 : Ans. whole numbers. In multiplying the 2nd, 6,40 and 3rd terms toge by multiplying by 1 1,60 ther, we observe the rules for multiplying fractions, and in divi 8,00:- =8,00 x 6400 ding by the 1st term, =$21,33+. the rules for division. 'Thus, in this example, we invert the terms of the divisor and multiply. 2. If 40%. cost £1], what will loz. cost? Ans. £1 5s 8d. · 3. If of a ship cost £273 2s 6d, what will of her cost? Ans. £227 12s 1d. 4. A mercer bought 31 pieces of silk each containing 244 yards. He paid 6s jd per yard: what does the whole come to? Ans. £25 14s 6 dt. 5. If 141b. of sugar cost $1], what will 61b. cost? Ans. $ 6. If of a yard of cloth cost of a dollar, what will 21 yards cost? Ans. $431 7. If 21b. of beef cost of a dollar, what will 3016. cost. Ans. $ 8. If 14} yards of cloth cost $19}, how much will 197 yards cost? Ans. $26. 9. If ,3 of a house cost $100,75, what would ,95 cost? Ans. $319,04+. 10. A man receives of his income and finds it equal to $3724,16: how much is his whole income? Ans. $ RULE OF THREE BY ANALYSIS. § 144. The solution of questions in the Rule of Three by analysis consists in finding the ratio of two of the given terms, and multiplying this ratio by the other term. The ratio of two of the terms will generally express the value or cost of a single thing. EXAMPLES 1. If 3 barrels of flour cost $24, what will 7 barrels cost? By dividing the $24 by 3 we get the 1 o. 2 st the OPERATION. cost of 1 barrel. For, if $24 will buy 3 3)24 barrels, it is plain that of it will buy 1 barrel. This, multiplied by 7, gives $56 8X7=56 the cost of 7 barrels. Ans. $56. Q. In what does the solution of questions by analysis consist ? What does the ratio of two of the terms express? If this ratio be multiplied by the other term what is the product? If 6 oranges cost 12 cents, how much will 8 cost? If 3 apples cost 1 cent, how much will 7 cost? What is the ratio of 3 to 1? If 9 yards of cloth cost $27, what will 15 yards cost ? If 15 bushels of wheat cost $30, what will 50 bushels cost ? 2. If in 29 days a man travels 58 miles, how far will he travel in 30 days ? Ans. 60. OPERATION. 3. If 6 men consume 1 barrel of flour in 30 days, how much would 48 men consume? It is evident that i of a barrel would be the amount consumed by 1 man; hence, 48 times 1 is the 1x48=8. amount consumed by 48 men. Ans. 8. 4. If į of a barrel of flour cost of a dollar, what will cost? Ans. $1. *°5. If I walk 84 miles in 3 days, how far should I walk at the same rate in 9 ? Ans. 252. 6. If 8lb. of sugar cost $1,28, how much will 1316. cost? What is 16x13? Ans. $ 7. If of a piece of cloth cost $8,25, what will cost? Ans. $24,75. 8. If 300 barrels of flour cost $570, what will 200 cost? What is x570? Ans. 9. If of a barrel of cider cost î of a dollar, what will cost ? What is 2.xi? Ans. $18 OF QUESTIONS REQUIRING TWO STATEMENTS. § 145. The answer to each of the above questions has been found by a single statement. Questions, however, frequently occur in which two or more statements are necessary. In most Arithmetics, such questions are arranged under a rule called Compound Proportion, or the Double Rule of Three. They can, however, be answered by the rules already given. EXAMPLES. 1. If a family of 6 persons expend $300 in 8 months, how much will serve a family of 15 persons for 20 months ? First question. If OPERATION. $300 will support a persons. persons. $ family of 6 persons 6 : 15 :: 300 : Ans. for 8 months, how 15 many dollars will sup 6)4500 port 15 persons for Ans. $750 the same time? Second question. If $750 will support a family of 15 persons for 8 months, how much will serve them for 20 months ? : Ans. OPERATION. 20 2. Jf 16 men build 18 feet of wall in 12 days, how many men must be employed to build 72 feet in 8 days, working at the same rate ? The first question OPERATION. is, how long would it feet. feet. days. days. take the 16 men to 18 72 :: 12 Ans. build the 72 feet of 72 wall. 24 It is evident that 18 84 feet of wall, is to 72 18)864(48 days. feet, as 12 days, the 72 time necessary to build 18 feet, is to 48 144 days, the time neces 144 sary to build 72 feet. men. The second ques OPERATION. tion is, if 16 men can days. days. men. build 72 feet of wall 8 48 16 : Ans. in 48 days, how many 48 men are necessary to 128 build it in 8 days? 64 Make 16 men the 8)768( third term. Then as the same work is to Ans. 96 men. be done in less time, more men will be necessary; therefore, the fourth term will be greater than the third, and hence 8 days are placed in the first term (see § 142). 3. If a man travel 217 miles in 7 days, travelling 6 hours a day, how far would he travel in 9 days, if he travelled 11 hours a day. |