20. What is the value of 20001b. of standard gold, the eagle, or $ 10 piece, weighing lOpwt. 18gr.? 21. If 4£ yards of cloth cost $9.75, what will 13£ yards cost? Ans. $ 29.25. 22. What is the length of a rectangle whose contents are 1 sq. ft. and whose breadth is 2£ inches? Ans. 57§ inches. 23. If -J^ of a ship cost 51£., what are ^ of her worth? Ans. 10£. 18s. 6fd. 24. If the moon moves 13° 10' 35" in one day, in what time does she perform one revolution? Ans. 27da. 7h. 43m.-f 25. If 71b. of sugar cost £ of a dollar, what are 121b. worth? Ans. $1.28f. 26. If $1.75 will buy 71b. of loaf-sugar, how much will $ 213.50 buy? * Ans. 8cwt. 2qr. 41b. 27. If 7 ounces of gold are worth 30£., what is the value of 71b. lloz. Ans. 407£. 2s. lOfd. 28. A friend borrowed of me $ 500 for 6 months; how long ought he to lend me $ 600, to requite the favor? 29. If the penny loaf weighs 7oz. when flour is $ 8 per barrel, how much should it weigh when flour is $ 7.50 per barrel? Ans. 7-^ ounces. 30. If a regiment of soldiers, consisting of 1000 men, are to be clothed, each suit to contain 3J yards of cloth that is 11 yards wide, and to be lined with flannel 1 £ yards wide, how many yards will it take to line the whole? Ans. 5625yd. 31. If by working 14 hours per day C. Sjmmons can plant half of a field in 9 days, in what time will he plant the remainder, working 10 hours per day at the same rate each hour? Ans. 12§ days. 32. If 75 gallons of water fall into a cistern containing 500 gallons, and 40 gallons run out, in an hour, in what time will it be filled?" Ans. 14h. 17m. 8fsec. 33. How many dozen pairs of gloves, at $0.56 per pair, can be bought for $ 120.96? Ans. 18 dozen. 34. A certain cistern has three pipes; the first will empty it in 20 minutes, the second in 40 minutes, and the third in 75 minutes; in what time would they all empty it? Ans. 1 lm. 19^§sec. 35. A can mow a certain field in 5 days, and B can mow it in 6 days; in what time would both of them together mow it? Ans. 2T8r days. 36. A wall, which was to be built 32 feet high, was raised 8 feet by 6 men in 12 days; how many men must be employed to finish the wall in 6 days? 37. A can build a boat in 20 days, but with the assistance of C he can do it in 12 days; in what time would C do it alone? Ans. 30 days. 38. In a fort there are 700 men provided with 1840001b. of provisions, of which each man consumes 51b. a week; how long can they subsist? Ans. 52 weeks 4 days. 39. If 25 men have f of a pound of beef each, three times in a week, how long will 31501b. last them? Ans. 56 weeks. 40. How many tiles 8 inches square will lay a floor 20 feet long and 16 feet wide? Ans. 720. 41. How many stones 10 inches long, 9 inches broad, and 4 inches thick, would it require to build a wall 80 feet long, 20 feet high, and 2£ feet thick? Ans. 17280 stones. 42. If there be paid for 1 ton 7cwt. 3qr. 201b. of coal $ 9.50, what will 13 tons 5cwt. 2qr. cost? 43. If 61.3 pounds of tea cost $ 44.9942, what is the price per pound? Ans. $ 0.734. 44. What is the value of .15 of a hogshead of lime, at $ 2.39 per hogshead? Ans. $ 0.3585. 45. If .75 of a ton of hay cost $ 15, what is it per ton? Ans. $ 20. 46. How many yards of carpeting that is half a yard wide will cover a room that is 30 feet long and 18 feet wide? Ans. 120 yards. 47. If a man perform a journey in 15 days, when a day is 12 hours long, in how many days will he do it when a day is but 10 hours long? 48. If 450 men are in a garrison, and their provisions will last them but 5 months, how many must leave the garrison that the same provisions may be sufficient to supply the remaining men 9 months? Ans. 200 men. 49. The hour and minute hands of a watch are together at 12 o'clock; when will they next be together? Ans. lh. 5m. 27^-sec 50. A and B can perform a piece of work in 5-j*T days, B and C in 6§ days, and A and C in 6 days; in what lime would each of them perform the work alone, and how long would it take them to do the work together? Ans. A would do the work in 10 days; B, in 12 days; C, in 15 days; A, B, and C together, in 4 days. 339. To divide a number or quantity into parts, which are proportional to given numbers. Ex. 1. Divide $250 into two parts which shall be one to the other as 2 to 3. Ans. $ 100 and $ 150. Operation. Since the parts 2 -(- 3 = 5 are to be propor 5 : 3 :: $ 250 : $ 150, the greater part, ) . tional to 2 and 3, 5 : 2 : : $ 250 : $ 100, the less part, f Ans. *h.ose ??m TM.5' r'' it is evident that the sum of the two numbers, 5, will have the same ratio to the greater of them, 3, as the amount to be divided, $ 250, has to the greater of the required parts; and that the sum, 5, will have the same ratio to the less number, 2, as the $ 250 has to the less of the required parts; we therefore make two statements, and then find the required term of each proportion as in Art. 337. Hence, As the sum of the given numbers is to any one of them, so is the whole quantity to be divided to the part corresponding to the number used as the second term. Note. — This application of proportion is sometimes called Distributive or Partitive Proportion. Examples. 2. A farmer divides between his three sons 246A. 1R. 32p. of land, sharing it between them as the numbers 3, 4, and 5. What were the shares? Ans. 61A. 2R. 18p.; 82A. OR. 24p.; 102A. 2R. 30p. 3. Divide 319 into four parts, that shall be to each other as the numbers 4£, 6£, 6-f. and 7. Ans. 55f |?; 855W; 86fff; 91 j§?. 4. Standard gold for coinage consists of 9 parts of pure gold, and 1 part alloy. Allowing the alloy to be silver and copper in equal parts, how much pure gold, silver, and copper are contained in a double eagle, its weight being loz. lpwt. 12gr.? Ans. 19pwt. 8fgr. gold; lpwt. lfgr. silver; lpwt. l*gr. copper. 5. The half-dollar of the United States coinage weighs 192 grains Troy, and consists of 9 parts pure silver and 1 part of copper. How much pure silver and how much copper in 20 half-dollars? Ans. 7oz. 4pwt. silver; 16pwt. copper. 6. Divide $ 600 between three men, so that the second man shall receive one third more than the first, and the third man shall receive two thirds more than the second. 7. A, B, and C freight a steamer; A puts on board 98 tons, B 86 tons, and C 64 tons. Owing to danger of being wrecked, there were thrown overboard while at sea 93 tons. What should be the number of tons lost by each? Ans. A, 36| tons; B, 32£ tons; and C, 24 tons. 8. A and B start together by railroad from Chicago for Galena; A travels by freight train at the rate of 15 miles per hour, and B by passenger train at the rate of 25 miles per hour. C leaves Galena for Chicago at the same time by express train, whose velocity is at the rate of 32 miles per hour. Allowing the distance between the two places to be 160 miles, how far from Chicago will A and B each be, when C passes them? Ans. A, 51j\ miles; B 70^ miles. COMPOUND PROPORTION. 340i A Compound Proportion is an expression of equality between a coripound and a simple ratio. Thus, g [ y | : : 60 : 63, is a compound proportion. Compound proportion is employed in the solution of such questions as would require two or more statements in Simple Proportion. Ex. 1. If 8 men spend $ 32 in 13 weeks, what will 24 men spend in 52 weeks? Ans. $ 384. Operation. In stating the Extreme. Mean. question, we make 80 jt ~> Mean. Extreme. Ji nn .... „ men :24 men |. . « 30 • ft * 32' wmcn ls of 13 weeks : 52 weeks ) .. the same kind as o , the required term, J? . , * the third term. t I X 32 _ Then,takingofthe $ X id ~~ reimininT forms two of the same kind, 8 men and 24 men, we inquire whether the answer depending on these alone must be greater or less than the third term; and since it must be greater, because 24 men will spend more than 8 men in the same time, we make 24 men the second term, and 8 men the first term. Again, we take the two remaining terms, and make 52 weeks the second term, and 13 weeks the first, since the same number of men would spend more in 52 weeks than in 13 weeks. We then find the continued product of the second and third terms, and divide by the product of the first terms. By Analysis. — If 8 men spend $32 in 13 weeks, 1 man will spend = $4 in 13 weeks, and 24 men will spend $4 X 24 = $'JG in 13 weeks. If 24 men spend $96 in 13 weeks, in 1 week they will spend $ ff, and in 52 weeks, %\\ x 52 = $ 384, Ans. By Ratio. — The ratio of 8 : 24 == \; the ratio of 13 : 52 «= \; $ 32 -5- J X | = $ 384, Ans. Note. — To have solved the question by simple proportion, two statements would have been required, which would have produced the following proportions : — 8 men :24 men :: $32 : $96. 13 weeks : 52 weeks : : $96 : $384. Rule. — Make that number which is of the same kind as the answer required the third term of a proportion. Of the remaining numbers, take any two, that are of the same kind, and consider whether an answer depending upon these alone would be greater or less than the third term, and place them as directed in Simple Proportion. Then take any other two, and consider whether an answer depending only upon them would be greater or less than the third term, and arrange them accordingly; and so on until all are used. Multiply the product of the second terms by the third, and divide the result by the product of the first terms. The quotient will be the fourth term, or answer. Examples. 2. If a man can travel 117 miles in 30 days, travelling 9 hours a day, how far can he go in 20 days, travelling 12 hour? a day? Ans. 104 miles. 3. If 6 men in 16 days of 9 hours each build a wall 20 feet long, 6 feet high, and 4 feet thick, in how many days of 8 hours each will 24 men build a wall 200 feet long, 8 feet high, and 6 feet thick? 4. If $100 gain $ 6 in one year, how much would $500 gain in four months? Ans. $ 10. 5. If $ 100 gain $ 6 in one year, what must be the sum to gain $ 10 in 4 months? Ans. $ 500. |