OPERATION. a proportion, or state the question, we make the 243 miles the third term, because it is of the same kind as the required fourth term, and as from the nature of the question the latter must be greater than the third term, we make the greater of the other two numbers the second term, and the less the first ; and, then, the product of the means divided by the given extreme gives the required extreme (Art. 336). BY ANALYSIS. — If a man travel 243 miles in 9 days, will in 1 day travel f of 243 miles 27 miles; then, if he travel 27 miles in 1 day, in 24 days he will travel 24 times 27 miles 648 miles, the answer, as before. BY RATIO. — 9: 24 § ; 243 miles • 648 miles, Ans. 2. If 15 yards of cloth cost $ 48.90, what will 5 yards cost ? Ans. $ 16.30. We state the question by making 15 : 5 :: $ 4 8.9 0 : $ $ 48.90 the third term, because it is of the same kind as the required term. $ x 48.90 Then, since the answer must be less = 16.3 0. 15 than $ 48.90, because 5 yards will cost less than 15 yards, we make 5 yards, 3 Ans. $16.30. the less of the two numbers, the second term, and 15 yards the first ; and proceed as in the first example, except that we abridge the work by cancellation. BY ANALYSIS. - If 15 yards cost $ 48.90, 1 yard will cost it of $ 48.90 - $ 3.26; then, if 1 yard cost $3.26, 5 yards will cost 5 times $ 3.26 :$ 16.30. RULE. Write the given number that is of the same kind as the required fourth term, or answer, for the third term of the proportion. Of the other two numbers write the larger for the second term, and the less for the first, when the answer should exceed the third term ; but write the less for the second term, and the larger for the first, when the answer should be less than the third term. Multiply the second and third terms together, and divide their product by the first; or divide the third term by the ratio of the first term to the second. NOTE 1. · When the first and second terms are of different denominations, they must be reduced to the same denomination; and when the third term is a compound number, it must be reduced to the lowest denomination mentioned in it. The answer will be of the same denomination as the third term. NOTE 2. To shorten the operations, factors common to the dividend and divisor may be cancelled. NOTE 3. - The pupil should perform these questions by analysis, as well as by proportion, and introduce cancellation when it will abbreviate the operation. EXAMPLES. 3. If 16 acres of land cost $ 720, what will 197 acres cost ? Ans. $ 8865. 4. If $ 8865 buy 197 acres, how many acres may be bought for $ 720? 5. What will 84hhd. of molasses cost, if 15hhd. can be purchased for $ 175.95 ? Ans. $ 985.32. 6. If $ 100 gain $ 6 in 12 months, how much would it gain in 40 months ? Ans. $ 20. 7. If a certain vessel has provisions sufficient to last a crew of 10 men 45 days, how long would the provisions last if the vessel were to ship 5 new hands? Ans. 30 days. 8. If 7 and 9 were 12, what, on the same supposition, would 8 and 4 be ? 9. If 9 men can perform a certain piece of labor in 17 days, how long would it take 3 men to do it ? Ans. 51 days. 10. If 3 men can perform a piece of labor in 51 days, how many must be added to the number to perform the labor in 17 days? Ans. 6. 11. A rectangular piece of land containing an acre is 5} rods in breadth. What is its length ? Ans. 29/1 rods. 12. If $ 100 gain $ 6 in a year, how much will $ 850 gain? 13. If $ 100 gain $ 6 in a year, how much would be sufficient to gain $ 32 in a year? Ans. $ 533.331. 14. If 20 gallons of water weigh 1671b., what will 180 gallons weigh? Ans. 1503lb. 15. If a staff 3 feet long cast a shadow of 2 feet, how high is that steeple whose shadow is 75 feet ? Ans. 1123 feet. 16. If 5 cwt. be carried 36 miles for $4.75, how far might it be carried for $ 160 ? Ans. 12121% miles. 17. If 100 workmen can perform a piece of work in 12 days, how many men are sufficient to perform the work in 8 days? Ans. 150. 18. If 1 of a yard cost zo of a dollar, what will of a yard cost ? Ans. $ 0.48. 19. What must be paid for 21A, 3R. 20p. of land, if 36A. 3R. cost $ 1260 ? Ans. $ 750. 20. What is the value of 2000lb. of standard gold, the eagle, or $10 piece, weighing 10pwt. 18gr. ? 21. If 41 yards of cloth cost $ 9.75, what will 131 yards cost? Ans. $ 29.25. 22. What is the length of a rectangle whose contents are 1 sq. ft. and whose breadth is 24 inches ? Ans. 573 inches. 23. If te of a ship cost 51£., what are of her worth ? Ans. 10£. 185. 64d. 24. If the moon moves 13° 10' 35" in one day, in what time does she perform one revolution ? Ans. 27da. 7h. 43m. + 25. If 7lb. of sugar cost of a dollar, what are 121b. worth ? Ans. $ 1.284. 26. If $ 1.75 will buy 7lb. of loaf-sugar, how much will $ 213.50 buy? Ans. 8cwt. 2qr. 4lb. 27. If 7 ounces of gold are worth 30£., what is the value of 7lb. lloz. Ans. 407 £. 2s. 104d. 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 four is $ 8 per barrel, how much should it weigh when flour is $ 7.50 per barrel ? Ans. 775 ounces. 30. If a regiment of soldiers, consisting of 1000 men, are to be clothed, each suit to contain 3 yards of cloth that is 1% yards wide, and to be lined with flannel 14 yards wide, how many yards will it take to line the whole ? Ans. 5625yd. 31. If by working 14 hours per day C. Simmons 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. 84sec. 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. 11m. 1933sec. 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. 241 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 184000lb. of provisions, of which each man consumes 5lb. a week ; how long can they subsist ? Ans. 52 weeks 4 days. 39. If 25 men have is of a pound of beef each, three times in a week, how long will 3150lb. 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. 20lb. 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 OPERATION 50. A and B can perform a piece of work in 5 days, B and C in 6 days, and A and C in 6 days; in what time 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. Since the parts 2 + 3 = 5 are to be propor5:3 :: $ 250 : $ 150, the greater part, ļ Ans. whose sum is 5, tional to 2 and 3, 5 : 2 :: $ 250 : $ 100, the less part, 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. 61 A. 2R. 18p. ; 82A. OR. 24p. ; 102A. 2R. 30p. 3. Divide 319 into four parts, that shall be to each other as the numbers 41, 61, 63, and 7. Ans. 55734 ; 857757 ; 86739 ; 91384. 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. 1pwt. 12 gr.? Ans. 19pwt. 8 gr. gold: 1pwt. 14gr silver; lpwt. 15gr. copper. |