3. If 8 horses eat 70 bushels of oats in 5 weeks, in what time will 7 horses eat 40 bushels? 40 bushels will last 8 as long as 70 bushels, or 48 of 5 bu. bu. wks. wks. weeks. This is expressed by the proportion 70: 40:5 40 bushels will last 7 horses as long as it will last 8 horses, or of 48 of 5 weeks. This is expressed by the proportion h. h. 7:8 = 48 of 5 weeks: weeks. As the answer does not depend upon either of the ratios alone, but upon both combined, the two ratios 70: 40, and 7:8 may be written together, and then reduced as in the above 70: 40) examples; thus, ; which being reduced, 7: 8) = 5: - gives 49: 325 weeks: weeks. -- As questions in simple proportion have three terms given to find a fourth, so in compound proportion five, seven, or some other odd number of terms, are given to find a sixth, or an eighth term, &c. RULE FOR COMPOUND PROPORTION. Write that number for the third term which is of the same kind as the answer. Then take the other quantities in pairs, or two of a kind, and arrange them as in simple proportion. Reduce the compound proportion thus formed to the form of a simple proportion, and find the fourth term as in simple proportion. 4. If 7 men build 84 rods of wall in 12 days, by working 12 hours a day, how many men can build 100 rods in 5 days, working 10 hours a day? 7 men is put for the 3d term, being of the same kind as the answer. It will take more men to build 100 rods than to build 84; therefore 100 rods should be put for the 2d term. It will take more men to build it in 5 days than in 12 days; therefore 12 days must be the 2d term. It will take more men to build it by working 10 hours a day than by working 12 hours a day; therefore 12 hours must be the 2d term of the ratio. Solve the following questions both by proportion and by analysis. (47 and 95.) 5. If 14 men build 84 rods of wall in 3 days, how long will it take 20 men to build 300 rods? 6. If 24 horses eat 126 bushels of oats in 36 days, how many bushels will 32 horses eat in 48 days? 7. If 4 men build a wall 10 ft. long, 6 ft. high, 2 ft. thick, in 6 days, how long will it take 12 men to build one 100 ft. long, 8 ft. high, and 3 ft. thick? 8. If $200 gain $12 in 12 months, what will $500 gain in 8 months? 9. If 7 men, working 9 hours a day, dig a ditch 210 ft. long, 3 ft. wide, and 4 ft. deep, in 4 days, in what time will 35 men, working 12 hours a day, dig a ditch 420 feet long, 4 feet wide, and 5 feet deep? 10. If 4 men can build 381 rods of wall in 34 days, how long will it take 9 men to build 12311⁄2 rods? 11. If 75 men, working 10 hours a day, in 25 days grade 175 rods of rail-road, how many men will it take to grade 448 rods in 40 days, working 12 hours per day? 12. If the carriage of 10 boxes of sugar, weighing each 420 lb., 15 miles, cost $5, what would the carriage of 15 boxes, each weighing 450 lb., cost for 60 miles? 13. A person undertook to perform a piece of work in 8 days with 15 men, but at the end of 6 days, found of it unfinished; how many more men must he employ to finish it at the set time? 14. If a footman, when the days are 14 hours long, can travel 276 miles in 16 days, in how many days can he travel 852 miles, when the days are but 12 hours long? 15. A garrison of 500 men have provisions for 15 weeks, at the rate of 18 oz. per day to each man; how many men will the same provisions maintain for 10 weeks, allowing each man only 12 oz. per day? 16. If a bar of iron 6 feet long, 3 inches broad, and 2 inches thick, weighs 72 pounds, what will a bar weigh that is 4 feet long, 2 inches wide, and 1 inch thick? 17. If 248 men in 5 days of 11 hours each can dig a trench 230 yards long, 3 wide, and 2 deep, in how many days of 9 hours long will 24 men dig a trench 420 yards long, 5 wide, and 3 deep? 18. The par value of the pound sterling being $43, how many dollars will pay a debt of £450, the rate of exchange on England being at a premium of 8 per cent.? at 94 per cent.? 19. How many pounds sterling are equal to $1000, the premium of exchange being 8 per cent.? 9 per cent.? 131. The terms of a proportion may be distinguished into causes and effects. Thus, in the last Art., example 11th, men, days and hours may be regarded as causes, and rods as the effect produced by those causes. So, in example 12th, boxes, pounds and miles may be classed together as causes, and the money expended as the effect. In example 15th, all the terms may be classed as causes; the effect produced upon the men who consumed the provision not being expressed in numbers. 1. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick, in 16 days, by working 10 hours per day, in how many days will 24 men build a wall 200 feet long, 8 feet high, and 6 feet thick, by working 12 hours per day? The statement of the above question by proportion is as follows: In this example 6 men, 16 days, and 10 hours, are the first set of causes, and 24 men, days, and 12 hours, the second; 20 feet, feet, and 4 feet, are the first set of effects, and 200 feet, 8 feet, and 6 feet, the second. Since in every proportion the product of the extreme terms is equal to the product of the mean terms, we see, by the above statement, that the product of the first set of causes multiplied by the second set of effects, which are the numbers constituting the mean terms, is equal to the product of the second set of causes multiplied by the first set of effects, which are the numbers constituting the extreme terms. That is, the two products of each cause, or set of causes, by the opposite effect, or set of effects, are equal to each other. If, therefore, the terms are arranged so that each set of causes shall be in the same column with the opposite effects, we shall, by making the numbers in the column that contains the blank or unknown quantity factors of a divisor, and those in the other column factors of a dividend, obtain the answer. The terms may be arranged thus: 6 men 2 16 days 10 hours 3 12 hours 10 200 feet 20 feet 8 feet 6 feet 6 feet 4 feet causes. effects. We first write the first set of causes on the left of a vertical line, toward the top, and the corresponding set of effects on the opposite side of the line, toward the bottom. We then write the second set of causes opposite to the first set, on the right of the vertical line, and the second set of effects on the left of the vertical line, opposite to the first. The numbers on the left of the line are factors of the dividend, and those on the right, containing the blank, are factors of the divisor. After cancelling equal factors in the divisor and dividend, we multiply the remaining factors and perform the division. 200366 days. 2. If I pay $40 for the carriage of 5 cwt. 150 miles, what must I pay for carrying 75 cwt. 64 miles? The mixed numbers may be changed to improper fractions. Transposing the denominators of fractional quantities to the opposite column, is the same as multiplying each column by the same numbers, which does not alter their relation to each other. (74.) NOTE. In writing down the terms of the proportion, either cause or either effect may be first written; it will make no difference in the result. It is generally most convenient to write the terms just as they occur in the question, taking care that each cause and its effect be on opposite sides of the vertical line. "All the terms acting, producing, or consuming, are CAUSES; viz., men, horses, me, capital, length, breadth, thickness, or parts of a compound, &c. EFFECTS are the result or consequence of said causes; viz., work, wages, interest, superficial and solid contents, &c. By fixing these distinctions in the memory, the student will soon be able to apply the criterion with ease and certainty.” — PLAIN CALCULATor, BY JOERRES. 3. If 6 oxen in 8 days eat 5 acres of grass, will serve 12 oxen 96 days? 4. If the interest of $500 for 3 be the interest of $1020 for 5 cent. ? how many acres years be $105, what would years, at the same rate per 5. Six men, by working 8 days, 10 hours a day, can do of a piece of work; in how many days will ten men do of the same work, working 9 hours a day? 6. If a rectangular cistern, 7 feet long, 5 feet wide, and 6 feet deep, hold 13125 pounds of water, how much will a cistern hold that is 10 feet long, 7 feet wide, and 9 feet deep? 7. What principal at 6 per cent. will yield $15.50 interest, in 21 years? 8. In what time will the interest of $500.75 amount to $35.75, at 6 per cent. ? 9. At what per cent. must $1000 be put on interest, in order to yield $192.50 in 3 years? 10. How much flour at $6.87 per barrel, must be given for 150 bushels of corn at $0.621 per bushel? 11. If a staff 5 feet long cast a shadow 84 feet long, how high is the steeple which, at the same time, casts a shadow of 175 feet? NOTE. Let the pupil perform examples in both simple and compound proportion, by this method, till he can perform them with facility. |