692. When the terms of a proportion are considered in the relation of cause and effect, the operations are the same as when considered in the relation of magnitude. (Art. 689.) Solve the following by either or both the preceding methods: 3. Bought 41 yd. of flannel for $16.40; how much would 83 yd. cost? 4. Bought 18 kilos of ginger for $8.50; how much will 10 kilos cost? 5. If a stage goes 84 kilometers in 12 hours, how far will it go in 15 hours? 6. If 26 horses eat 72 hektoliters of oats in a week, how many hektoliters will 25 horses eat in the same time? 7. If a railroad car runs 125 kilometers in 5 hours, how far will it run in 12 hours? 8. If 9 ounces of silver will make 4 tea spoons, how many spoons will 25 pounds of silver make? 9. If 5 yd. of cloth are worth $273, what are 504 yd. worth? 10. If 60 men can build a house in 90 days, how long will it take 15 men to build it? 11. What will 49 yd. velvet cost, if 71⁄2 yd. cost £7 18s. 4d.? 12. At 7s. 6d. per ounce, what is the value of a silver pitcher weighing 9 oz. 13 pwt. 8 gr. ? 13. If 405 yd. linen cost £69 7s. 6d., what cost 243 yd? 14. If A can saw a cord of wood in 6 hours, and B in 10 hours, how long will it take both together to saw a cord? 15. A cistern has 3 stop-cocks, the first of which will empty it in 10 min.; the second, in 15 min.; and the third, in 30 min.; how long will it take all of them together to empty it? 16. A man and a boy together can mow an acre of grass in 4 hours; the man can mow it alone in 6 hours; how long will it take the boy to mow it? 17. If the interest of $675.25 is $55.625 for 1 year, how much will be the interest of $2368.85 ? 18. What must be the length of a board which is 9 in. wide, to make a square foot? 19. If 98 yds. carpeting 14 yard wide will cover a floor, how many yardsyd. wide will it take to cover it? COMPOUND PROPORTION. 693. Compound Proportion is an equality between a compound ratio and a simple one. Thus, 8:4) 123, is a compound proportion. That is, 8×6:4×3: 12:3; for, 8×6×3=4×3× 12. It is read, "The ratio of 8 into 6 is to 4 into 3, as 12 to 3." WRITTEN EXERCISES. 694. 1. If 4 men earn $60 in 10 days, how much can 6 men earn in 8 days? OPERATION. 4 m. : 6 m. 10 d. : 8 d. } :: $60 Ans. EXPLANATION.-Since the answer is to be money, we make $60 the third term. We then arrange the other numbers in pairs, two of a kind, placing them according as the answer would be greater or less than the third term, if it depended on each pair alone. Now, as 6 m. can earn more than 4 m., we place the larger for the second term and the smaller for the first. Again, as they will earn less in 8 d. than in 10 d., we place the smaller for the second term, and the larger for the first. Reducing the compound ratio to a simple one, we have, 4 x 10 6 x 8 :: 60: Ans. Dividing the prod. of the means by the extreme, cancelling, etc. 695. From the preceding example we have the following RULE.-I. Make that number which is of the same kind as the answer, the third term. II. Then take the other numbers in pairs, or two of a kind, and arrange them as in simple proportion. (Art. 687.) III. Multiply the second and third terms together, and divide the product by the product of the first terms. The quotient will be the answer. PROOF.-If the product of the first and fourth terms equals that of the second and third terms, the work is right. NOTES.-1. The terms of each couplet in the compound ratio must be reduced to the same denomination, and the third term to the lowest denomination contained in it, as in Simple Proportion. 2. In Compound Proportion, all the terms are given in couplets or pairs of the same kind, except one. This is called the odd term, or demand, and is always the same kind as the answer. 3. Problems in Compound Proportion may also be solved by Analysis and by Simple Proportion. Take the preceding example. BY ANALYSIS.-If 4 men can earn $60 in 10 d., 1 man can earn in the same time, of $60, which is $15, and 6 men can earn 6 times 15 or $90. Again, if 6 men earn $90 in 10 d., in 1 d. they can earn is $9; and in 8 d. they can earn 8 times 9, or $72, Ans. of $90, which BY SIMPLE PROPORTION.-4 m. : 6 m. : $60: x, or $90. 2. If 8 men can clear 30 acres of land in 63 days, working 10 hours a day, how many acres can 10 men clear in 72 days, working 12 hours a day? NOTE.-When the vertical form of cancellation is used, the antecedents must be placed on the left of the line, and the consequents with the odd term on the right. 3. If a man can walk 192 miles in 4 days, traveling 12 hours a day, how far can he go in 24 days, traveling 8 hours a day? 4. If 8 men can make 9 rods of wall in 12 days, how many men will it require to make 36 rods in 4 days? 5. If 5 men make 240 pair of shoes in 24 days, how many men will it require to make 300 pair in 15 days? 6. If 60 lbs. of meat will supply 8 men 15 days, how long will 72 lbs. last 24 men? 7. If 12 men can reap 80 acres of wheat in 6 days, how long will it take 25 men to reap 200 acres? 8. If 18 horses eat 128 bushels of oats in 32 days, how many bushels will 12 horses eat in 64 days? 9. If 8 men can build a wall 20 ft. long, 6 ft. high, and 4 ft. thick, in 12 days, how long will it take 24 men to build one 200 ft. long, 8 ft. high, and 6 ft. thick? 10. If 8 men reap 36 acres in 9 days, working 9 hours per day, how many men will it take to reap 48 acres in 12 days, working 12 hours per day? 11. If $100 gain $6 in 12 months, how long will it take $400 to gain $18. Ans. 9 mo. 12. If $200 gain $12 in 12 mo., what will $400 gain in 9 mo.? 13. If 6 men can dig a drain 20 rods long, 6 feet deep, and 4 feet wide, in 16 days, working 9 hours each day, how many days will it take 24 men to dig a drain 200 rods long, 8 feet deep, and 6 feet wide, working 8 hours per day? 14. If 3 lbs. of yarn will make 10 yards of cloth 1 yard wide, how many pounds will be required to make a piece 100 yards long, and 14 yd. wide? 15. A general wished to remove 80000 lbs. of provision from a fortress in 9 days, and it was found that in 6 days 18 men had carried away but 15 tons; how many men would be required to carry the remainder in 3 days? 16. If a man travels 130 miles in 3 days, when the days are 14 hours long, how long will it take him to travel 390 miles when the days are 7 hours long? 17. If the price of 10 oz. of bread is 5d., when corn is 4s. 2d. per bushel, what must be paid for 3 lbs. 10 oz. when corn is 5s. 5d. per bushel? 18. If 6 journeymen make 132 pair of boots in 4 weeks, working 5 days a week, and 12 hours per day, how many pair will 18 men make in 13 weeks, working 41 days per week, and 11 hours per day? PARTITIVE PROPORTION. 696. Partitive Proportion is dividing a number into two cr more parts having a given ratio to each other. ORAL EXERCISES. 697. 1. Charles and Robert divided 28 pears between them in the ratio of 3 to 4; how many had each? ANALYSIS. Since Charles had 3 parts as often as Robert had 4, both had 3+4, or 7 equal parts. Hence, Charles had and Robert ‡ of 28. Now of 28 are 12, and are 16. Therefore, Charles had 12, and Robert 16 pears. PROOF.-12 pears + 16 pears = 28 pears. 2. Divide 35 into 2 such parts that one shall be to the other as 3 to 2. 3. Divide 42 cts. into two such parts that one shall be to the other as 2 to 4. 4. A farmer had 56 acres of which he made 2 pastures in the ratio of 3 to 5; how many acres were in each ? 5. A man bought a cow and a calf for $50; the cow was worth 4 times as much as the calf; what was the value of each? 6. Divide $72 into two such parts that one shall be to the other as 4 to 8. WRITTEN EXERCISES. 698. To divide a number into two or more parts which shall have a given ratio to each other. 1. A and B divided $145 in the ratio of 2 to 3; how much had each ? SOLUTION.-The sum of the proportional parts is to each separate part as the number to be divided is to each man's share. That is, 5 (2+3) is to 2 as $145 to A's share. Again, 5 is to Hence, the 3 as $145 to B's share. OPERATION. 52 $145 : A's s. 5:3: $145: B's s. ($145 × 2)÷5 = $58, A's s. ($145x3)-5= $87, B's s. |