COMPOUND PROPORTION. 739. A Compound Proportion is a proportion in which one or both ratios are compound. 1. The product of the simple ratios of the first couplet equals the product of the simple ratios of the second couplet. For, the value of a compound ratio is the product of the simple ratios, and these compound ratios are equal, since a proportion expresses the equality of ratios. Thus, from the second of the above proportions, we have 15% = 1×12. 2. The product of all the terms in the extremes equals the product of all the terms in the means. = For, from the nature of proportion, we have from the proportion above, &X1% =×12, and clearing of fractions, we have 3x5X8X12 =4×6×6×10, which by examination, we see is the product of the extremes equal to the product of the means. 3. Any term in either extreme equals the product of the means divided by the product of the other terms in the extremes. For, since from the proportion above we have 3×5×8×12=4×6 4×6×6×10 ×6×10, we will have 3and similarly for any other 5X8X12 term in either extreme. = 4. Any term in either mean equals the product of the extremes divided by the product of the other terms in the means. For, from the above proportion, we have 3×5×8×12=4×6×6× 3x5X8X12 10, hence 4= and similarly for any other term in the 6×6×10 means. EXAMPLES FOR PRACTICE. Find the term denoted by x in each of the following: APPLICATION OF COMPOUND PROPORTION. 741. Compound Proportion is used in the solution. of problems in which the required term depends on a compound ratio. 742. The Unknown Quantity in simple proportion depends upon the relation of one pair of similar quantities; in compound proportion it depends upon two or more pairs of similar quantities. NOTE.-Problems in compound proportion may be solved by two or more simple proportions, or by analysis. 1. If 6 men earn $90 in 5 days, how much will 8 men earn in 9 days? that they can earn in 5 days, as 9 is to 5; hence the sum that 8 men can earn in 9 days is to $90 (the sum that 6 men will earn in 5 days) as 8:6 and 95; hence we have the proportion, The sum: $90 :: from which, by Prin. 3, we have, The sum = $90×8×9 6x5 { 8:61 9:5 or $216. Rule.-I. Put the required quantity for the first term and the similar known quantity for the second term, and form ratios with each pair of similar quantities for the second couplet, as if the result depended upon each pair and the second term. II. Find the required term by dividing the product of the means by the product of the fourth terms. NOTES.-1. Teachers may put the unknown quantity in the fourth term instead of the first, if they prefer it. The method of solution will be the same in principle, and the rule can be readily changed to correspond with it. 2. Pupils should be required to solve both ways, and to give the rule for both methods. 2. If 47 horses eat 94 bundles of hay in 36 days, how many bundles will 57 horses eat in 48 days? Ans. 152. 3. If 25 yd. of muslin 13yd. wide cost $7.25, what cost 27 yd. of the same quality, 14 yd. wide? Ans. $8.70. 4. If 1053 bricks, 8 in. long, 4 in. wide, are required for a walk 39 ft. long, 6 ft. wide, how many bricks will be required for a walk 144 ft. long and 8 ft. wide? Ans. 5184 bricks. 5. If 20 pipes, each delivering 18 gal. a minute, fill a cistern in 3 h. 24 min., how many pipes, each delivering 12 gal. a minute, will fill a cistern twice as large in 4 h. 15 min.? Ans. 48 pipes. 6. A farmer has a bin 7 ft. long, 5 ft. wide, and 4 ft. deep, which holds 112 bu. of corn; how deep must he make another which is 20 ft. long and 9 ft. wide, so that it may hold 864 bushels? Ans. 6 ft. 7. How many days will it take 15 men to cut 810 cords of wood, working 9 hours a day, if 13 men can cut 364 cords in 14 days, working 12 hours a day? Ans. 36 days. 8. Required the cost of 192 loaves of bread, each loaf weighing 7 oz., when flour is worth $12 a barrel, if 315 loaves, weighing 6 oz. each, cost $16.20, when flour is $9 a barrel. Ans. $15.36. 9. If $7486.50 be paid for a farm of 150 A. 150 P., what will be the cost of 90 A. 75 P., if 6 acres of the latter be worth 5 of the former? Ans. $3739.37. 10. How many men will be required to dig a trench 450 rods long, 18 ft. wide, and 10 ft. deep, in 18 days, if 45 men can dig a trench 180 rods long, 15 ft. wide, and 9 ft. deep, in 12 days? Ans. 100 men. 11. If a cistern 28 ft. long, 14 ft. wide, 11 ft. deep, hold 512 barrels of water, how many barrels of water will a cistern hold that is 21 ft. long, 8 ft. deep, and 11 ft. wide? Ans. 219 bar. 12. If 13 men can cut 364 cords of wood in 14 days by working 12 hours a day, how many hours a day must 15 men work to cut 810 cords in 36 days? Ans. 9 hours. 13. If 24 pipes, each delivering 6 gal. a minute, fill a cistern 8 ft. long, 6 ft. wide, and 5 ft. deep, in 1234 min., how many pipes, each flowing 8 gal. a minute, will fill a cistern 10 ft. long, 7 ft. wide, and 9 ft. deep, in 21 minutes? Ans. 27 pipes. 14. What cost 54 planks 35 ft. long, 28 in. wide, and 5 in. thick, if 42 planks 36 ft. long, 25 in. wide, and 7 in. thick, cost $178 when lumber was worth 4 more per foot? Ans. $138.444. 15. The first couplet of a compound proportion is made up of the ratios 5 : 15 and 4 : 16, and the first ratio of the second couplet is 4: 12; what is the second ratio, if the antecedent is 11? Ans. 11: 44. 16. The first couplet of a compound proportion is made up of the ratios 5 : 15 and 4 : 16, and the first ratio of the second couplet is 4:12; what is the other ratio, if the antecedents of the second couplet are as 2 to 7? Ans. 14: 56. 17. If 6 compositors in 18 days of 12 hours each, set up 27 sheets of 24 pages each, 45 lines on a page and 48 letters in a line, in how many days, 10 hours long, can 7 compositors set up, in the same type, 35 sheets, 16 pages each, 51 lines to a page, 45 letters in a line? Ans. 17 days. 18. The second couplet of a compound proportion consists of the simple ratios 8: 10 and 14: 16, and the antecedents of the first couplet are as 9: 7, and the second consequent of that couplet is 8; required the ratios of the first couplet. Ans. 18: 45 and 14: 8. 19. The first couplet of a compound proportion consists of the simple ratios 7 : 11 and 8 : 14, and the consequents of the second couplet are as 7: 11, and the first antecedent of that couplet is 9; required the ratios of the second couplet. Ans. 9 21 and 28: 33. 20. If 27 men in 18 days of 10 hours each dig a ditch 180 rods long, 6 ft. wide, and 3 ft. deep, of 5 degrees of hardness, how many days of 9 hours each will it take 45 men to dig a ditch 300 rods long, 8 ft. wide, and 4 ft. deep, of 71⁄2 degrees of hardness? Ans. 531 days. PARTITIVE PROPORTION. 743. Partitive Proportion is the process of separating a number into parts which bear certain relations to each other. 744. There are several cases arising from the various relations which may exist between the parts into which a number is divided. NOTE.-The method of solution is analytical, and no rule is given. CASE I. 745. When one part is a number more or less than another. 1. Divide 48 into two parts so that the first may be 12 more than the second. SOLUTION.-The 2d part plus 12 equals the 1st part, which, added to the 2d part, equals 2 times the 2d part, plus 12, which equals 48; if twice the 2d, plus 12 equals 48, twice the 2d part equals 48 minus 12, or 36, and once the 2d part equals of 36, or 18, and the second part plus 12 equals 18 plus 12, or 30. OPERATION. 2 times 2d+12=48 = 2. A and B have $20,000, and A has $1500 more than B; what is the fortune of each? Ans. A, $10750; B, $9250. 3. A man divided $50,000 among his three sons, giving the first $15,000 more than the second, and the second $5000 less than the third; how much did each receive? Ans. 1st, $25,000; 2d, $10,000; 3d, $15,000. 4. A and B had the same number of shares of Erie; A sold 60 shares and B bought 54 shares, and they then together had 144 shares; how many had each at first? Ans. 75. 5. Four young men, A, B, C, and D, started to Europe with $5000; A had $57 more than B, C had $65 less than D, D had $98 more than B; how much money had each? Ans. A $1260; B, $1203; C, $1236; D, $1301. CASE II. 746. When one part is a number of times another or a fractional part of another. |