15. If a man spends $230 in the three spring months, at the same rate per day how much will he spend in a year? Ans. $912.50. 16. If 12 men build a wall in 24 days, how long will it take 60 men to build it at the same rate? Ans 4 da. NOTE. Here it is evident that the time in which 60 men do t, is to 24 days, the time in which 12 men do it, as 12 men is to 60 men. 17. If 28 men mow a field of grass in 12 days, how many men will be required to mow it in 8 days? Ans. 42 men. 18. What is the height of a staff which casts 41 ft. of shadow, if a staff 6 ft. 9 in. high cast a shadow 13 ft. 8 in. ? Ans. 20 ft. Ans. $7.50. 19. If 13 bu. of corn cost $6.25, what will 16a. cost at the same price per bushel? 20. If of a barrel of flour cost of an eagle, how many dollars will 15 barrels cost? Ans. $91.20. 21. If a person do a piece of work in 142 days, working 9 hours per day, in what time will he do it, working 63 hours a day? Ans. 189 days. 22. If 96 bushels of oats keep 42 horses 8 days, how long will 168 bushels keep them? Ans. 14 da. 23. If 4 A. 120 P. of land cost $437, what will 16 A. 80 P cost at a rate of $25 less an acre? Ans. $1105.50. 24. If 17 men can mow a field in 9 days, how long would it take to reap half of it if 5 men refuse to labor? Ans. 63 days. 25. A failed and could pay only 75 cts. on each dollar he owed; how much did C receive, whom he owed $1968? Ans. $1476. 26. If a three-cent loaf weigh 9 ounces when flour is $6 a arrel, how much should it weigh when flour is $8 a barrel? Ans. 62 oz. 27. A lent me $560 for 10 months; how end him $800 to reciprocate the favor? long should I Ans. 7 mo. 28. A has grain worth $1.12 a bushel, and B has flour worth $6.25 a barrel; now if in an exchange A puts his grain at $1.25 a bushel, what should B charge for his flour? Ans. $6.943 29. Two cog-wheels, one having 28 and the other 20 cogs, run together; in how many revolutions of the larger wheel will the smaller gain 12 revolutions? Ans. 30. 30. A garrison of 2400 men has provisions sufficient to last them 20 days, at the rate of 11⁄2 lb. a day; how large a reinforcement could be received for the time if the allowance be reduced to 15 oz. a day? Ans. 1440 men COMPOUND PROPORTION. 566. A Compound Proportion is a proportion in which one or both ratios are compound. (4:12) (5:10) 7:145 26:185 :: 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 are equal, since a proportion expresses the equality of ratios. Thus, from the second of the above proportions we have,××ͧ• 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, XX; and clearing of fractions, we have 4x7x10x18 =5×6×12×14, 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 4×7×10×18=5×6 5×6×12×14 X12×14, we will have 4and similarly for any other term in either extreme. " 7x10x18 4. Any term in either mean equals the product of the ex tremes divided by the product of the other terms in the means. For, from the above proportion, we have 4×7×10×18=5×6×12 x14, hence 5(4×7×10×18)÷(6×12×14); and similarly for any other term in the means. Find the term denoted by x in each of the following: APPLICATION OF COMPOUND PROPORTION. 568. Compound Proportion is used in the solution of problems in which the required term depends on a compound ratio. 569. In simple proportion the unknown quantity depends upon the relation of one pair of similar quantities; in compound proportion the unknown quantity 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 4 men can earn $24 in 7 days, how much can 14 men earn in 12 days? OPERATION. 14:4 The sum: $24::12:7 The sum= $24 × 14 × 12 =$144, Ans. SOLUTION.-It is evident that the sum 14 men can earn in a given time is to the sum that 4 men can earn in that time as 14 to 4, and also the sum that they can earn in 12 days is to the sum that they can earn in 7 days as 12 is to 7; hence the sum that earn in 12 days is to $24 (the sum that 4 men will earn 14:4 and 12:7; hence we have the proportion, the sum : frein which, by Prin. 3, we have the_sum = 14 men can in 7 days), as 14:4 $24:: 12:7 OPERATION. 3 2 12 14 × $24-$114 77 4 ANALYSIS.-If 4 men earn $24 in 7 da., 1 man will earn of $24, and 14 men will earn of $24. If 14 men earn of $24 in 7 da., in 1 day they will earn of of $24, and in 12 da. they will earn 12 of 14 of $24, which, by cancelling and multiplying, equals $144. NOTE.-The analysis may be abbreviated thus: If 4 men earn $24, 14 men will earn 14 of $24. If they earn it in 7 da., in 12 da. they will earn of 4 of $24 = $144. 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. WRITTEN EXERCISES. 2. If 36 men earn $324 in 18 days, how much will 42 men earn in 27 days? Ans. $567. 3. If 58 cows eat 29 bundles of hay in 25 days, how many cows will eat 35 bundles in 14 days? Ans. 125 cows. 4. If $600 in 4 yr. 6 mo. at 6% gain $162 interest, how much will $800 gain in 6 yr. 4 mo. at 8%? Ans. $405. 5. If 12 men in 35 days build a wall 140 rd. long, 6 ft. high, how many men can in 40 days build a wall of the same thickness 144 rods long, 5 ft. high? Ans. 9 men. 6. If 18 carpenters build a house in 45 days, working 12h. a day, in how many days would 36 carpenters have built it, working 10 h. a day? Ans. 27 days. 7. If 28 men dig a trench 120 rods long, 15 ft. wide, and 12 feet deep, how many men will dig a trench 360 rods long, 9 feet wide, and 10 feet deep? Ans. 42 men. 8. If 8 yd. of muslin, 14 yd. wide, cost 10 yd. of the same quality, 14 yd. wide? $1.25, what cost Ans. $1.40. 9. If 35 horses can eat a lot of grain in 36 days, in what time will 3 times as much grain be consumed, if 5 horses are added when the grain is Ans. 104ğ da. eaten? 10. If 32 men can build 60 rods of wall in 15 days, in what time can they build 75 rods, if 8 men leave when 40 rode have been built? Ans. 214 du 11. If 147 loaves of bread, weighing 6 oz. each, cost $8.10, when flour is $7 a barrel, what cost 96 loaves, of 7 oz. each, when flour is worth $9 a barrel? Ans. $7.93+. 12. If 17 plank, 35 ft. long, 28 in. wide, and 6 in. thick, cost $68, what cost 40 plank, 32 ft, long, 25 in. wide, and 7 in. thick, lumber worth more per foot? Ans. $182.85. 13. If from a dairy of 24 cows, each giving 18 qt. of milk daily, 10 cheeses of 60 lb. each are made in 12 weeks, how many cows will produce 40 cheeses of 75 lb. in 9 weeks, if they give 12qt. each? Ans. 240 cows. 14. If 24 men, in 15 da. of 12 h. each, dig a trench 300 rd. long, 5 yd. wide, ft. deep, in what time can 45 men, working 10 h. a day, dig a trench 125 rd. long, 15 ft. wide, 8 ft. deep? Ans. 5 da. 15. If 50 men can build 50 rods of wall in 75 days, how many men will be required to build 80 rods of wall as thick and as high in 40 days? Ans. 180 men. INSURANCE. 570. Insurance is a contract of indemnity for loss o damage within a given time. It is of two kinds: Property Insurance and Personal Insurance. 571. Property Insurance is security against loss by fire or transportation. Insuring anything is called "taking a risk." 572. Property Insurance is of two kinds: Fire Insurance and Marine Insurance. 573. Fire Insurance is security against loss by fire; Marine Insurance is security against loss by navigation. 574. The Insurer or Underwriter is the party or company taking the risk. The Insured or Assured is the party protected. 1. The Policy is the written agreement or contract between the insurers and the insured. 2. The Premium is the sum charged for insurance; it is a certain rate per cent. of the amount insured. Partitive Proportion is on page 379, Conjoined Proportion on page 384, and Medial Proportion on page 387. |