31. A rectangular lot containing 20 sq. rd. is 4 rd. wide. How long is it? SOLUTION. sq. rd. sq. rd. rd. :: rd. 4 : 20 1 : (5) EXPLANATION.-(See Figure, 209.) Four sq. rd. along one side are 1 rod wide. Therefore, 4 sq. rd. are to the 20 sq. rd., as the width of the 4 sq. rd. is to width of the 20 sq. rd. 32. If 2 mo. boarding cost $50, what would 15 mo. boarding cost, at the same rate? Ans. $314.65. 33. A lot 40 rd. long and 4 rd. wide contains an acre. How long must an acre lot be, if 8 rd. wide? Ans. 20 rd. 34. If sound goes 6160 ft. in 5 sec., how far does it go in 16 sec.? Ans. 18480 ft. 35. If the moon moves on the average 52° 42′ 20′′ in 4 da., in what time would it make a complete circuit of the earth? Ans. 27.32 da.+. 36. If a clock loses 1 min. 45 sec. in 36 hr., how much time will it lose in 2 wk.? Ans. 16 min. 20 sec. Ans. $12. Ans. $30. 37. If $175 gain $21, what do $100 gain? Ans. $.0032. Ans. $400. 41. What does $1 earn? Ans. $.40. 42. If the 13 loaf weigh 15 oz. when flour is $8 per bbl., what should it weigh at $12 per bbl.? Ans. 10 oz. 43. What should it weigh at $6 per bbl.? Ans. 20 oz. short was the gallon measure? 44. A 40 gal. barrel is filled with a defective gallon measure, and thus appears to hold 42 gal. 45. If of a barrel of cider cost $1, what is the cost of of a barrel ? Ans. $141. How much Ans. 2 gi. Compound Proportion. Compound Proportion is used when the solution of a problem depends upon two or more pairs of conditions. Oral Exercises EXAMPLE. If $50 gain $2 in 8 mo., how much will $25 gain in 10 mo. ? SOLUTION.-If $50 gain $2 in 8 mo., $25 will in the same time gain less, in the ratio of 25 to 50, or 25 = 1; of $2 = $1. If $25 gain $1 in 8 mo., in 10 mo. it will gain more in the ratio of 10 to 8, or 10 = 5; 5 of $1 = $11. Therefore, if $50 gain $2 in 8 mo., $25 will gain $14 in 10 mo. PROBLEMS. 1. If 8 men mow 32 A. in 3 da., how many men can mow 48 A. in 12 da. ? 2. If 10 loaves are sufficient for 20 men 1 da., how many loaves will supply 40 men 2 da. ? 3. If 5 horses eat 10 tons of hay in 12 mo., how many tons will 8 horses consume in 6 mo. ? 4. If 3 boys earn $7 in 4 da., how much can 9 boys earn in 20 da.? 5. If 400 bu. coal are required to supply 8 fires 5 mo., how much coal would 2 fires consume in 10 mo. ? 6. If 5 weeks' boarding of 6 men cost $150, what will 7 wk. boarding of 5 men cost? 7. If 10 men can build a wall 30 ft. long in 5 da., how long will it take 16 men to build a similar wall 96 ft. long? 8. If 25 oranges are worth 50 lemons, and 6 lemons are worth 244, how much are 15 oranges worth? 9. If 10 men, by working 10 hr. per da., can build a boat in 8 da., how many da. would it take 12 men to do the work, if they wrought only 8 hr. per da. ? 10. If a 5 loaf weighs 10 oz. when flour is $8 a bbl., how much should an 8 loaf weigh, when flour is $5 a bbl.? Written Exercises+ EXAMPLE.-If 16 men in 20 da. 10 hr. long can build 5 mi. of railroad, in how many da. 12 hr. long can 64 men build 30 miles of railroad? 64 × 12 × 5: 16 × 10 × 30 :: 20 : (25) EXPLANATION.-We write 20 da, for 3d term, because our answer, or 4th term, is required in da. Since 64 men will perform the work in less time than 16 men, our 4th term must be less than the 3d, and we therefore write 64 men for our 1st term and 16 men for the 2d. Since it would take a shorter time to do a piece of work by working 12 hr. per day than by working 10 hr. per day, our 4th term must be less than the 3d, and we therefore write 12 hr. for the 1st term and 10 hr. for the 2d. Since it would take more time to build 30 mi. of road than 5, the 4th must be greater than the 3d term, and we therefore write 5 mi. for the 1st and 30 mi. for the 2d term. We then reduce the compound ratio to a simple one (269), and find the 4th term by Rule (284), or by (281, RULE I.) 285. RULE.-Write as the third term that number which is of the same kind as the required answer. Then, of the other terms, form a compound ratio, being careful to write the terms in pairs, and to state each pair of terms, as if the value of the 4th term depended on that pair alone. After which, find the fourth term in accordance with the Principle of Proportion (281). This and all similar examples may be solved by Analysis. Thus, SOLUTION.-If 16 men can build 5 mi. (or any other number of miles) of railroad in 20 da., 64 men will build the same amount in the same part of 20 da. that 16 is of 64, that is of 20 da., or 5 da.; and if they can build 5 mi. in 5 da., they can build 30 miles in § of 30 da. 30 da. And if they can build it in 30 da., by working 10 hr. per da., they can, by working 12 hr. per day, build it in 19, or & of 30 da., or 25 da., Ans. = This and similar examples may also be solved by Cause and Effect. Thus, We have now to find the number of days in the Sec. cause. And since this is a proportion, the product of the extremes equals the product of the means, and we therefore can readily fill the blank ( ) by Rule I, or II. (281). PROBLEMS. 1. If 6 men build a wall 3 rd. long in 5 da., how many men would build a wall 15 rd. long in 10 da. ? Ans. 2. 1f 24 cows eat 36 bu. corn meal in 12 da., how many bu. will last 20 cows 60 da. ? Ans. 150 bu. 3. If $75 gain $12 in 2 yr., what sum will gain $150 in 18 mo. ? Ans. $1250. Ans. $7680. 4. If 16 men earn $640 in 26 da., what will 48 men earn in 104 da. ? 5. If 3 men in 18 da. earn $162, how many dollars will 9 men earn in 12 da., with like wages? Ans. $324. 6. If 10 persons spend $600 in 8 mo., how much would 8 persons spend in 12 mo., at the same rate? Ans. $720. 7. If 7 persons spend $273 in 2 mo., in how many mo. would 5 persons spend $650? Ans. 6 mo. 8. If $250 gain $15 in 1 yr., in how many months will $750 gain $37.50? Ans. 10 mo. 9. If $300 gain $9 in 3 mo., what will gain $50 in 10 mo. Ans. $500. 10. If 8 men dig a ditch 6 rd. long in 10 da., how many men would dig 27 rd. long in 12 da.? Ans. 30 men. 11. If 8 men dig a ditch 6 rd. long in 10 length of time would 30 men dig 27 rd. long? 12. If 8 men dig a ditch 6 rd. long in would 30 men dig in 12 da. ? 10 da., in what Ans. 12 da. da., how far Ans. 27 rd. 13. If 12 horses eat 18 bu. of oats in 6 da., how many bu. would last 20 horses 30 da.? Ans. 150 bu. Ans. 40 da. 14. If 6 horses eat 4 bu. oats in 3 da., how many days will 100 bu. last 10 horses? 15. If 4 horses eat 12 bu. of oats in 6 da., how many horses will require 75 bu. in 30 da. ? Ans. 5 horses. 16. If 11 men mow 45 A. of grass in 6 da. of 10 hr. each, how many men will be required to mow 81 A. in 12 da. of 11 hr. each ? Ans. 9 men. 17. If 11 men mow 45 A. in 6 da. of 10 hr. each, how many hr. per da. must 9 men mow, to finish 81 A. in 12 da.? Ans. |