2. If 6 men can do a piece of work in 5 days, in how many days can 10 men do the work ? Process. We find that the answer, or x, will be less than the third quantity, for 10 men will not require so long a time as 6 men. We have, therefore : Analysis. 10 : 6 Or, 6 : 10 = x :5. If 6 men = 5 days, 10x 30 days; x = 3 da. 10 men = 3 days. 5 : X. 30. 1 man = THE RULE OF THREE. 1. Let x represent the required term or answer. 2. With x and the quantity of like denomination form a ratio. 3. Compare the two terms of the ratio, and determine from the conditions of the problem whether x is greater or less than the other term. 4. With the two given like quantities form a ratio equal to the first. 5. Express the equality of the ratios, and apply formula A or B, as the case may require. PROBLEMS. NOTE.—Solve the following problems both by analysis and proportion. Suggestion : Let x represent the fourth term. 2. How long will it take 24 men to do a piece of work that 8 men can do in 12 days ? 3. How far can a certain load be carried for $34, if $64 will carry it 100 miles ? 4. If 231 men have provisions for 8 mo., how long will the same provisions last 308 men ? 5. If 95 cents will buy one bushel of wheat, how many bushels will $11.75 buy ? 6. A man owes $2500, and can pay only $1000. How much does he pay on a dollar ? 7. If 28 yd. of oil-cloth, .875 yd. wide, cover a certain floor, how many yards 1.25 yd. wide will cover the same floor? 8. If 22 bu. 3 pk. of corn be produced on one acre, how many acres will produce 546 bu. ? 9. If two men earn $72 in 6 da., how much will 30 men earn in the same time? 10. If 121 tons of hay cost $180.25, what will 811 tons cost? Suggestion : Let x represent the third term in the following problems. 11. A regiment of 960 men has provisions for 40 days. How long will it last if the regiment is reinforced by 240 men ? 12. A field can be mowed in 4 days of 11 hours each ; how many days of 9 hours each will it take? 13. At the time when a man 5 ft. 9 in. in height casts a shadow 4 ft. 6 in. long, what is the height of a tree that casts a shadow 52 ft. 6 in. long ? 14. If a locomotive runs 964 miles in 31 hours, how many miles will it run in 57 hours ? 15. A wheel makes 75 revolutions in 5 min. How many does it make in an hour? 16. A. can do a piece of work in 6 days, B. can do it in 7 days. If B.'s wages are $2.10 per day, how much should A. receive per day ? 17. If a 5-cent loaf of bread weighs 8 ounces when flour is worth $5, what should such a loaf weigh when flour is at $6 ? 18. / yd. cost $. Find the cost of yd. 5 i 19. If a cistern containing 3000 gal. leak 1 gal. 2 qt. a min., how long will it take to empty it? 20. If 42 yd. of carpet 2 ft. 3 in. wide are required for a room, how many yd. of carpet 2 ft. 4 in, wide will be required ? a Suggestion: Let a represent the second term in the following problems. a How many 21. If a train, at the rate of is of a mile per min., requires 31 hr. to make a certain distance, how long will it require at the rate of % of a mile a min. ? 22. If a train travels t of a mile in 18 sec., how many miles an hour does it travel ? 23. A. gains 4 yd. on B. in running 30 yd. yd. will he gain while B. is running 971 yd. ? 24. If a man spends $276 in the three summer months, how much will he spend in a year at the same rate per day? 25. If 28 men mow a field of grass in 12 days, how many men will be required to mow it in 8 days? 26. If 17 men can mow a field in 9 days, how long would it take to mow half of it if 5 men refuse to labor ? 27. If 141 yd. of cloth cost $191, how much will 197 yd. cost? 28. If of a ship costs £273 2s. 6d., what will be of her cost? 29. If 2; gal. of molasses cost 65 cents, what will 3} hhd. cost? 30. If a steeple 150 ft. high casts a shadow 210 ft., what is the length of the shadow cast, at the same time, by a staff 5 ft. high? Suggestion : Let x represent the first term in the following problems. 31. If the interest of $600 for 6 mo. is $15, what principal will gain $64 in the same time? 32. If 151 yd. of silk that is tyd. wide will make a dress, how many yards of muslin that is 18 yd. wide will be required to line it? 33. If I borrow $500 and keep it 1 yr. 4 mo., for how long . a time should I lend $240 as an equivalent for the favor ? 32 34. A butcher in selling meat sells 1411 oz. for a pound. How much does he cheat a customer who buys of him to the amount of $30? 35. In what time can a man pump 64 hhd. of water if he can pump 12 hhd. in 2 hr. 15 min. ? 36. How many men can do in 24 days a piece of work which would employ 40 men 6 days? 37. If 1 of 1 of 64 bbl. of beef cost $78, how much will { of ã of 37 bbl. cost? 38. If 450 tiles, each 12 in. square, will pave a cellar, how many tiles that are 9 in. by 8 in. will pave the same? 39. If a distance of 48 miles is represented on a map by 14 in., what distance is represented on the same map by 7} in.? 40. Twenty-four men in 30 days can finish a piece of work. After 16 days 11 men quit work. In how many days can the rest finish the work ? a 5 3 COMPOUND PROPORTION. A Compound ratio indicates the product of two or more simple ratios ; for instance, * x is a compound ratio, being the product of the simple ratios and }, written 3 : A Compound proportion has one of its ratios compound. ILLUSTRATION. If 5 men build a wall 6 ft. high in 7 days of 8 hr., in how many days of 9 hr. can 10 men build a wall 11 ft. high ? Process. : The second ratio is simply 7 da. : x da. We write 9: 8, for 9 hr. per day require fewer days than 8 hr. } 10: 5 9: 8 4 5 x 11 x 8 x 7 By formula A, x = 154 = 529 days. 10 X 6 X 9 2 7 x 5 x 11 x 8 By Analysis : 1 X 10 X 6 X 9 Since 5 men require 7 da., 10 men require a shorter time, i. e., 1% of 7 da. ; since 6 ft., etc. 527 da. PROBLEMS. 1. If 6 men can mow 24 acres of grass in 2 days, by working 10 hours per day, how many days will it take 7 men to mow 56 acres by working 12 hrs. per day? 2. If 4 men mow 15 A. in 5 da. of 14 hr., in how many da. of 13 hr. can 7 men mow 191 A. ? 3. If 810 bricks, 8 in. long and 4 in. wide, are required for a walk 36 ft. long and 5 ft. wide, how many bricks will be required for a walk 66 ft. long and 4 ft. wide ? 4. If the interest on $640 for 4 yr. 6 mo. is $172.8, what is the interest on $820 for 2 yr. 8 mo. at the same rate ? 5. If it requires 275 yd. of cloth i yd. wide to make 75 garments, how many yards of cloth 14 yd. wide will it require to make 215 such garments ? 6. If it costs $2.40 to carry 20 cwt. 50 miles, what will it cost to carry 40 cwt. 40 miles at the same rate ? 7. If 12 candles, 8 weighing a pound, last from 5 o'clock to 11, how many candles, 6 weighing a pound, will last from 7 o'clock to 11 ? 8. A farmer owning 25 horses traded them for sheep. If 3 horses are worth 12 cows, 6 cows are worth 42 pigs, and 25 |