which they depend increase and decrease; whilst in the three last questions the answers increase and decrease as the quantities upon which they depend decrease and increase. On this account, some authors make two divisions, Rule of Three Direct and Rule of Three Inverse. But since all questions classed under those two heads are solved by the same method, we shall not make that distinction. 269. Another class of questions, differing from those in (§ 267), in having more than three quantities given are known under the name of Double Rule of Three, but which might more correctly be named Rule of Five, &c., since five, &c., quantities are given, and a sixth, &c., is to be found. The following is an instance :— Ex. 1. If 36 acres of land be mown by 6 men in 5 days, how many acres can be mowed by 40 men in 16 days? 6 men in 5 days mow 36 acres, .. 1 man in 5 days mows 36, or 6 acres, .. 1 man in 1 day mows g acre, ..40 men in 1 day mow 8 40×6 or 48 acres, 1 ..40 men in 16 days mow 16 × 48, or 768 acres. Ex. 2. If 25 men, in 12 days of 9 hours each, can dig a trench 50 yards long, 4 feet broad, and 6 feet deep, how many men would be required to dig a trench 100 yards long, 3 feet broad, and 4 feet deep, in 18 days of 10 hours each? days. hrs. yds. feet. feet. in 12 of 9 a trench 50 lg. 4 bd. 6 dp. is dug by 25 men. This very simple and natural process is easily condensed into the following form, when the pupil has acquired practice. It is observed that the last term is of the same kind as the quantity required. days. hrs. ⚫ yds. feet. feet. in 12 of 9 a trench 50 lg. 4 bd. 6 dp. is dug by 25 men. 12 × 9 × 25 50×4×6 =3×5, or 15 men. 1 1 1 Ex. 3. If the sixpenny loaf weighs 5 lbs. when wheat is at 5.75s. per bushel, what must be paid for 524 lbs. of bread, when wheat is at 18.5s. per bushel? ...when wheat is 5.75s. per bushel, 5 lbs. of bread cost 6d. 6d. 5.75×5 270. MISCELLANEOUS EXERCISES IN THE RULE OF THREE. 1. If 17 workmen perform 119 yards of work, how much will 12 workmen perform in the same time? 2. If I lend a friend £500 for 9 months, how long ought he to lend me £300 to requite my kindness? 3. How many yards of paper, 27 inches wide, will hang a room that measures 50 feet round and 9 feet high? 4. What is the value of 144 cwt. of cheese, if 9 cwt. be worth £23. 6s. 6d. ? 5. How many tons of iron can I procure for £121. 5s. 4d., at the rate of £12. 4s. 4d. per ton? 6. If 6 cwt. 3 qrs. 12 lbs. of flour cost £9, what would 4 cwt. 2 qrs. cost. 7. If 5 acres are rented for £4. 13s. 4d., how much of like land may be rented for £70. 10s. 6d. ? 8. If the penny loaf weighed 14 ounces when wheat was 4s. per bushel, what must it weigh when wheat is 6s. 9d. per bushel? 9. If 14 horses eat 56 bushels of oats in 16 days, how many bushels will serve 20 horses 24 days? 10. What must be the breadth of a piece of ground which is 14 yards long, so that it may be as large as a piece 40 yards long and 43 yards broad? 11. An iron beam, 24 feet long, 24 feet broad, and 13 inches thick, weighs 28800 lbs. What must be the length of a beam whose breadth is 3 feet, thickness 8 inches, and weight 12428 lbs. ? 12. If 4 steam engines of 8-horse power, when employed 5 days a week, and 12 hours a day, raise through a given height 128 four-bushel sacks of wheat, weighing 64 lbs. per bushel, in what time will 9 engines of 10-horse power, when employed 3 days in a week, and 10 hours per day, raise through 8 times the former elevation 196 three-bushel sacks of wheat, weighing 65 lbs. a bushel? 13. If 8 horses can plough, in 5 days, 56 acres, how many acres will 35 horses plough in 8 days? 14. A captain of a ship has 10 months' provisions for 40 men. How long would those provisions last 32 men? 15. If the shilling loaf weighs 3 lb. 6 oz. when flour sells at £1. 13s. 6d. per quarter, how much should it weigh when flour sells at £1. 7s. 6d. per quarter? 16. Provisions in a fortified town are found to last 10000 soldiers 6 months; but a reinforcement being required, so that the provisions will last only two months, the number of soldiers to be added to the garrison is required. 17. If £1. 7s. be the wages of 4 men for 7 days, what will be the wages of 14 men for 10 days? 18. If 5 compositors, in 16 days of 14 hours long, can compose 20 sheets, of 24 pages in each sheet, 50 lines in a page, and 40 letters in a line, in how many days of 7 hours each can 10 compositors compose 40 sheets, 16 pages in a sheet, 60 lines in a page, and 50 letters in a line? 19. If the carriage of 4 cwt. of goods, for 50 miles, cost £1. 2s. 9d., how far can I have 144 cwt. carried, for the same money? 20. If 30 men perform a piece of work in 11 days, how many will accomplish another piece, six times as large, in a fourth of the time? 21. If 540 tiles, each 12 in. long and 12 in. broad, will pave a floor, how many will the same require when the tiles are 10 in. long and 8 in. broad? 22. The length of a tower, which casts a shadow 160 feet in length is required, knowing that at the same time a post 4 ft. 6 in. casts a shadow of 15 ft. 8 in. ? 23. A coat requires in making 24 yards of cloth, yard broad. How much would be required if the cloth was 13 yards broad? 24. A company of gold seekers extracted 475 lbs. 8 oz. 12 dwts. 16 grs. of gold in 9 yrs. 207 days 7 hrs. 50 min., when working 11 hours per day. How much will be extracted by the same company in 7 yrs. 67 days 9 hrs. 36 min., working 12 hrs. a day? 25. A garrison of 6000 men has provisions for 6 months, allowing every man 18 oz. per day; a reinforcement of 1500 men is sent, and the provisions are to last 10 months. What must be each man's allowance ? 26. Two pumping engines, in 8 days of 11 hours each, have lowered the level of a lake 6 feet. How many hours a day would three engines, similar to the first, be at work to produce the same effect in 5 days? 27. What is the cost of 13 packs of wool, each 2 cwt. 1 qr. 19 lbs., at the rate of £21. 10s. for 3 cwt. 17 lbs. ? 28. A person completes a journey of 160 miles in 3 days, travelling 11 hours a day. In how many days would he complete 1000 miles, travelling 15 hours a day, at the same rate? 29. If 9 masons can erect a wall, in 20% days of 9 hours each, how long would it take 3 masons to do 33 as much, reckoning 12 hours to the day? 30. If the rent of 46 ac. 3 ro. 14 p. be £100, what will be the rent of 35 ac. 2 ro. 10 p. ? 31. Suppose a greyhound makes 27 springs while a hare makes 25, and the springs are of equal length. Now, if the hare be 50 springs before the hound, in how many springs will he overtake her? 32. If the value of 1 lb. avoirdupois be £4. Os. 24d., how many shillings must be paid for 1 lb. Troy? 33. If a canal, 1500 yards long, 6 yards wide, and 3 yards deep, be dug in 30 days, of 12 hours each, by 280 men, how many men would be required to dig another canal, 3 miles long, 4 yards wide, and 2 yards deep, in 125 days of 10 hours each? 34. If 14 lbs. of sugar be sold for 9s. 4d., when the cost of 1 cwt. is £3. 7s. 6d., what would be the cost of 1 cwt. when 11 lbs. are sold for 7s. 14d.? 35. If 4 artillerymen can fire a gun 48 times, and 5 men 52 times in an hour, how much more time will be required for firing 21216 shots from 26 guns, when there are 4 men to a gun than when there are 5 men? 36. In what time would the wind move from the pole to the equator, at the rate of 23 miles per hour, the distance being 6214 miles? 37. If 63 lbs. of tea cost £20. 10s. 6d., what will 70 lbs. cost of a different quality, 9 lbs. of the former being equal in value to 10 lbs. of the latter? 38. If the circumference of the driving wheel of a locomotive be 16 feet, how many revolutions will it make between Bristol and Exeter, the distance being 75 miles? 39. If 135 lbs. of thread are woven into a piece of linen, 86 yards long and yard broad, what would be the length of another piece, woven out of 116 lbs. of thread, and which is 1 yards broad? 40. If yard of velvet cost £3, what will yard cost? 41. The making of 43 yds. 1 qr. 8 in. of cloth cost £48. 16s. 9d. What will be the cost of 77 yds. 2 qrs. 9 in., at the same rate? 42. To clothe 25000 men, it takes 125000 yards of cloth, § yard wide. How many yards of cloth, yard wide, are wanted to clothe 3840 men? |