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OBS. 1. Each remainder is of the same denomination as the dividend from which it arose. (Art. 113. Obs. 1.)

2. Reducing compound numbers to lower denominations may, with propristy, be called Reduction by Multiplication ; reducing them to higher denominations, Reduction by Division. The former is often called Reduction Descending; the latter, Reduction Ascending. They mutually prove each other.

EXAMPLES FOR PRACTICE.

1. In 136 rods and 2 yards, how many feet? Operation.

Proof. rods. yds.

3)2250 ft. 2136 2

54)750 yds. 54 yds. 1 r.

2 682

11)1500 68

136 r. 4 rem.=2 yards. 750 yds.

Now 136 r. 2 yds. is the 3 ft. 1 yd.

given number. 2250 ft. Ans. 2. In £71, 138. 63d., how many farthings ? 3. In £90, 7s. 8d., how many farthings? 4. In £295, 18s. 3 d., how many farthings ? 5. In 95 guineas, 17s. 93d., how many farthings? 6. How many pounds, shillings, &c., in 24651 farthings? 7. How many pounds, shillings, &c., in 415739 farthings? 8. How many guineas, &c., in 67256 pence? 9. In £36, 4s., how many six-pences ? 10. In £75, 12s. 6d., how many three-pences ? 11. Reduce 29 lbs, 7 oz. 3 pwts. to grains. 12. Reduce 37 lbs. 6 oz. to pennyweights. 13. Reduce 175 lbs. 4 oz. 5 pwts. 7 grs. to grains. 14. Reduce 12256 grs. to pounds, &c. 15. Reduce 42672 pwts. to pounds, &c. 16. In 15 cwt. 3 qrs. 21 lbs., how many pounds ? 17. In 17 tons 12 cwt. 2 qrs., how many ounces?

QUEST.-Obs. Of what denomination is each remainder ? What may reducing compound numbers to lower denominations be called ? To higher denominations ?

Which of the fundamental rules is employed by the former? Which by the latter ?

18. In 52 tons 3 cwt., how many pounds ?
19. In 140 tons, how many drams?
20. In 16256 ounces,

how

many hundred weight, &c. ? 21. In 267235 pounds, how many tons, &c. ? 22. In 563728 drams, how many tons, pounds, &c. ? 23. Reduce 95 pounds (apothecaries' weight) to drams. 24. Reduce 130 pounds to scruples. 25. Reduce 6237 drams (apothecaries' weighi) to pounds, &c 26. Reduce 25463 scruples to ounces, &c. 27. How many feet in 27 miles ? 28. How many inches in 45 leagues ? 29. How many yards in 3000 miles ? 30. In 290375 feet, how many

miles ? 31. In 1875343 inches, how many leagues ? 32. In 15 m. 5 fur. 31 r., how many rods ? 33. In 1081080 inches, how many miles, &c. ? 34. How many feet in the circumference of the earth ? 35. How many nails in 160 yards ? 36. How many quarters in 1000 English ells ? 37. In 102345 nails, how many yards, &c. ? 38. In 223267 nails, how many French ells ? 39. In 634 yards, 3 qrs.,

nails? 40. In 28 hhds. 15 gals. wine measure, how many quarts ? 41. In 5 pipes, 1 hhd., how many gallons 42. In 3 tuns, 1 hhd. 10 gals., how many gills ? 43. In 12256 pints, how many barrels, wine measure ? 44. In 475262 gills, how many pipes, &c. ? 45. In 50 hhds. 1 bbl. 10 gals., how many gills, wine measure 9 46. In 45 bbls., how many pints, beer measure ? 47. How many barrels of beer in 25264 pints ? 48. How many hogsheads of beer in 136256 quarts ? 49. How many pints in 45 hhds. 10 gals. of beer ? 50. In 15 bushels, 1 peck, how many quarts ? 51. In 763 bushels, 3 pecks, how many quarts ? 52. In 56 quarters,5 bushels, how many pints? 53. In 45672 quarts, how many bushels, &c. ? 54. In 260200 pints, how many quarts ?

how many

55. Reduce 25 days, 6 hours to minutes.
56. Reduce 365 days, 6 hours to seconds.
57. Reduce 847125 minutes to weeks.
58. Reduce 5623480 seconds to days.
59. How many seconds in a solar

year · ? 60. How many seconds in 30 years, allowing 365 days 6 hours to a year?

61. How many years of Sabbaths are there in 70 years ?.
62. In 110 degrees, 20 minutes, how many seconds ?
63. In 11 signs, 45 degrees, how many seconds ?
64. In 7654314 seconds, how many degrees ?
A5. In 1000000000 minutes, how many signs ?
66. Reduce 1728 sq. rods, 23 yds. 5 feet to feet.
67. Reduce 100 acres, 37 rods to square

feet. 68. Reduce 832590 sq. rods to sq.

inches. 69. Reduce 25363896 sq. feet to acres, &c. 70. In 150 cubic feet, how many inches ? 71. In 97 yds. 15 ft., how many cubic inches ? 72. In 49 cords, 23 feet, how many cubic inches ? 73. In 84673 cubic inches, how many

feet? 74. In 49216 cubic feet, how many

cords? 75. In 65 tons of round timber, how many cubic inches ? 76. In 4562100 cubic inches, how many tons of hewn timber?

APPLICATIONS OF REDUCTION.

283. To reduce Troy to Avoirdupois weight.

First reduce the given pounds, ounces, &c., to grains ; then divide by the number of grains in a dram, and the quotient will be the answer in drams. (Art. 252.)

Obs. If the answer is required to be in pounds and a fraction of a pound, divide the grains by 7000.

Ex. 1. In 175 pounds Troy, how many pounds avoirdupois ?

Solution.175X12 X 20 X24=1008000 grs., and 1008000 grs.+2744=36864 drams, or 144 lbs. avoirdupois. Ans.

QUEST.-283. How is Troy weight reduced to avoirdupois ?

2. In 700 lbs. Troy of silver, how many pounds avoirdupois ? 3. In 840 lbs. 6 oz. 10 pwts., how many pounds, &c., avoirdu

pois ?

4. An apothecary bought 1000 lbs. of opium by Troy weight, and sold it by avoirdupois : how many pounds did he lose ?

5. A merchant bought 1500 pounds of lead Troy weight, and sold it by avoirdupois : how many pounds did he lose ?

284. To reduce Avoirdupois to Troy weight.

First reduce the given pounds, ounces, dc., to drams, then multiply by the number of grains in a dram, and the product will be the answer in grains. (Art. 252.)

Obs. 1. When the given example contains pounds only, we may multiply them by 7000, and the product will be grains.

2. If the answer is required to be in pounds and a fraction of a pound, divide the grains by 5760.

6. In 32 lbs. avoirdupois, how many pounds Troy ?

Solution.--32X16X16X2711=224000 grs., and 224000 grs. =38 lbs. 10 oz. 13 pwts. 8 grs. Ans.

7. In 48 lbs. avoirdupois, how many pounds Troy?

8. A merchant bought 100 lbs. 10 oz. of tea avoir. and sold it by Troy weight: how many pounds did he gain ?

9. A druggist bought 1260 lbs. of alum avoirdupois, and retailed it by Troy weight: how many more pounds did he sell than he bought ?

285. The area of a floor, a p.ece of land, or any surface which has four sides and four right-angles, is found by multiplying its length and breadth together.

Note.-1. The area of a figure is the superficial contents or space contained within the line or lines, by which the figure is bounded. It is reckoned in square inches, feet, yards, rods, &c.

2. A figure which has four sides and four right-angles, like the following diagram, is called a Rectangle or Parallelogram.

QUEST.-284. How is avoirdupois weight reduced to Troy? 285. How do you find the area or superficial contents of a surface having four sides and four right-angles ? Note. What is meant by the term area ? How is it reckoned? What is a figure which has four sides and four right-angles called ?

10. How many square yards of carpeting will it take to cover a room, 4 yards long and 3 yards wide ?

Suggestion.-Let the given room be represented by the subjoined figure, the length of which is divided into 4 equal parts, and the breadth into 3 equal parts, which we will call linear yards. Now it is plain that the room will contain as many square yards as there are squares in the given figure. But the number of squares in the figure is equal to the number of equal parts (linear yards) which its length contains, repeated as many times as there are equal parts (linear yards) in its breadth; that is, it is equal to 4X3, or 12. Ans. 12 yds.

11. How many sq. feet in a floor, 20 feet long, 18 feet wide ? 12. How many acres in a field, 50 rods long, 45 rods wide ?

13. How many square yards in a ceiling, 35 feet long and 28 feet wide ?

14. How many acres in a farm, 420 rods long and 170 rods wide ?

15. What is the area of a square field, whose sides are 80 rods in length ?

16. How many yards of carpeting, a yard wide, will it take to cover a floor 18 feet square.

17. How many yards of plastering are required to cover four sides of a room, 18 ft. long, 15 feet wide, and 9 ft. high ?

18. How many square yards of shingling will cover both sides of a roof, whose rafters are 20 feet, and whose ridge pole is 25 feet long ?

286. The cubical contents, or solidity of boxes of goods, piles of wood, &c., are found by multiplying the length, breadth, and thickness together.

19. How many cubic feet in a box 5 feet long, 4 feet wide and 3 feet deep?

Solution.--5X4=20, and 20X3=60. Ans. 60 cu. ft.

QUEST.286. How are the cubical contents of a box of goods, a pile of wood, &c, found ?

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