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are as many pounds as 20 is contained times in 102, which is £5, Ans. £5, 2s. 7d. 3 far.

and 2s. over.
Operation.
4)4927 far.
12)1231d. 3 far. over.
20)102s. 7d. over.

We first reduce the given farthings to pence, the next higher denomination, by dividing them by 4, because 4 far. make 1d. (Art. 247.) Next we reduce the pence to shillings by diAns. £5, 2s. 7d. 3. far. viding them by 12, because 12d. make 1s. Finally, we reduce the shillings to pounds by dividing them by 20, because 20s. make £1. The last quotient with the several remainders, constitute the answer.

£5, 2s. over.

Note.-2. The last example is exactly the reverse of the first; that is, lower denominations are reduced to higher, which is done by successive divisions. 281. From the preceding illustrations we derive the following

GENERAL RULE FOR REDUCTION.

I. To reduce compound numbers to lower denominations. Multiply the highest denomination given, by that number which it takes of the next lower denomination to make ONE of this higher; to the product, add the number expressed in this lower denomination in the given example. Proceed in this manner with each successive denomination, till you come to the one required.

II. To reduce compound numbers to higher denominations. Divide the given denomination by that number which it takes of this denomination to make ONE of the next higher. Proceed in this manner with each successive denomination, till you come to the one required. The last quotient, with the several remainders, will be the answer sought.

282. PROOF.-Reverse the operation; that is, reduce back the answer to the original denominations, and if the result corresponds with the numbers given, the work is right.

QUEST.-How are farthings reduced to pence? Why divide by 4? How reduce pence to shillings? Why? How reduce shillings to pounds? Why? 281. How are compound numbers reduced to lower denominations? How to higher denominations? 282. How is Reduction proved?

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 propriety, 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?

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2. In £71, 13s. 64d., how many farthings? 3. In £90, 7s. 8d., how many farthings?

4. In £295, 18s. 33d., how many farthings?

5. In 95 guineas, 17s. 94d., 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?

168

REDUCTION.

[SECT

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' weight) to pound 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., how
40. In 28 hhds. 15 gals. wine

many nails?

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 measu 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 quarts, 5 bushels, how many pints?
53. In 45672 quarts, how many bushels, &c.?
54. In 260200 pints, how many quarts?

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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 1 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? 65. 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 39216 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 tim

APPLICATIONS OF REDUCTION.

283. To reduce Troy to Avoirdupois weight.

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

OBS. If the answer is required to be in pounds and a fraction of a p divide the grains by 7000.

Ex. 1. In 175 pounds Troy, how many pounds avoirdupo Solution.-175 × 12 × 20 × 24=1008000 grs., and 100 grs.2736864 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., avoirdupois?

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, &c., 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.-32×16×16×272=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 piece 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?

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