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23. In 2 yards how many quarters? In 3? In 5? In 10 ? In 12 ? In 15? In 16? In 20 ? In 24? In 26 ? In 30 ?

24. In 3 ells English how many quarters? In 7 ? In 12 ?

25. In 12 quarters how many yards? In 16 ? In 19? In 24 ? In 30? In 36 ? In 38? In 40 ? In 41 ? In 47 ?

26. In 6 quarters how many nails? In 6? In 7 ? In 8 ? In 12? In 14? In 16? In 20? In 22 ? In 30 ?

27. In 24 nails how many quarters? In 30? In 36 ? In 40 ? In 46 ? In 50? In 70 ? În 100? In 150 ? In 160 ?

28. In 100 pence how many shillings? In 144? In 160 ? 29. In 140 shillings how many pounds? In 150? In 160 ? 30. In 96 ounces how many lbs. ? In 100? In 200 ? In 300 ? 31. In 56lbs. how many quarters ?' In 84? In 112? In 200 ?

32. In 8 quarters how many cwt.? In 12 ? In 16? In 28 ? In 44? In 48 ? In 50 ? In 60 ? In 66 ? In 70 ? In 80 ?

33. In 40cwt. how many tons? In 80 ? In 100? In 120 ? In 180 ? In 200? In 800? In 850 ? In 866 ? In 1000 ?

34. In 48qrs. how many dwt.? In 72? In 96 ? In 144?
35. In 9 scruples how many drams? In 12 ? In 18 ?
36. In 48 ounces, troy, how many lbs.? In 96? In 144 ?

37. In 12 quarters how many lbs. ? In 16 ? In 28 ? In 40 ? In 48 ? In 56 ? In 60 ? In 100? In 120: In 130 ? In 150 ?

38. In 25 quarters how many ells English? In 30? In 60 ? In 70 ? In 100 ? In 110 ? In 120 ? In 125? In 150 ?

39. In 36 inches how many feet? In 48? In 60 ? In 144 ? In 180? In 190? In 100? In 120 ? In 130 ? In 140 ? In 150 ?

40. In 12 feet how many yards? In 15 ? In 24? In 30 ? In 48? In 60 ? In 66 ? In 90 ? In 100 ? In 120 ? In 140 >

41. In 80 rods how many furlongs? In 120 ? In 720 ?

42. In 24 furlongs how many miles ? In 32 ? In 48? In 72 ? In 80 ? In 90 ? In 96 ? In 100 ? In 200 ? In 300 ? In 400 ?

43. In 3 weeks how many days ? In 4? In 5 ? In 6? In 7 ? In 8? In 9 ? In 12 ? In 14? In 20 ? In 25 ? In 30 ? In 50 ?

44. In 5 years how many months ? In 3? In 4? In 8 ? In 9? In 10 ? In 12 ? In 24? In 30 ? In 32 ? In 36 ? In 40 ?

45. In 2 quarts how many pints? In 4? In 6 ? In 8 ? In 10? In 20 ? In 50 ? In 60 ? In 190? In 120 ? In 150 ?

46. In 5 years how many months ? In 3? In 4? In 8? In 9? In 10? In 12 ? In 24? In 26? In 28 ? In 30? In 32 ?

47. In 2 quarts how many pints? In 4? In 6 ? In 8? In 10? In 20? In 50 ? In 60 ? In 70 ? In 75 ? In 80 ? In 90 ?

48. In 3 gallons how many quarts? In 4? In 5? In 6 ? In 8? In 9 ? In 10? In 20? In 40 ? In 50 ? In 100 ? In 200 ?

49. In 2 hogsheads how many gallons? In 3? In 4? In 5 ? In 6? In 7 ? In 8? In 10? In 12? In 14? In 20 ? In 24 ?

50. In 3 pipes how many hogsheads? In 5 ? In 7 ? In 8 ? In 10? In 12? In 20? In 24? In 26 ? In 30 ? In 40 ?

51. In 2 tuns how many hogsheads! In 5? In 7? In 8 ? In 9? In 10? In 12 ? In 14? In 16? In 20 ? In 24?

52. In 2 bushels how many pecks ? In 3? In 5 ? In 6 ? In 3? In 10 ? In 15 ? In 20 ? In 60? In 70? In 80 ?

53. In 3 pecks how many quarts? In 4? In 5? In 10 ? In 12? In 15? In 20 ? In 25? In 30 ? In 40 ? In 50 ? In 60 ?

54. In 2 feet how many square inches ? In 4? In 5 ? In 10? In 20 ? In 50 ? In 60 ? In 70? In 80 ? In 90 ? Ip 100 ?

55. In 3 square yards how many feet? In 4? In 6? In 7 ? In 8? In 10? In 20 ? In 30 ? In 40 ? In 50 ? In 60 ?

56. In 5 roods how many rods? In 10? In 20? In 30 ? 57. In 2 acres how many roods? In 5 ? In 10 ? In 20 ? In 30? In 40 ? In 50 ? In 100? In 120 ? In 150 ? In 160 ?

58. In 3 acres how many rods? In 5 ? In 10 ? In 50 ? 59. In 5 minutes how many seconds? In 10 ? In 20 ?

60. In 2 hours how many minutes ? In 3? In 4? In 5 ? In 8 ? In 12 ? In 15 ? In 20? In 30 ? In 35 ? In 40 ?

61. In 3 days how many hours? In 5? In 6? In 8? In 10 ? In 15 ? 'In 20 ? In 25 ? In 50 ? In 100 ? In 120 ?

62. In 2 cents how many mills? In 3? In 4 ? In 6 ? In 8? In 9 ? In 12 ? In 16? In 18? In 25 ? In 35 ? In 45 ?

63. In 20 mills how many cents? In 30 ? In 40 ? In 60 ? In 70 ? In 80 ? In 90 ? In 100 ? In 120 ? In 150 ? In 160 ?

64. In 3 dimes how many cents? In 5 ? In 7 ? In 9 ? 65. In 30 cents how many dimes? In 80 ? In 100 ? In 200 ?

66. In 2 dollars how many cents ? In 5 ? In 6? In 7 ? In 10? In 20? In 30? In 50 ? In 100? In 120 ? In 130 ?

67. In 300 cents how many dollars? In 400 ? In 500 ? In 600? In 800 ? In 900 ? In 1000? In 10000? In 100000 ?

68. In 5 eagles how many dollars? In 6? In 10 ? In 20 ? In 25 ? In 35? In 75 ? In 100? In 120 ? In 200 ? In 300 ?

69. In 20 dollars how many eagles ? In 30 ? In 50 ? In 70 ? In 100 ? In 200 ? In 1000? In 1500? In 100000 ? 70. In 4 yards how many nails? In 8 ? In 10 ? In 12 ? 71. In 2 Ells English how many yards ? In 3 ? In 4 ? In 5 ?

72. What is the difference between 3 square feet, and 3 feet square ? 73. How many inches round one foot square ? 2 feet ? 3 feet? The student will now perceive that the object of

Reduction is to change numbers of one denomination to another without losing their value.

It consists of two parts, Descending, and Ascending. The former is performed by Multiplication, the latter by Division.

Reduction Descending teaches to bring numbers of a higher denomination to a lower ; as, to bring pounds into shillings, or tons into hundred weights.

Reduction Ascending teaches to bring numbers of a lower denomination into a higher; as, to bring farthings into pence, or shillings into pounds.

REDUCTION DESCENDING.

EXAMPLE.

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1. In 48£. 125. 7d. 2qr, how many farthings? £. 48 12 7 2 In this example, we multiply the 48£. by 20, 20

because it takes 20 shillings to make a pound;

and, to this product, we add the 12s. in the 972 + 12

question. Then multiply by 12, because it 12

takes 12 pence to make one shilling ; and, to 11671 +7 the product, we add the 7 pence in the ques4

tion. We then multiply by 4, the number of 46686 +2

farthings in a penny, and, to the product, we

add the 2 farthings, and the work is done. From the above example and illustration, we deduce the following

RULE.

Multiply the highest denomination given by so many of the next less, as will make one of that greater ; and so proceed, until it is brought to the denomination required, observing to bring in the lower denominations to their respective places.

NOTE 1. To multiply by a }, we divide the multiplicand by 2, and to multiply by a }, we divide by 4.

NOTE 2. — The answers to Reduction Descending will be found in the questions of Reduction Ascending.

2. In 127£. 15s. 8d. how many farthings ?
3. In 28£. 195. 11d. 3qr. how many farthings ?
4. In 378£. how many pence ?
5. How many grains in 281b. 11oz. 12dwt. 15gr. troy ?
6. In 171b. 12dwt. troy, how many pennyweights ?

7. If a silver tankard weigh 3lb. 11oz. how many grains will it be ?

8. How many scruples in 231bs., apothecaries' weight ?

9. If a load of hay weigh T. 16cwt. 2qr. 18lbs. how many ounces will it be ?

10. Required the number of drachms in a hogshead of sugar, weighing 2T. 17cwt. Sqr. 161b. 15oz. 13dr. ?

11. In 57yds. how many nails ?
12. In 83947 E.E. 4qr. how many nails ?
13. In 2263 E.F. 2qr. how many quarters ?
14. How many feet in 79 miles ?
15. How many inches in 396 furlongs ?

16. How many inches from Haverhill to Boston, the distance being 30 miles ?

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17. How many barleycorns will it take to reach round the world ?

18. In 403m. 7fur. 35rd. 2yd. Oft. Oin. Ibar. how many barleycorps ? (19. In 413Le. 2m. 2fur. 38rd. lyd. Oft. 7in. how many inches ? 20. In 144m. 1fur. 8rd. lyd. 1ft. how many feet? 21. How many inches in 1051yd. 2ft. 5in. 22. In 3576fur. 12rd. Byd. how many yards ? 23. How many square feet in 25 acres ? 24. How many square rods in 365 square miles ?

25. The surface of the earth contains 196563942 square miles. What would it be in square inches ?

26. Required the number of feet in 10A. 3R. 38p. 6yd. 5ft. 72in.

27. In 2R. Op. 24yd. 3ft. how many inches ? 28. In 1A. 3R. 34p. 27yd. 4ft. 54in. how many inches ? 29. In 17 cords of wood how many inches ? 30. In 19 tons of round timber how many inches ? 31. How many cubic feet of wood in 128 cords ? 32. In 4899hhd. 4galls. 3qt. how many quarts ? 33. In 1224 tuns, 1p. 1hhd. 19galls. 1qt. Opt. lgi. how many gills ?

34. How many pints in 790p. Ohhd. 58galls. Oqt. 1pt. ? 35. In 460 butts, 1bhd. 31galls. of beer, how many galls. ? 36. In 36hhd. 26galls. 3qt. 1pt. how many pints ? 37. In 16 tons of round timber, how many inches ?

38. How many seconds since the deluge, it being 2348 years B. C.?

39. How many days did the last war continue, it having commenced June 18, 1812, and ended Feb. 17, 1815 ?

40. How many pecks in 676 chaldrons ?
41. In 657 cents how many mills ?
42. In 3165 dimes how many mills ?
43. In 63 dollars how many cents ?
44. In 27 eagles, how many mills ?

REDUCTION ASCENDING.

EXAMPLE.

1. In 76789 farthings, how many pounds ?

Ans. 79£. 19s. 9 d. 4)76789

We first divide by 4, because it takes 4 12)19197 - 1

farthings to make a penny. We then

divide by 12, because pence is the next 20) 1599-9

higher denomination. Lastly we divide 79£. 19s. 9fd.

by 20, it being the number of shillings in

a pound.

From the preceding illustration and example we deduce the following

RULE.

Divide the lowest denomination given by that number, which it takes of that denomination to make one of the next higher ; so proceed, until it is brought to the denomination required.

NOTE 1. To divide by 5}, we multiply the multiplicand by 2, and divide the product by 11 ; to divide 16, we multiply by 2 and divide by 33 ; and to divide by 272], we multiply by 4 and divide by 1089.

NOTE 2. The answers to Reduction Ascending are the questions in Reduction Descending.

2. In 122672 farthings, how many pounds ? 3. In 27839 farthings, how many pounds ? 4. In 90720 pence, how many pounds ? 5. In 166863 grains, how many pounds troy ? 6. How many pounds in 4092 pennyweights ? 7. How many pounds troy in 22560 grains ? 8. In 6624 scruples, how many pounds ? 9. In 137376 ounces, how many tons ? 10. In 1660157 drachms, how many tons ? 11. How many yards in 912 nails ? 12. Required the ells English in 1678956 nails. 13. Required the ells Flemish in 6791 quarters. 14. Required the miles in 417120 feet. 15. Required the furlongs in 3136320 inches. 16. Required the miles in 1900800 inches. 17. How many degrees in 4755801600 barleycorns ? 18. How many miles in 76789567 barleycorns ? 19. How many leagues in 78653167 inches ? 20. Required the miles in 761116 feet. 21. Required the yards in 37865 inches. 22. Required the furlongs in 786789 yards. 23. How many acres in 1089000 square feet ? 24. How many square miles in 37376000 square rods ?

25. How many square miles in 789,103,900,894,003,200 square inches ?

26. In 478675 square feet, how many acres ? 27. In 3167856 square inches, how many roods ? 28. How many acres in 12345678 square inches ? 29. How many cords in 3760128 cubic inches ? 30. How many tons of round timber in 1313280 cubic inches ? 31. How many cords in 16384 cubic feet? 32. How many hogsheads of wine in 1234567 quarts ? 33. How many tuns of wine in 9877001 gills ? 84. In 796785 pints of wine how many pipes ? 35. How many butts of beer in 49765 gallons ? 36. In 15767 pints, how many hogsheads of beer?

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