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56. 814 pence = how many farthings ? 57. 628 weeks = how many days? 58. 476 acres = how many roods ? 59. 1437 pecks = how many quarts ? Note.— The reduction of numbers from a higher denomination to a lower is called REDUCTION DESCENDING. The last seven examples are illustrations of it, as are several of the following.

60. 46 bu. 2 pk. 5 qt. = how many quarts ?

SOLUTION. — Since 1 bushel = 4 pecks, 46 bushels must equal 46 times 4 pecks, which is equivalent to 4 times 46 pecks. Hence, multiplying 46 pecks by 4, and adding the 2 pecks gives 186 pecks as the value of 46 bu. 2 pk.

But since 1 peck = 8 quarts, 186 pecks must equal 186 times 8 quarts, which is equivalent to 8 times 186 quarts. Hence, multiplying 186 quarts by 8, and adding in the 5 quarts, gives 1493 quarts as the value of 186 pk. 5 qt., or of 46 bu. 2 pk. 5 qt.

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71. How many square rods are there in a road 1537 rd. long, and 4 rd. wide ?

72. How many square feet are there in the four walls of a room 18 ft. long, 16 ft. wide, and 9 ft. high ?

73. How many cubic feet are there in a pile of wood 43 ft. long, 4 ft. wide, and 4 ft. high?

74. How many cubic inches are there in a block of wood 17 in. long, 9 in. wide, and 8 in. high?

75. How many cubic feet are there in a wall 328 ft. long, 3 ft. thick, and 6 ft. high?

49. Multiplication by Factors.

(a.) When the multiplier is the product of factors, we may find the required product by multiplying by one of the factors, then this product by another, and so on until all the factors are used.

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ILLUSTRATION. - Since 18 = 6 times 3, 18 times a number must equal 6 times 3 times the number, wbich may be found by multiplying by 3, and the product thus obtained by 6. 5943. (1. What is the product of 5943 multiplied by

36? 53487

SOLUTION. — Since 36 = 4 X 9, we may find the product

of 5943 multiplied by 36 by multiplying by 9, and that 213948

product by 4, as in the work written opposite. (b.) Find the following products. 2. 9376 x 32.

12. 25.79 x 15. 3. 6794 x 12.

13. 32.67 x 25. 4. 8723 x 24.

14. 4.289 X 48. 5. 5876 x 42.

15. 347.6 x 16. 6. 8967 x 64.

16. 2974 x 36. 7. 4358 x 63.

17. 3598 x 42. 8. 2761 x 49.

18. 4861 x 54. 9. 9547 x 18.

19. 31.87 x 56. 10. 3254 x 21.

20. .0059 x 35. 11. 6795 x 45.

21. 479.5 X 81.

(c.) The most important application of the foregoing principle is made when the multiplier contains 10, 100, 1000, or some other power of 10 as a factor.

ILLUSTRATIONS. - To multiply by 800, or 8 times 100, we may first multiply by 100 by removing the decimal point two places to the right (see 14), and then multiply that product by 8; or we may simply multiply by 8 and remove the point two places further to the right.

To multiply by 7000, we merely multiply by 7 and remove the point three places to the right.

22. What is the product of 694.7 X 4000 ? Solution. — Since 4000 = 4 times 1000, we may find

694.7 the required product by multiplying 694.7 by 4, and

4000 removing the point three places to the right, as illus

2778800. trated in the written work opposite. (d.) Find the following products. 23. 5794 x 20.

33. 356.7 x 4000. 24. 36.85 x 700.

34. .079 x 50000. 25. 4.396 x 40.

35. 81.23 x 90000. 26. 5764 x 3000.

36. .0547 x 800000. 27. 8342 x 500.

37. 52.78 x 500. 28. 8427 X 60.

38. 327.1 X 9000. 29. .0049 X 3000.

39. 4358 x 600. 30. 73648 x 60.

40. .329 X 5000000. 31. 4957.3 X 300.

41. 6275 X 4000000. 32. 2796 X 8000.

42. 5279 x 300000.

50. Multiplication by Large Numbers.

(a.) The product of two numbers is equal to the sum of the products obtained by multiplying one of them by the parts into which the other may be divided. (See illustration under 3d method of proof.)

(b.) This principle and the one illustrated in 49, c, are most frequently applied when the multiplier is a large number.

ILLUSTRATIONS.— To multiply a number by 28, we add 20 times the number to 8 times the number. To multiply a number by 576, we add together 500 times the number, 70 times the number, and 6 times the number.

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1. What is the product of 58.67 x 635?

Solution. — 635 times 58.67 = 600 times 58.67 plus 30 times 58.67 plus 5 times 58.67; to find which, we first multiply by 5, then by 30, then by 500, and add the several products together, as in the written work opposite.

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NOTE. — The attentive student will observe that the right-hand figure of each product after the first stands under the second figure of the preceding product. 2. 6497 x 83.

12. 2457 x 257. 3. 2578 x 61.

13. 3869 X 209. 4. 4379 x 23.

14. 3427 x 234. 5. 5847 x 45.

15. 5868 x 918. 6. 6948 x 96.

- 16. 6258 X 308. 7. 5387 x 48.

17. 2794 x 437. 8. 6526 x 59.

18. 3865 x 2536. 9. 3984 x 235.

19. 6978 x 3215. 10. 4295 x 320.

20. 2587 x 4208. 11. 7328 x 170.

21. 6842 x 3007. 22. How much will 24 acres of land cost at $143.75 per acre ?

23. How many quarts are there in 19 casks, each containing 237 quarts ?

24. How many square rods are there in a field 48 rods long and 36 rods wide ?

25. How many square feet are there in a house lot 137 feet long and 98 feet wide ?

26. I bought 75 casks of oil, each containing 186 gallons. How many gallons did I buy?

27. I bought a pile of wood 37 ft. long, 4 ft. wide, and 5 ft. high. How many cubic feet did it contain ?

28. How many cubic inches are there in a box 37 in. long, 29 in. wide, and 23 in, high ?

29. My garden is 96 ft. long and 87 ft. wide, and I wish to build a tight board fence 13 ft. high around it. IIow many square feet of boards will it contain ?

30. In a certain cornfield there are 149 rows, with 136 hills in each row. How many hills are there in the field ?

31. A ship's crew of 38 men have provisions enough to allow each man 28 ounces per day for 75 days. How many ounces have they?

32. I bought 37 barrels of flour at $8.375 per bbl., and sold the lot for $369.63. What was my gain?

33. How much must be paid for 18 pounds of tea at 46 cents per pound, and 9 pounds of coffee at 18 cents per pound ?

34. I bought 86 hogsheads of molasses at $73.285 per hogshead. How much did it cost me ?

35. 23 sq. ft. = how many square inches?
36. 3 T. 15 cwt. 2 qr. 13 lb. = how many pounds ?
87. £58 13s. 6d. 2 far. = how many farthings ?
38. 15 w. 3 da. 5 h. = how many hours ?

39. How much will 15 hogsheads of wine, each containing 63 gallons, cost at $1.42 per gallon ?

40. I bought 17 pounds of sugar at $.13 per pound, 16 gallons of molasses, at $.37 per gallon, and 23 pounds of tea at $.42 per pound. What was the amount of my purchase ?

41. I bought a lot of land 89 ft. long and 53 ft. wide, at $1.48 per square foot. How much did it cost ?

42. I sold 7 houses at $2478 each, receiving in payment 147 acres of land at $97.25 per acre, and the rest in money. How much money did I receive ?

43. I bought 42 cords of oak wood at $6 per cord, 28 cords of maple at $5.50 per cord, and 24 cords of pine at $3.50 per cord. What was the amount of my purchase ?

44. A silversmith bought 13 lb. 5 oz. 15 dwt. of silver at $.05 per dwt. How much did it cost him?

45. I bought 328 casks of nails, each containing 100 pounds, at $.05 per pound, and sold 129 casks at $.07 per pound, and the rest at $.08 per pound. How much did I gain !

46. How many cubic feet in a block 47 ft. long, 35 ft. wide, and 27 ft. high?

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