ing written out the work as at first performed, to begin at the left hand, thus : 75.69 7 529.83 7 times 7 tens are 49 tens; but as there are 52 tens in the product, 3 tens must have come from the product of the lower denominations. 3 tens = 30 units, and adding to this the 9 units written in the units place, we find there ought to be 39 units in the product. 7 times 5 units are only 35 units; hence, if the work be right, 4 units must have come from the product of the lower denominations. 4 units = 40 tenths, and adding to this the 8 tenths written in the tenths' place of the product, we find that there ought to be 48 tenths in the product. 7 times 6 tenths are only 42 tenths; hence, if the product be right, 6 tenths must have come from the product of the hundredths. 6 tenths hundredths, and adding to this the 3 hundredths written in the hundredths' place, we find that there ought to be 63 hundredths in the product; and as 7 times 9 hundredths are 63 hundredths, we may infer that the work is correct. £. d. 3 9 7 8 s. 72 s.; 27 16 8 8 times £3 = £24. But as in the written product there are £27, £3 must have come from the lower denominations. £3 60 s., and adding to this the 16 s. written in the shillings of the product, we find that there ought to be 76 shillings in the product. 8 times 9 s. = therefore, if the work be right, 4 shillings must have come from the lower denominations. 4 s. = 48 d., and adding to this the 8 d. already written, we find there ought to be 56 d. in the product; and as 8 times 7 d. = 56 d., we infer that the work is correct. If the computer should find by multiplying numbers in one method a result different from that obtained by multiplying the same numbers in some other method, he may be sure that there is an error in one opera. tion or the other, and he should examine his work patiently till he finds it. No person who is willing to allow an error to pass undetected can be a good arithmetician. (See 54.) 79. Problems. — Multiplier a single Figure. Note. — This article includes reduction descending. See Note after 36th example.) 1. What is the product of 84687 X 4 ? 10. 1 pound Avoirdupois of distilled water contains 27.7015 cubic inches. How many cubic inches will 8 pounds. contain ? Reasoning Process. - If 1 pound contains 27.7015 cubic inches, 8 pounds will contain 8 times 27.7015 cubic inches, which can be found by multiplying it by 8. 11. What will 7 acres of land cost at $94.839 per acre ? 12. 1 pound Troy of distilled water contains 22.794377 cubic inches. How many cubic inches will 8 pounds contain? 13. How many cubic inches are there in 7 cubic feet? 14. How much will it cost to build 9 miles of railroad at $19783.27 per mile ? 15. How much will 3 farms cost at $3879.86 each ? 16. How many feet would a man walk in 6 days at the rate of 56487 feet per day ? 17. How many miles would a ship sail in 9 weeks, if she sails at the rate of 1198.47 per week ? 18. How many inches are there in 4 miles, there being 63360 inches in 1 mile ? 19. How many pounds are there in 5 loads of hay, each weighing 2794 pounds ? 20. How many acres in 7 lots, each containing 24.74386 acres ? 21. How many square rods in a road 754 rods long and 4 rods wide ? Reasoning Process. — Since a space 1 rod long and 1 rod wide contains 1 square rod, a space 754 rods long and I rod wide must contain 754 square rods, and a space 754 rods long and 4 rods wide must con tain 4 times 754 square rods, which may be found by multiplying 754 by 4. (See 40, Note.) 22. How many square feet in a walk 796 feet long and 9 feet wide? 23. How many square feet in a wall 437 feet long and 6 feet high? 24. Mr. Haven's garden is 124 feet long and 97 feet wide, and is enclosed by a tight board fence 5 feet high. How many square feet of boards are there in the fence ? Suggestion to the Student. — Draw a plan of the garden. 25. Mr. Haven laid out a gravel walk, 4 feet wide, within the fence, and extending all around the garden. How many square feet did it contain ? Suggestion to the Student. — Draw a plan of the garden and walk. 26. What will 49.67 barrels of apples cost at $3 per barrel ? Reasoning Process. — If i barrel costs 3 dollars, 49.67 barrels will cost 49.67 times 3 dollars, which is equivalent to 3 times 49.67 dollars. The work would be written thus : $49.67 = Multiplicand. Multiplier. 3 = $149.01 = Product. Another Reasoning Process. - It is evident that if each barrel should cost a dollar, all would cost as many dollars as there are barrels bought, which in this instance would be $49.67. But if, at $1 per bbl., they cost $49.67, at $3 per bbl., they would cost 3 times $49.67. The work may be written thus : 49.67 bbl. at $3 per bbl. $ 49.67 = cost at $1 per bbl. 3 times $49.67 = $149.01 = cost at $3 per bbl. = Ans. 27. What will 3749 lb. of saleratus cost at .08 per lb. ? 28. What will 178.69 bbl. of flour cost at $7 per bbl. ? 29. What will 27.96 firkins of butter cost at $9 per firkin? 30. What will be the weight of 3794 cannon balls, each ball weighing 8 lb. ? 31. If a soldier eats 4 lb. of meat in a week, how many pounds will 2896 soldiers eat in the same time? 32. What will 4736 casks of raisins cost at $7 per cask ? 33. If a vessel sails uniformly at the rate of 9 miles per hour, how far will she sail in 476 hours ? 34. How many bushels in 1487 barrels, if each barrel holds 3 bushels ? 35. How much will 3479 window weights weigh, if each weighs 6 pounds ? 36. 82 bu. 3 pk. 5 qt. 1 pt. = how many pints ? Reasoning Process. Since there are 4 pecks for every bushel, there must be 4 times as many pecks as bushels, or, in this case, 4 times 82 pecks, which are 328 pecks, and adding 3 pecks to this gives 331 pecks as the value of 82 bu. 3 pk. Since there are 8 quarts for every peck, there must be 8 times as many quarts as there are pecks, or, in this case, 8 times 331 quarts, which are, &c. Another Form. — Since 1 bushel = 4 pecks, 82 bushels must equal 82 times 4 pecks, equivalent to 4 times 82 pecks, which are 328 pecks, and adding 3 pecks to this gives 331 pecks, as the value of 82 bu. 3 pk. Since 1 peck = 8 quarts, 331 pecks must equal 331 times 8 quarts, equivalent to 8 times 331 quarts, which are, &c. NOTE. — Questions like the above, requiring that the value of numbers of one denomination shall be expressed in terms of some lower denomination, are called questions in Reduction Descending; but as they do not differ at all from other questions in multiplication, they require no separate treatment. In performing them, there is no need of writing the multipliers, as they may be known from the table of the weight or measure used. In reducing to any denomination of which there are units already expressed, it will usually be more convenient to add those units at the time we make the multiplication. METHOD OF WRITING THE WORK. 82 bu. 3 pk. 5 dt. 1 pt. The following abbreviated form may be adopted after the above is perfectly familiar: 82 bu. 3 pk. 5 qt. 1 pt. 331 pk. 2653 qt. 5307 pt. = Ans. 37. 97 bu. 2 pk. 7 qt. 1 pt. = how many pints ? 38. 187 bu. 1 pk. 6 mt. O pt. 3 gi. how many gills ? 39. 238 gal. 3 qt. 1 pt. 2 gi. = how many gills ? 40. 596 gal. 2 qt. 1 pt. 3 gi. = how many gills ? 41. 9 lb 113 63 2 = how many scruples ? 42. 8 th 33 73 1? = how many scruples ? 43. 937 le. 1 m. 5 fur. = how many furlongs ? 44. 286 le. 2 m. 7 fur. = how many furlongs ? 45. What will 37 bu. 2 pk. 3 qt. 1 pt. of cherries cost at 4 cents per pint ? 46. What will 114 bu. 3 pk. 2 qt. of wheat cost at 1 cent per gill? 47. What will 17 gal. 3 qt. 1 pt. 3 gi. of oil cost at 5 cents per gill? 48. What will 93 gal. 2 qt. 1 pt. 2 gi. of brandy cost at 8 cents per gill? 49. What is the product of £22 18 s. 8 d. 2 qr. by 7 ? Explanation. — 7 times 2 qr. = 14 qr., which (since 4 qr. = 1 d.) equal as many pence as there are times 4 in 14, which are 3 times, with a remainder of 2. Hence, 14 qr. = 3 d. 2 qr. 7 times 8 d. 56 d., and 3 d. added 59 d., which (since 12 d. = 1 s.) equal as many shillings as there are times 12 d. in 59 d., which are 4 times, with a remainder of 11. Hence, 59 d. = 4s. 11 d. 7 times 18 s. = 126 s., and 4 s. added = 130 s., which (since 208 |