Several other sets of factors of 168 may be used, and give the same product. Every similar example may be solved in like Hence, manner. RULE. Multiply the multiplicand by one of the factors of the multiplier, and that product by another factor, and so on until all the factors in the set have been taken; the last product will be the true product. 52. Multiply 743 by 42, i. e. by 7 and 6. 54. 839 × 54= how many? 56. 469876 X 81=? 57. 478969 X 1728 =? 58. 5387469 × 96 =? 59. 987462 X 49 =? Ans. 31206. Ans. 45306. Ans. 827658432. 62. To multiply by 10, 100, 1000, or 1 with any number of ciphers annexed: RULE. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number so formed will be the product. NOTE. The reason of the rule is obvious. Annexing a cipher removes each figure in the multiplicand one place toward the left, and thus its value is made ten fold (Art. 15). 60. Multiply 74 by 10. 64. 59874 X 1000000000=? Ans. 740. Ans. 8690000. Ans. 7698400000. 63. To multiply by 20, 50, 500, 25000, or any similar number: RULE. Multiply by the significant figures, and to the product annex as many ciphers as there are ciphers at the right of the significant figures of the multiplier. 61. Rule for multiplying by a composite number? 62. How is a number multipled by 10? By 100? Why? 63. How is a number multiplied by 20? Why? 65. Multiply 756 by 30. OPERATION. 756 30 22680 Ans. 22680. This is upon the principle of Art. 61. The factors of 30 are 3 and 10. Having multiplied by 3, the product is multipled by 19 by annexing 0 (Art. 62). 64. To multiply when there are ciphers at the right of both multiplicand and multiplier: RULE. Multiply the significant figures of the multiplicand by those of the multiplier, and then annex as many ciphers to the product as there are ciphers at the right of both factors. 70- Multiply 8000 by 900. OPERATION. 8000 900 Ans. 7200000 The factors of 8000 are 8 and 1000, and those of 900 are 9 and 100. Now, as it is immaterial in what order the factors are taken (Art. 59, Note), first multiply 8 by 9, then mul tiply this product by 1000 (Art. 62), and this product by 100. 71. Multiply 730000 by 2900. OPERATION. 730000 2900 657 146 Product, 211 7 0 0 0 0 0 0 64. Rule when there are ciphers at the right of both factors? The reason? Ans. 2268000000. 72. Multiply 840 by 2700000, 73. 7693000 X 569000=? 65. To multiply when there are ciphers between the significant figures of the multiplier: RULE. Multiply only by the significant figures of the multiplier, taking care to set the first figure of each partial product directly under the figure of the multiplier which gives that product. 66. To multiply by 9, 99, or any number of 9's: RULE. Annex as many O's to the multiplicand as there are 9's in the multiplier, and from the number so formed subtract the multiplicand; the remainder will be the product sought. 78. Multiply 234 by 99. OPERATION. 2 3 4 0 0 234 100 times the multiplicand. 23166= 99 times the multiplicand, Ans. 79. Multiply 3746 by 999. 80. Multiply 427 by 9999. Ans. 3742254. 65. Rule for multiplying when there are ciphers between the significant figures of the multiplier? The reason? 66. To multiply by 9? By 99? Rule? Reason? 67. To multiply by 13, 14, 15, 16, 17, etc. : RULE. Multiply by the right-hand figure of the multiplier, set the product under the multiplicand, ONE PLACE FURTHER TO THE RIGHT, and add. In a similar manner multiply by 102, 1005, 10009, etc. 83. Multiply 2463 by 102. 84. Multiply 3248 by 104. By 1004. By 1008. 68. To multiply by 21, 31, etc.: RULE. Multiply by the left-hand figure of the multiplier, set the product under the multiplicand, ONE PLACE FURTHER TO THE LEFT, and add. 86. Multiply 34264 by 81. By 41. By 61. In like manner multiply by 201, 301, 6001, etc. 87. Multiply 4237 by 501. Ans. 2122737. 88. Multiply 34265 by 801. By 4001. By 30001. 67. To multiply by 13? By 15? By 102? By 1005? Reason? 68. To multiply by 21? By 31? By 501? Reason? Why better than the common method! MISCELLANEOUS EXAMPLES IN MULTIPLICATION. 1. What cost 11 pounds of beef at 9 cents per pound? Ans. 99 cents. 2. What cost 98 tons of hay at $15 per ton? Ans. $1470. 3. In one hogshead of wine are 63 gallons; how many gallons in 75 hogsheads? 4. In a certain house are 75 rooms, in each room four windows, and in each window 12 panes of glass; how many panes of glass in the house? 5. The earth, in its annual revolution, moves 19 miles in a second; how far will it move in an hour, there being 60 seconds in a minute, and 60 minutes in an hour? 6. Light moves 192000 miles in a second; how far will it move in an hour? 7. How many yards of cloth in 10 bales, each bale containing 25 pieces, and each piece 24 yards? 8. If 12 men do a piece of work in 7 days, in how many days can 1 man do 5 times as much work? 9. Multiply forty-three million, seven hundred and four thou sand, eight hundred and sixteen, by forty-two thousand and eight. 10. A man bought 24 city lots at $365 each; what did they all cost him? = 11. Multiplicand=4632; multiplier 4008; product=? 12. Multiplier=3333; multiplicand=4444; product=? EXAMPLES IN THE FOREGOING PRINCIPLES. 1. Two men start from the same place, and travel in the same direction, one at the rate of 56 miles and the other 75 miles per day, how far apart are they at the end of 43 days? 2. Had the men named in Ex. 1 traveled in opposite directions, how far apart would they have been in 56 days? 3. Bought 58 tons of hay for $600 and sold it for $12 per ton; did I gain or lose? How much? 4. Bought 25 horses for $125 each, and 14 pairs of oxen at $87 a pair; what did I pay for all? |