49. Divide $18306.25 into seven hundred and twenty-five equal parts. 50. Bought 7 yards of cloth for 16s. .4d.: what did it cost per yard? 51. A man travelled 265 mi. 6 fur. 16 rd. in 12 days: how far did he travel in one day? 52. If 569 A. 2 R. 23 P. be equally divided among 9 persons, how much will 5 of them have? 53. The annual income of a gentleman is $10000: how much is that per day, counting 365 days to the year? 54. What number multiplied by 9999 will give the product 987551235? 55. A gentleman owning an estate of $75000, gave one-fourth of it to his wife, and the remainder was divided equally among his five children: how much did each receive? 56. The expenditure of the United States for 1853 was $54026818: how much would that be per day, allowing 365 days to the year? 57. If 28 yards of cloth cost one hundred and thirty-three dollars, what will one yard cost? 58. If I pay $637.50 for 51 yards of cloth, what is the price per yard? 59. The city of New York, in 1850, had 104 periodical publications, with an aggregate circulation of 78747600 copies : what was the average circulation of each? 60. Bought 19 bushels of wheat for $30.875: what was the cost of one bushel? 61. How long will 9125 loaves of bread last 5 families, if each family consumes 5 loaves a day? 62. The product of two numbers is 7207272072, and the multiplier 9009: what is the multiplicand? 63. How many rings, each weighing 4 dwt. 12 gr., can be made from 10 oz. 11 dwt. 12 gr, of gold? 64. If iron is worth 2 cents a pound, how much can be bought for $67.50? 65. If 14 sticks of hewn timber measure 12 T. 38 ft. 118 in., how much does each stick contain? 66. In 1850, Pennsylvania manufactured 285702 tons of pig iron, and employed 9285 hands: what was the average product of each hand? 67. The number of college libraries in the United States in 1850, was 213, containing 942321 volumes: what would be the average number of volumes in each? 68. Supposing the sun to make a complete revolution in 365 days, what is his daily velocity of rotation? 69. How many dozen spoons can be made from 3 lb. 11 oz. of silver, allowing 15 pwt. 16 gr. to each spoon? 70. A gentleman bought a piece of cloth, containing 48 yards, at $3.25 per yard. At what price per yard must he sell the cloth to gain $60? 71. If £75 18s. 9d. will pay 5 men for their weekly labor, what amount would pay 18 men at the same rate? 72. How many revolutions does the wheel of a railcar make in going a distance of 75 miles, supposing the wheel to be 9 ft. 6 in. in circumference? 73. There is a certain number, which being divided by 3, the quotient multiplied by 4, and 6 added to the product, will give 18 what is the number? 74. If a farm, containing 512 acres, costs $28672, how much does it cost per acre? 75. If 288120 be divided into 432 equal parts, what will be the value of one of the parts? 76. Light moves with immense rapidity. It comes from the sun to the earth in 8 minutes, the distance being 96 millions of miles how far does it move in 1 second? CONTRACTIONS IN MULTIPLICATION. 77. Contractions in Multiplication are short methods of finding products when the multipliers are particular numbers. 78. To multiply by 25. OPERATION. 4) 35600 1. Multiply the number 356 by 25. ANALYSIS.-If we annex two ciphers to the multiplicand, we multiply it by 100 (Art. 61): this product is four times too great: for the multiplier is but one-fourth of 100; hence, to multiply by 25, Annex two ciphers to the multiplicand, and divide the re 8900 3. Multiply 6741 by 25. 4. Multiply 3074 by 25. 79. When the multiplier contains a fractional unit. 1. What is the product of 15, multiplied by 3? ANALYSIS.-The multiplicand is to be taken 3 and one-fifth times: taking it one-fifth times, which is done by dividing by 5, gives 3, which we write in the unit's place: then, taking it 3 times, gives 45, and the sum 48 is the product; hence, OPERATION. 15 31 3 45 48 Ans. Take such a part of the multiplicand as the fraction is of 1; then multiply by the integral number, and the sum of the products will be the required product. Examples. 1. Multiply 327 by 81. 4. Multiply 1272 by 127. 5. Multiply 9824 by 2721. 6. Multiply 3828 by 731. 77. What are contractions in Multiplication?-78. How do you multiply by 25?-79. How do you multiply when the multiplier contains a fractional unit? 80. To multiply by 12). 1. Multiply the number 286 by 12. ANALYSIS.-Since 12 is one-eighth of 100, Annex two ciphers to the multiplicand, and divide the result by 8. Examples. 1. Multiply 384 by 124. 2. Multiply 476 by 12. OPERATION. 8) 28600 3575 3. Multiply 14800 by 124. 4. Multiply 670418 by 121. 81. To multiply by 331. 1. Multiply the number 975 by 33. ANALYSIS. Annexing two ciphers to the multiplicand, multiplies it by 100: but the multiplier is one-third of 100: hence, Annex two ciphers, and divide the result by 3. Examples. 1. Multiply 1679252 by 331. 2. Multiply 1480724 by 331. OPERATION. 3) 97500 32500 3. Multiply 10675512 by 33. 4. Multiply 4442172 by 333. 82. To multiply by 125. 1. Multiply the number 1125 by 125. ANALYSIS.—Annexing three ciphers to the multiplicand, multiplies it by 1000: but 125 is but one-eighth of one thousand: hence, OPERATION. 8) 1125000 Annex three ciphers, and divide the result by 8. 1. Multiply 59264 by 125. Examples. 2. Multiply 17593408 by 125. 140625 3. Multiply 1940812 by 125. 4. Multiply 140588 by 125. 80. How do you multiply by 12?-81 How do you multiply by 8817-82. How do you multiply by 125? Applications. 83. The analysis of a practical question, in Multiplication, requires that the multiplier be an abstract number; and then the unit of the product will be the same as the unit of the multiplicand. 84. To find the cost of several things, when we know the price of one, and the number of things: 1. What will six yards of cloth cost, at 8 dollars a yard? ANALYSIS.-Six yards of cloth will cost 6 times as much as 1 yard. Since 1 yard of cloth costs 8 dollars, 6 yards will cost 6 times 8 dollars, which are 48 dollars; therefore, 6 yards of cloth, at 8 dollars a yard, will cost 48 dollars: hence, The cost of any number of things is equal to the price of a single thing multiplied by the number of things. We have seen that the product of two numbers will be the same (that is, contain the same number of units), whichever be taken for the multiplicand (Art. 59). Hence, in practice, we may multiply the two factors together, taking either for the multiplier, and then assign the proper unit to the product We generally take the less number for the multiplier. 85. To find the cost, when the price is an aliquot part of a dollar. NOTE. For definition of Aliquot part, see Art. 98. OPERATION. 4) 45.00 1. Find the cost of 45 bushels of apples, at 25 cents a bushel. ANALYSIS.-If the price were 1 dollar a bushel, the cost would be as many dollars as there are bushels. But the price is 25 cents = of a dollar; hence, the cost will be one-fourth as many dollars as there are bushels; that is, as many dollars as 4 is contained times in 45, which is 11, and 1 dollar over. This is reduced to cents by adding two ciphers; then dividing again by 4, we have the entire cost: hence, $11.25 83. What does the analysis of a practical question in Multiplication require? What then follows?-84. How do you find the cost of several things when you know the price of a single thing?-85. How do you find the cost when the price is an aliquot part of a dollar? |