places in both factors. If the number of figures in the product be less than the decimal places in both factors, prefix ciphers to the left to supply the defect. 1. Multiply 75 by,8. Operation. 75 ,8 Ans. =60,0 EXAMPLES. Multiplying by a fraction is taking a certain part of the multiplicand for the product consequently, multiplying a whole number by a certain part of a whole number, as 8 tenths, produces a product less than the whole number: in this example, the number of decimal places in the factors being one, therefore we point off one figure in the product, and the answer is 60. Likewise multiplying one decimal fraction by another, produces a fraction smaller than either of the factors. 2. Multiply 4,25 by 3,6. 3,6 2550 1275 Ans. 15,300 In this example, the whole number of decimal places in both multiplier and multiplicand is three, therefore we point off three figures in the product. 3. Multiply 5,34 by,008. ,008 In this example, the number of ,04272 figures in the product is less than the decimal places in both factors; the defect must be supplied by prefixing a cipher; that is, placing it at the left hand. 4. Multiply 36,5 by 7,27. 5. Multiply 3,92 by 196. Ans. 265,355. Ans. 768,32. Ans. 28,399112. Ans. 1836,88305. Ans. ,000081. 9. Multiply 25 dollars by 25 cents, and what is the product? 10. What cost 8,75 yards of cloth, at Ans. $6,25. $3,96 per yard? Ans. $34,65. 11. What cost 18,75 barrels of flour, at $6,75 per barrel ? Ans. $126, 56c. 25m. 12. What is the value of 18,25lbs. of butter, at $,125 Ans. $2 28c. 128 m. per pound? 13. At ,03cts. profit on a dollar, what is the profit on $18,75? Ans. 56cts. 2m. 14. Multiply 135 dollars by $,06 or cents. Ans. $8,10. 15. Multiply $14,56 by $1,25. 16. Multiply 3672 by,85. Ans. $18,20. Ans. 3121,2. 17. Multiply 235,45 dollars by ,007, or 7 mills. Ans. 1dol. 64cts. 815m. 18. Multiply $,95 or 95cts. by $,125, or 12cts. 5m. Ans. 11cts. 85m. 19. How much is ,5 of 138. Ans. 69. Ans. $89,70. Note. To multiply by 10, 100, 1000, &c., remove the decimal point as many places to the right as the multiplier Division of decimals differs from the division of whole numbers only in pointing off the decimal places. We have seen, in Multiplication of decimals, that the decimal places in the product must be equal to the number of decimal places in both factors counted together. So in division of decimals, the number of decimal places in the divisor and quotient counted together, must be equal to the number of decimal places in the dividend; because the divisor and quotient are the factors which produce the dividend. RULE. 1. Divide as in whole numbers, and from the right hand in the quotient point off as many figures for decimals, as the decimal places in the dividend exceed those in the divisor. 2. If the places in the quotient be not so many as the rule requires, supply the deficiency by prefixing ciphers. 3. If the divisor has more decimal places than the dividend, annex as many ciphers as you please to the dividend so as to make it equal at least to the divisor. You may also annex ciphers to the remainder, if any, and carry on the quotient to any degree of exactness; but the ciphers annexed must be counted as so many decimals of the dividend. EXAMPLES. 1. Divide 89,756 by,8. ,8)89,7560 We divide as in whole numbers, and there being a remainder, we annex a cipher and divide: there are now four decimal places in the dividend and one in the divisor. We therefore, by the rule, point off three figures in the quotient for decimals, which makes the number of decimal places in the divisor and quotient counted together, equal to the number of decimal places in the dividend. 2. Divide,36792 by 4,2. 4,2),36792(,0876 336 319 294 252 252 and quotient counted places in the dividend. In this example, there are five decimal places in the dividend, and only one in the divisor; therefore we must point off four figures in the quotient. Now, because there are only three figures in the quotient, we place a cipher on the left, and the decimal places in the divisor together, are equal to the decimal 3. Divide 44,98 by 1,3. 9. Divide $256,125 by 12,5. 10. Divide $510, by $1256. Ans. 34,6. Ans. 17,825. Ans. ,00812 x. Ans. 1,1955. Ans. ,00031 x. Ans. ,0645 X. Ans. $20,49. Ans. $,40605 X. 11. If 8,75 yards of cloth cost $34,65, what is that a yard? Ans. $3,96. 12. Bought 18,75 barrels of flour for $126,5625; how much was that a barrel? Ans. $6,75cts. 13. If 148,5lbs. of butter cost $18,5625, what will 1lb. cost? Ans. $,125 124cts. 14. At $1,79 per barrel, how many barrels of cider can be bought for $270,29 ? Ans. 151. 15. If a bushel of wheat cost $1,87, how many bushels can be bought for $28,985 ? Ans. 15,5bu. 15}bu. 16. How many times 6,25 dollars can I have in 1235 dollars? Ans. 197,6. Note. To divide by 10, 100, 1000, &c., remove the decimal point in the dividend as many places towards the left hand as there are ciphers in the divisor. To reduce a Vulgar Fraction to its equivalent decimal. RULE. Annex ciphers to the numerator of the given fraction, and divide by the denominator, the quotient will be the decimal required, which must contain as many decimal places as there are ciphers annexed to the numerator. If there are not so many figures in the quotient, make up the deficiency by placing ciphers on the left. EXAMPLES. 1. Reduce to its equivalent decimal. Operation. 8)7,0 0 0 ,87 5 Answer. 2. Reduce to a decimal fraction. Operation. ,55555 We might continue annexing ciphers to this remainder, and carry on the quotient still lower; but were it carried to an infinite num ber of figures, we should never arrive at a complete quotient.* 3. Reduce to a decimal fraction. 4. Reduce to a decimal. Ans.,5. Ans.,75. 5. What decimal is equal to ? Ans.,2. Ans. ,91666+. * Such fractions are called circulating, or repeating decimals. To reduce quantities of different denominations to a decimal of the highest denomination. RULE. 1. Reduce the given denominations to a Vulgar Fraction, as taught in Problem 7, page 93; then reduce the Vulgar Fraction to its equivalent decimal. EXAMPLES. 1. Reduce 8s. 7d. 2qrs. to the decimal of a pound. 960) 414,00000(,43125 Ans. 3840 3000 2880 1200 960 2400 1920 4800 4800 8s. 7d. 2qrs. Note. This Rule will give the true decimal; but the following Rule is much easier, and therefore the best for practice. See this example performed by Rule 2. Rule 2. Write the denominations below each other, placing the lowest name at the top; then divide each denomination (beginning at the top,) by that number which makes one of the next higher, and the last quotient will be the decimal sought. Perform the preceding example by this Rule. 12/7,5 2'08,625 ,43125 In dividing by the several divisors, we annex as many ciphers to each dividend as are necessary. Thus we annex 1 cipher to the farthings, and divide by the number of qrs. in a penny, and so on with each denomination. |