places from the right as there are decimal places in both the multiplicand and multiplier. Example. Multiply 62.53 by .124. Explanation. We write the two numbers so that the two figures at the extreme right of both the multiplicand and multiplier are in the vertical line. Multiplying as with whole nimbers we get the result as shown. To point off the decimal places in the product, we add the number of places in the multiplicand, 2, to the number of places in the multiplier, 3, to get 5. In the answer, therefore, we will point off 5 places beginning at the right and place the decimal point in front of the fifth figure. 61. Division of Decimals. To divide decimal numbers we proceed as with whole numbers, forgetting about the decimal point for the moment. If there are not sufficient figures in the dividend so that the division may be made, we simply add ciphers to the right of the decimal point. This does not change the value of the number. We may carry the operation as far as we please, adding ciphers if necessary. To locate the decimal point in the quotient we point off as many places from the right as there are more decimal places in the dividend than in the divisor. Example. Divide 528.01 by 9.5. Explanation. This example will give no difficulty until we come to locate the decimal point. There are 2 places in the dividend and 1 place in the divisor. Subtracting the number of places in the divisor from the number in the dividend, 2 minus 1, gives 1 place to point off from the right in the quotient. In the last division there is a remainder, but since this is less than half of the divisor, we neglect it. Example. Divide 2 by 1.43. 1.43 ) 2.0000 ( 1.39 Ans. 1 43 570 1410 Explanation. We must add ciphers to the dividend before we can divide. We place a decimal point after the 2 and add the ciphers to the right. We add as many ciphers as are necessary to get the required number of places in the quotient. If we desired to carry the division farther, we would add more ciphers. To locate the decimal point in the quotient, we subtract the number of places in the divisor from the number in the dividend, 4-2=2, and point off two places in the answer. 62. Accuracy of Results. When we were working with common fractions, we could always express a remainder in division by means of a fraction. You will notice that in the above examples in division we might continue to add ciphers to the dividend and carry the division out as far as we please. There will always be a remainder, but the further we carry the operation the more accurate will be the result. For all practical problems we must put a limit to this or the labor involved will be out of all proportion to the accuracy obtained. For all practical problems four places to the right of the decimal point are all that are necessary, and for most cases two or three places are enough. When more than four, or three, or two, as the case may be, occur, we may neglect all of the figures to the right of the fourth, or third, or second, providing the next figure is less than five. If the next figure is more than five, we add one to the last figure that we count and neglect all the others. If the first figure that we neglect happens to be five, we either do or do not add one as our judgment dictates. For example, the number 52.345278 may be written to the fourth decimal place as 52.3453-, The - sign following the number means that the real value is something less than the value given. Taking another example, the decimal 0.02534 may be written to the fourth decimal place as 0.0253+. The + sign means that the real value is something more than that given. It is not always necessary to write the + and signs as has been done here, but it is frequently done if it makes the work clearer. 63. Changing a Decimal Fraction to a Common Fraction. To change a decimal fraction to a common fraction it is only necessary to supply the denominator and to reduce the resulting fraction to its lowest terms. Example. Reduce .625'' to a common fraction. Explanation. We first supply the denominator, which will be 1, with as many ciphers following as there are decimal places to the right of the decimal point, or 1000. Reducing this fraction to its lowest terms by dividing by 5 three times, gives the result. 64. Changing a Common Fraction to a Decimal Fraction. To change a common fraction to a decimal fraction, place a decimal point after the numerator and add as many ciphers after it as are needed, usually four or less, depending upon the number of decimal places desired in the quotient. Then divide the numera tor by the denominator and point off as many places in the quotient as there are ciphers added in the dividend. It is not possible to reduce every common fraction to an exactly equivalent decimal fraction. Fractions having only the prime numbers 2 and 5 in the denominator may be reduced to an exactly equivalent decimal. However, we may reduce any common fraction with sufficient exactness for all practical purposes. Example. Change 16" to a decimal fraction. 16 ) 7.0000.4375 Ans. 64 60 48 120 80 |