say, 5×4=20, and 3 carried, for 5 × 6, make 23; 3 is set down and 2 carried; the rest of the multiplication by 5 proceeds in the usual way. Then, in multiplying the multiplicand by 4, we add 2 to the product, because 18 is nearer 20 than 10, and, therefore, it is nearer the truth to carry 2 than 1; and this result is placed under the former one, so that the last figures are under one another, since they both express ten thousandths. Proceed in the same manner with regard to the other figures of the multiplier, not neglecting to arrange the partial products, so that their last figures may stand in the same column. The sum is found as usual; four places are marked off from the right for decimals, and the last is omitted, since only three decimal places are required. Ex. Multiply 763.05403678956 by 254.4630578, so that the decimal in the product may contain five figures, or be correct to .00001. 763.05403678956 8750364.452 152610807358 38152701839 3052216147 305221614 45783242 2289162 38152 5341 610 194169.063465 In this example, the units' place of the multiplier reversed, is under the millionth of the multiplicand, and the other figures are easily set down, then proceed as in the last example. Ex. Multiply .681472 by .01286, so that the decimal in the product may contain five figures. .681472 68210.0 6814 1363 545 41 .008763 Here a cipher is placed below the sixth place of the multiplicand, because the multiplier contains no integer. Ciphers are set to the right of the product, to make up the decimal required. Ex. Multiply 2.656419 by 1.723, correct to 6 places. 2.6564190 3232327.1 26564190 18594933 531284 79692 5313 797 53 8 4.5776270 The multiplier has been carried out so as to ensure correctness to six decimal places. 200. EXERCISES ON THE MULTIPLICATION OF DECIMALS. 1. Find the product of 24.36 and 8; 97.25 and 100; 75.05 and 1000. 2. 25.75 x 3.5; 45.05 x 15.05; 60.007 x.027; .03054 x .023. 3. Multiply 764.30456 by 75.74063; 80096.208034 by 6.00007 4. .345 × 46; 89.634 × 25; .0523 ×.74; .6935 × 2.36. 5. Multiply 362.405 by .003; 8457.8 by 87.5; 8.46314 by 5.032064. DIVISION OF DECIMAL S. 201. We shall consider the three following cases :1st, when the dividend is a decimal and the divisor an integer. 2nd, when the dividend is an integer and the divisor a decimal. 3rd, when both divisor and dividend are decimals. 202. First, when the dividend is a decimal and the divisor an integer. Divide 487.83 by 21. 21)487.83(23.23 42 67 63 48 42 63 63 We find, first, the quotient of 48 tens, and 21, which is 2 tens; secondly, the quotient of 67 units and 21, which is 3 units; thirdly, the quotient of 48 tenths and 21 is 2 tenths; therefore, the decimal point must be placed before the tenths; lastly, the quotient of 63 hundredths and 21 is 3 hundredths. Therefore, when the dividend alone contains decimals, proceed as in common division, setting the decimal point after the tenths' place of the dividend has been brought down. 203. Secondly, when the dividend is integer, and the divisor a decimal. Therefore, if the divisor only is a decimal quantity, annex to the right of the dividend as many ciphers as there are decimals in the divisor, and divide as in common division; to the last remainder ciphers are added to get decimals, and the process is carried on to the approximation required. 204. Thirdly, when both divisor and dividend are decimal. 12.58164.7—12·5816 4.7 2.6769. 4.7 12,5816=2.6769; or thus: 12.58160 In the first method we have multiplied both dividend and divisor, by such number as will make the longest decimal integer, and the division is performed as before, annexing ciphers to obtain decimals in the quotient. Or, by the second method, which is an abbreviation of the first, divide as whole numbers, attaching ciphers to the dividend when necessary; then the quotient will have a number of decimal places equal to the excess of the number of decimal places of the dividend above that in the divisor. 205. This last law applies generally to every case in the division of decimals. 206. If the divisor be a recurring decimal, it will be more convenient to reduce it to a vulgar fraction; the dividend had better be unaltered when a recurring decimal. 207. There is also a method for shortening the labour of the division of two numbers, each consisting of many figures. This method, which we are going to investigate, depends upon this fact that every figure of the quotient is obtained by dividing the two or three last figures of the several partial dividends by the two first figures of the divisor. Thus, the quotient will be correct if we operate upon the two or three first figures of every partial dividend. We subjoin both operations. The common method will explain the reason of the contracted method. 208. Let the quotient of 1234.569 and 27.35894 be required, correct to three decimal places. 27.35894)1234.56|90(45.124 27.35894)1234.56900000(45.124 The operation on the left hand is performed according to the common method; therefore, we shall not make any remark upon it. Begin by making the number of decimal figures of the dividend equal to that of the divisor, by adding two ciphers, and as three decimal places are required in the quotient, annex three more ciphers to the dividend, then cut off six figures from the right of the dividend, since there are seven in the divisor; but as the divisor is not contained in the dividend. two figures of it are omitted, which is expressed by placing dots below them; divide now 123456 by 27358, the quotient is 4; multiply 27358 by 4, and add to the product 4, proceeding from 4 times the last figure cut off, or 36, which is nearer 40 than 30, the remainder is 14020; instead of adding the next figure of the dividend to it, we omit the 8 from the divisor, which is denoted by placing a point below it, and we divide 14020 by 2735, the quotient is 5, multiply 2735 by 5, and add to the product 4, proceeding from 40, the product of the omitted figure and 5, the remainder is 341; we now reject 5 from the |