12. Multiply 248 thousandths by 10000. CASE II. 326. When the number of decimal places in the multiplier and multiplicand is large, the number of decimals in the product must also be large. But decimals below the fifth or sixth order, express so small parts of a unit, that when obtained, they are commonly rejected. It is therefore desirable to avoid the unnecessary labor of obtaining those which are not to be used. 17. It is required to five places of decimals. multiply 1.3569 by .36742, and retain First Operation. 1.3569 .36742 .40707 8141 4 949 83 27138 .49855 2198 Ans. It is evident from the nature of decimal notation, that if the partial product of each figure in the multiplier is advanced one place to the right instead of the left, the operation will correspond with the descending scale, and at the same time will give the true product. (Art. 86. Obs.) But since only five decimals are required, those on the right of the perpendicular are useless. Our present object is to show how the answer can be obtained without them. Contraction. 1.356 9 .3674 2 .4070 7 814 1 95 0 54 3 .4985 5 Ans. Beginning at the right hand, we will first multiply the multiplicand by the tenths' figure of the multiplier, and place the first figure of the partial product under the figure multiplied. In obtaining the second partial product, (i. e. multiplying by 6,) it is plain we may omit the right hand figure of the multiplicand, for, if multiplied, its product will fall to the right of the perpendicular line, and therefore will not be used. But if we multiply 9 into 6, the product will be 54; consequently there would be 5 to carry to the next product; we therefore carry 5 to 36, which makes 41. Again, in the third partial product, (i. e. in multiplying by 7,) we may omit the two right hand figures of the multiplicand; for, their product will fall to the right of the perpendicular line. But by recurring to the rejected figures, it will be seen that the product of 7 into 6 is 42, and 6 to carry make 48; we therefore add 5 to the product of 7 into 5, because 48 is nearer 50 than 40; consequently it is nearer the truth to carry 5 than to carry 4. In the fourth partial product we may omit the three right hand figures, and in the fifth or last, the four right hand figures. 18. Multiply .2356 by .3765, and retain 4 decimals in the product. Operation. .2356 .3765 Multiplying as before, the first figure of the partial product must be set in the fifth order, or one place to the right of the figure multiplied; for, there are 4 decimals in the multiplicand and the one by which we multiply makes 5. (Art. 324.) But since we wish to retain only 4 decimals in the product, we may omit this figure, carrying 2 to the next product. Proceed in the same manner with the other figures in the multiplier. Finally, the sum of the partial products which are retained, is the answer required. Hence, 14 1 .0887 Ans. 327. To multiply decimals and retain only a given number of decimal figures in the product. Count off in the multiplicand as many decimal places less one, as are required in the product. Then beginning at the right hand figure counted off, multiply the multiplicand by the tenths or first decimal figure of the multiplier, and set the first figure of the partial product one place to the right of the figure multiplied, increasing it by the nearest number of tens that would arise from the QUE ST.-327. How multiply decimals, and retain a given number of figures in the product? rejected figure if multiplied. Next multiply by the second decimal figure, omitting the next right hand figure of the multiplicand and carrying as before. Proceed in the same manner with all the figures of the multiplier whose product will come under the decimal places counted off, omitting an. .additional figure on the right of the multiplicand, as you multiply by each successive figure, and set the first figure of each partial product under that of the preceding. Finally, from the sum of the partial products, cut off the required number of decimals, and the result will be the answer. OBS. 1. In order to determine where to place the decimal point in the product, we have only to observe that the product of the right hand figure of the multiplicand into the tenths of the multiplier is of the order denoted by the sum of the orders of the two figures multiplied; (Art. 324;) and when the multiplier is tenths it is of the order next lower than the figure multiplied. For this reason the first partial product is set one place to the right of the figure multiplied. But since we count off one decimal less than is required in the product, the right hand figure in the sum of the partial products must consequently be the right hand decimal place in the answer. 2. If the multiplier contains units, tens, hundreds, &c., in multiplying by the units, we must begin one figure to the right of those counted off, and set the first figure of the partial product under the figure multiplied. In multiplying by the tens, we must begin two figures to the right of those counted off, and set the first figure of the partial product under that of the units; in multiplying by the hundreds, we must begin three figures to the right, and set the first figure of the partial product under that of the preceding, &c. This will bring the same orders under each other. 19. Multiply .72543414 by .24826421 retaining 5 decimal places in the product. Operation. .7254'3414 .2482 6421 1450 9 290 2 58 0 14 .1800 9 Ans. Having counted off 4 decimals in the multiplicand, increase the product of 2 into 4 by 1, because the product of the 3 rejected into 2 is nearer 10 than 0. Set the 9 one place to the right of the figure multiplied. The 4 in the last partial product, is the number which would be carried to this order, if the 7 were multiplied by 6. 20. Multiply 67.1498601 by 92.4023553 retaining four deci mals in the product. Operation. 67.149'8601 92.402 3553 6043.487 4 134 299 7 26 859 9 134 3 20 1 34 3 6204.805 1 Ans. In this operation we multiply first by the tens figure of the multiplier, beginning two places to the right of those counted off in the multiplicand. It is immaterial as to the result whether we multiply by the tenths first, or by the units, tens, or hundreds, provided we set the first figure of the partial product in its proper place. (Art. 327. Obs. 2.) 21. Multiply .863541 by .10983 retaining 5 decimal places. 22. Multiply 1.123674 by 1.123674 retaining 6 decimal places. 23. Multiply .26736 by .28758 retaining 4 decimal places. 24. Multiply .1347866 by .288793 retaining 7 decimal places. 25. Multiply .681472 by .01286 retaining 5 decimal places. 26. Multiply .053407 by .047126 retaining 6 decimal places. 27. Multiply .3857461 by .0046401 retaining 6 decimal places. DIVISION OF DECIMAL FRACTIONS. 328. Ex. 1. How many bushels of oats, at .2 of a dollar a bushel, can you buy for .84 of a dollar? Analysis. Since 2 tenths of a dollar will buy 1 bushel, 84 hundredths of a dollar will buy as many bushels, as 2 tenths is contained times in 84 hundredths. Now .84; and .2=1, 20 or 2. (Art. 191.) And = 4, or 41. But, (Art. 311,) 44.2, which is the answer required. Operation. .2).84 We divide as in whole numbers, and point off 4.2 Ans. one decimal figure in the quotient. OBS. The reason for pointing off one decimal figure in the quotient may be thus explained. We have seen in the multiplication of decimals, that the product has as many decimal figures, as the multiplier and multiplicand. (Art. 324.) Now since the dividend is equal to the product of the divisor and quotient, (Art. 112,) it follows that the dividend must have as many decimals as the divisor and quotient together; consequently, as the dividend has two decimals, and the divisor but one, we must point off one in the quotient. In like manner it may be shown universally, that 329. The quotient must have as many decimal figures, as the decimal places in the dividend exceed those in the divisor; that is, the decimal places in the divisor and quotient together, must b equal in number to those in the dividend. 2. What is the quotient of 3.775 divided by 2.5? 3. What is the quotient of .0072 divided by 2.4. Operation. 2.4).0072(.003 Ans. 72 make up the deficiency. Ans. 1.51. Since the dividend has three decimals more than the divisor, the quotient must have three decimals. But as it has but one figure, we prefix two ciphers to it to OBS. It will be noticed that 3, the first figure of the quotient, denotes thousandths; also the product of 2, the units figure of the divisor, into the first quotient figure, is written under the thousandths in the dividend. Hence, The first figure of the quotient is of the same order, as that figure of the dividend under which is placed the product of the units of the divisor into the first quotient figure. 330. From the preceding illustrations we deduce the following general RULE FOR DIVISION OF DECIMALS. Divide as in whole numbers, and point off as many figures for decimals in the quotient, as the decimal places in the dividend exceed those in the divisor. If the quotient does not contain figures enough, supply the deficiency by prefixing ciphers. PROOF.-Division of Decimals is proved in the same manner as Simple Division. (Art. 121.) OBS. 1. When the number of decimals in the divisor is the same as that in the dividend, the quotient will be a whole number. QUEST.-330. How are decimals divided? How point off the quotient? How is division of decimals proved? |