If the divisor have more figures than the dividend, zeros may, of course, be annexed to the dividend; but then that portion of the work in which these zeros are brought into use cannot be depended upon, unless the last figure of the dividend be strictly accurate. When such is not the case, therefore, the overplus figures of the divisor should be cut off before commencing the division, if incorrect figures are to be excluded from the quotient. 1. As an example, let that at page 127 be taken; that is, let the value of 721.17562÷2·257432 be required to as many decimals as can be depended upon. As no decimals are to be 2.257432)721-1756(319.4672 brought down to be annexed 6772296 the operation is continued till the divisor is reduced to the single figure 2, and the work ends. This divisor 2, of the last dividend 6, gives only 2 for quotient; because if we try 3 we find that 1 must be carried from the divisor figure last rejected; so that the product would be 7. It is probable, however, that 3 is nearer the truth than 2; but the last figure of the quotient found in this way cannot, in general, be depended upon as strictly true. Suppose, for instance, that the final 2 in the divisor should be in strictness 2; then, what is now a 6 at the close of the work should be a 5, and the quotient figure 2 would be correct; but if the final 2 in the divisor should be 2- or 3=1.66, &c., then the 6 at the end should be a 7, and 3 would be the correct quotient figure; so that here, as in contracted multiplication, the last decimal in the result may err by a unit. 3. Divide 18.75, in which the 5 is not strictly correct, by 2.01747. 650 1168 1935 1625 3108 2925 183 163 20 19 Here, instead of adding three zeros to the dividend, as we should do, or conceive to be done, if the last figure of the dividend were strictly accurate, we cut three figures from the divisor, and proceed as below, taking care, in multiplying by the first quotient figure 9, to carry what arises from the dismissed figures of the divisor, namely 7. If the final decimal 5 of the dividend had been quite accurate, the operation would then have been as here annexed, and the quotient may be considered as perfectly accurate, as far as four places of decimals; namely, 9.2938. 2:01,747)18-75(9.29 1816 1 2.01747)18.75(9.29383 1815723 From the illustrations now given, you can find no difficulty in multiplying and dividing decimals, which are not in themselves strictly correct in the final figures, so as to secure the greatest possible accuracy in your results. The subject is one of very great importance, and it therefore deserves your careful attention. In the following exercises the final figure of each decimal is supposed to be more or less inaccurate, except when otherwise stated. 21. Perform Ex. 7 on the supposition that the final figure 2 of the divisor is strictly accurate. 22. Perform Ex. 20 on the supposition that the final figure 6 of the dividend is strictly accurate. 23. Perform Examples 3 and 12 on the supposition that each dividend is strictly accurate. 24. The old wine-gallon contained 231 cubic inches; the new or imperial gallon contains 277.274 cubic inches, the third decimal, however, 4, being a little too great: it is required to find how many imperial gallons are contained in the old wine-hogshead of 63 wine-gallons, old measure. (91.) Application of Decimals to Concrete Quantities. The application of Decimals to Concrete Quantities, is so like the application of whole numbers and common fractions, as to render any distinct rules here unnecessary: it will be sufficient to present to you a few examples, worked at length, as specimens of the operations. Ex. 1. Find the value of 761£. The work is in the margin, and consists, as in common reduction, in simply reducing pounds to shillings, pence, and farthings. The answer is 15s. 24d., and the decimal, 56 of a farthing, or 15s. 2 d. nearly. 2. How much is 37 of 15s. 5d.? Here, 15s. 5d. = 185d., and 185d.x37 68.45d. = 5s. 84d., and 8, that is, of a farthing, or 5s. 8d. nearly. 3. What decimal of £3 7s. is £1 2s. 3d.? Here, as in fractions, the two quantities must be re 20 20 67 22 268 ) 89(3321 804 86 •35 £ 804 12)3 d. 56 536 2,0),2.25 8. 24 •1125 £ 3.35)1.1125(-3321 1005 1075 70 670 30 duced to a common denomination, and then the latter divided by the former: it is both methods is given in the 2,0),7 s. 4. What decimal of £5 is £3 17s. 63d.? Here, the shortest way is to proceed according to the second of the above methods, and to reduce the 17s. 6d. to the decimal of a £, as in the margin, and then to divide by the £5. The denomination, £, here placed against dividend and divisor, might have been omitted, since the quotient is the same abstract number whether dividend and divisor be concrete or not. 4)3 12)6.75 2,0)1,7.5625 £5)3.878125£ 5. Reduce 2s. 9 d. to the decimal of 7s. 93d. Here, it would seem, that the best way is to reduce first to the lowest denomination, farthings, and then to divide the former quantity by the latter: it is plain, however, that the two may be a little simplified, by dividing each by 3: thus, 28. 9 d. 11 d. 45 f. 4 = = 7s. 9 d. 28. 74d. 125f. = 25; we have, 775625 5)9 5)1.8 .36 therefore, merely to turn the fraction,, into a decimal, by actual division, as in the margin; so that 2s. 93d. is 36, that is, 36 hundredths of 7s. 93d. 6. Reduce 7 drams to the decimal of 1 lb. avoirdupois. Ex. 9. •28 £ 1.4 112 28 •375)-392(1£ 17 20 •340(08. 12 4.08(11 d. nearly Required the values of the following decimals, &c. 1. 09375 acres. 3. 4625 tons. 2. 3.6285 degrees. 4. 4375 shillings. |