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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(319.467 2 brought down to be annexed

6772296 to the remainders, the final 2 in the dividend is suppressed

439460 as useless. The first figure 3

225743 of the quotient is multiplied into the entire divisor as it

213717 stands; for the next quotient

203169 figure 1, the divisor is curtailed by a figure, and a dot

10548 is put under this figure 2 to

9030 remind us of this: for the quotient figure 9, the cor

1518 responding divisor is 2.2574,

1354 another dot being put under the 3, to imply that it is no

164 longer to be used, except for

158 what is carried from it: as 3 x 9=27, the figure carried

6 from it is 3.

In this manner 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 5=1.66, &c., then the 6 at the

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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. 2. Divide 7.66858 by

31

•0325)7.66858(235.956 100

650 3.25 Here, 31=3.25, and = .0325; 100

1168 and as this decimal is strictly correct,

975 the operation should proceed without contraction till the final figure 8 in

1935 the dividend has been used; after

1625 which, contraction must commence, as in the margin.

3108 3. Divide 1875, in which the 5 is not 2925 strictly correct, by 2.01747.

Here, instead of adding three zeros to 183 the dividend, as we should do, or conceive 163 to be done, if the last figure of the dividend were strictly accurate, we cut three 20 figures from the divisor, and proceed as 19 below, taking care, in multiplying by the first quotient figure 9, to carry what arises

1 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 bave 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

2.01747)1875(9.29383 1816

1815723

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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.

Exercises. 1. 31•782 = 4.817.

11. 1708.4592.00024. 2. 2490•3048::573286. 12. •3412·8·4736. 3. 2.149- 500.78.

13. 75.347 = .3829. 4. 47•298; 6.029.

14. 1:10:473654. 5. 4650.75= 325.

15. 5.474558-31. 6. 8.6134:7.3524.

16. 1045. 7. 16.804379; 3:142. 17. 2387.64378. 8. 673.1489;:41432.

18. 14:3589; 7854. 9. 2:7182818:3•1415927. 19. 2972160=31773.244. 10. 00128-8.192.

20. 103.936 = 1059.108. 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.

761

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Ex. 1. Find the value of •761£. Ex. 1. Ex. 2. The work is in the margin, and

185 d. consists, as in common reduction, 20

37 in simply reducing pounds to shillings, pence, and farthings. The 15•220 s. 1295 answer is 158. 2 d., and the deci- 12

555 mal, •56 of a farthing, or 158. 2 d. nearly.

2.64 d. 68.45 d. 2. How much is •37 of 15s. 5d. ? 4

4 Here, 158. 5d. = 185d., and 185d. x.37 = 68.45d. = 58. 8 d., 2.56 f. 1.80 f. and 8, that is, of a farthing, or 58. 8£d. nearly.

3. What decimal of £3 7s. is £1 2s. 3d. ? Here, as in fractions, the

£.

£. $. d. two quantities must be re

3 7 1 2 3 duced to a common denomi

20

20 nation, and then the latter divided by the former : it is

67

22 matter of choice, whether you

4

4 bring them to the lowest denomination mentioned or to

268 ) 89(3321 the highest : the work by

804 both methods is given in the 2,0) 78. margin: in the first method,

86 the quantities are reduced to •35 £

804 threepences ; in the second, to pounds : in the second, the 12)3 d.

56 78. is converted into 35£;

536 the 3d. into •258., to which 2,0) 2.25 8. the 2s. is prefixed; and then,

24 the entire number of shil

•1125 £ lings, namely, 2.258., brought into •1125£; so that the proposed quan- 3.35)1.1125(3321 tities, in the denomination, pounds,

1005 are 3.35£ and 1.1125£; the latter, divided by the former, gives •3321,

1075 true to the nearest unit in the last

1005 decimal; the 1 is a little too great, but the error would have been

70 greater if 0 had been put instead.

670

30

4. What decimal of £5 is £3 178. 6 d. ?

4)3 Here, the shortest way is to proceed according to the second of the above 12) 6.75 methods, and to reduce the 178. 6 d. to the decimal of a £, as in the margin, 2,0)1,7.5625 and then to divide by the £5. The denomination, £, here placed against dividend and £5)3.878125£ divisor, might have been omitted, since the quotient is the same abstract number whether 775625 dividend and divisor be concrete or not.

5. Reduce 28. 93d. to the decimal of 7s. 9 d.

Here, it would seem, that the best way is to 5)9 reduce first to the lowest denomination, farthings, and then to divide the former quantity by the 5)1.8 latter: it is plain, however, that the two may be a little simplified, by dividing each by 3: thus, .36 28. 9 d.

111d. 45 f. 78. 9d. 2s. 7.d.

25; we have,

125f. therefore, merely to turn the fraction, s, into a decimal, by actual division, as in the margin ; so that 28. 9 d. is 36, that is, 36 hundredths of 78. 9 d. 6. Reduce 7 drams to the decimal

Ex. 9. of 1 lb. avoirdupois.

•28 £ 7

1.4 Here,

= .027344. 16 x 16 7. Reduce 14 min. to the decimal

112 of a day.

28 14 Here,

= 72 = .0097. 375) 392(1£ 60 X 24

375 8. Find the value of .0125 lb. troy. Here, .0125 x 12 x 20 dwt. =3 dwt.

17 9. If ; yard cost ,5£, what will

20 1% yard cost ? Here, s = •375; 25 = 28, and

•340(0 s. 13 = 1.4 ; i. 375 : 1.4 :: •28£

12 •28 x 1.4

£1 Os. 11d. •375

4:08(11 d. nearly nearly.

Exercises. Required the values of the following decimals, &c. 1. •09375 acres.

2. 3•6285 degrees. 3. •4625 tons.

4. .4375 shillings.

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