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In this example the purpose is to find how many times $2.25 is contained in $15.375. There are more decimal places in the dividend than in the divisor. The first thing that suggests itself, is to reduce the divisor to the same denomination as the dividend, that is, to mills or thousandths. This is done by annexing a zero, thus, $2.250. The ques. tion is now, to find how many times 2250 mills are contain: ed in 15375 mills. . It is not important whether the .poin' be taken away or not.

15375 (2250
13500

6.83 + gals. Ans.
18750
18000

7500
6750

750

Instead of reducing the divisor to mills or thousandths, we may reduce the dividend to cents or hundredths, thus, $15.375 are 1537.5 cents. The question is now, to find how many times 225 cents are contained in 1537.5 cents. This is now the same as the case where there were deci' mals in the dividend only, the divisor being a whole number.

1537.5 (225
1350

6.83 + gals. Ans. as before.
1875
1800

750
075

75 If 3.15 bushels of oats will keep a horse 1 week, how many wceks will 37.5764 bushels keep him ?

The question is, to find how many times 3.15 is contained in 37.5764.' The dividend contains ten thousandths. The divisor is 31500 ten thousandths.

375764 (31500
31500

11.929 + weeks. Ans.
60764
31500

292640
283500

91400
63000

284000
283500

500

Instead of reducing the divisor to ten-thousandths, we may reduce the dividend to hundredths. 37.5764 are 3757.64 hundredths of a bushel. The decimal .64 in this, is a fraction of an hundredth.

3.15 are 315 hundredths. Now the question is, to find how many times 315 hundredths are contained in 3757.64 hundredths.

3757.64 (315
315

11.929 + weeks. Ans. as before.
607
315

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From the two last examples we derive the following rule for division : When the dividend contains more decimal places

than the divisor : Reduce them both to the same denomination, and divide as in whole numbers.

N. B. There are two ways of reducing them to the same denomination. First, the divisor may be reduced to the same denomination as the dividend, by annexing zeros, and taking away the points from both. Secondly, the dividend, may be reduced to the same denornination as the divisor, by taking away the point from the divisor, and removing it in the dividend towards the right as many places as there are in the divisor. The second method is preferable.

The same result may be produced by another mode of reasoning. The quotient must be such a number, that being multiplied with the divisor will reproduce the dividend Now a product must have as many decimal places as there are in the multiplier and multiplicand both. Consequently the decimal places in the divisor and quotient together must be equal to those in the dividend. In the last example there were four decimal places in the dividend and two in the divisor ; this would give two places in the quotient. Then a zero was annexed in the course of the division, which made three places in the quotient. The rule may be expressed as follows:

Divide as in whole numbers, and in the result point off as many places for decimals as those in the dividend exceed those in the divisor. If zeros are annexed to the dividend, count them as so many decimals in the dividend. If there are not so many places in the result as are required, they must be. supplied by writing zeros on the left.

Division in decimals, as well as in whole numbers, may be expressed in the form of common fractions.

What part of .5 is .3? Ans..
What part of .08 is .05 ? Ans..
What part of .19 is .43 ? Ans. is
What part of .3 is .07 ?

To answer this, .3 must be reduced to hundredths. .3 is .30, the answer therefore is .

What part of 14.035 is 3.8 ?
3.8 is 3.800, the answer therefore is 38.00

In fine, to express the division of one number by another, when either or both contain decimals, reduce them both to the

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lowest denomination mentioned in either, and then write the divisor under the dividend, as if they were whole numbers.

*

Circulating Decimals.

5 6 333

XXIX. There are some common fractions which cannot be expressed exactly in decimals. If we attempt to change

to decimals for example, we find .3333, &c. there is always å remainder 1, and the same figure 3 will always be repeated however far we may continue it. At each division we approximate ten times nearer to the true value, and yet we can never obtain it. '&=.1666, &c.; this begins to repeat at the second figure. l =.545454, &c.; this repeats two figures. In the division the remainders are alternately 6 and 5.

.168168, &c. ; this repeats three figures, and the remainders are alternately 56, 227, and 272. Some do not begin to repeat until after two or three or more places. It is evident that whenever the same remainder recurs a second time, the quotient figures and the same remainders will repeat over again in the same order. In the last example for instance, the number with which we commenced was 56 ; we annexed a zero and divided ; this gave a quotient 1, and a remainder 227 ; we annexed another zero, and the quotient was 6, and the remainder 272 ; we annexed another zero, and the quotient was 8, and the remainder 56, the number we commenced with. If we annex a zero to this, it is evident that we shall obtain the same quotient and the same remainder as at first, and that it will continue to repeat the same three figures for ever.

It is evident that the number of these remainders, and consequently the number of figures which repeat, must be one less than the number of units in the divisor. If the fraction is 4, there can be only six different remainders ; after this number, one of them must necessarily recur again, and then the figures will be repeated again in the same or der.

1 (7
10
7.1428571, &c.

It commences with 1 for the 30

dividend, then annexing zeros, 28

the remainders are 3, 2, 6, 4, 5,

which are all the numbers below 20

7; then comes 1 again, the num14

ber with which it commenced,

and it is evident the whole will be 60

repeated again in the same order. 56

Decimals which repeat in this

way are called circulating deci 40

mals. 35

50 49

10
7

3 Whenerer we find that a fraction begins to repeat, we may write out as many places as we wish to retain, without the trouble of dividing.

As it is impossible to express the value of such a fraction by a decimal exactly, rules have been invented by which operations may be performed on them, with nearly as much accuracy as if they could be expressed; but as they are long and tedious, and seldom used, I shall not notice them. Suf ficient accuracy may always be attained without them.

I shall show, however, how the true value of them may always be found in common fractions. The fraction reduced to a decimal, is .1111

&c. Therefore, if we wish to change this fraction to a common fraction, instead of calling it io, do, or judo, which will be a value too small, whatever number of figures we take, we must call it . This is exact, because it is the fraction which produces the decimal. If we have the fraction .2222.. &c. it is plain that this is twice as much the other, and must be called. If} be reduced to a decimal, it produces .2222

&c. If we have .3333 ..&c. this being three times an

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