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Second Method.
8567923 the given circulating decimal.

85 the terminate part which is subtracted.
8567838 the numerator of the answer.

856 7 8 3 8 Ans. Note.-1. The reason of this operation may be shown thus: 8567923= 8500000+67923. Now 8500000—85+-67923 is equal to 8567923–85.

2. It is evident that the required denominator is the same as that of the periodical part; (Art. 357;) for, the denominator of the periodical part is the least common multiple of the two denominators. Hence,

358. To reduce a mixed circulating decimal to a common fraction.

Change both the terminate and periodical part to common fractions separately, and their sum will be the answer required.

Or, from the given mixed periodical, subtract the terminate part, and the remainder will be the numerator required. The denominator is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part.

PROOF.Change the common fraction back to a decimal, and if the result is the same as the given circulating decimal, the work is right.

13. Reduce .138 to a common fraction. Ans. lb, or .
14. Reduce .53 to a common fraction.
15. Reduce .5925 to a common fraction.
16. Reduce .583 to a common fraction.
17. Reduce .0227 to a common fraction.
18. Reduce .4745 to a common fraction.
19. Reduce .5925 to a common fraction.
20. Reduce .008497133 to a common fraction.

CASE III. -Dissimilar periodicals reduced to similar and conter minous ones.

359. In changing dissimilar periods, or repetends, to similai and conterminous ones, the following particulars require attention.

1. Any terminate decimal may be considered as interminate by annexing ciphers continually to the numerator. Thus .46.460000, &c.=,460.

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2. Any pure periodical may be considered as mixed, by taking the given period for the terminate part, and making the given period the interminate part. Thus 465.46+.0046, &c.

3. A single period may be regarded as a compound periodical. Thus .3 may become 33, or .333; so.63 may be made .6333, or .63333, &c.

4. A single period may also be made to begin at a lower order, regarding its higher orders as terminate decimals. Thus .3 may be made .33, or .3333, &c.

5. Compound periods may also be made to begin at a lower order. Thus .36 may be changed to .363, or .36363, &c.; or by extending the number of places .479 may be made .47979, 4197979, &c.; or making both changes at once .532 may be changed to .5325325, &c. Hence,

360. To make any number of dissimilar periodical decimals similar.

Move the points, so that each period shall begin at the same order as the period which has the most figures in its terminate part.

21. Change 6.814, 3.26, and .083 to similar and conterminous periods. Operation.

Having made the given periods simi6.81456.81481481 lar, the next step is to make them con3.26=3.26262626

terminous. Now as one of the given .083=0.08333333

periods contains 3 figures, another 2,

and the other 1, it is evident the new periodical must contain a number of figures which is some multiple of the number of figures in the different periods ; viz: 3, 2, and 1. But the least common multiple of 3, 2, and 1 is 6; therefore the new periods must at least contain 6 figures. Hence,

361. To make any number of dissimilar periodical decimals, similar and conterminous.

First make the periods similar ; (art. 360 ;) then extend the figures of each to as many places, as there are units in the least common multiple of the NUMBER of periodical figures contained in each of the given decimals. (Art. 176.)

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27. Change 46.162, 5.26, 63.423, .486, and 12.5, to similar and conterminous periodicals.

Operation. 46.162=46.16216216 The numbers of periodical figures in 5.26 =

5.26262626 the given decimals are 3, 2, 3, and 1; 63.423=63.42342342 and the least common multiple of

.486= 0.48666666 them is 6. Therefore the new periods 12.5 = 12.50000000

must each have 6 figures. 23. Make .27, .3, and .045 similar and conterminous. 24. Make 4.321, 6.4263, and .6 similar and conterminous.

ADDITION OF CIRCULATING DECIMALS.

Dissimilar.

Ex. 1. What is the sum of 17.23+41.2476+8.61+1.5+. 35.423 ?

Operation.

Sim. & Conterminous. 17.23 = 17.2323232 First make the given decimals sim41.2476=41.2476476 ilar and conterminous. (Art. 361.) 8.61 8.6161616 Then add the periodical parts as in = 1.5000000

simple addition, and since there are 35.423 =35.4232323

6 figures in the period, divide their Ans. 104.0i93648 sum by 999999; for this would be

its denominator, if the sum of the periodicals were expressed by a common fraction. (Art. 355.) Setting down the remainder for the repeating decimals, carry the quotient 1 to the next column, and proceed as in addition of whole numbers. Hence,

1.5

362. We derive the following general

RULE FOR ADDING CIRCULATING DECIMALS.

First make the periods similar and conterminous, and find their sum as in Simple Addition. Divide this sum by as many 9s as there are figures in the period, set the remainder under the figures added for the period of the sum, carry the quotient to the next column, and proceed with the rest as in Simple Addition,

Obs. If the remainder has not so many figures as the period, ciphers must be prefixed to make up the deficiency.

2. What is the sum of 24.i32+2.23+85.24 +67.6 ?
3. What is the sum of 328.126+81.23+5.624 +61.6 ?
4. What is the sum of 31.62+7.824+8.392+.027 ?
5. What is the sum of 462.34+60.82+71.164 +.35?
6. What is the sum of 60.25+.34+6.435+.45+45.24 ?
7. What is the sum of 9.814+1.5+87.26+0.83+124.09 ?

8. What is the sum of 3.6+78.3476+735.3 + 375+27+ 187.4 ?

9. What is the sum of 5391.35+72.38+187.21+4.2965+ 217.8496+42.1767.523+58.30048 ?

10. What is the sum of .162+134.09+2.93 +97.26+3.769230 +99.083+1.5+.814 ?

SUBTRACTION OF CIRCULATING DECIMALS.

Ex. 1. From 52.86 take 8.37235.
Operation.

We first make the given decimals simi52.86=52.86868

lar and conterminous, then subtract as in 8.37235= 8.37235 whole numbers. But since the period in 44.49632 the lower line is larger than that above

it, we must borrow 1 from the next higher order. This will make the right hand figure of the remainder one less than if it was a terminate decimal. Hence,

363. We derive the following general

RULE FOR SUBTRACTING CIRCULATING DECIMALS.

Make the periods similar and conterminous, and subtract as in whole numbers. If the period in the lower line is larger than that above it, diminish the right hand figure of the remainder by 1.

Obs. The reason for diminishing the right hand figure of the remainder by 1, if the period in the lower line is larger than that above it, may be explained thus:

When the period in the lower line is larger than that above it, we must evidently borrow 1 from the next higher order. Now if the given decimals were extended to a second period, in this period the lower number would also be

larger than that above it, and therefore we must borrow 1. But having bor. rowed 1 in the second period, we must also carry one to the next figure in the lower line, or, what is the same in effect, diminish the right hand figure of the remainder by 1.

2. From 85.62 take 13.76432. Ans. 71.86193. 3. From 476.32 take 84.7697. 4. From 3.8564 take .0382. 5. From 46.123 take 41.3. 6. From 801.6 take 400.75. 7. From 4.7824 take .87. 8. From 1419.6 take 1200.9. 9. From .634852 take .02i. 10. From 8482.42i take 6031.035.

MULTIPLICATION OF CIRCULATING DECIMALS.
Ex. 1. What is the product of 36 into .25 ?
Operation.

We first reduce the given periodi.86=38=

cals to common fractions; (Art. 358 ;) .25=+=**. then multiply them together. (Art. Now tX33=

219.) Finally, we reduce the product But =.092 Ans. to a periodical decimal. Hence,

92 990

364. We derive the following general

RULE FOR MULTIPLYING CIRCULATING DECIMALS.

First reduce the given periodicals to common fractions, and multiply them together as usual. (Art. 219.) Finally, reduce the product to decimals and it will be the answer required.

Obs. If the numerators and denominators have common factors, the operation may be contracted by canceling those factors before the multiplication is performed. (Art. 221.)

2. What is the product of 37.23 into .26 ? Ans. 9.928.
3. What is the product of .i23 into .. ?
4. What is the product of .245 into 7.3 ?
5. What is the product of 24.6 into 15.1 ?
6. What is the product of 48.23 into 16.13 ?

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