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Rule.-I. Make the repetends similar if they are not so.

II. Add as in finite decimals, increasing the right hand term by the amount which would be added to it if the circulates were expanded, and make a repetend in the sum similar to those above.

2. Add 2.34, 3.0i, .756, 6.05. Ans. 12.1595140. 3. Add 72.43, 2.012, 65.13, 18 576. Ans. 158.1536. 4. Add 18.96, 5.73, 17.671, 4.ig, Ans 46.55692419. 5. Add 8.29301, 47, 7.00), 3.9236. Ans. 19.6. 6. Add 5 0549367, 1.53, 8.0763, 4.5. Ans. 19.2. 7. Add .3467, .0543, .08,.065, .4.

Ans. 1. 8. Add 63.45, 14.572, 8.1243715, 2.7354. Ans. 88.8.

SUBTRACTION OF CIRCULATES. 324. Subtraction of Circulates is the process of finding the difference between two circulates.

1. From 6.04 take 2.057.

SOLUTION.—Having made the repetends OPERATION. similar, we subtract as in finite decimals, observing to diminish the right hand term by

6.04=6.0460460 unity, since this would be necessary if the cir- 2.057= 2.0575757 culate were expanded, and we have 3.9884702.

3.9384702 Rule.-I. Make the repetends similar if they are not so.

II. Subtract as in finite decimals, diminishing the right hand term of the remainder by 1, when it would be necessary if the circulates were expanded, and make a repetend in the result similar to those above. 2. Subtract 5.62 from 20.5478.

Ans. 14.92. 3. Subtract 4.2296 from 12.37.

Ans. 8.14. 4. Subtract 71.3 from 74.325.

Ans. 3.012. 5. Subtract .296 from

Ans. .592. 6. Subtract .437465 from 141.

Ans. . 548. 7. Subtract 1.7836290 from 10.0563. Ans. 8.27. 8. Subtract 79.3650 from 88.5317460. Ans. 9.16.

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MULTIPLICATION OF CIRCULATES. 325. Multiplication of Circulates is the process of inding a product when one or both terms are circulates. 1. Multiply .2546 by 4.63.

OPERATION. SOLUTION.—4.63 equals 4.61. Multiplying by .6 and carrying to the right hand term

.2546

as much as would be necessary if the repetend were continued, we have.15278; multiplying by 4 in the same man

.152787878 ner, we have 1.0185; multiplying by .Of we have 1.018585858 .008488215; making these partial products similar,

.008488215 and adding, we have 1.179861952.

1.179861952 Rule.-I. If the multiplier contains a repetend, reduce it to a common fraction.

II. Multiply as in finite decimals, adding to the right hand term of each partial product the amount necessary if the repetend were expanded.

III. Make the partial products similar and find their sum.

Find the value of 2. 8.23 x 4.839.

Ans. 39.964. 3. .952380 x .763.

Ans. 0.12. 4. 16.204 x 32.75.

Ans. 530.810446. 5. 6.217 x 1.53.

Ans. 9.5330663997. 6. 4.923076 x.481.

Ans. 2.370. 7. 8.594 x 6.290.

Ans. 54.0678132. 8. .9625668449197860 x .75.

Ans. .72.

DIVISION OF CIRCULATES. 326. Division of Circulates is the process of finding a quotient when one or both terms are circulates. 1. Divide .95698 by.376.

OPERATION. SOLUTION.-If we make the repetends .37676 .95698 similar and subtract the finite part of each

376 956 repetend from the whole repetend, the remainders will be numerators of fractions

37300) 94742(2.54 having a common denominator, Art. 320,

746 Dividing the one by the other, we have 2.54.

2014 etc.

Rule.- Make the repetends similar, subtract the finite part from the entire repetend, omit the dots, and use the results for the dividend and divisor.

Note. When the divisor is not a circulate, divide as in finite decimals, bringing down the figures of the repetend instead of ciphers.

Find the value of 2. .0929 - .36.

Ans. .25. 3. 39.964 = 4.839.

Ans. 8.25. 4. 4.956-.75.

Ans. 6.6087542. 5. 3.97348=.2083.

Ans. 19.072. 6. 7.714285-952380.

Ans. 8.1. 7. 54.0678132 -8.594.

Ans. 6.290. 8. Divide .72 by .75. Ans. .9625668449197860.

GREATEST COMMON DIVISOR OF DECIMALS.

327. The Greatest Common Divisor of two or more decimals, either finite or infinite, is the greatest decimal that will exactly divide them. 1. Find the greatest common divisor of .375 and .423.

OPERATION. SOLUTION.–We make the two cir

.37575751.4234231 culates similar, and subtract the finite part, which reduces them to

3757572 42342301 fractions having a common denomi

3757572 nator. (Art. 320.) We then find the greatest common divisor of their

3813264 4766588 numerators, 1638, which is the nu

55692

501228 9 merator of the G. C. D., the denomi

49140 245702 nator being of the same denomina

6552

26208 4 tion as the original dividend and

6552 16384 divisor; hence the G.C.D. is.0001638.

gry go=.0001638, G. C. D. Rule.- Reduce the decimals to a common denominator, find the G. C. D. of their numerators, write the result over the common denominator, and reduce the resulting fraction to a decimal.

Note.-The G. C. D. can be found by reducing the decimals to common fractions, and applying the rule given in Art. 255, but the process here given is generally less tedious and more direct.

1638 9999990

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LEAST COMMON MULTIPLE OF DECIMALS.

328. The Least Common Multiple of two or more decimals is the least number that will exactly contain each of them.

1. Find the L. C. M. of .327, i.oii and .075. SOLUTION.—We reduce the

OPERATION. circulates to fractions having

1.32727 1.01110 .07575 a common denominator, as in

3
10

0 the previous case. The least common multiple of these 3, 32724 101100 07575 numerators is 275699700, 4 10908 3 3700 2 525 which is the numerator of the L. C. M., the denominator

25 2727 8425 2525 being the common denomina 101 2727

101 tor of the fractions. Reduc

27 337 ing 275699700, the L. C. M., to whole numbers and deci- 3X 4X 25X101 x 27 x 337=275699700 mals, we have 2757 2. the L. 275593700=2757.2727, L. C. M. C. M. Hence the

=2757.2 Rule.Reduce the decimals to a common denominator, find the L. C. M. of their numerators, write the result over the common denominator, and reduce the resulting fraction to a decimal.

Note.—The L. C. M. may be found by reducing the decimals to common fractions and applying the rule given in Art. 256 ; but the process here given is often more direct.

Find the L. C. M.
2. Of 42.123 and 45.6.

Ans. 33698.4. 3. .., .545, and .787.

Ans. 78. 4. Of 8.4, 5.27 and 16.185.

Ans. 971.i. 5. Of .6857142, 1.44, .3, and .35.

Ans. 100.8. 6. Of 6.6, 7.46, 9.35, and 10.054. Ans. 992745.6.

PRINCIPLES OF CIRCULATES. 329. These Principles of Circulates will be found to embrace some interesting and practical properties.

1. A common fraction whose denominator contains no other prime factors than 2 and 5, can be reduced to a simple decimal.

Since 2 and 5 are factors of 10, if we annex as many ciphers to the numerator as there are 2's or 5's in the denominator, the numerator will then be exactly divisible by the denominator. Therefore, etc.

2. The number of places in the simple decimal to which a" common fraction may be reduced is equal to the greatest number of 2's or 5's in the denominator. .

For, to make the numerator contain the denominator we must annex a cipher for every 2 or 5 in the denominator, and the number of places in the quotient, which is the decimal, will equal the number of ciphers annexed. Therefore, etc.

3. Every common fraction, in its lowest terms, whose denominator contains other prime factors than 2 or 5, will give an interminate decimal.

For, since 2 and 5 are the only factors of 10, if the denominator contains other prime factors, the numerator with ciphers annexed will not exactly contain the denominator, hence the division will not terminate and the result will be an interminate decimal. Therefore, etc.

4. Every common fraction which does not give a simple decimal gives a circulate.

In reducing there cannot be more different remainders than there are units in the denominator; hence if the division be continued, a remainder must occur which has already been used, and hence we shall have a series of remainders and dividends like those already used, therefore the terms of the quotient will be repeated.

5. A common fraction whose denominator contains 2's or 5.s with other prime factors will give a mixed circulate, and the number of places in the non-repeating part will equal the greatest number of 2's or 5's in the denominator. This principle is evident from Prins. 2 and 4, and may be illustrated

1
100

5

which : Tì= 22x5x7=22x5x7x100 5x100 = I whi will evidently give a mixed repetend, the repeating part beginning at the third decimal place.

6. The number of figures in a repetend cannot exceed the number of units in the denominator of the common fraction which produces it, less one.

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