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9. Reduce 123 to an equal vulgar fraction.

Earplanation.
OPERATIon. - -
- We have here one repeating fi-
First 900 denominator. gure, and two in the finite part;

Then 123 — 12 = 111 numerator therefore to one 9 I subjoin two ciphers, making 900 for the deno

111 37 minator. Then from 123 the given

Therefore 900 - 300 the frac- j repetend, I subtract iš the

- - finite part, which gives l l l for the

tion required. numerator; the fraction is then reduced to its lowest terms,

10. Reduce .5925 to an equivalent vulgar fraction. Thus 9990 denominator. and 5925 – 5 = 5920 numerator.

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11. Reduce 27 and 53 to vulgar fractions. Also, and #

12. Reduce 345, .i.284, and 43310 to vulgar fractions.

252. To make dissimilar repetends similar and conterminous.

RULE I. Consider which of the given repetends begins the farthest from unity, and continue each of the other repetends to as many places from unity, putting a dot over the right hand figure of each; this will make them similar.

II. Continue all the repetends to as many more figures as are equal to the least common multiple of the several numbers of places in all the repetends, and place a dot over the last, or right hand figure; this will make the repetends conterminous.

13. Given the following dissimilar repetends 32i, 5.47, 3.2, .123, and 2.39, to make them similar and conterminous.

Earplanation.

Here .123 is the repetend which begins

- .." farthest from unity; I therefore continue 32i = .32132133 all the other repetends to the third place, - - - over which I put a dot. Now one of the 5.47 = 5.47474747 repetends contains 3 places, one contains - 5. 3. 2, and two contain each 1, and the least

3.2 . – 3.22222222 common multiple of 3, 2, and 1, is 6 ; .123 = . 12312312 wherefore I continue the repeating figures

- - - , in each repetend to 6 places farther, and 2.39 = 2,39393939 place a dot over the last, y

OPERATION. dissimilar. sim. and conterm.

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14. Given 123.4%, 3.91%, 3013, and 9.3s.5, to make them similar and conterminous.

dissimilar. sim. and conterm.

123.45 = 1234555555555555
3.91% = 3.9f21212121213
3013 = 30.1230.12301.23

9.385 = 9.3853S53s.53853 15. Make 12.384 and 23; similar and conterminous. Ans. 12.3843 and 2.3444. 16. Make 1234 and 5.033219 similar and conterminous. Ans. 12341444 and 5.04331943. 17. Make 3.5, 45, and 1.084, similar and conterminous.

253. To find whether the decimal equivalent to afty given vulgar fraction be finite or infinite; and if infinite, to find how many places the repetend will consist of

Rule I. Reduce the given fraction to its lowest terms, then divide the denominator of the new fraction by 10, 5, or 2, as long as division by either can be made.

II. If by this division the denominator be reduced to unity, the decimal will be finite, consisting of as many places as you performed divisions.

III. But if after such division the denominator, viz. the last quotient, be greater than unity, divide 9999, &c. by the said last quotient till nothing remains; the number of nines made use of will be equal to the number of figures in the repetend, which will begin after as many places of figures as there were divisions by 10, 5, or 2.

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(2) (2) (2) (2) Then, 16 . . . 8 . . . 4. . . 2 ... 1; therefore the given fraction produces a finite decimal, consisting of four places, viz. 210 3

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112OT 16 1 II 1 19. Is the decimal equivalent to 7700 finite or infinite 2 if in

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finite, where does the repetend begin, and how many places does it consist of?

I 111 IOI First reduced to its lowest terms is * 7700 700

(10) (5) (2) - Then, 700. .. 70. ... 14. .. 7; wherefore 7).999999 in which 1428.57° sir nines are used before the work terminates. Now since 3 divisions (by 10, 5, and 2) have taken place, there will be 3 finite places; and since there are sir nines employed in the division by the last quotient 7, there will be six circulating figures, beginning at

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20. Are the decimals equivalent to o, o, --, and ++,

6O 12 114 1050 finite or infinite, how many places does each consist of, and what are the particulars?

254. ADDITION OF CIRCULATING DECIMALs.

Rule I. Make the repetends which are to be added together similar and conterminous, (Art. 252.)

II. On the right hand of each repetend place two or three of the repeating figures, and add them together for the purpose of carrying.

III. Carry the tens from the left of the sum of these figures to the right hand row of figures in the repetends, and add up the whole as in finite decimals; then mark as many figures of the sum for a repetend as there are in each repetend added'.

* The reason of this rule is sufficiently plain; for it is evident, that all the repetends to be added must be made similar and conterminous (if they are not so already) before the operation commences: and since these repetends may be continued indefinitely, and that the sum of the right hand figures of the first repetend would, in that case, be increased by the number carried from the left hand figures of the second, and the sum of the right hand figures of the second by the number carried from the right hand figures of the third, and so on; and that these carryings would be always the same, as each arises from the addition of the same figures; it follows, that, in order to have the true repetend in the sum, the right hand figure of that repetend must be increased by the number

ExAMPLEs.

1. Add 274 123456 + 43 + 456; + 987. 4.1334 together.

OPERATIon.
similar and
dissimilar. conterminous. fg.
27 = 2.7777777 777
123456 = 12.3456456 456
45 = 4.54545i 545
456% = 4567,7777; 777
987. = 987.0000000 000
.1333 = .133423i 234

sum 1459.4791700 carry 2 to the 4.

Earplanation. The repetends being made similar and conterminous, the numbers marked fig. on the right are a few of the first figures of each repetend, and are added,

only to find what is to be carried to the 4.

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3. Add 13 + 35 + 12.3 + 123.4 + 4.3% together. Sum 141,675.

4. Add 17.64 + 2.8 + 423 + 1.83 + 54.9 together. Sum 81.604.

carried from the left; or, (which is the same,) by carrying from the numbers marked fog. (as in the 1st and 2d examples) to the said right hand figure of the repetend.

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RULE I. Make the given repetends similar and contermimous, and place the less number under the greater.

II. Subtract as in finite decimals; observing, that if the lower repetend be greater than the upper, the right hand figure of the remainder must be made less by 1 than it would be were the expressions finite".

ExAMPLEs.

1. From 12345 take 21.53%.

OPERATION. similar and dissimilar. conterminous. 123.45 = 123.4555555

21.532 = 21.5325325
Difference 101.9230236

2. From 374.123 take 40.373.

OPERATION. sumilar and elissimilar. conterminous. Erplanation. ', 'a' oo loo Here the repetend to be sub374.123 = 374.133123 tracted is the greater, the right -a – • e hand figure of the difference is 40.379 = 40.379999 therefore decreased by 1.

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3. From 39.2178 take 17.68. Diff. 21.530949i.
4. From 1.8 take 10047. Diff. 2953.
5. From 85.6% take 13.86433. Diff. 71.85193.
6. From 10,0413 take 264. Diff. 9.7%66948.

m The reason of this rule will be plain from the preceding note.

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