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3. What is the quotient of 1561.275 divided by 24.3 ?

Ans. 64.25. 4. What is the quotient of 0.264 divided by 0.2?

Ans. 1.32. 5. What is the quotient of 3.52375 divided by 3.355 ? 6. What is the quotient of 1001.25 divided by 2.25?

- ABRIDGED DIVISION OF DECIMALS.

42. If we divide 0.30679006 by 0.27610603, by the last rule, our work will be as follows:

OPERATION.
0.27610603)0.30679006(1.1111313

27610603
306840310
27610603
30734270
27610603
31236670
27610603
36260670
27610603
86500670
828 31809
36688610
27610603
90780070
82831809

77948261 By simply inspecting the above work, it is obvious that all that part of the work which is on the right of the vertical line, can in no way affect the accuracy of our quotient figures.

By the following rule we may perform the work of division so as to exclude all that part of the work on the right of the vertical line, thereby shortening the work, and still obtaining as accurate a result as by the last rule. To contract the work in division of decimals, we have this

RULE. . Proceed as in the last rule, until we reach that point of the work where it would be necessary to annex ciphers to the remainder. Then, instead of annexing a cipher to the remainder, . omit the right-hand figure of the divisor, and we shall obtain the next figure of the quotient ; and thus continue, at each suc. cessive figure of the quotient to omit the right-hand figure of the divisor until there is but one figure in the remainder.

Examples. 1. What is the quotient of 365.424907 divided by 0.263803? 0.263303)365.424907(1385.21892

263803
1016219
791409
2248100
2110424

1376767
1319015

57752
52761
4991
2638
2353
2110

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2. What is the quotient of 0.123456 divided by 1.912478 ?

OPERATION.
1.912478)0.123456(0.064552

114748

8708
7650
1058
956

102

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,

3. What is the quotient of 0.5260000 divided by 0.5260202?

Ans. 0.9999616. 4. What is the quotient of 7.45678 divided by 4.56789 ?

Ans. 1.63244. 5. What is the quotient of 7.632038 divided by 3.716048? 6. What is the quotient of 2 divided by 15.314865 ? 7. What is the quotient of 0.926954 divided by 0.3547898 ? 8. What is the quotient of 13.75992 divided by 6.76897 ?

42. To change a vulgar fraction into an equivalent deci. mal fraction.

It is obvious that the rule under Art. 33, will apply to this case, by considering all the denominate values as decreasing regularly in a tenfold ratio. Hence, this

RULE. Annex a cipher to the numerator, and then divide by the denominator ; to the remainder annex another cipher, and again divide by the denominator: and so continue until there is no

remainder, or until we have obtained as many decimal figures as may be desired. The successive quotients will be the successive decimal figures required.

. Examples. 1. What decimal fraction is equivalent to it?

16)100(0.0625 Ans.

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43. It will often happen, as in examples 2 and 5, under Art. 42, page 58, that the process will never terminate, in which case there is no decimal value which is accurately equal to the vulgar fraction.

Since we constantly multiplied the remainders by 10, it fol. lows that whenever the denominator of the vulgar fraction contains no prime factors different from those which compose 10, viz. 2 and 5, then the decimal value will terminate. But in all other cases, the decimal expression must consist of an infinite number of figures.

Hence, to determine whether a given vulgar fraction can be accurately expressed in decimals, we have this

RULE. Decompose the denominator of the vulgar fraction into its prime factors, (by rule under Art. 6,) then, if there are no primo factors different from 2 and 5, the vulgar fraction can be accu. rately expressed by decimals ; but if it contain different fac. tors, it can not be accurately expressed in decimals.

Examples. 1. Can the vulgar fraction kā be accurately expressed in decimals ?

In this example we find that 386=2x193; so that the de. nominator contains the prime factor 193, which is different from 2 or 5; consequently, záā can not be accurately express. ed in decimals.

2. Can the vulgar fraction 17 be accurately expressed by decimals?

Ans. It can not. 3. Can the vulgar fractions, having for denominators 640, be expressed in decimals accurately? . Ans. They can.

44. When a vulgar fraction can be accurately. expressed in decimals, we may determine the number of decimal places by the following

:' RULE. Decompose the denominator into its prime factors, (by rule under Art. 6,) which factors can not differ from 2 and 5, (by rule under Art. 43.) The highest exponent of 2 or 5 will be the number of decimal places sought.

Examples. 1. How many places of decimals will be required to express to?

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