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Is 5/12 = .4166666666 a repeating decimal expansion even though the entire solution does not repeat and only the 6 repeats?

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Oct 27, 2010
Convert Repeating Decimal to Fraction
by: Staff

The question:

Is 5/12 = .4166666666 a repeating decimal expansion even though the entire solution does not repeat and only the 6 repeats?



The answer:

5/12 does in fact = .4166666666...
(the 6 repeats endlessly)


You can determine the true fraction for any repeating decimal:

I am going to show you how to begin with your decimal (.4166666666...), then solve for the fraction 5/12.


Begin with your decimal

x = .4166666666...


Multiply each side of the equation by 100

100 * x = 100 * .4166666666...

100x = 41.66666666...


Subtract the original value of x from 100x

. . . . . . .100x = 41.66666666...
. . . . . . . . - x = - .4166666666...
. . . . . . .____ ____________
. . . . . . . 99x = 41.25


Divide each side of the equation by 99

99x = 41.25

99x/99 = 41.25/99

x = 41.25/99


Since you now know that x = 41.25/99, you only need to reduce that fraction to its simplest form, and you are finished:

x = 41.25/99


Multiply the fraction 41.25/99 by the fraction 100/100

(Multiplying by 100/100 does not change the value of the fraction 41.25/99. 100/100 = 1. Multiplying anything by 1 does not change its value)

x = (41.25/99)(100/100)

x = 4125/9900

4125 = 5 * 825
9900 = 12 * 825

Therefore:

x = 4125/9900 = (5 * 825)/(12 * 825)


Notice that the 825 in the numerator and the 825 in the denominator cancel one another

x = (5)/(12)

x = 5/12



The final answer is:

x = 5/12 = .4166666666...






Thanks for writing.


Staff
www.solving-math-problems.com



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