# repeating decimal expansion

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

### Comments for repeating decimal expansion

 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 decimalx = .4166666666...Multiply each side of the equation by 100100 * x = 100 * .4166666666...100x = 41.66666666...Subtract the original value of x from 100x. . . . . . .100x = 41.66666666.... . . . . . . . - x = - .4166666666.... . . . . . .____ ____________. . . . . . . 99x = 41.25Divide each side of the equation by 9999x = 41.2599x/99 = 41.25/99x = 41.25/99Since you now know that x = 41.25/99, you only need to reduce that fraction to its simplest form, and you are finished:x = 41.25/99Multiply 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 = 4125/99004125 = 5 * 8259900 = 12 * 825Therefore:x = 4125/9900 = (5 * 825)/(12 * 825)Notice that the 825 in the numerator and the 825 in the denominator cancel one anotherx = (5)/(12)x = 5/12The final answer is:x = 5/12 = .4166666666...Thanks for writing.Staff www.solving-math-problems.com