# Characterization of Repeating and Terminating Decimals

a. Which fractions a/b have terminating decimals?

b. Which fractions a/b have repeating decimals?

c. Sometimes (for example 1/12 ) only part of the decimal repeats while in other cases (for example 1/7 ) the whole decimal repeats. Assuming a/b is a reduced fraction, how can you tell which of these two scenarios will occur? Explain.

*d.* Marcus says he computed and found

0.0588235294

Josie looks at this answer for a second or two and tells Marcus that he is mistaken. Josie did NOT calculate any of the numbers in this decimal. How might Josie have known MarcusÃ¢?? calculation is incorrect?

Â© BrainMass Inc. brainmass.com December 24, 2021, 9:56 pm ad1c9bdddfhttps://brainmass.com/math/fractions-and-percentages/characterization-repeating-terminating-decimals-426904

## SOLUTION This solution is **FREE** courtesy of BrainMass!

a. Fractions with terminating decimals are fractions in which the only primes dividing the denominators (when reduced) are 2 and 5.

b. All other fractions have repeating decimals.

c. The whole decimal repeats precisely when the denominator is a factor of 10^n-1, where n is the period of repetition. In particular, if the denominator is even (as it is for 1/12), only part of the decimal repeats.

d. I don't understand what this question is asking for. What is Marcus computing? If he is computing the decimal expansion of 1/17, he is correct up to the number of places he has calculated. However, if this is his final answer, the fraction must have been 294117647/5000000000, which is unlikely.

Â© BrainMass Inc. brainmass.com December 24, 2021, 9:56 pm ad1c9bdddf>https://brainmass.com/math/fractions-and-percentages/characterization-repeating-terminating-decimals-426904