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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


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 Marcusâ?? calculation is incorrect?

Solution Preview

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 ...

Solution Summary

In these problems we state which fractions yield repeating and terminating decimal expansions as well as which yield purely repeating decimals, i.e. with the period of repetition starting immediately after the decimal point.