Find a rule to decide whether a given rational number has terminating or non-terminating decimal expansion by looking at its denominator.

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Solution

Any rational number with its denominator in the form of $2^{m} \times 5^{n}$ , where $m, n$ are positive integers have terminating decimal expansion.
Also, any rational number with its denominator in the form other than $2^{m} \times 5^{n}$ , where $m, n$ are positive integers have non-terminating decimals.
In this way, just by looking at the denominator of a rational number, we can say whether it will have terminating or non-terminating decimal expansions.

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