Question

Identify a non-terminating repeating decimal.

• A. $\frac{24}{1600}$
• B. $\frac{171}{800}$
• C. $\frac{123}{2^2\times 5^3}$
• D. $\frac{145}{2^3\times 5^2\times 7^2}$

Answer 1

The correct option is D $\frac{145}{2^3\times 5^2\times 7^2}$

We know that, a fraction will have terminating decimal expansion only if its denominator is in the form of $2^m\times 5^n$.

Let's check the denominator of each option:

$\left(i\right)\frac{24}{1600}$

$1600=2\times 2\times 2\times 2\times 2\times 2\times 5\times 5=2^6\times 5^2$

Hence, the denominator is $2^6\times 5^2$

$\left(ii\right)\frac{171}{800}$

$800=2\times 2\times 2\times 2\times 2\times 5\times 5=2^5\times 5^2$

Hence, the denominator is $2^5\times 5^2$

$\left(iii\right)\frac{123}{2^3\times 5^3}$

Hence, the denominator is $2^3\times 5^3$

$\left(iv\right)\frac{145}{2^3\times 5^2\times 7^2}$

Hence, the denominator is $2^3\times 5^2\times 7^2$

Only option D does not have a denominator of the form $2^m\times 5^n$.

Hence, it will have a non-terminating decimal expansion. Further, the expansion will be repeating.

Hence, option D is correct.