Without performing long division, state whether $\frac{13}{3125}$ has a terminating or a non-terminating repeating decimal expansion. Write the expansion if it terminates.

easy 2 marks Class 10 · Mathematics · Real Numbers

Answer

It terminates.

\frac{13}{3125} = 0.00416

The correct answer is It terminates. $\frac{13}{3125} = 0.00416$ — $\frac{13}{3125} = \frac{13 \times 32}{5 ^{5} \times 2 ^{5}} = \frac{416}{100000} = 0.00416$

Step-by-step solution

A fraction in its lowest terms has a terminating decimal expansion exactly when its denominator can be written in the form $2^{n} 5^{m}$ , where $n$ and $m$ are whole numbers.
Factorise the denominator: $3125 = 5^{5}$ .
This is of the form $2^{0} \times 5^{5}$ , so the decimal expansion terminates.
To write it out, turn the denominator into a power of ten by multiplying numerator and denominator by $2^{5} = 32$ .
$\frac{13}{3125} = \frac{13 \times 32}{5 ^{5} \times 2 ^{5}} = \frac{416}{100000} = 0.00416$

Common mistake: Checking the denominator before reducing the fraction to its lowest terms. The test only applies once every common factor has been cancelled.

Concept tested: Condition for a terminating decimal expansion

Published 19 August 2026 · How this solution was prepared and checked

← All Real Numbers questions

Without performing long division, state whether 312513​ has a terminating or a non-terminating repeating decimal expansion. Write the expansion if it terminates.

Step-by-step solution

Without performing long division, state whether $\frac{13}{3125}$ has a terminating or a non-terminating repeating decimal expansion. Write the expansion if it terminates.