Question

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`3/8`

Answer 1

Given: The rational number is `3/8`.

Stepwise calculation: To determine if the decimal expansion of `3/8` terminates or recurs, consider the denominator in its simplest form.

The denominator is 8, which can be factorised into prime factors as 8 = 2³.

Since the denominator 8 is of the form 2^m × 5ⁿ where m = 3 and n = 0, it fits the condition for a terminating decimal expansion.

A rational number `p/q` in lowest terms has a terminating decimal if and only if the prime factorisation of q involves only 2s and/or 5s.

Therefore, `3/8` will have a terminating decimal expansion, without performing the long division.