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.

`47/225`

Answer 1

To determine whether the rational number `47/225` has a terminating decimal expansion or a non-terminating recurring decimal expansion, we need to analyse the denominator after the fraction is in simplest form.

A rational number `p/q` in simplest form will have a terminating decimal expansion if and only if the denominator q is of the form 2^m × 5ⁿ, where m and n are non-negative integers.

Now, consider the denominator:

225 = 15²

= 3 × 5² 

= 3² × 5²

Because the denominator contains a prime factor 3 other than 2 and 5, the decimal expansion of `47/225` will be non-terminating and recurring.

Thus, the decimal expansion of `47/225` is non-terminating recurring.