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.

`11/48`

Answer 1

Given: The rational number is `11/48`.

Stepwise calculation:

1. Factorize the denominator 48 into its prime factors 48 = 2⁴ × 3.

2. For a rational number `p/q` to have a terminating decimal expansion, the denominator q in simplest form must only have the prime factors 2 and/or 5.

3. Since 48 has a prime factor of 3 in addition to 2, it is not solely composed of 2s and/or 5s.

4. Therefore, `11/48` will have a non-terminating recurring decimal expansion.

`11/48` has a non-terminating recurring decimal expansion without performing long division.