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.

`13/350`

Answer 1

To determine whether the rational number `13/350` has a terminating decimal expansion or a non-terminating recurring decimal expansion, we look at the prime factorization of the denominator (in simplest form).

• The denominator is 350.
• Prime factorize: 350 = 2 × 5² × 7.

A rational number has a terminating decimal expansion if and only if the denominator after simplification is of the form 2^m × 5ⁿ, where m and n are non-negative integers.

Since the denominator 350 contains a prime factor 7 besides 2 and 5, the decimal expansion of `13/350` will be non-terminating and recurring.

Hence, `13/350` has a non-terminating recurring decimal expansion.