Question

Which of the following has a terminating decimal expansion?

Options

• `32/91`

• `19/80`

• `23/45`

• `25/42`

Answer 1

`bb(19/80)`

Explanation:

A rational number `p/q` in lowest terms has a terminating decimal expansion exactly when q’s prime factors are only 2 and/or 5 (q = 2^m × 5ⁿ).

Check the options (already in simplest form):

`32/91`, denominator 91 = 7 × 13 (contains primes other than 2 or 5) → nonterminating.

`19/80`, denominator 80 = 2⁴ × 5 → terminating. (correct)

`23/45`, denominator 45 = 3² × 5 (contains 3) → nonterminating.

`25/42`, denominator 42 = 2 × 3 × 7 (contains 3 and 7) → nonterminating.