Which of the following rational numbers can be expressed as a repeating decimal: 7/8 , 5/12

Question

, 17/25
, or 31/40
?(1 point)

Answer 1

To determine which of the given rational numbers can be expressed as a repeating decimal, we first need to consider the denominators of the fractions after they are simplified. A rational number can be expressed as a terminating decimal if the prime factorization of its denominator only contains the prime factors 2 and/or 5. If there are other prime factors, the decimal representation will be repeating.

Let's analyze each fraction:

1. 7/8:

   • The denominator (8) can be factored as \(2^3\).
   • Since the only prime factor is 2, \(7/8\) can be expressed as a terminating decimal.

2. 5/12:

   • The denominator (12) can be factored as \(2^2 \times 3\).
   • Since there is a prime factor of 3, \(5/12\) can be expressed as a repeating decimal.

3. 17/25:

   • The denominator (25) can be factored as \(5^2\).
   • Since the only prime factor is 5, \(17/25\) can be expressed as a terminating decimal.

4. 31/40:

   • The denominator (40) can be factored as \(2^3 \times 5\).
   • Since the only prime factors are 2 and 5, \(31/40\) can be expressed as a terminating decimal.

Based on this analysis, the only rational number from the list that can be expressed as a repeating decimal is:

5/12.