Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40

Question

?(1 point)

Answer 1

To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check their denominators when the fractions are in simplest form. A fraction expressed in simplest form will have a denominator that contains only the prime factors 2 and/or 5 if it can be expressed as a terminating decimal. If it has other prime factors, it will be a repeating decimal.

1. 7/8: The denominator is 8, which is \(2^3\). Since it only contains the prime factor 2, it is a terminating decimal.

2. 5/12: The denominator is 12, which factors into \(2^2 \times 3^1\). The presence of the factor 3 means this fraction is a repeating decimal.

3. 17/25: The denominator is 25, which is \(5^2\). Since it only contains the prime factor 5, it is a terminating decimal.

4. 31/40: The denominator is 40, which factors into \(2^3 \times 5^1\). It contains only the prime factors 2 and 5, so it is a terminating decimal.

From the analysis, the only fraction that can be expressed as a repeating decimal is 5/12.