Question

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

Answer 1

A rational number can be expressed as a repeating decimal if its denominator (in lowest terms) has prime factors other than 2 and 5.

Let's analyze each of the given fractions:

1. **7/8**: The denominator is 8, which is \(2^3\). This means it can be expressed as a terminating decimal.

2. **5/12**: The denominator is 12, which factors as \(2^2 \times 3\). The presence of 3 means it cannot be expressed as a terminating decimal, so it is a repeating decimal.

3. **17/25**: The denominator is 25, which is \(5^2\). This means it can be expressed as a terminating decimal.

4. **31/40**: The denominator is 40, which factors as \(2^3 \times 5\). This means it can also be expressed as a terminating decimal.

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