Question

Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/30

Answer 1

To determine which of the given rational numbers can be expressed as a repeating decimal, we need to look at their denominators when expressed in simplest form. A rational number can be expressed as a terminating decimal if its denominator (in simplest form) has only the prime factors 2 and/or 5. If there are any other prime factors, the decimal will be repeating.

Let's check the denominators:

1. **\(7/8\)**: The denominator is 8, which is \(2^3\). Since it only has the prime factor 2, \(7/8\) can be expressed as a terminating decimal.

2. **\(5/12\)**: The denominator is 12, which factors into \(2^2 \times 3\). Since it has the prime factor 3, \(5/12\) cannot be expressed as a terminating decimal and will be a repeating decimal.

3. **\(17/25\)**: The denominator is 25, which is \(5^2\). Since it only has the prime factor 5, \(17/25\) can be expressed as a terminating decimal.

4. **\(31/30\)**: The denominator is 30, which factors into \(2 \times 3 \times 5\). Since it has the prime factor 3, \(31/30\) cannot be expressed as a terminating decimal and will be a repeating decimal.

Therefore, the rational numbers that can be expressed as repeating decimals are \(5/12\) and \(31/30\).