Question

Which of the following rational numbers can be expressed as a repeating decimal: 7/8 , 5/12 , 17/25 , or 31/40 ?(1 point)%0D%0A

Answer 1

To determine which of the given rational numbers can be expressed as a repeating decimal, we need to look at their denominators in simplest form and analyze their prime factors.

A rational number will have a terminating decimal if the only prime factors of its denominator (when the fraction is in simplest form) are 2 and/or 5. If there are any other prime factors, the decimal will be repeating.

Let's analyze each option:

1. **7/8**: The denominator is 8, which factors to \(2^3\). Since all the prime factors are 2, \(7/8\) has a terminating decimal.

2. **5/12**: The denominator is 12, which factors to \(2^2 \times 3\). Since there is a prime factor of 3, \(5/12\) has a repeating decimal.

3. **17/25**: The denominator is 25, which factors to \(5^2\). Since all the prime factors are 5, \(17/25\) has a terminating decimal.

4. **31/40**: The denominator is 40, which factors to \(2^3 \times 5\). Since all the prime factors are 2 and 5, \(31/40\) has a terminating decimal.

In conclusion, the only fraction among the options that can be expressed as a repeating decimal is **5/12**.