Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check their denominators.
For a rational number to have a repeating decimal representation, its denominator must contain at least one prime factor that is not a factor of 10.
The prime factorization of the denominators are as follows:
7/8: The denominator, 8, has a prime factorization of $2^3$.
5/12: The denominator, 12, has a prime factorization of $2^2 \cdot 3$.
17/25: The denominator, 25, has a prime factorization of $5^2$.
31/40: The denominator, 40, has a prime factorization of $2^3 \cdot 5$.
Among the given rational numbers, only 5/12 and 31/40 have denominators that contain prime factors other than 2 or 5. Therefore, 5/12 and 31/40 can be expressed as repeating decimals.
To determine which of the rational numbers can be expressed as a repeating decimal, we need to consider their decimal representations. To do this, we divide the numerator by the denominator for each fraction.
Let's go through the options:
1. For 7/8: Divide 7 by 8.
7 ÷ 8 = 0.875
The decimal representation terminates, so it is not a repeating decimal.
2. For 5/12: Divide 5 by 12.
5 ÷ 12 = 0.416666...
The decimal representation repeats, so it is a repeating decimal.
3. For 17/25: Divide 17 by 25.
17 ÷ 25 = 0.68
The decimal representation terminates, so it is not a repeating decimal.
4. For 31/40: Divide 31 by 40.
31 ÷ 40 = 0.775
The decimal representation terminates, so it is not a repeating decimal.
Therefore, out of the options given, only 5/12 can be expressed as a repeating decimal.