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 rational numbers can be expressed as a repeating decimal, we need to convert each fraction into decimal form.
7/8:
The denominator, 8, is not divisible by any prime number other than 2.
Therefore, this fraction can be expressed as a terminating decimal: 7/8 = 0.875.
5/12:
The denominator, 12, is divisible by both 2 and 3.
Since 5 and 12 are relatively prime, we expect a repeating decimal.
We perform long division to find 5/12 = 0.416666...
17/25:
The denominator, 25, is not divisible by any prime number other than 5.
Therefore, this fraction can be expressed as a terminating decimal: 17/25 = 0.68.
31/40:
The denominator, 40, is divisible by both 2 and 5.
Since 31 and 40 are relatively prime, we expect a repeating decimal.
We perform long division to find 31/40 = 0.775.
Thus, the rational numbers that can be expressed as a repeating decimal are 5/12 and 31/40.