Which fraction has a repeating decimal as its decimal expansion?
3/25
3/16
3/11
3/8
The Correct Answer and Explanation is :
Among the given fractions, the one with a repeating decimal expansion is 3/11.
Explanation:
Decimal Expansions of Fractions:
When dividing two integers to convert a fraction into a decimal, you can either get a terminating or a repeating decimal. A terminating decimal occurs when the division results in a finite number of digits after the decimal point. A repeating decimal, on the other hand, occurs when the decimal digits continue infinitely in a repeating pattern.
The type of decimal expansion depends on the denominator of the fraction when it is in its simplest form. Specifically:
- A terminating decimal occurs when the denominator, after removing any common factors with the numerator, has only the prime factors 2 or 5. This is because powers of 2 or 5 divide evenly into powers of 10, which makes the division terminate.
- A repeating decimal occurs when the denominator has prime factors other than 2 or 5, as these do not divide evenly into powers of 10. As a result, the decimal repeats.
Check each fraction:
- 3/25:
- 25 = 5² (denominator has only the prime factor 5).
- Decimal expansion is terminating: 3 ÷ 25 = 0.12.
- 3/16:
- 16 = 2⁴ (denominator has only the prime factor 2).
- Decimal expansion is terminating: 3 ÷ 16 = 0.1875.
- 3/11:
- 11 is a prime number and does not divide powers of 10 evenly.
- Decimal expansion is repeating: 3 ÷ 11 = 0.272727…, with “27” repeating.
- 3/8:
- 8 = 2³ (denominator has only the prime factor 2).
- Decimal expansion is terminating: 3 ÷ 8 = 0.375.
Conclusion:
The fraction 3/11 has a repeating decimal because its denominator contains a prime factor (11) other than 2 or 5, resulting in the repeating decimal 0.272727…