Question

Assertion (A): \[\frac{31}{90}\] is a nonterminating, repeating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a repeating decimal, if q has at least one factor other than 2 and 5.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. The fraction is in lowest terms and \[90 = 2 \times 3^{2} \times 5\] has a factor other than 2 and 5. Therefore, its decimal expansion is nonterminating and repeating.

Step 2 – Reason: R is true. A rational number whose denominator has a factor other than 2 or 5 has a nonterminating repeating decimal expansion.

Step 3 – Link: Since the denominator 90 contains the factor 3, R correctly explains A.