Question

Assertion (A): \[\frac{19}{500}\] is a terminating decimal.

Reason (R): The rational number \[\frac{p}{q}\] is a terminating decimal, if \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

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. \[500 = 2^{2} \times 5^{3}\] and thus \[\frac{19}{500}\] has a denominator whose prime factors are only 2 and 5, so its decimal expansion terminates.

Step 2 – Reason: R is true. It gives the criterion that a rational number has a terminating decimal when its denominator has the form \[q = (2^{m} \times 5^{n})\] for some whole numbers m and n.

Step 3 – Link: The denominator 500 satisfies this criterion, so R correctly explains A.