Question

Which of the following rational numbers has terminating decimal expansion?

• A. \(\frac{37}{3^2\times5}\)
• B. \(\frac{21}{2^3\times5^2}\)
• C. \(\frac{17}{7^2}\)
• D. \(\frac{89}{2^2\times3^2}\)

Answer 1

The Correct Option is B

Step 1: Recall the condition for a rational number to have a terminating decimal expansion.

A rational number \( \frac{p}{q} \) (in its simplest form) has a terminating decimal expansion if and only if the denominator \( q \) has no prime factors other than 2 or 5.

Step 2: Analyze each option.

• (1) \( \frac{37}{3^2 \times 5} \): The denominator is \( 3^2 \times 5 \), which includes the prime factor 3. Hence, this does not have a terminating decimal expansion.
• (2) \( \frac{21}{2^3 \times 5^2} \): The denominator is \( 2^3 \times 5^2 \), which contains only the prime factors 2 and 5. Hence, this has a terminating decimal expansion.
• (3) \( \frac{17}{7^2} \): The denominator is \( 7^2 \), which includes the prime factor 7. Hence, this does not have a terminating decimal expansion.
• (4) \( \frac{89}{2^2 \times 3^2} \): The denominator is \( 2^2 \times 3^2 \), which includes the prime factor 3. Hence, this does not have a terminating decimal expansion.

Final Answer: The rational number with a terminating decimal expansion is \( \mathbf{\frac{21}{2^3 \times 5^2}} \), which corresponds to option \( \mathbf{(2)} \).