Question

Which of the following fractions has terminating decimal expansion?

• A. \(\frac{14}{2^0 \times 3^2}\)
• B. \(\frac{9}{5^1 \times 7^2}\)
• C. \(\frac{8}{2^2 \times 3^2}\)
• D. \(\frac{15}{2^2 \times 5^3}\)

Hint:

Always simplify the fraction first before checking the prime factors of the denominator. Sometimes, prime factors other than 2 or 5 in the denominator might cancel out with factors in the numerator.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A rational number (a fraction) has a terminating decimal expansion if and only if its denominator, in the simplest form, can be expressed as a product of powers of 2 and 5 only (i.e., in the form \(2^n 5^m\), where n and m are non-negative integers).

Step 2: Detailed Explanation:
Let's analyze each option after simplifying the fraction:
(A) \(\frac{14}{2^0 \times 3^2} = \frac{14}{1 \times 9} = \frac{14}{9}\). The denominator has a factor of 3. Not terminating.
(B) \(\frac{9}{5^1 \times 7^2} = \frac{9}{5 \times 49}\). The denominator has a factor of 7. Not terminating.
(C) \(\frac{8}{2^2 \times 3^2} = \frac{8}{4 \times 9} = \frac{2}{9}\). The denominator has a factor of 3. Not terminating.
(D) \(\frac{15}{2^2 \times 5^3} = \frac{3 \times 5}{2^2 \times 5^3} = \frac{3}{2^2 \times 5^2}\). The denominator is in the form \(2^n 5^m\). Therefore, this fraction has a terminating decimal expansion.

Step 3: Final Answer:
The fraction with a terminating decimal expansion is \(\frac{15}{2^2 \times 5^3}\).