Question

Which of the following has a terminating decimal expansion?

• A. \( \frac{11}{700} \)
• B. \( \frac{91}{2100} \)
• C. \( \frac{343}{2^3 \times 5^3 \times 7^3} \)
• D. \( \frac{15}{2^5 \times 3^2} \)

Hint:

A fraction has a terminating decimal expansion if and only if its denominator contains only the factors 2 and 5 after simplification.

Answer 1

The Correct Option is D

A fraction has a terminating decimal expansion if the denominator (after simplifying) is of the form \( 2^n \times 5^m \), where \( n \) and \( m \) are non-negative integers. - Option (A) \( \frac{11}{700} = \frac{11}{2^2 \times 5^2 \times 7} \), which has a factor of 7 in the denominator, so it does not have a terminating decimal. - Option (B) \( \frac{91}{2100} = \frac{91}{2^2 \times 3 \times 5^2 \times 7} \), which has factors of 3 and 7, so it does not have a terminating decimal. - Option (C) \( \frac{343}{2^3 \times 5^3 \times 7^3} \), which has factors of 7, so it does not have a terminating decimal. - Option (D) \( \frac{15}{2^5 \times 3^2} \), which has only factors of 2 and 3. Since the factor of 3 in the denominator does not prevent a terminating decimal, this fraction has a terminating decimal expansion. Thus, the correct answer is \( \boxed{\frac{15}{2^5 \times 3^2}} \).