Question

Which of the following rational numbers has terminating decimal expansion?

• A. \(\frac{7}{40}\)
• B. \(\frac{11}{35}\)
• C. \(\frac{5}{21}\)
• D. \(\frac{2}{15}\)

Answer 1

The Correct Option is A

For a rational number to have a terminating decimal expansion, its denominator, when reduced to its simplest form, must only have 2 and 5 as its prime factors.  
Now, let's check the prime factors of the denominators of each option:  

\(\frac{7}{40}\): The prime factorization of 40 is \(2^3 \times 5\). Since the denominator contains only 2 and 5, the decimal expansion will terminate.
\(\frac{11}{35}\): The prime factorization of 35 is \(5 \times 7\). Since it contains a factor of 7, its decimal expansion will not terminate.
\(\frac{5}{21}\): The prime factorization of 21 is \(3 \times 7\). Since it contains factors other than 2 and 5, its decimal expansion will not terminate.
\(\frac{2}{15}\): The prime factorization of 15 is \(3 \times 5\). Since it contains a factor of 3, its decimal expansion will not terminate.

So, the correct option is (A): \(\frac{7}{40}\)