Question

Among \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\) the non-terminating decimal is

• A. \(\frac{1}{2}\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{4}\)
• D. \(\frac{1}{5}\)

Answer 1

The Correct Option is B

Step 1: A fraction \( \dfrac{p}{q} \) (in lowest terms) has a terminating decimal only if the prime factorization of the denominator \( q \) contains no primes other than 2 or 5.

Step 2: Analyze each option:

\( \dfrac{1}{2} \): Denominator is 2 ⇒ Prime factor is 2 ⇒ Terminating

\( \dfrac{1}{3} \): Denominator is 3 ⇒ Prime factor is 3 ⇒ Non-terminating

\( \dfrac{1}{4} \): Denominator is 4 = \( 2^2 \) ⇒ Prime factor is 2 ⇒ Terminating

\( \dfrac{1}{5} \): Denominator is 5 ⇒ Prime factor is 5 ⇒ Terminating

Step 3: Only \( \dfrac{1}{3} \) contains a prime factor other than 2 or 5, so it is a non-terminating decimal.

Step 4: Decimal form of \( \dfrac{1}{3} \) is 0.333... or \( 0.\overline{3} \), which is non-terminating repeating.

The correct option is (B): \(\frac{1}{3}\)