To determine which fraction has a terminating decimal representation, we need to examine the prime factors of the denominators of the given fractions.

Understanding Terminating Decimals

A rational number represented as a fraction $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$, and the fraction is in its simplest form) has a terminating decimal representation if and only if the prime factorization of the denominator $q$ contains only the prime numbers 2 and 5.

Analyzing Fraction Denominators

Let's analyze each option:

• Option 1: 3/8
  • The denominator is 8.
  • The prime factorization of 8 is $2^3$.
  • Since the only prime factor is 2, this fraction has a terminating decimal representation.
• Option 2: 1/6
  • The denominator is 6.
  • The prime factorization of 6 is $2 \times 3$.
  • The presence of the prime factor 3 (other than 2 or 5) means this fraction has a non-terminating (repeating) decimal representation.
• Option 3: 2/7
  • The denominator is 7.
  • The prime factorization of 7 is $7^1$.
  • The presence of the prime factor 7 means this fraction has a non-terminating (repeating) decimal representation.
• Option 4: 4/9
  • The denominator is 9.
  • The prime factorization of 9 is $3^2$.
  • The presence of the prime factor 3 means this fraction has a non-terminating (repeating) decimal representation.

Conclusion

Based on the analysis, only the fraction 3/8 has a denominator whose prime factorization consists solely of powers of 2. Therefore, 3/8 is the number with a terminating decimal representation.