To determine which fraction represents a terminating decimal, we need to examine the denominators of each fraction when they are in their simplest form. A fraction will represent a terminating decimal only if its denominator has no prime factors other than 2 or 5.

Let's analyze the given options:

1. $\frac{1}{3}$: The denominator is 3. Since 3 is a prime number and is not 2 or 5, $\frac{1}{3}$ is a repeating decimal.

2. $\frac{3}{7}$: The denominator is 7. Since 7 is a prime number and is not 2 or 5, $\frac{3}{7}$ is a repeating decimal.

3. $\frac{6}{19}$: The denominator is 19. Since 19 is a prime number and is not 2 or 5, $\frac{6}{19}$ is a repeating decimal.

4. $\frac{9}{8}$: The denominator is 8. Since 8 is $2^3$ (a power of 2), $\frac{9}{8}$ is a terminating decimal.

Thus, the correct answer is $\frac{9}{8}$, which represents a terminating decimal.

Would you like any further details or clarification?

Here are some related questions you might find interesting:

1. How do you convert a fraction to a decimal?
2. Why do only denominators with factors of 2 or 5 give terminating decimals?
3. What is a repeating decimal, and how is it expressed?
4. Can fractions with large denominators still represent terminating decimals?
5. How do you determine if a decimal is repeating or terminating from a division?

Tip: For any fraction, if its denominator only contains 2's and 5's in its prime factorization, it will represent a terminating decimal.