Identifying Rational Numbers with Terminating Decimals

To solve this, we need to understand two conditions:

• Rational Number: A number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
• Terminating Decimal: A decimal number that ends after a finite number of digits. A rational number $\frac{p}{q}$ (in its simplest form) results in a terminating decimal if and only if the prime factors of the denominator $q$ are exclusively 2s and/or 5s.

Analyzing the Options

Let's examine each option based on these criteria:

• Option 1: $\frac{1}{3}$
  • Is it rational? Yes, it's in the form $\frac{p}{q}$.
  • Is it a terminating decimal? The denominator is $3$. The prime factor is $3$. Since $3$ is not $2$ or $5$, this fraction results in a repeating decimal ($0.333...$).
• Option 2: $\frac{4}{5}$
  • Is it rational? Yes, it's in the form $\frac{p}{q}$.
  • Is it a terminating decimal? The denominator is $5$. The only prime factor is $5$. Therefore, this fraction results in a terminating decimal ($0.8$).
• Option 3: $\sqrt{3}$
  • Is it rational? No. $\sqrt{3}$ is an irrational number, meaning it cannot be expressed as a simple fraction $\frac{p}{q}$. Its decimal representation is non-terminating and non-repeating.
• Option 4: $\pi$
  • Is it rational? No. $\pi$ is a famous irrational number. Its decimal representation is non-terminating and non-repeating.

Conclusion

Based on the analysis, only the number $\frac{4}{5}$ satisfies both conditions: it is a rational number and it results in a terminating decimal.