Understanding Terminating Decimals from Fractions

A terminating decimal is a decimal that ends after a finite number of digits. When a fraction is converted to a decimal, it will be a terminating decimal if and only if the prime factorization of the denominator (in its simplest form) contains only the prime numbers 2 and 5.

Analyzing Each Fraction Option

Let's examine the denominator of each given fraction to determine its prime factors. Remember, the fraction must be in its simplest form first, but in this case, all the given fractions are already in their simplest form.

1. Fraction: $\frac{5}{8}$
   Denominator: 8
   Prime factorization of 8: $8 = 2 \times 2 \times 2 = 2^3$
   The prime factors of the denominator are only 2. Since the prime factors are only 2 and 5 (in this case, just 2), this fraction will result in a terminating decimal.
2. Fraction: $\frac{1}{3}$
   Denominator: 3
   Prime factorization of 3: $3 = 3^1$
   The prime factor of the denominator is 3. Since 3 is not 2 or 5, this fraction will result in a non-terminating, repeating decimal.
3. Fraction: $\frac{1}{7}$
   Denominator: 7
   Prime factorization of 7: $7 = 7^1$
   The prime factor of the denominator is 7. Since 7 is not 2 or 5, this fraction will result in a non-terminating, repeating decimal.
4. Fraction: $\frac{2}{11}$
   Denominator: 11
   Prime factorization of 11: $11 = 11^1$
   The prime factor of the denominator is 11. Since 11 is not 2 or 5, this fraction will result in a non-terminating, repeating decimal.

Based on the analysis of the denominators' prime factors, only the fraction $\frac{5}{8}$ has a denominator whose prime factors are exclusively 2 and 5.

Conclusion on Terminating Decimals

A fraction $\frac{a}{b}$ (in simplest form) is a terminating decimal if the prime factorization of $b$ is of the form $2^m 5^n$, where $m$ and $n$ are non-negative integers. If the prime factorization of $b$ contains any prime factor other than 2 or 5, the decimal representation will be non-terminating and repeating.

Comparing this rule to the given options:

The fraction that results in a terminating decimal is $\frac{5}{8}$.

Revision Table: Fractions and Decimal Types

Additional Information: Converting Terminating Decimals

To convert a terminating decimal to a fraction, write the decimal as a fraction with a denominator that is a power of 10 (10, 100, 1000, etc.), and then simplify the fraction. For example, $0.75 = \frac{75}{100} = \frac{3}{4}$.

To convert a fraction that results in a terminating decimal, divide the numerator by the denominator. For $\frac{5}{8}$, we perform the division:

$5 \div 8 = 0.625$

This is clearly a terminating decimal.