Advanced Fractions & Decimals

Method: Divide the numerator by the denominator using long division. Formula: $$\text{Decimal} = \frac{\text{numerator}}{\text{denominator}}$$

Example 1: $\frac{3}{8} = 3 \div 8 = 0.375$. This is a terminating decimal (ends after 3 digits).

Example 2: $\frac{1}{3} = 0.333\ldots = 0.\overline{3}$. This is a repeating decimal. The bar over 3 means it repeats forever.

Example 3: $\frac{5}{6} = 0.8333\ldots = 0.8\overline{3}$. Notice the 8 doesn't repeat, only the 3 repeats.

Tip: If the denominator is a power of 10, it's easy: $\frac{3}{10} = 0.3$, $\frac{47}{100} = 0.47$, $\frac{125}{1000} = 0.125$.

Terminating decimals: If the denominator's prime factors are only 2 and 5, the decimal terminates. Examples: $\frac{1}{2} = 0.5$, $\frac{3}{4} = 0.75$, $\frac{7}{20} = 0.35$.

Repeating decimals: If the denominator has other prime factors, the decimal repeats. Examples: $\frac{1}{3}$, $\frac{2}{7}$, $\frac{5}{9}$ all produce repeating decimals.

Long division steps: 1) Set up division, 2) Add decimal point and zeros, 3) Divide as usual, 4) Continue until pattern emerges or decimal terminates.