1Cademy - Simplifying $$\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}$$

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Procedure to Simplify a Complex Rational Expression by Writing it as Division

Example

Simplifying $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}$

Simplify the complex fraction:

$\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}$

Because fraction bars act as grouping symbols, the numerator and denominator expressions must each be fully simplified before the overall division is performed.

Step 1 — Simplify the numerator (LCD = 6). Rewrite each fraction with denominator $6$ : $\frac{1}{3} = \frac{2}{6}$ and $\frac{1}{6}$ already has denominator $6$ . Add: $\frac{2}{6} + \frac{1}{6} = \frac{3}{6}$ .

Step 2 — Simplify the denominator (LCD = 6). Rewrite each fraction with denominator $6$ : $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$ . Subtract: $\frac{3}{6} - \frac{2}{6} = \frac{1}{6}$ .

Step 3 — Rewrite as a division problem: $\frac{3}{6} \div \frac{1}{6}$ .

Step 4 — Multiply the first fraction by the reciprocal of the second: $\frac{3}{6} \cdot \frac{6}{1}$ .

Step 5 — Simplify. The $6$ in the denominator and the $6$ in the numerator cancel, leaving $3$ .

The result is $3$ . This example illustrates the importance of treating fraction bars as grouping symbols: the addition in the numerator and the subtraction in the denominator must each be completed before the complex fraction can be rewritten as a division problem.

Updated 2026-04-21

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OpenStax Elementary Algebra

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