Complex Fractions | Definition, Simplification & Examples | Study.com - Free Printable

Let's solve the expression step by step:

The given expression is:

$$
\frac{\frac{1}{2} + \frac{2}{3}}{\frac{3}{4} + \frac{5}{6}} \times \frac{12}{12}
$$

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Step 1: Simplify the numerator and denominator separately

#### Numerator: $\frac{1}{2} + \frac{2}{3}$

Find a common denominator. The least common denominator (LCD) of 2 and 3 is 6.

$$
\frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}
$$

$$
\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}
$$

---

#### Denominator: $\frac{3}{4} + \frac{5}{6}$

LCD of 4 and 6 is 12.

$$
\frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12}
$$

$$
\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}
$$

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Step 2: Now the fraction becomes:

$$
\frac{\frac{7}{6}}{\frac{19}{12}} \times \frac{12}{12}
$$

Dividing fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

So:

$$
\frac{7}{6} \div \frac{19}{12} = \frac{7}{6} \times \frac{12}{19}
$$

Now simplify:

$$
\frac{7}{6} \times \frac{12}{19} = \frac{7 \times 12}{6 \times 19} = \frac{84}{114}
$$

Simplify $\frac{84}{114}$:

Both divisible by 6:

$$
\frac{84 \div 6}{114 \div 6} = \frac{14}{19}
$$

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Step 3: Multiply by $\frac{12}{12}$

Note that:

$$
\frac{12}{12} = 1
$$

So multiplying by 1 doesn't change the value.

$$
\frac{14}{19} \times 1 = \frac{14}{19}
$$

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✔ Final Answer:

$$
\boxed{\frac{14}{19}}
$$

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Summary:

We simplified the complex fraction by:
- Adding the fractions in the numerator and denominator,
- Dividing the two results,
- Multiplying by 1 (since $\frac{12}{12} = 1$),
- And simplified to get the final answer.

✔ Answer: $\boxed{\frac{14}{19}}$