Solve 7/3 / [(3 1/9) / \{1 3/4 - 1 5/6 + (1 2/3 - 3/4 - 1/6)\}]

Divide $\frac{7}{3}$ by $\frac{\frac{3\times 9+1}{9}}{\frac{1\times 4+3}{4}-\frac{1\times 6+5}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}}$ by multiplying $\frac{7}{3}$ by the reciprocal of $\frac{\frac{3\times 9+1}{9}}{\frac{1\times 4+3}{4}-\frac{1\times 6+5}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}}$.

$$\frac{7\left(\frac{1\times 4+3}{4}-\frac{1\times 6+5}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Multiply $1$ and $4$ to get $4$.

$$\frac{7\left(\frac{4+3}{4}-\frac{1\times 6+5}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Add $4$ and $3$ to get $7$.

$$\frac{7\left(\frac{7}{4}-\frac{1\times 6+5}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Multiply $1$ and $6$ to get $6$.

$$\frac{7\left(\frac{7}{4}-\frac{6+5}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Add $6$ and $5$ to get $11$.

$$\frac{7\left(\frac{7}{4}-\frac{11}{6}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Least common multiple of $4$ and $6$ is $12$. Convert $\frac{7}{4}$ and $\frac{11}{6}$ to fractions with denominator $12$.

$$\frac{7\left(\frac{21}{12}-\frac{22}{12}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Since $\frac{21}{12}$ and $\frac{22}{12}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(\frac{21-22}{12}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Subtract $22$ from $21$ to get $-1$.

$$\frac{7\left(-\frac{1}{12}+\frac{1\times 3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Multiply $1$ and $3$ to get $3$.

$$\frac{7\left(-\frac{1}{12}+\frac{3+2}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Add $3$ and $2$ to get $5$.

$$\frac{7\left(-\frac{1}{12}+\frac{5}{3}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Least common multiple of $12$ and $3$ is $12$. Convert $-\frac{1}{12}$ and $\frac{5}{3}$ to fractions with denominator $12$.

$$\frac{7\left(-\frac{1}{12}+\frac{20}{12}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Since $-\frac{1}{12}$ and $\frac{20}{12}$ have the same denominator, add them by adding their numerators.

$$\frac{7\left(\frac{-1+20}{12}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Add $-1$ and $20$ to get $19$.

$$\frac{7\left(\frac{19}{12}-\frac{3}{4}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Least common multiple of $12$ and $4$ is $12$. Convert $\frac{19}{12}$ and $\frac{3}{4}$ to fractions with denominator $12$.

$$\frac{7\left(\frac{19}{12}-\frac{9}{12}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Since $\frac{19}{12}$ and $\frac{9}{12}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(\frac{19-9}{12}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Subtract $9$ from $19$ to get $10$.

$$\frac{7\left(\frac{10}{12}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Reduce the fraction $\frac{10}{12}$ to lowest terms by extracting and canceling out $2$.

$$\frac{7\left(\frac{5}{6}-\frac{1}{6}\right)}{3\times \frac{3\times 9+1}{9}}$$

Since $\frac{5}{6}$ and $\frac{1}{6}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\times \frac{5-1}{6}}{3\times \frac{3\times 9+1}{9}}$$

Subtract $1$ from $5$ to get $4$.

$$\frac{7\times \frac{4}{6}}{3\times \frac{3\times 9+1}{9}}$$

Reduce the fraction $\frac{4}{6}$ to lowest terms by extracting and canceling out $2$.

$$\frac{7\times \frac{2}{3}}{3\times \frac{3\times 9+1}{9}}$$

Express $7\times \frac{2}{3}$ as a single fraction.

$$\frac{\frac{7\times 2}{3}}{3\times \frac{3\times 9+1}{9}}$$

Multiply $7$ and $2$ to get $14$.

$$\frac{\frac{14}{3}}{3\times \frac{3\times 9+1}{9}}$$

Multiply $3$ and $9$ to get $27$.

$$\frac{\frac{14}{3}}{3\times \frac{27+1}{9}}$$

Add $27$ and $1$ to get $28$.

$$\frac{\frac{14}{3}}{3\times \frac{28}{9}}$$

Express $3\times \frac{28}{9}$ as a single fraction.

$$\frac{\frac{14}{3}}{\frac{3\times 28}{9}}$$

Multiply $3$ and $28$ to get $84$.

$$\frac{\frac{14}{3}}{\frac{84}{9}}$$

Reduce the fraction $\frac{84}{9}$ to lowest terms by extracting and canceling out $3$.

$$\frac{\frac{14}{3}}{\frac{28}{3}}$$

Divide $\frac{14}{3}$ by $\frac{28}{3}$ by multiplying $\frac{14}{3}$ by the reciprocal of $\frac{28}{3}$.

$$\frac{14}{3}\times \frac{3}{28}$$

Multiply $\frac{14}{3}$ times $\frac{3}{28}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{14\times 3}{3\times 28}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{14}{28}$$

Reduce the fraction $\frac{14}{28}$ to lowest terms by extracting and canceling out $14$.

$$\frac{1}{2}$$

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