Evaluate: (/(7) 12+/(9) 16-1)-(/(51) 24-2)

Expression: $$( \frac { 7 } { 12 } + \frac { 9 } { 16 } - 1 ) - ( \frac { 51 } { 24 } - 2 )$$

Least common multiple of $12$ and $16$ is $48$. Convert $\frac{7}{12}$ and $\frac{9}{16}$ to fractions with denominator $48$.

$$\frac{28}{48}+\frac{27}{48}-1-\left(\frac{51}{24}-2\right)$$

Since $\frac{28}{48}$ and $\frac{27}{48}$ have the same denominator, add them by adding their numerators.

$$\frac{28+27}{48}-1-\left(\frac{51}{24}-2\right)$$

Add $28$ and $27$ to get $55$.

$$\frac{55}{48}-1-\left(\frac{51}{24}-2\right)$$

Convert $1$ to fraction $\frac{48}{48}$.

$$\frac{55}{48}-\frac{48}{48}-\left(\frac{51}{24}-2\right)$$

Since $\frac{55}{48}$ and $\frac{48}{48}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{55-48}{48}-\left(\frac{51}{24}-2\right)$$

Subtract $48$ from $55$ to get $7$.

$$\frac{7}{48}-\left(\frac{51}{24}-2\right)$$

Reduce the fraction $\frac{51}{24}$ to lowest terms by extracting and canceling out $3$.

$$\frac{7}{48}-\left(\frac{17}{8}-2\right)$$

Convert $2$ to fraction $\frac{16}{8}$.

$$\frac{7}{48}-\left(\frac{17}{8}-\frac{16}{8}\right)$$

Since $\frac{17}{8}$ and $\frac{16}{8}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{7}{48}-\frac{17-16}{8}$$

Subtract $16$ from $17$ to get $1$.

$$\frac{7}{48}-\frac{1}{8}$$

Least common multiple of $48$ and $8$ is $48$. Convert $\frac{7}{48}$ and $\frac{1}{8}$ to fractions with denominator $48$.

$$\frac{7}{48}-\frac{6}{48}$$

Since $\frac{7}{48}$ and $\frac{6}{48}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{7-6}{48}$$

Subtract $6$ from $7$ to get $1$.

$$\frac{1}{48}$$