# How do you simplify 7 1/2 - (1/9 + 2/3) div 3/2?

May 19, 2018

$7 \frac{1}{2} - \left(\frac{1}{9} + \frac{2}{3}\right) \div \frac{3}{2} = \frac{377}{54}$

#### Explanation:

Simplify:

$7 \frac{1}{2} - \left(\frac{1}{9} + \frac{2}{3}\right) \div \frac{3}{2}$

Follow the order of operations as indicated by the acronym PEMDAS:

Parentheses/brackets
Multiplication and Division in order from left to right.
Addition and Subtraction in order from left to right.

Simplify the parentheses.

$\left(\frac{1}{9} + \frac{2}{3}\right)$

In order to add or subtract fractions, they must have the same denominator, called the least common denominator (LCD).

The LCD is $9$. Multiply $\frac{2}{3}$ by $\frac{3}{3}$ to get an equivalent fraction with a denominator of $9$. Since $\frac{3}{3} = 1$, the numbers will change, but the value remains the same.

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

$\frac{1}{9} + \frac{6}{9} =$

$\frac{7}{9}$

Rewrite the expression.

$7 \frac{1}{2} - \frac{7}{9} \div \frac{3}{2}$

Carry out the division next. Since we are dividing by a fraction, we invert it and multiply.

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

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

$7 \frac{1}{2} - \frac{14}{27}$

Convert $7 \frac{1}{2}$ to an improper fraction by multiplying the denominator by the whole number and adding the numerator. Place the result over the denominator of $2$.

$\frac{\left(2 \times 7 + 1\right)}{2} - \frac{14}{27}$

Simplify.

$\frac{15}{2} - \frac{14}{27}$

The denominators $2$ and $27$ do not have a common multiple, so to get the LCD, we multiply the denominators.

LCD$=$$2 \times 27 = 54$

Multiply $\frac{15}{2}$ by $\frac{27}{27}$, and multiply $\frac{14}{27}$ by $\frac{2}{2}$ to get equivalent fractions with $54$ as the denominator.

$\frac{15}{2} \times \frac{27}{27} - \frac{14}{27} \times \frac{2}{2}$

$\frac{15 \times 27}{2 \times 27} - \frac{14 \times 2}{27 \times 2}$

Simplify.

$\frac{405}{54} - \frac{28}{54} = \frac{377}{54}$