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

Simplify:

`7 1/2-(1/9+2/3)-:3/2`

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

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

Simplify the parentheses.

`(1/9+2/3)`

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

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

`1/9+2/3xx3/3=`

`1/9+6/9=`

`7/9`

Rewrite the expression.

`7 1/2-7/9-:3/2`

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

`7 1/2-7/9xx2/3`

`7 1/2-(7xx2)/(9xx3)`

`7 1/2-14/27`

Convert `7 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`.

`((2xx7+1))/2-14/27`

Simplify.

`15/2-14/27`

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

LCD`=``2xx27=54`

Multiply `15/2` by `27/27`, and multiply `14/27` by `2/2` to get equivalent fractions with `54` as the denominator.

`15/2xx27/27-14/27xx2/2`

`(15xx27)/(2xx27)-(14xx2)/(27xx2)`

Simplify.

`405/54-28/54=377/54`