Question

How do you simplify `\frac{((a+b)\div a)-\frac{b}{a+b}}{a+b}+\frac{\frac{b}{a}-\frac{a-b}{a+b}}{a-b}`?

Answer 1

`-b/(a(a-b))`

(Get ready! It's a long one.)

Explanation:

Rewrite the expression.

`(((a+b)/a)-b/(a+b))/(a+b)+(b/a-(a-b)/(a+b))/(a-b)`

Point your focus to the fractions in the numerators. Add each set of fractions by multiplying by either `a` or `a+b`.

`((a+b)^2/(a(a+b))-(b(a+b))/(a(a+b)))/(a+b)+((b(a+b))/(a(a+b))-(a(a-b))/(a(a+b)))/(a-b)`

`(((a+b)^2-b(a+b))/(a(a+b)))/(a+b)+((b(a+b)-a(a-b))/(a(a+b)))/(a-b)`

Simplify the complex fractions by multiplying by `1/(a+b)` or `1/(a-b)`.

`(1/(a+b))(((a+b)^2-b(a+b))/(a(a+b)))+(1/(a-b))((b(a+b)-a(a-b))/(a(a+b)))`

`((a+b)^2-b(a+b))/(a(a+b)^2)+(b(a+b)-a(a-b))/(a(a+b)(a-b))`

Simplify.

`(a+b-b)/(a(a+b))+(b(a+b)-a(a-b))/(a(a+b)(a-b))`

`a/(a(a+b))+(b(a+b)-a(a-b))/(a(a+b)(a-b))`

Multiply `a/(a(a+b))` by `(a-b)/(a-b)` to combine the fractions.

`(a(a-b)-b(a+b)-a(a-b))/(a(a+b)(a-b))`

Simplify.

`(-b(a+b))/(a(a+b)(a-b))`

`-b/(a(a-b))`