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

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Answer 1 · 2017-11-14T17:03:51

$-b/(a(a-b))$

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

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.

$(-b(a+b))/(a(a+b)(a-b))$

$-b/(a(a-b))$

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}\frac{((a+b)\div a)-\frac{b}{a+b}}{a+b}+\frac{\frac{b}{a}-\frac{a-b}{a+b}}{a-b}?