Simplify (a-2b)/(3b)-(b-2a)/(3a)

$\frac{a - 2 b}{3 b} - \frac{b - 2 a}{3 a}$

To write $\frac{a - 2 b}{3 b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{a - 2 b}{3 b} \cdot \frac{a}{a} - \frac{b - 2 a}{3 a}$

To write $- \frac{b - 2 a}{3 a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{a - 2 b}{3 b} \cdot \frac{a}{a} - \frac{b - 2 a}{3 a} \cdot \frac{b}{b}$

Write each expression with a common denominator of $3 b a$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a - 2 b}{3 b}$ by $\frac{a}{a}$ .

$\frac{(a - 2 b) a}{3 b a} - \frac{b - 2 a}{3 a} \cdot \frac{b}{b}$

Multiply $\frac{b - 2 a}{3 a}$ by $\frac{b}{b}$ .

$\frac{(a - 2 b) a}{3 b a} - \frac{(b - 2 a) b}{3 a b}$

Reorder the factors of $3 b a$ .

$\frac{(a - 2 b) a}{3 a b} - \frac{(b - 2 a) b}{3 a b}$

Combine the numerators over the common denominator.

$\frac{(a - 2 b) a - (b - 2 a) b}{3 a b}$

Simplify the numerator.

Apply the distributive property.

$\frac{a \cdot a - 2 b a - (b - 2 a) b}{3 a b}$

Multiply $a$ by $a$ .

$\frac{a^{2} - 2 b a - (b - 2 a) b}{3 a b}$

Apply the distributive property.

$\frac{a^{2} - 2 b a + (- b - (- 2 a)) b}{3 a b}$

Multiply $- 2$ by $- 1$ .

$\frac{a^{2} - 2 b a + (- b + 2 a) b}{3 a b}$

Apply the distributive property.

$\frac{a^{2} - 2 b a - b \cdot b + 2 a b}{3 a b}$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$\frac{a^{2} - 2 b a - (b \cdot b) + 2 a b}{3 a b}$

Multiply $b$ by $b$ .

$\frac{a^{2} - 2 b a - b^{2} + 2 a b}{3 a b}$

Add $- 2 b a$ and $2 a b$ .

Move $b$ .

$\frac{a^{2} - b^{2} - 2 a b + 2 a b}{3 a b}$

Add $- 2 a b$ and $2 a b$ .

$\frac{a^{2} - b^{2} + 0}{3 a b}$

Add $a^{2} - b^{2}$ and $0$ .

$\frac{a^{2} - b^{2}}{3 a b}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$\frac{(a + b) (a - b)}{3 a b}$