Algebra Examples

$\frac{a - 2 b}{15 a} + \frac{b - a}{20 b}$

Step 1

To write $\frac{a - 2 b}{15 a}$ as a fraction with a common denominator, multiply by $\frac{4 b}{4 b}$ .

$\frac{a - 2 b}{15 a} \cdot \frac{4 b}{4 b} + \frac{b - a}{20 b}$

Step 2

To write $\frac{b - a}{20 b}$ as a fraction with a common denominator, multiply by $\frac{3 a}{3 a}$ .

$\frac{a - 2 b}{15 a} \cdot \frac{4 b}{4 b} + \frac{b - a}{20 b} \cdot \frac{3 a}{3 a}$

Step 3

Write each expression with a common denominator of $60 a b$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{a - 2 b}{15 a}$ by $\frac{4 b}{4 b}$ .

$\frac{(a - 2 b) (4 b)}{15 a (4 b)} + \frac{b - a}{20 b} \cdot \frac{3 a}{3 a}$

Step 3.2

Multiply $4$ by $15$ .

$\frac{(a - 2 b) (4 b)}{60 a b} + \frac{b - a}{20 b} \cdot \frac{3 a}{3 a}$

Step 3.3

Multiply $\frac{b - a}{20 b}$ by $\frac{3 a}{3 a}$ .

$\frac{(a - 2 b) (4 b)}{60 a b} + \frac{(b - a) (3 a)}{20 b (3 a)}$

Step 3.4

Multiply $3$ by $20$ .

$\frac{(a - 2 b) (4 b)}{60 a b} + \frac{(b - a) (3 a)}{60 b a}$

Step 3.5

Reorder the factors of $60 b a$ .

$\frac{(a - 2 b) (4 b)}{60 a b} + \frac{(b - a) (3 a)}{60 a b}$

Step 4

Combine the numerators over the common denominator.

$\frac{(a - 2 b) (4 b) + (b - a) (3 a)}{60 a b}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\frac{(a \cdot 4 - 2 b \cdot 4) b + (b - a) \cdot 3 a}{60 a b}$

Step 5.2

Move $4$ to the left of $a$ .

$\frac{(4 \cdot a - 2 b \cdot 4) b + (b - a) \cdot 3 a}{60 a b}$

Step 5.3

Multiply $4$ by $- 2$ .

$\frac{(4 \cdot a - 8 b) b + (b - a) \cdot 3 a}{60 a b}$

Step 5.4

Apply the distributive property.

$\frac{4 a b - 8 b \cdot b + (b - a) \cdot 3 a}{60 a b}$

Step 5.5

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

Tap for more steps...

Step 5.5.1

Move $b$ .

$\frac{4 a b - 8 (b \cdot b) + (b - a) \cdot 3 a}{60 a b}$

Step 5.5.2

Multiply $b$ by $b$ .

$\frac{4 a b - 8 b^{2} + (b - a) \cdot 3 a}{60 a b}$

Step 5.6

Apply the distributive property.

$\frac{4 a b - 8 b^{2} + (b \cdot 3 - a \cdot 3) a}{60 a b}$

Step 5.7

Move $3$ to the left of $b$ .

$\frac{4 a b - 8 b^{2} + (3 \cdot b - a \cdot 3) a}{60 a b}$

Step 5.8

Multiply $3$ by $- 1$ .

$\frac{4 a b - 8 b^{2} + (3 \cdot b - 3 a) a}{60 a b}$

Step 5.9

Apply the distributive property.

$\frac{4 a b - 8 b^{2} + 3 b a - 3 a \cdot a}{60 a b}$

Step 5.10

Multiply $a$ by $a$ by adding the exponents.

Tap for more steps...

Step 5.10.1

Move $a$ .

$\frac{4 a b - 8 b^{2} + 3 b a - 3 (a \cdot a)}{60 a b}$

Step 5.10.2

Multiply $a$ by $a$ .

$\frac{4 a b - 8 b^{2} + 3 b a - 3 a^{2}}{60 a b}$

Step 5.11

Add $4 a b$ and $3 b a$ .

Tap for more steps...

Step 5.11.1

Move $b$ .

$\frac{- 8 b^{2} + 4 a b + 3 a b - 3 a^{2}}{60 a b}$

Step 5.11.2

Add $4 a b$ and $3 a b$ .

$\frac{- 8 b^{2} + 7 a b - 3 a^{2}}{60 a b}$

Step 6

Simplify with factoring out.

Tap for more steps...

Step 6.1

Factor $- 1$ out of $- 8 b^{2}$ .

$\frac{- (8 b^{2}) + 7 a b - 3 a^{2}}{60 a b}$

Step 6.2

Factor $- 1$ out of $7 a b$ .

$\frac{- (8 b^{2}) - (- 7 a b) - 3 a^{2}}{60 a b}$

Step 6.3

Factor $- 1$ out of $- (8 b^{2}) - (- 7 a b)$ .

$\frac{- (8 b^{2} - 7 a b) - 3 a^{2}}{60 a b}$

Step 6.4

Factor $- 1$ out of $- 3 a^{2}$ .

$\frac{- (8 b^{2} - 7 a b) - (3 a^{2})}{60 a b}$

Step 6.5

Factor $- 1$ out of $- (8 b^{2} - 7 a b) - (3 a^{2})$ .

$\frac{- (8 b^{2} - 7 a b + 3 a^{2})}{60 a b}$

Step 6.6

Simplify the expression.

Tap for more steps...

Step 6.6.1

Rewrite $- (8 b^{2} - 7 a b + 3 a^{2})$ as $- 1 (8 b^{2} - 7 a b + 3 a^{2})$ .

$\frac{- 1 (8 b^{2} - 7 a b + 3 a^{2})}{60 a b}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{8 b^{2} - 7 a b + 3 a^{2}}{60 a b}$