Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

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

Step 1.2

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

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

Step 1.3

Move $2$ to the left of $2 a + b$ .

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(a - b) (a + b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Reorder the factors of $(a - b) (a + b)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $2$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 1$ by $3$ .

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

Step 5.5

Add $4 a$ and $3 a$ .

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

Step 5.6

Subtract $3 b$ from $2 b$ .

$\frac{7 a - b}{(a + b) (a - b)}$