Algebra Examples

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

Step 1

Combine the numerators over the common denominator.

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

Step 2

Subtract $a$ from $2 a$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $b \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Add $a \cdot 2$ and $a$ .

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Step 6.1

Reorder $a$ and $2$ .

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

Step 6.2

Add $2 \cdot a$ and $a$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Write each expression with a common denominator of $6 b$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

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

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

Step 9.2

Multiply $3$ by $2$ .

$\frac{3 a \cdot 3}{6 b} + \frac{2 a}{3 b} \cdot \frac{2}{2}$

Step 9.3

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

$\frac{3 a \cdot 3}{6 b} + \frac{2 a \cdot 2}{3 b \cdot 2}$

Step 9.4

Multiply $2$ by $3$ .

$\frac{3 a \cdot 3}{6 b} + \frac{2 a \cdot 2}{6 b}$

Step 10

Combine the numerators over the common denominator.

$\frac{3 a \cdot 3 + 2 a \cdot 2}{6 b}$

Step 11

Simplify the numerator.

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Step 11.1

Factor $a$ out of $3 a \cdot 3 + 2 a \cdot 2$ .

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Step 11.1.1

Factor $a$ out of $3 a \cdot 3$ .

$\frac{a (3 \cdot 3) + 2 a \cdot 2}{6 b}$

Step 11.1.2

Factor $a$ out of $2 a \cdot 2$ .

$\frac{a (3 \cdot 3) + a (2 \cdot 2)}{6 b}$

Step 11.1.3

Factor $a$ out of $a (3 \cdot 3) + a (2 \cdot 2)$ .

$\frac{a (3 \cdot 3 + 2 \cdot 2)}{6 b}$

Step 11.2

Multiply $3$ by $3$ .

$\frac{a (9 + 2 \cdot 2)}{6 b}$

Step 11.3

Multiply $2$ by $2$ .

$\frac{a (9 + 4)}{6 b}$

Step 11.4

Add $9$ and $4$ .

$\frac{a \cdot 13}{6 b}$

Step 12

Move $13$ to the left of $a$ .

$\frac{13 a}{6 b}$