Algebra Examples

$\frac{a + 2}{4 b} - \frac{a - 1}{10 b} + \frac{a - 3}{5 b}$

Step 1

To write $\frac{a + 2}{4 b}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{a + 2}{4 b} \cdot \frac{5}{5} - \frac{a - 1}{10 b} + \frac{a - 3}{5 b}$

Step 2

To write $- \frac{a - 1}{10 b}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a + 2}{4 b} \cdot \frac{5}{5} - \frac{a - 1}{10 b} \cdot \frac{2}{2} + \frac{a - 3}{5 b}$

Step 3

Write each expression with a common denominator of $20 b$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{a + 2}{4 b}$ by $\frac{5}{5}$ .

$\frac{(a + 2) \cdot 5}{4 b \cdot 5} - \frac{a - 1}{10 b} \cdot \frac{2}{2} + \frac{a - 3}{5 b}$

Step 3.2

Multiply $5$ by $4$ .

$\frac{(a + 2) \cdot 5}{20 b} - \frac{a - 1}{10 b} \cdot \frac{2}{2} + \frac{a - 3}{5 b}$

Step 3.3

Multiply $\frac{a - 1}{10 b}$ by $\frac{2}{2}$ .

$\frac{(a + 2) \cdot 5}{20 b} - \frac{(a - 1) \cdot 2}{10 b \cdot 2} + \frac{a - 3}{5 b}$

Step 3.4

Multiply $2$ by $10$ .

$\frac{(a + 2) \cdot 5}{20 b} - \frac{(a - 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 4

Combine the numerators over the common denominator.

$\frac{(a + 2) \cdot 5 - (a - 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{a \cdot 5 + 2 \cdot 5 - (a - 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.2

Move $5$ to the left of $a$ .

$\frac{5 \cdot a + 2 \cdot 5 - (a - 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.3

Multiply $2$ by $5$ .

$\frac{5 \cdot a + 10 - (a - 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.4

Apply the distributive property.

$\frac{5 a + 10 + (- a - - 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.5

Multiply $- 1$ by $- 1$ .

$\frac{5 a + 10 + (- a + 1) \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.6

Apply the distributive property.

$\frac{5 a + 10 - a \cdot 2 + 1 \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.7

Multiply $2$ by $- 1$ .

$\frac{5 a + 10 - 2 a + 1 \cdot 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.8

Multiply $2$ by $1$ .

$\frac{5 a + 10 - 2 a + 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.9

Subtract $2 a$ from $5 a$ .

$\frac{3 a + 10 + 2}{20 b} + \frac{a - 3}{5 b}$

Step 5.10

Add $10$ and $2$ .

$\frac{3 a + 12}{20 b} + \frac{a - 3}{5 b}$

Step 5.11

Factor $3$ out of $3 a + 12$ .

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Step 5.11.1

Factor $3$ out of $3 a$ .

$\frac{3 (a) + 12}{20 b} + \frac{a - 3}{5 b}$

Step 5.11.2

Factor $3$ out of $12$ .

$\frac{3 a + 3 \cdot 4}{20 b} + \frac{a - 3}{5 b}$

Step 5.11.3

Factor $3$ out of $3 a + 3 \cdot 4$ .

$\frac{3 (a + 4)}{20 b} + \frac{a - 3}{5 b}$

Step 6

To write $\frac{a - 3}{5 b}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 (a + 4)}{20 b} + \frac{a - 3}{5 b} \cdot \frac{4}{4}$

Step 7

Write each expression with a common denominator of $20 b$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{a - 3}{5 b}$ by $\frac{4}{4}$ .

$\frac{3 (a + 4)}{20 b} + \frac{(a - 3) \cdot 4}{5 b \cdot 4}$

Step 7.2

Multiply $4$ by $5$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $3$ by $4$ .

$\frac{3 a + 12 + (a - 3) \cdot 4}{20 b}$

Step 9.3

Apply the distributive property.

$\frac{3 a + 12 + a \cdot 4 - 3 \cdot 4}{20 b}$

Step 9.4

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

$\frac{3 a + 12 + 4 \cdot a - 3 \cdot 4}{20 b}$

Step 9.5

Multiply $- 3$ by $4$ .

$\frac{3 a + 12 + 4 \cdot a - 12}{20 b}$

Step 9.6

Add $3 a$ and $4 a$ .

$\frac{7 a + 12 - 12}{20 b}$

Step 9.7

Subtract $12$ from $12$ .

$\frac{7 a + 0}{20 b}$

Step 9.8

Add $7 a$ and $0$ .

$\frac{7 a}{20 b}$