Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Combine $\frac{1}{5}$ and $a$ .

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

Step 1.3

Combine $\frac{1}{5}$ and $b$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

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

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

Step 1.6

Combine $3 b$ and $\frac{5}{5}$ .

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

Step 1.7

Combine the numerators over the common denominator.

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

Step 1.8

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

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

Step 1.9

Combine $- a$ and $\frac{5}{5}$ .

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

Step 1.10

Combine the numerators over the common denominator.

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

Step 1.11

Rewrite $\frac{a + b + 3 b \cdot 5 - a \cdot 5}{5}$ in a factored form.

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Step 1.11.1

Multiply $5$ by $3$ .

$\frac{\frac{a + b + 15 b - a \cdot 5}{5}}{- a + \frac{4}{5} \cdot (a + b)}$

Step 1.11.2

Multiply $5$ by $- 1$ .

$\frac{\frac{a + b + 15 b - 5 a}{5}}{- a + \frac{4}{5} \cdot (a + b)}$

Step 1.11.3

Subtract $5 a$ from $a$ .

$\frac{\frac{- 4 a + b + 15 b}{5}}{- a + \frac{4}{5} \cdot (a + b)}$

Step 1.11.4

Add $b$ and $15 b$ .

$\frac{\frac{- 4 a + 16 b}{5}}{- a + \frac{4}{5} \cdot (a + b)}$

Step 1.11.5

Factor $4$ out of $- 4 a + 16 b$ .

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Step 1.11.5.1

Factor $4$ out of $- 4 a$ .

$\frac{\frac{4 (- a) + 16 b}{5}}{- a + \frac{4}{5} \cdot (a + b)}$

Step 1.11.5.2

Factor $4$ out of $16 b$ .

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

Step 1.11.5.3

Factor $4$ out of $4 (- a) + 4 (4 b)$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Combine $\frac{4}{5}$ and $a$ .

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

Step 2.3

Combine $\frac{4}{5}$ and $b$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Factor $4$ out of $4 a + 4 b$ .

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Step 2.5.1

Factor $4$ out of $4 a$ .

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

Step 2.5.2

Factor $4$ out of $4 b$ .

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

Step 2.5.3

Factor $4$ out of $4 (a) + 4 (b)$ .

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

Step 2.6

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

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

Step 2.7

Combine $- a$ and $\frac{5}{5}$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Rewrite $\frac{- a \cdot 5 + 4 (a + b)}{5}$ in a factored form.

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Step 2.9.1

Multiply $5$ by $- 1$ .

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

Step 2.9.2

Apply the distributive property.

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

Step 2.9.3

Add $- 5 a$ and $4 a$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $- a + 4 b$ .

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Step 4.1

Factor $- a + 4 b$ out of $4 (- a + 4 b)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

$\frac{4}{5} \cdot 5$

Step 5

Cancel the common factor of $5$ .

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Step 5.1

Cancel the common factor.

$\frac{4}{5} \cdot 5$

Step 5.2

Rewrite the expression.

$4$