Basic Math Examples

$\frac{m^{2} + 5 a m}{3 c} \div \frac{2 m^{2} - 50 a^{2}}{6 m - 30 a}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{m^{2} + 5 a m}{3 c} \cdot \frac{6 m - 30 a}{2 m^{2} - 50 a^{2}}$

Step 2

Simplify terms.

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Step 2.1

Factor $m$ out of $m^{2} + 5 a m$ .

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Step 2.1.1

Factor $m$ out of $m^{2}$ .

$\frac{m \cdot m + 5 a m}{3 c} \cdot \frac{6 m - 30 a}{2 m^{2} - 50 a^{2}}$

Step 2.1.2

Factor $m$ out of $5 a m$ .

$\frac{m \cdot m + m (5 a)}{3 c} \cdot \frac{6 m - 30 a}{2 m^{2} - 50 a^{2}}$

Step 2.1.3

Factor $m$ out of $m \cdot m + m (5 a)$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{6 m - 30 a}{2 m^{2} - 50 a^{2}}$

Step 2.2

Cancel the common factor of $6 m - 30 a$ and $2 m^{2} - 50 a^{2}$ .

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Step 2.2.1

Factor $2$ out of $6 m$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m) - 30 a}{2 m^{2} - 50 a^{2}}$

Step 2.2.2

Factor $2$ out of $- 30 a$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m) + 2 (- 15 a)}{2 m^{2} - 50 a^{2}}$

Step 2.2.3

Factor $2$ out of $2 (3 m) + 2 (- 15 a)$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m - 15 a)}{2 m^{2} - 50 a^{2}}$

Step 2.2.4

Cancel the common factors.

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Step 2.2.4.1

Factor $2$ out of $2 m^{2}$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m - 15 a)}{2 (m^{2}) - 50 a^{2}}$

Step 2.2.4.2

Factor $2$ out of $- 50 a^{2}$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m - 15 a)}{2 (m^{2}) + 2 (- 25 a^{2})}$

Step 2.2.4.3

Factor $2$ out of $2 (m^{2}) + 2 (- 25 a^{2})$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m - 15 a)}{2 (m^{2} - 25 a^{2})}$

Step 2.2.4.4

Cancel the common factor.

$\frac{m (m + 5 a)}{3 c} \cdot \frac{2 (3 m - 15 a)}{2 (m^{2} - 25 a^{2})}$

Step 2.2.4.5

Rewrite the expression.

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 m - 15 a}{m^{2} - 25 a^{2}}$

Step 2.3

Factor $3$ out of $3 m - 15 a$ .

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Step 2.3.1

Factor $3$ out of $3 m$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 (m) - 15 a}{m^{2} - 25 a^{2}}$

Step 2.3.2

Factor $3$ out of $- 15 a$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 (m) + 3 (- 5 a)}{m^{2} - 25 a^{2}}$

Step 2.3.3

Factor $3$ out of $3 (m) + 3 (- 5 a)$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 (m - 5 a)}{m^{2} - 25 a^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $25 a^{2}$ as ${(5 a)}^{2}$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 (m - 5 a)}{m^{2} - {(5 a)}^{2}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m$ and $b = 5 a$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 (m - 5 a)}{(m + 5 a) (m - (5 a))}$

Step 3.3

Multiply $5$ by $- 1$ .

$\frac{m (m + 5 a)}{3 c} \cdot \frac{3 (m - 5 a)}{(m + 5 a) (m - 5 a)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $m + 5 a$ .

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Step 4.1.1

Factor $m + 5 a$ out of $m (m + 5 a)$ .

$\frac{(m + 5 a) m}{3 c} \cdot \frac{3 (m - 5 a)}{(m + 5 a) (m - 5 a)}$

Step 4.1.2

Cancel the common factor.

$\frac{(m + 5 a) m}{3 c} \cdot \frac{3 (m - 5 a)}{(m + 5 a) (m - 5 a)}$

Step 4.1.3

Rewrite the expression.

$\frac{m}{3 c} \cdot \frac{3 (m - 5 a)}{m - 5 a}$

Step 4.2

Cancel the common factor of $3$ .

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Step 4.2.1

Factor $3$ out of $3 c$ .

$\frac{m}{3 (c)} \cdot \frac{3 (m - 5 a)}{m - 5 a}$

Step 4.2.2

Cancel the common factor.

$\frac{m}{3 c} \cdot \frac{3 (m - 5 a)}{m - 5 a}$

Step 4.2.3

Rewrite the expression.

$\frac{m}{c} \cdot \frac{m - 5 a}{m - 5 a}$

Step 4.3

Multiply $\frac{m}{c}$ by $\frac{m - 5 a}{m - 5 a}$ .

$\frac{m (m - 5 a)}{c (m - 5 a)}$

Step 4.4

Cancel the common factor of $m - 5 a$ .

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Step 4.4.1

Cancel the common factor.

$\frac{m (m - 5 a)}{c (m - 5 a)}$

Step 4.4.2

Rewrite the expression.

$\frac{m}{c}$