Basic Math Examples

$\frac{25 m - 175}{5 m - 40} \div \frac{m^{2} - 49}{m^{2} - 3 m - 40}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{25 m - 175}{5 m - 40} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $25 m - 175$ and $5 m - 40$ .

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Step 2.1.1

Factor $5$ out of $25 m$ .

$\frac{5 (5 m) - 175}{5 m - 40} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.2

Factor $5$ out of $- 175$ .

$\frac{5 (5 m) + 5 \cdot - 35}{5 m - 40} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.3

Factor $5$ out of $5 (5 m) + 5 (- 35)$ .

$\frac{5 (5 m - 35)}{5 m - 40} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.4

Cancel the common factors.

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Step 2.1.4.1

Factor $5$ out of $5 m$ .

$\frac{5 (5 m - 35)}{5 (m) - 40} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.4.2

Factor $5$ out of $- 40$ .

$\frac{5 (5 m - 35)}{5 m + 5 \cdot - 8} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.4.3

Factor $5$ out of $5 m + 5 \cdot - 8$ .

$\frac{5 (5 m - 35)}{5 (m - 8)} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.4.4

Cancel the common factor.

$\frac{5 (5 m - 35)}{5 (m - 8)} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.1.4.5

Rewrite the expression.

$\frac{5 m - 35}{m - 8} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.2

Factor $5$ out of $5 m - 35$ .

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Step 2.2.1

Factor $5$ out of $5 m$ .

$\frac{5 (m) - 35}{m - 8} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.2.2

Factor $5$ out of $- 35$ .

$\frac{5 m + 5 \cdot - 7}{m - 8} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 2.2.3

Factor $5$ out of $5 m + 5 \cdot - 7$ .

$\frac{5 (m - 7)}{m - 8} \cdot \frac{m^{2} - 3 m - 40}{m^{2} - 49}$

Step 3

Factor $m^{2} - 3 m - 40$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 40$ and whose sum is $- 3$ .

$- 8, 5$

Step 3.2

Write the factored form using these integers.

$\frac{5 (m - 7)}{m - 8} \cdot \frac{(m - 8) (m + 5)}{m^{2} - 49}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $49$ as $7^{2}$ .

$\frac{5 (m - 7)}{m - 8} \cdot \frac{(m - 8) (m + 5)}{m^{2} - 7^{2}}$

Step 4.2

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 $b = 7$ .

$\frac{5 (m - 7)}{m - 8} \cdot \frac{(m - 8) (m + 5)}{(m + 7) (m - 7)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $m - 7$ .

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Step 5.1.1

Factor $m - 7$ out of $5 (m - 7)$ .

$\frac{(m - 7) \cdot 5}{m - 8} \cdot \frac{(m - 8) (m + 5)}{(m + 7) (m - 7)}$

Step 5.1.2

Factor $m - 7$ out of $(m + 7) (m - 7)$ .

$\frac{(m - 7) \cdot 5}{m - 8} \cdot \frac{(m - 8) (m + 5)}{(m - 7) (m + 7)}$

Step 5.1.3

Cancel the common factor.

$\frac{(m - 7) \cdot 5}{m - 8} \cdot \frac{(m - 8) (m + 5)}{(m - 7) (m + 7)}$

Step 5.1.4

Rewrite the expression.

$\frac{5}{m - 8} \cdot \frac{(m - 8) (m + 5)}{m + 7}$

Step 5.2

Cancel the common factor of $m - 8$ .

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Step 5.2.1

Cancel the common factor.

$\frac{5}{m - 8} \cdot \frac{(m - 8) (m + 5)}{m + 7}$

Step 5.2.2

Rewrite the expression.

$5 \frac{m + 5}{m + 7}$

Step 5.3

Combine $5$ and $\frac{m + 5}{m + 7}$ .

$\frac{5 (m + 5)}{m + 7}$