Pre-Algebra Examples

$\frac{p^{2} - 10 p + p q - 10 q}{2 p^{2} - 2 q^{2}} \div \frac{p - 10}{7 p - 7 q}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{p^{2} - 10 p + p q - 10 q}{2 p^{2} - 2 q^{2}} \cdot \frac{7 p - 7 q}{p - 10}$

Step 2

Simplify the numerator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$\frac{(p^{2} - 10 p) + p q - 10 q}{2 p^{2} - 2 q^{2}} \cdot \frac{7 p - 7 q}{p - 10}$

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{p (p - 10) + q (p - 10)}{2 p^{2} - 2 q^{2}} \cdot \frac{7 p - 7 q}{p - 10}$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $p - 10$ .

$\frac{(p - 10) (p + q)}{2 p^{2} - 2 q^{2}} \cdot \frac{7 p - 7 q}{p - 10}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $2$ out of $2 p^{2} - 2 q^{2}$ .

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Step 3.1.1

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

$\frac{(p - 10) (p + q)}{2 (p^{2}) - 2 q^{2}} \cdot \frac{7 p - 7 q}{p - 10}$

Step 3.1.2

Factor $2$ out of $- 2 q^{2}$ .

$\frac{(p - 10) (p + q)}{2 (p^{2}) + 2 (- q^{2})} \cdot \frac{7 p - 7 q}{p - 10}$

Step 3.1.3

Factor $2$ out of $2 (p^{2}) + 2 (- q^{2})$ .

$\frac{(p - 10) (p + q)}{2 (p^{2} - q^{2})} \cdot \frac{7 p - 7 q}{p - 10}$

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 $a = p$ and $b = q$ .

$\frac{(p - 10) (p + q)}{2 (p + q) (p - q)} \cdot \frac{7 p - 7 q}{p - 10}$

Step 4

Cancel the common factor of $p - 10$ .

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Step 4.1

Cancel the common factor.

$\frac{(p - 10) (p + q)}{2 (p + q) (p - q)} \cdot \frac{7 p - 7 q}{p - 10}$

Step 4.2

Rewrite the expression.

$\frac{p + q}{2 (p + q) (p - q)} (7 p - 7 q)$

Step 5

Cancel the common factor of $p + q$ .

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Step 5.1

Cancel the common factor.

$\frac{p + q}{2 (p + q) (p - q)} (7 p - 7 q)$

Step 5.2

Rewrite the expression.

$\frac{1}{2 (p - q)} (7 p - 7 q)$

Step 6

Multiply $\frac{1}{2 (p - q)}$ by $7 p - 7 q$ .

$\frac{7 p - 7 q}{2 (p - q)}$

Step 7

Factor $7$ out of $7 p - 7 q$ .

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Step 7.1

Factor $7$ out of $7 p$ .

$\frac{7 (p) - 7 q}{2 (p - q)}$

Step 7.2

Factor $7$ out of $- 7 q$ .

$\frac{7 (p) + 7 (- q)}{2 (p - q)}$

Step 7.3

Factor $7$ out of $7 (p) + 7 (- q)$ .

$\frac{7 (p - q)}{2 (p - q)}$

Step 8

Cancel the common factor of $p - q$ .

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Step 8.1

Cancel the common factor.

$\frac{7 (p - q)}{2 (p - q)}$

Step 8.2

Rewrite the expression.

$\frac{7}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{2}$

Decimal Form:

$3.5$

Mixed Number Form:

$3 \frac{1}{2}$