Algebra Examples

$\frac{\frac{1}{4 n} - \frac{1}{2}}{\frac{n}{5} - \frac{1}{20 n}}$

Step 1

Multiply the numerator and denominator of the fraction by $20 n$ .

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Step 1.1

Multiply $\frac{\frac{1}{4 n} - \frac{1}{2}}{\frac{n}{5} - \frac{1}{20 n}}$ by $\frac{20 n}{20 n}$ .

$\frac{20 n}{20 n} \cdot \frac{\frac{1}{4 n} - \frac{1}{2}}{\frac{n}{5} - \frac{1}{20 n}}$

Step 1.2

Combine.

$\frac{20 n (\frac{1}{4 n} - \frac{1}{2})}{20 n (\frac{n}{5} - \frac{1}{20 n})}$

Step 2

Apply the distributive property.

$\frac{20 n \frac{1}{4 n} + 20 n (- \frac{1}{2})}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $4 n$ .

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Step 3.1.1

Factor $4 n$ out of $20 n$ .

$\frac{4 n (5) \frac{1}{4 n} + 20 n (- \frac{1}{2})}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.1.2

Cancel the common factor.

$\frac{4 n \cdot 5 \frac{1}{4 n} + 20 n (- \frac{1}{2})}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.1.3

Rewrite the expression.

$\frac{5 + 20 n (- \frac{1}{2})}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{5 + 20 n \frac{- 1}{2}}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.2.2

Factor $2$ out of $20 n$ .

$\frac{5 + 2 (10 n) \frac{- 1}{2}}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.2.3

Cancel the common factor.

$\frac{5 + 2 (10 n) \frac{- 1}{2}}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.2.4

Rewrite the expression.

$\frac{5 + 10 n \cdot - 1}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.3

Multiply $- 1$ by $10$ .

$\frac{5 - 10 n}{20 n \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.4

Cancel the common factor of $5$ .

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Step 3.4.1

Factor $5$ out of $20 n$ .

$\frac{5 - 10 n}{5 (4 n) \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.4.2

Cancel the common factor.

$\frac{5 - 10 n}{5 (4 n) \frac{n}{5} + 20 n (- \frac{1}{20 n})}$

Step 3.4.3

Rewrite the expression.

$\frac{5 - 10 n}{4 n \cdot n + 20 n (- \frac{1}{20 n})}$

Step 3.5

Raise $n$ to the power of $1$ .

$\frac{5 - 10 n}{4 (n^{1} n) + 20 n (- \frac{1}{20 n})}$

Step 3.6

Raise $n$ to the power of $1$ .

$\frac{5 - 10 n}{4 (n^{1} n^{1}) + 20 n (- \frac{1}{20 n})}$

Step 3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 - 10 n}{4 n^{1 + 1} + 20 n (- \frac{1}{20 n})}$

Step 3.8

Add $1$ and $1$ .

$\frac{5 - 10 n}{4 n^{2} + 20 n (- \frac{1}{20 n})}$

Step 3.9

Cancel the common factor of $20 n$ .

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Step 3.9.1

Move the leading negative in $- \frac{1}{20 n}$ into the numerator.

$\frac{5 - 10 n}{4 n^{2} + 20 n \frac{- 1}{20 n}}$

Step 3.9.2

Cancel the common factor.

$\frac{5 - 10 n}{4 n^{2} + 20 n \frac{- 1}{20 n}}$

Step 3.9.3

Rewrite the expression.

$\frac{5 - 10 n}{4 n^{2} - 1}$

Step 4

Factor $5$ out of $5 - 10 n$ .

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Step 4.1

Factor $5$ out of $5$ .

$\frac{5 (1) - 10 n}{4 n^{2} - 1}$

Step 4.2

Factor $5$ out of $- 10 n$ .

$\frac{5 (1) + 5 (- 2 n)}{4 n^{2} - 1}$

Step 4.3

Factor $5$ out of $5 (1) + 5 (- 2 n)$ .

$\frac{5 (1 - 2 n)}{4 n^{2} - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $4 n^{2}$ as ${(2 n)}^{2}$ .

$\frac{5 (1 - 2 n)}{{(2 n)}^{2} - 1}$

Step 5.2

Rewrite $1$ as $1^{2}$ .

$\frac{5 (1 - 2 n)}{{(2 n)}^{2} - 1^{2}}$

Step 5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 n$ and $b = 1$ .

$\frac{5 (1 - 2 n)}{(2 n + 1) (2 n - 1)}$

Step 6

Cancel the common factor of $1 - 2 n$ and $2 n - 1$ .

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Step 6.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{5 (- 1 (- 1) - 2 n)}{(2 n + 1) (2 n - 1)}$

Step 6.2

Factor $- 1$ out of $- 2 n$ .

$\frac{5 (- 1 (- 1) - (2 n))}{(2 n + 1) (2 n - 1)}$

Step 6.3

Factor $- 1$ out of $- 1 (- 1) - (2 n)$ .

$\frac{5 (- 1 (- 1 + 2 n))}{(2 n + 1) (2 n - 1)}$

Step 6.4

Reorder terms.

$\frac{5 (- 1 (2 n - 1))}{(2 n + 1) (2 n - 1)}$

Step 6.5

Cancel the common factor.

$\frac{5 (- 1 (2 n - 1))}{(2 n + 1) (2 n - 1)}$

Step 6.6

Rewrite the expression.

$\frac{5 \cdot (- 1)}{2 n + 1}$

Step 7

Multiply $5$ by $- 1$ .

$\frac{- 5}{2 n + 1}$

Step 8

Move the negative in front of the fraction.

$- \frac{5}{2 n + 1}$