Pre-Algebra Examples

$\frac{5 - \frac{5}{w}}{5 - \frac{5}{w - 1}}$

Step 1

Cancel the common factor of $5 - \frac{5}{w}$ and $5 - \frac{5}{w - 1}$ .

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Step 1.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 1 - \frac{5}{w}}{5 - \frac{5}{w - 1}}$

Step 1.2

Factor $5$ out of $- \frac{5}{w}$ .

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

Step 1.3

Factor $5$ out of $5 (1) + 5 (- \frac{1}{w})$ .

$\frac{5 (1 - \frac{1}{w})}{5 - \frac{5}{w - 1}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $5$ out of $5$ .

$\frac{5 (1 - \frac{1}{w})}{5 (1) - \frac{5}{w - 1}}$

Step 1.4.2

Factor $5$ out of $- \frac{5}{w - 1}$ .

$\frac{5 (1 - \frac{1}{w})}{5 (1) + 5 (- \frac{1}{w - 1})}$

Step 1.4.3

Factor $5$ out of $5 (1) + 5 (- \frac{1}{w - 1})$ .

$\frac{5 (1 - \frac{1}{w})}{5 (1 - \frac{1}{w - 1})}$

Step 1.4.4

Cancel the common factor.

$\frac{5 (1 - \frac{1}{w})}{5 (1 - \frac{1}{w - 1})}$

Step 1.4.5

Rewrite the expression.

$\frac{1 - \frac{1}{w}}{1 - \frac{1}{w - 1}}$

Step 2

Multiply the numerator and denominator of the fraction by $w (w - 1)$ .

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Step 2.1

Multiply $\frac{1 - \frac{1}{w}}{1 - \frac{1}{w - 1}}$ by $\frac{w (w - 1)}{w (w - 1)}$ .

$\frac{w (w - 1)}{w (w - 1)} \cdot \frac{1 - \frac{1}{w}}{1 - \frac{1}{w - 1}}$

Step 2.2

Combine.

$\frac{w (w - 1) (1 - \frac{1}{w})}{w (w - 1) (1 - \frac{1}{w - 1})}$

Step 3

Apply the distributive property.

$\frac{w (w - 1) \cdot 1 + w (w - 1) (- \frac{1}{w})}{w (w - 1) \cdot 1 + w (w - 1) (- \frac{1}{w - 1})}$

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $w$ .

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Step 4.1.1

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

$\frac{w (w - 1) \cdot 1 + w (w - 1) \frac{- 1}{w}}{w (w - 1) \cdot 1 + w (w - 1) (- \frac{1}{w - 1})}$

Step 4.1.2

Cancel the common factor.

$\frac{w (w - 1) \cdot 1 + w (w - 1) \frac{- 1}{w}}{w (w - 1) \cdot 1 + w (w - 1) (- \frac{1}{w - 1})}$

Step 4.1.3

Rewrite the expression.

$\frac{w (w - 1) \cdot 1 + (w - 1) \cdot - 1}{w (w - 1) \cdot 1 + w (w - 1) (- \frac{1}{w - 1})}$

Step 4.2

Cancel the common factor of $w - 1$ .

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Step 4.2.1

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

$\frac{w (w - 1) \cdot 1 + (w - 1) \cdot - 1}{w (w - 1) \cdot 1 + w (w - 1) \frac{- 1}{w - 1}}$

Step 4.2.2

Factor $w - 1$ out of $w (w - 1)$ .

$\frac{w (w - 1) \cdot 1 + (w - 1) \cdot - 1}{w (w - 1) \cdot 1 + (w - 1) w \frac{- 1}{w - 1}}$

Step 4.2.3

Cancel the common factor.

$\frac{w (w - 1) \cdot 1 + (w - 1) \cdot - 1}{w (w - 1) \cdot 1 + (w - 1) w \frac{- 1}{w - 1}}$

Step 4.2.4

Rewrite the expression.

$\frac{w (w - 1) \cdot 1 + (w - 1) \cdot - 1}{w (w - 1) \cdot 1 + w \cdot - 1}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $w - 1$ out of $w (w - 1) \cdot 1 + (w - 1) \cdot - 1$ .

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Step 5.1.1

Factor $w - 1$ out of $w (w - 1) \cdot 1$ .

$\frac{(w - 1) (w \cdot 1) + (w - 1) \cdot - 1}{w (w - 1) \cdot 1 + w \cdot - 1}$

Step 5.1.2

Factor $w - 1$ out of $(w - 1) \cdot - 1$ .

$\frac{(w - 1) (w \cdot 1) + (w - 1) (- 1)}{w (w - 1) \cdot 1 + w \cdot - 1}$

Step 5.1.3

Factor $w - 1$ out of $(w - 1) (w \cdot 1) + (w - 1) (- 1)$ .

$\frac{(w - 1) (w \cdot 1 - 1)}{w (w - 1) \cdot 1 + w \cdot - 1}$

Step 5.2

Multiply $w$ by $1$ .

$\frac{(w - 1) (w - 1)}{w (w - 1) \cdot 1 + w \cdot - 1}$

Step 6

Simplify the denominator.

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Step 6.1

Factor $w$ out of $w (w - 1) \cdot 1 + w \cdot - 1$ .

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Step 6.1.1

Factor $w$ out of $w (w - 1) \cdot 1$ .

$\frac{(w - 1) (w - 1)}{w ((w - 1) \cdot 1) + w \cdot - 1}$

Step 6.1.2

Factor $w$ out of $w \cdot - 1$ .

$\frac{(w - 1) (w - 1)}{w ((w - 1) \cdot 1) + w (- 1)}$

Step 6.1.3

Factor $w$ out of $w ((w - 1) \cdot 1) + w (- 1)$ .

$\frac{(w - 1) (w - 1)}{w ((w - 1) \cdot 1 - 1)}$

Step 6.2

Multiply $w - 1$ by $1$ .

$\frac{(w - 1) (w - 1)}{w (w - 1 - 1)}$

Step 6.3

Subtract $1$ from $- 1$ .

$\frac{(w - 1) (w - 1)}{w (w - 2)}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $w - 1$ to the power of $1$ .

$\frac{{(w - 1)}^{1} (w - 1)}{w (w - 2)}$

Step 7.2

Raise $w - 1$ to the power of $1$ .

$\frac{{(w - 1)}^{1} {(w - 1)}^{1}}{w (w - 2)}$

Step 7.3

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

$\frac{{(w - 1)}^{1 + 1}}{w (w - 2)}$

Step 7.4

Add $1$ and $1$ .

$\frac{{(w - 1)}^{2}}{w (w - 2)}$