Finite Math Examples

$\frac{a - b}{\frac{1}{a} - \frac{1}{b}}$

Step 1

Multiply the numerator and denominator of the fraction by $a b$ .

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Step 1.1

Multiply $\frac{a - b}{\frac{1}{a} - \frac{1}{b}}$ by $\frac{a b}{a b}$ .

$\frac{a b}{a b} \cdot \frac{a - b}{\frac{1}{a} - \frac{1}{b}}$

Step 1.2

Combine.

$\frac{a b (a - b)}{a b (\frac{1}{a} - \frac{1}{b})}$

Step 2

Apply the distributive property.

$\frac{a b a + a b (- b)}{a b \frac{1}{a} + a b (- \frac{1}{b})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $a$ .

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Step 3.1.1

Factor $a$ out of $a b$ .

$\frac{a b a + a b (- b)}{a (b) \frac{1}{a} + a b (- \frac{1}{b})}$

Step 3.1.2

Cancel the common factor.

$\frac{a b a + a b (- b)}{a b \frac{1}{a} + a b (- \frac{1}{b})}$

Step 3.1.3

Rewrite the expression.

$\frac{a b a + a b (- b)}{b + a b (- \frac{1}{b})}$

Step 3.2

Cancel the common factor of $b$ .

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Step 3.2.1

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

$\frac{a b a + a b (- b)}{b + a b \frac{- 1}{b}}$

Step 3.2.2

Factor $b$ out of $a b$ .

$\frac{a b a + a b (- b)}{b + b a \frac{- 1}{b}}$

Step 3.2.3

Cancel the common factor.

$\frac{a b a + a b (- b)}{b + b a \frac{- 1}{b}}$

Step 3.2.4

Rewrite the expression.

$\frac{a b a + a b (- b)}{b + a \cdot - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.1

Move $a$ .

$\frac{a \cdot a b + a b (- b)}{b + a \cdot - 1}$

Step 4.1.2

Multiply $a$ by $a$ .

$\frac{a^{2} b + a b (- b)}{b + a \cdot - 1}$

Step 4.2

Rewrite using the commutative property of multiplication.

$\frac{a^{2} b - a b \cdot b}{b + a \cdot - 1}$

Step 4.3

Multiply $b$ by $b$ by adding the exponents.

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Step 4.3.1

Move $b$ .

$\frac{a^{2} b - a (b \cdot b)}{b + a \cdot - 1}$

Step 4.3.2

Multiply $b$ by $b$ .

$\frac{a^{2} b - a b^{2}}{b + a \cdot - 1}$

Step 4.4

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

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Step 4.4.1

Factor $a b$ out of $a^{2} b$ .

$\frac{a b (a) - a b^{2}}{b + a \cdot - 1}$

Step 4.4.2

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

$\frac{a b (a) + a b (- b)}{b + a \cdot - 1}$

Step 4.4.3

Factor $a b$ out of $a b (a) + a b (- b)$ .

$\frac{a b (a - b)}{b + a \cdot - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Move $- 1$ to the left of $a$ .

$\frac{a b (a - b)}{b - 1 \cdot a}$

Step 5.2

Rewrite $- 1 a$ as $- a$ .

$\frac{a b (a - b)}{b - a}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $a - b$ and $b - a$ .

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Step 6.1.1

Factor $- 1$ out of $a$ .

$\frac{a b (- 1 (- a) - b)}{b - a}$

Step 6.1.2

Factor $- 1$ out of $- b$ .

$\frac{a b (- 1 (- a) - (b))}{b - a}$

Step 6.1.3

Factor $- 1$ out of $- 1 (- a) - (b)$ .

$\frac{a b (- 1 (- a + b))}{b - a}$

Step 6.1.4

Reorder terms.

$\frac{a b (- 1 (- a + b))}{- a + b}$

Step 6.1.5

Cancel the common factor.

$\frac{a b (- 1 (- a + b))}{- a + b}$

Step 6.1.6

Divide $a b \cdot (- 1)$ by $1$ .

$a b \cdot (- 1)$

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $- 1$ to the left of $a b$ .

$- 1 \cdot (a b)$

Step 6.2.2

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

$- a b$