Finite Math Examples

$\frac{\frac{a}{b} - \frac{c}{d}}{\frac{b}{a} - \frac{d}{c}}$

Step 1

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

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Step 1.1

Multiply $\frac{\frac{a}{b} - \frac{c}{d}}{\frac{b}{a} - \frac{d}{c}}$ by $\frac{b d a c}{b d a c}$ .

$\frac{b d a c}{b d a c} \cdot \frac{\frac{a}{b} - \frac{c}{d}}{\frac{b}{a} - \frac{d}{c}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

$\frac{b d a c \frac{a}{b} + b d a c (- \frac{c}{d})}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $b$ .

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Step 3.1.1

Factor $b$ out of $b d a c$ .

$\frac{b (d a c) \frac{a}{b} + b d a c (- \frac{c}{d})}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.1.2

Cancel the common factor.

$\frac{b (d a c) \frac{a}{b} + b d a c (- \frac{c}{d})}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.1.3

Rewrite the expression.

$\frac{d a c a + b d a c (- \frac{c}{d})}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

$\frac{d a^{2} c + b d a c (- \frac{c}{d})}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.6

Cancel the common factor of $d$ .

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Step 3.6.1

Move the leading negative in $- \frac{c}{d}$ into the numerator.

$\frac{d a^{2} c + b d a c \frac{- c}{d}}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.6.2

Factor $d$ out of $b d a c$ .

$\frac{d a^{2} c + d (b a c) \frac{- c}{d}}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.6.3

Cancel the common factor.

$\frac{d a^{2} c + d (b a c) \frac{- c}{d}}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.6.4

Rewrite the expression.

$\frac{d a^{2} c + b a c (- c)}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

$\frac{d a^{2} c + b a (- c^{2})}{b d a c \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.11

Cancel the common factor of $a$ .

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Step 3.11.1

Factor $a$ out of $b d a c$ .

$\frac{d a^{2} c + b a (- c^{2})}{a (b d c) \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.11.2

Cancel the common factor.

$\frac{d a^{2} c + b a (- c^{2})}{a (b d c) \frac{b}{a} + b d a c (- \frac{d}{c})}$

Step 3.11.3

Rewrite the expression.

$\frac{d a^{2} c + b a (- c^{2})}{b d c b + b d a c (- \frac{d}{c})}$

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Add $1$ and $1$ .

$\frac{d a^{2} c + b a (- c^{2})}{b^{2} d c + b d a c (- \frac{d}{c})}$

Step 3.16

Cancel the common factor of $c$ .

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Step 3.16.1

Move the leading negative in $- \frac{d}{c}$ into the numerator.

$\frac{d a^{2} c + b a (- c^{2})}{b^{2} d c + b d a c \frac{- d}{c}}$

Step 3.16.2

Factor $c$ out of $b d a c$ .

$\frac{d a^{2} c + b a (- c^{2})}{b^{2} d c + c (b d a) \frac{- d}{c}}$

Step 3.16.3

Cancel the common factor.

$\frac{d a^{2} c + b a (- c^{2})}{b^{2} d c + c (b d a) \frac{- d}{c}}$

Step 3.16.4

Rewrite the expression.

$\frac{d a^{2} c + b a (- c^{2})}{b^{2} d c + b d a (- d)}$

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

Add $1$ and $1$ .

$\frac{d a^{2} c + b a (- c^{2})}{b^{2} d c + b a (- d^{2})}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $a c$ out of $d a^{2} c + b a (- c^{2})$ .

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Step 4.1.1

Factor $a c$ out of $d a^{2} c$ .

$\frac{a c (d a) + b a (- c^{2})}{b^{2} d c + b a (- d^{2})}$

Step 4.1.2

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

$\frac{a c (d a) + a c (b (- c))}{b^{2} d c + b a (- d^{2})}$

Step 4.1.3

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

$\frac{a c (d a + b (- c))}{b^{2} d c + b a (- d^{2})}$

Step 4.2

Rewrite using the commutative property of multiplication.

$\frac{a c (d a - b c)}{b^{2} d c + b a (- d^{2})}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $b d$ out of $b^{2} d c + b a (- d^{2})$ .

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Step 5.1.1

Factor $b d$ out of $b^{2} d c$ .

$\frac{a c (d a - b c)}{b d (b c) + b a (- d^{2})}$

Step 5.1.2

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

$\frac{a c (d a - b c)}{b d (b c) + b d (a (- d))}$

Step 5.1.3

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

$\frac{a c (d a - b c)}{b d (b c + a (- d))}$

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reorder the terms.

$\frac{a c (a d - b c)}{b d (- a d + b c)}$

Step 6.2

Cancel the common factor of $a d - b c$ and $- a d + b c$ .

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Step 6.2.1

Factor $- 1$ out of $a d$ .

$\frac{a c (- (- a d) - b c)}{b d (- a d + b c)}$

Step 6.2.2

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

$\frac{a c (- (- a d) - (b c))}{b d (- a d + b c)}$

Step 6.2.3

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

$\frac{a c (- (- a d + b c))}{b d (- a d + b c)}$

Step 6.2.4

Rewrite $- (- a d + b c)$ as $- 1 (- a d + b c)$ .

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

Step 6.2.5

Cancel the common factor.

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

Step 6.2.6

Rewrite the expression.

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

Step 6.3

Simplify the expression.

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Step 6.3.1

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

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

Step 6.3.2

Move the negative in front of the fraction.

$- \frac{a c}{b d}$