Precalculus Examples

$s = \frac{P}{q + p (1 - q)}$

Step 1

Rewrite the equation as $\frac{P}{q + p (1 - q)} = s$ .

$\frac{P}{q + p (1 - q)} = s$

Step 2

Factor each term.

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Step 2.1

Apply the distributive property.

$\frac{P}{q + p \cdot 1 + p (- q)} = s$

Step 2.2

Multiply $p$ by $1$ .

$\frac{P}{q + p + p (- q)} = s$

Step 2.3

Rewrite using the commutative property of multiplication.

$\frac{P}{q + p - p q} = s$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$q + p - p q, 1$

Step 3.2

Remove parentheses.

$q + p - p q, 1$

Step 3.3

The LCM of one and any expression is the expression.

$q + p - p q$

Step 4

Multiply each term in $\frac{P}{q + p - p q} = s$ by $q + p - p q$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{P}{q + p - p q} = s$ by $q + p - p q$ .

$\frac{P}{q + p - p q} (q + p - p q) = s (q + p - p q)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $q + p - p q$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{P}{q + p - p q} (q + p - p q) = s (q + p - p q)$

Step 4.2.1.2

Rewrite the expression.

$P = s (q + p - p q)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$P = s q + s p + s (- p q)$

Step 4.3.2

Rewrite using the commutative property of multiplication.

$P = s q + s p - s p q$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $s q + s p - s p q = P$ .

$s q + s p - s p q = P$

Step 5.2

Subtract $s p$ from both sides of the equation.

$s q - s p q = P - s p$

Step 5.3

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

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Step 5.3.1

Factor $s q$ out of $s q$ .

$s q (1) - s p q = P - s p$

Step 5.3.2

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

$s q (1) + s q (- 1 p) = P - s p$

Step 5.3.3

Factor $s q$ out of $s q (1) + s q (- 1 p)$ .

$s q (1 - 1 p) = P - s p$

Step 5.4

Rewrite $- 1 p$ as $- p$ .

$s q (1 - p) = P - s p$

Step 5.5

Divide each term in $s q (1 - p) = P - s p$ by $s (1 - p)$ and simplify.

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Step 5.5.1

Divide each term in $s q (1 - p) = P - s p$ by $s (1 - p)$ .

$\frac{s q (1 - p)}{s (1 - p)} = \frac{P}{s (1 - p)} + \frac{- s p}{s (1 - p)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $s$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{s q (1 - p)}{s (1 - p)} = \frac{P}{s (1 - p)} + \frac{- s p}{s (1 - p)}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{q (1 - p)}{1 - p} = \frac{P}{s (1 - p)} + \frac{- s p}{s (1 - p)}$

Step 5.5.2.2

Cancel the common factor of $1 - p$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{q (1 - p)}{1 - p} = \frac{P}{s (1 - p)} + \frac{- s p}{s (1 - p)}$

Step 5.5.2.2.2

Divide $q$ by $1$ .

$q = \frac{P}{s (1 - p)} + \frac{- s p}{s (1 - p)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $s$ .

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Step 5.5.3.1.1.1

Cancel the common factor.

$q = \frac{P}{s (1 - p)} + \frac{- s p}{s (1 - p)}$

Step 5.5.3.1.1.2

Rewrite the expression.

$q = \frac{P}{s (1 - p)} + \frac{- 1 p}{1 - p}$

Step 5.5.3.1.2

Move the negative in front of the fraction.

$q = \frac{P}{s (1 - p)} - \frac{p}{1 - p}$

Step 5.5.3.2

To write $- \frac{p}{1 - p}$ as a fraction with a common denominator, multiply by $\frac{s}{s}$ .

$q = \frac{P}{s (1 - p)} - \frac{p}{1 - p} \cdot \frac{s}{s}$

Step 5.5.3.3

Write each expression with a common denominator of $s (1 - p)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{p}{1 - p}$ by $\frac{s}{s}$ .

$q = \frac{P}{s (1 - p)} - \frac{p s}{(1 - p) s}$

Step 5.5.3.3.2

Reorder the factors of $(1 - p) s$ .

$q = \frac{P}{s (1 - p)} - \frac{p s}{s (1 - p)}$

Step 5.5.3.4

Combine the numerators over the common denominator.

$q = \frac{P - p s}{s (1 - p)}$