Finite Math Examples

$\frac{3}{4} \cdot (2 r - 11) + \frac{1}{2} = \frac{5}{4}$

Step 1

Simplify $\frac{3}{4} \cdot (2 r - 11) + \frac{1}{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{3}{4} (2 r) + \frac{3}{4} \cdot - 11 + \frac{1}{2} = \frac{5}{4}$

Step 1.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.1

Factor $2$ out of $4$ .

$\frac{3}{2 (2)} (2 r) + \frac{3}{4} \cdot - 11 + \frac{1}{2} = \frac{5}{4}$

Step 1.1.2.2

Factor $2$ out of $2 r$ .

$\frac{3}{2 (2)} (2 (r)) + \frac{3}{4} \cdot - 11 + \frac{1}{2} = \frac{5}{4}$

Step 1.1.2.3

Cancel the common factor.

$\frac{3}{2 \cdot 2} (2 r) + \frac{3}{4} \cdot - 11 + \frac{1}{2} = \frac{5}{4}$

Step 1.1.2.4

Rewrite the expression.

$\frac{3}{2} r + \frac{3}{4} \cdot - 11 + \frac{1}{2} = \frac{5}{4}$

Step 1.1.3

Combine $\frac{3}{2}$ and $r$ .

$\frac{3 r}{2} + \frac{3}{4} \cdot - 11 + \frac{1}{2} = \frac{5}{4}$

Step 1.1.4

Multiply $\frac{3}{4} \cdot - 11$ .

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Step 1.1.4.1

Combine $\frac{3}{4}$ and $- 11$ .

$\frac{3 r}{2} + \frac{3 \cdot - 11}{4} + \frac{1}{2} = \frac{5}{4}$

Step 1.1.4.2

Multiply $3$ by $- 11$ .

$\frac{3 r}{2} + \frac{- 33}{4} + \frac{1}{2} = \frac{5}{4}$

Step 1.1.5

Move the negative in front of the fraction.

$\frac{3 r}{2} - \frac{33}{4} + \frac{1}{2} = \frac{5}{4}$

Step 1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 r}{2} - \frac{33}{4} + \frac{1}{2} \cdot \frac{2}{2} = \frac{5}{4}$

Step 1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{3 r}{2} - \frac{33}{4} + \frac{2}{2 \cdot 2} = \frac{5}{4}$

Step 1.3.2

Multiply $2$ by $2$ .

$\frac{3 r}{2} - \frac{33}{4} + \frac{2}{4} = \frac{5}{4}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3 r}{2} + \frac{- 33 + 2}{4} = \frac{5}{4}$

Step 1.5

Add $- 33$ and $2$ .

$\frac{3 r}{2} + \frac{- 31}{4} = \frac{5}{4}$

Step 1.6

Move the negative in front of the fraction.

$\frac{3 r}{2} - \frac{31}{4} = \frac{5}{4}$

Step 2

Move all terms not containing $r$ to the right side of the equation.

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Step 2.1

Add $\frac{31}{4}$ to both sides of the equation.

$\frac{3 r}{2} = \frac{5}{4} + \frac{31}{4}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{3 r}{2} = \frac{5 + 31}{4}$

Step 2.3

Add $5$ and $31$ .

$\frac{3 r}{2} = \frac{36}{4}$

Step 2.4

Divide $36$ by $4$ .

$\frac{3 r}{2} = 9$

Step 3

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 r}{2} = \frac{2}{3} \cdot 9$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{3} \cdot \frac{3 r}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 r}{2} = \frac{2}{3} \cdot 9$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 r) = \frac{2}{3} \cdot 9$

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Factor $3$ out of $3 r$ .

$\frac{1}{3} (3 (r)) = \frac{2}{3} \cdot 9$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 r) = \frac{2}{3} \cdot 9$

Step 4.1.1.2.3

Rewrite the expression.

$r = \frac{2}{3} \cdot 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2}{3} \cdot 9$ .

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $9$ .

$r = \frac{2}{3} \cdot (3 (3))$

Step 4.2.1.1.2

Cancel the common factor.

$r = \frac{2}{3} \cdot (3 \cdot 3)$

Step 4.2.1.1.3

Rewrite the expression.

$r = 2 \cdot 3$

Step 4.2.1.2

Multiply $2$ by $3$ .

$r = 6$