Pre-Algebra Examples

$\frac{2}{9} w = 1 + \frac{1}{6} w$

Step 1

Simplify $\frac{2}{9} w$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{2}{9} w = 1 + \frac{1}{6} w$

Step 1.2

Simplify by adding zeros.

$\frac{2}{9} w = 1 + \frac{1}{6} w$

Step 1.3

Combine $\frac{2}{9}$ and $w$ .

$\frac{2 w}{9} = 1 + \frac{1}{6} w$

Step 2

Combine $\frac{1}{6}$ and $w$ .

$\frac{2 w}{9} = 1 + \frac{w}{6}$

Step 3

Move all terms containing $w$ to the left side of the equation.

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Step 3.1

Subtract $\frac{w}{6}$ from both sides of the equation.

$\frac{2 w}{9} - \frac{w}{6} = 1$

Step 3.2

To write $\frac{2 w}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 w}{9} \cdot \frac{2}{2} - \frac{w}{6} = 1$

Step 3.3

To write $- \frac{w}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 w}{9} \cdot \frac{2}{2} - \frac{w}{6} \cdot \frac{3}{3} = 1$

Step 3.4

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{2 w}{9}$ by $\frac{2}{2}$ .

$\frac{2 w \cdot 2}{9 \cdot 2} - \frac{w}{6} \cdot \frac{3}{3} = 1$

Step 3.4.2

Multiply $9$ by $2$ .

$\frac{2 w \cdot 2}{18} - \frac{w}{6} \cdot \frac{3}{3} = 1$

Step 3.4.3

Multiply $\frac{w}{6}$ by $\frac{3}{3}$ .

$\frac{2 w \cdot 2}{18} - \frac{w \cdot 3}{6 \cdot 3} = 1$

Step 3.4.4

Multiply $6$ by $3$ .

$\frac{2 w \cdot 2}{18} - \frac{w \cdot 3}{18} = 1$

Step 3.5

Combine the numerators over the common denominator.

$\frac{2 w \cdot 2 - w \cdot 3}{18} = 1$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Factor $w$ out of $2 w \cdot 2 - w \cdot 3$ .

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Step 3.6.1.1

Factor $w$ out of $2 w \cdot 2$ .

$\frac{w (2 \cdot 2) - w \cdot 3}{18} = 1$

Step 3.6.1.2

Factor $w$ out of $- w \cdot 3$ .

$\frac{w (2 \cdot 2) + w (- 1 \cdot 3)}{18} = 1$

Step 3.6.1.3

Factor $w$ out of $w (2 \cdot 2) + w (- 1 \cdot 3)$ .

$\frac{w (2 \cdot 2 - 1 \cdot 3)}{18} = 1$

Step 3.6.2

Multiply $2$ by $2$ .

$\frac{w (4 - 1 \cdot 3)}{18} = 1$

Step 3.6.3

Multiply $- 1$ by $3$ .

$\frac{w (4 - 3)}{18} = 1$

Step 3.6.4

Subtract $3$ from $4$ .

$\frac{w \cdot 1}{18} = 1$

Step 3.7

Multiply $w$ by $1$ .

$\frac{w}{18} = 1$

Step 4

Multiply both sides of the equation by $18$ .

$18 \frac{w}{18} = 18 \cdot 1$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $18$ .

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Step 5.1.1.1

Cancel the common factor.

$18 \frac{w}{18} = 18 \cdot 1$

Step 5.1.1.2

Rewrite the expression.

$w = 18 \cdot 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $18$ by $1$ .

$w = 18$