Pre-Algebra Examples

$\frac{1}{4} a + \frac{1}{3} a + 8 = 22$

Step 1

Simplify $\frac{1}{4} a + \frac{1}{3} a + 8$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{4}$ and $a$ .

$\frac{a}{4} + \frac{1}{3} a + 8 = 22$

Step 1.1.2

Combine $\frac{1}{3}$ and $a$ .

$\frac{a}{4} + \frac{a}{3} + 8 = 22$

Step 1.2

To write $\frac{a}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{a}{4} \cdot \frac{3}{3} + \frac{a}{3} + 8 = 22$

Step 1.3

To write $\frac{a}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{a}{4} \cdot \frac{3}{3} + \frac{a}{3} \cdot \frac{4}{4} + 8 = 22$

Step 1.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{a}{4}$ by $\frac{3}{3}$ .

$\frac{a \cdot 3}{4 \cdot 3} + \frac{a}{3} \cdot \frac{4}{4} + 8 = 22$

Step 1.4.2

Multiply $4$ by $3$ .

$\frac{a \cdot 3}{12} + \frac{a}{3} \cdot \frac{4}{4} + 8 = 22$

Step 1.4.3

Multiply $\frac{a}{3}$ by $\frac{4}{4}$ .

$\frac{a \cdot 3}{12} + \frac{a \cdot 4}{3 \cdot 4} + 8 = 22$

Step 1.4.4

Multiply $3$ by $4$ .

$\frac{a \cdot 3}{12} + \frac{a \cdot 4}{12} + 8 = 22$

Step 1.5

Combine the numerators over the common denominator.

$\frac{a \cdot 3 + a \cdot 4}{12} + 8 = 22$

Step 1.6

Reorder $a$ and $4$ .

$\frac{3 \cdot a + 4 \cdot a}{12} + 8 = 22$

Step 1.7

Add $3 \cdot a$ and $4 \cdot a$ .

$\frac{7 a}{12} + 8 = 22$

Step 2

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

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Step 2.1

Subtract $8$ from both sides of the equation.

$\frac{7 a}{12} = 22 - 8$

Step 2.2

Subtract $8$ from $22$ .

$\frac{7 a}{12} = 14$

Step 3

Multiply both sides of the equation by $\frac{12}{7}$ .

$\frac{12}{7} \cdot \frac{7 a}{12} = \frac{12}{7} \cdot 14$

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 $\frac{12}{7} \cdot \frac{7 a}{12}$ .

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Step 4.1.1.1

Cancel the common factor of $12$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{12}{7} \cdot \frac{7 a}{12} = \frac{12}{7} \cdot 14$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 a) = \frac{12}{7} \cdot 14$

Step 4.1.1.2

Cancel the common factor of $7$ .

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Step 4.1.1.2.1

Factor $7$ out of $7 a$ .

$\frac{1}{7} (7 (a)) = \frac{12}{7} \cdot 14$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 a) = \frac{12}{7} \cdot 14$

Step 4.1.1.2.3

Rewrite the expression.

$a = \frac{12}{7} \cdot 14$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{12}{7} \cdot 14$ .

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Step 4.2.1.1

Cancel the common factor of $7$ .

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Step 4.2.1.1.1

Factor $7$ out of $14$ .

$a = \frac{12}{7} \cdot (7 (2))$

Step 4.2.1.1.2

Cancel the common factor.

$a = \frac{12}{7} \cdot (7 \cdot 2)$

Step 4.2.1.1.3

Rewrite the expression.

$a = 12 \cdot 2$

Step 4.2.1.2

Multiply $12$ by $2$ .

$a = 24$