Finite Math Examples

$\frac{1}{10} \cdot (a + 3) = \frac{1}{4} \cdot (2 - a)$

Step 1

Simplify $\frac{1}{10} \cdot (a + 3)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{10} \cdot (a + 3) = \frac{1}{4} \cdot (2 - a)$

Step 1.2

Simplify by adding zeros.

$\frac{1}{10} \cdot (a + 3) = \frac{1}{4} \cdot (2 - a)$

Step 1.3

Apply the distributive property.

$\frac{1}{10} a + \frac{1}{10} \cdot 3 = \frac{1}{4} \cdot (2 - a)$

Step 1.4

Combine $\frac{1}{10}$ and $a$ .

$\frac{a}{10} + \frac{1}{10} \cdot 3 = \frac{1}{4} \cdot (2 - a)$

Step 1.5

Combine $\frac{1}{10}$ and $3$ .

$\frac{a}{10} + \frac{3}{10} = \frac{1}{4} \cdot (2 - a)$

Step 2

Simplify $\frac{1}{4} \cdot (2 - a)$ .

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Step 2.1

Apply the distributive property.

$\frac{a}{10} + \frac{3}{10} = \frac{1}{4} \cdot 2 + \frac{1}{4} (- a)$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $4$ .

$\frac{a}{10} + \frac{3}{10} = \frac{1}{2 (2)} \cdot 2 + \frac{1}{4} (- a)$

Step 2.2.2

Cancel the common factor.

$\frac{a}{10} + \frac{3}{10} = \frac{1}{2 \cdot 2} \cdot 2 + \frac{1}{4} (- a)$

Step 2.2.3

Rewrite the expression.

$\frac{a}{10} + \frac{3}{10} = \frac{1}{2} + \frac{1}{4} (- a)$

Step 2.3

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

$\frac{a}{10} + \frac{3}{10} = \frac{1}{2} - \frac{a}{4}$

Step 3

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

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Step 3.1

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

$\frac{a}{10} + \frac{3}{10} + \frac{a}{4} = \frac{1}{2}$

Step 3.2

To write $\frac{a}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a}{10} \cdot \frac{2}{2} + \frac{a}{4} + \frac{3}{10} = \frac{1}{2}$

Step 3.3

To write $\frac{a}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{a}{10} \cdot \frac{2}{2} + \frac{a}{4} \cdot \frac{5}{5} + \frac{3}{10} = \frac{1}{2}$

Step 3.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{a}{10}$ by $\frac{2}{2}$ .

$\frac{a \cdot 2}{10 \cdot 2} + \frac{a}{4} \cdot \frac{5}{5} + \frac{3}{10} = \frac{1}{2}$

Step 3.4.2

Multiply $10$ by $2$ .

$\frac{a \cdot 2}{20} + \frac{a}{4} \cdot \frac{5}{5} + \frac{3}{10} = \frac{1}{2}$

Step 3.4.3

Multiply $\frac{a}{4}$ by $\frac{5}{5}$ .

$\frac{a \cdot 2}{20} + \frac{a \cdot 5}{4 \cdot 5} + \frac{3}{10} = \frac{1}{2}$

Step 3.4.4

Multiply $4$ by $5$ .

$\frac{a \cdot 2}{20} + \frac{a \cdot 5}{20} + \frac{3}{10} = \frac{1}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{a \cdot 2 + a \cdot 5}{20} + \frac{3}{10} = \frac{1}{2}$

Step 3.6

Reorder $a$ and $5$ .

$\frac{2 \cdot a + 5 \cdot a}{20} + \frac{3}{10} = \frac{1}{2}$

Step 3.7

Add $2 \cdot a$ and $5 \cdot a$ .

$\frac{7 a}{20} + \frac{3}{10} = \frac{1}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{3}{10}$ from both sides of the equation.

$\frac{7 a}{20} = \frac{1}{2} - \frac{3}{10}$

Step 4.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{7 a}{20} = \frac{1}{2} \cdot \frac{5}{5} - \frac{3}{10}$

Step 4.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$\frac{7 a}{20} = \frac{5}{2 \cdot 5} - \frac{3}{10}$

Step 4.3.2

Multiply $2$ by $5$ .

$\frac{7 a}{20} = \frac{5}{10} - \frac{3}{10}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{7 a}{20} = \frac{5 - 3}{10}$

Step 4.5

Subtract $3$ from $5$ .

$\frac{7 a}{20} = \frac{2}{10}$

Step 4.6

Cancel the common factor of $2$ and $10$ .

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Step 4.6.1

Factor $2$ out of $2$ .

$\frac{7 a}{20} = \frac{2 (1)}{10}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $2$ out of $10$ .

$\frac{7 a}{20} = \frac{2 \cdot 1}{2 \cdot 5}$

Step 4.6.2.2

Cancel the common factor.

$\frac{7 a}{20} = \frac{2 \cdot 1}{2 \cdot 5}$

Step 4.6.2.3

Rewrite the expression.

$\frac{7 a}{20} = \frac{1}{5}$

Step 5

Multiply both sides of the equation by $\frac{20}{7}$ .

$\frac{20}{7} \cdot \frac{7 a}{20} = \frac{20}{7} \cdot \frac{1}{5}$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{20}{7} \cdot \frac{7 a}{20}$ .

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Step 6.1.1.1

Cancel the common factor of $20$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{20}{7} \cdot \frac{7 a}{20} = \frac{20}{7} \cdot \frac{1}{5}$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 a) = \frac{20}{7} \cdot \frac{1}{5}$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $7 a$ .

$\frac{1}{7} (7 (a)) = \frac{20}{7} \cdot \frac{1}{5}$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 a) = \frac{20}{7} \cdot \frac{1}{5}$

Step 6.1.1.2.3

Rewrite the expression.

$a = \frac{20}{7} \cdot \frac{1}{5}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Factor $5$ out of $20$ .

$a = \frac{5 (4)}{7} \cdot \frac{1}{5}$

Step 6.2.1.2

Cancel the common factor.

$a = \frac{5 \cdot 4}{7} \cdot \frac{1}{5}$

Step 6.2.1.3

Rewrite the expression.

$a = \frac{4}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{4}{7}$

Decimal Form:

$a = 0.\overline{571428}$