Solve for a 1/10(a+2)=1/4(2-a)

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

Step 1

Simplify $\frac{1}{10} \cdot (a + 2)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{a}{10} + \frac{1}{2 (5)} \cdot 2 = \frac{1}{4} \cdot (2 - a)$

Cancel the common factor.

$\frac{a}{10} + \frac{1}{2 \cdot 5} \cdot 2 = \frac{1}{4} \cdot (2 - a)$

Rewrite the expression.

$\frac{a}{10} + \frac{1}{5} = \frac{1}{4} \cdot (2 - a)$

Step 2

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

Apply the distributive property.

$\frac{a}{10} + \frac{1}{5} = \frac{1}{4} \cdot 2 + \frac{1}{4} (- a)$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{a}{10} + \frac{1}{5} = \frac{1}{2 (2)} \cdot 2 + \frac{1}{4} (- a)$

Cancel the common factor.

$\frac{a}{10} + \frac{1}{5} = \frac{1}{2 \cdot 2} \cdot 2 + \frac{1}{4} (- a)$

Rewrite the expression.

$\frac{a}{10} + \frac{1}{5} = \frac{1}{2} + \frac{1}{4} (- a)$

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

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

Step 3

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

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

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

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

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

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

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

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

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

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

Multiply $10$ by $2$ .

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

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

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

Multiply $4$ by $5$ .

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

Combine the numerators over the common denominator.

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

Reorder $a$ and $5$ .

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

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

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

Step 4

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

Subtract $\frac{1}{5}$ from both sides of the equation.

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

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

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

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

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

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

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

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

Multiply $2$ by $5$ .

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

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

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

Multiply $5$ by $2$ .

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

Combine the numerators over the common denominator.

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

Subtract $2$ from $5$ .

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

Step 5

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

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

Step 6

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $20$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{7} (7 a) = \frac{20}{7} \cdot \frac{3}{10}$

Cancel the common factor of $7$ .

Factor $7$ out of $7 a$ .

$\frac{1}{7} (7 (a)) = \frac{20}{7} \cdot \frac{3}{10}$

Cancel the common factor.

$\frac{1}{7} (7 a) = \frac{20}{7} \cdot \frac{3}{10}$

Rewrite the expression.

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

Simplify the right side.

Simplify $\frac{20}{7} \cdot \frac{3}{10}$ .

Cancel the common factor of $10$ .

Factor $10$ out of $20$ .

$a = \frac{10 (2)}{7} \cdot \frac{3}{10}$

Cancel the common factor.

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

Rewrite the expression.

$a = \frac{2}{7} \cdot 3$

Combine $\frac{2}{7}$ and $3$ .

$a = \frac{2 \cdot 3}{7}$

Multiply $2$ by $3$ .

$a = \frac{6}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{6}{7}$

Decimal Form:

$a = 0.\overline{857142}$

Solve for a 1/10*(a+2)=1/4*(2-a)

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Solve for a 1/10(a+2)=1/4(2-a)