Solve for a 1/10(a+1)=1/6(2-a)

$\frac{1}{10} \cdot (a + 1) = \frac{1}{6} \cdot (2 - a)$

Step 1

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

Rewrite.

$0 + 0 + \frac{1}{10} \cdot (a + 1) = \frac{1}{6} \cdot (2 - a)$

Simplify by adding zeros.

$\frac{1}{10} \cdot (a + 1) = \frac{1}{6} \cdot (2 - a)$

Apply the distributive property.

$\frac{1}{10} a + \frac{1}{10} \cdot 1 = \frac{1}{6} \cdot (2 - a)$

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

$\frac{a}{10} + \frac{1}{10} \cdot 1 = \frac{1}{6} \cdot (2 - a)$

Multiply $\frac{1}{10}$ by $1$ .

$\frac{a}{10} + \frac{1}{10} = \frac{1}{6} \cdot (2 - a)$

Step 2

Simplify $\frac{1}{6} \cdot (2 - a)$ .

Apply the distributive property.

$\frac{a}{10} + \frac{1}{10} = \frac{1}{6} \cdot 2 + \frac{1}{6} (- a)$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{a}{10} + \frac{1}{10} = \frac{1}{2 (3)} \cdot 2 + \frac{1}{6} (- a)$

Cancel the common factor.

$\frac{a}{10} + \frac{1}{10} = \frac{1}{2 \cdot 3} \cdot 2 + \frac{1}{6} (- a)$

Rewrite the expression.

$\frac{a}{10} + \frac{1}{10} = \frac{1}{3} + \frac{1}{6} (- a)$

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

$\frac{a}{10} + \frac{1}{10} = \frac{1}{3} - \frac{a}{6}$

Step 3

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

Add $\frac{a}{6}$ to both sides of the equation.

$\frac{a}{10} + \frac{1}{10} + \frac{a}{6} = \frac{1}{3}$

To write $\frac{a}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{a}{10} \cdot \frac{3}{3} + \frac{a}{6} + \frac{1}{10} = \frac{1}{3}$

To write $\frac{a}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{a}{10} \cdot \frac{3}{3} + \frac{a}{6} \cdot \frac{5}{5} + \frac{1}{10} = \frac{1}{3}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a}{10}$ by $\frac{3}{3}$ .

$\frac{a \cdot 3}{10 \cdot 3} + \frac{a}{6} \cdot \frac{5}{5} + \frac{1}{10} = \frac{1}{3}$

Multiply $10$ by $3$ .

$\frac{a \cdot 3}{30} + \frac{a}{6} \cdot \frac{5}{5} + \frac{1}{10} = \frac{1}{3}$

Multiply $\frac{a}{6}$ by $\frac{5}{5}$ .

$\frac{a \cdot 3}{30} + \frac{a \cdot 5}{6 \cdot 5} + \frac{1}{10} = \frac{1}{3}$

Multiply $6$ by $5$ .

$\frac{a \cdot 3}{30} + \frac{a \cdot 5}{30} + \frac{1}{10} = \frac{1}{3}$

Combine the numerators over the common denominator.

$\frac{a \cdot 3 + a \cdot 5}{30} + \frac{1}{10} = \frac{1}{3}$

Reorder $a$ and $5$ .

$\frac{3 \cdot a + 5 \cdot a}{30} + \frac{1}{10} = \frac{1}{3}$

Add $3 \cdot a$ and $5 \cdot a$ .

$\frac{8 \cdot a}{30} + \frac{1}{10} = \frac{1}{3}$

Cancel the common factor of $8$ and $30$ .

Factor $2$ out of $8 \cdot a$ .

$\frac{2 (4 \cdot a)}{30} + \frac{1}{10} = \frac{1}{3}$

Cancel the common factors.

Factor $2$ out of $30$ .

$\frac{2 (4 \cdot a)}{2 (15)} + \frac{1}{10} = \frac{1}{3}$

Cancel the common factor.

$\frac{2 (4 \cdot a)}{2 \cdot 15} + \frac{1}{10} = \frac{1}{3}$

Rewrite the expression.

$\frac{4 \cdot a}{15} + \frac{1}{10} = \frac{1}{3}$

$\frac{4 a}{15} + \frac{1}{10} = \frac{1}{3}$

Step 4

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

Subtract $\frac{1}{10}$ from both sides of the equation.

$\frac{4 a}{15} = \frac{1}{3} - \frac{1}{10}$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{4 a}{15} = \frac{1}{3} \cdot \frac{10}{10} - \frac{1}{10}$

To write $- \frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 a}{15} = \frac{1}{3} \cdot \frac{10}{10} - \frac{1}{10} \cdot \frac{3}{3}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{10}{10}$ .

$\frac{4 a}{15} = \frac{10}{3 \cdot 10} - \frac{1}{10} \cdot \frac{3}{3}$

Multiply $3$ by $10$ .

$\frac{4 a}{15} = \frac{10}{30} - \frac{1}{10} \cdot \frac{3}{3}$

Multiply $\frac{1}{10}$ by $\frac{3}{3}$ .

$\frac{4 a}{15} = \frac{10}{30} - \frac{3}{10 \cdot 3}$

Multiply $10$ by $3$ .

$\frac{4 a}{15} = \frac{10}{30} - \frac{3}{30}$

Combine the numerators over the common denominator.

$\frac{4 a}{15} = \frac{10 - 3}{30}$

Subtract $3$ from $10$ .

$\frac{4 a}{15} = \frac{7}{30}$

Step 5

Multiply both sides of the equation by $\frac{15}{4}$ .

$\frac{15}{4} \cdot \frac{4 a}{15} = \frac{15}{4} \cdot \frac{7}{30}$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{15}{4} \cdot \frac{4 a}{15}$ .

Cancel the common factor of $15$ .

Cancel the common factor.

$\frac{15}{4} \cdot \frac{4 a}{15} = \frac{15}{4} \cdot \frac{7}{30}$

Rewrite the expression.

$\frac{1}{4} (4 a) = \frac{15}{4} \cdot \frac{7}{30}$

Cancel the common factor of $4$ .

Factor $4$ out of $4 a$ .

$\frac{1}{4} (4 (a)) = \frac{15}{4} \cdot \frac{7}{30}$

Cancel the common factor.

$\frac{1}{4} (4 a) = \frac{15}{4} \cdot \frac{7}{30}$

Rewrite the expression.

$a = \frac{15}{4} \cdot \frac{7}{30}$

Simplify the right side.

Simplify $\frac{15}{4} \cdot \frac{7}{30}$ .

Cancel the common factor of $15$ .

Factor $15$ out of $30$ .

$a = \frac{15}{4} \cdot \frac{7}{15 (2)}$

Cancel the common factor.

$a = \frac{15}{4} \cdot \frac{7}{15 \cdot 2}$

Rewrite the expression.

$a = \frac{1}{4} \cdot \frac{7}{2}$

Multiply $\frac{1}{4}$ by $\frac{7}{2}$ .

$a = \frac{7}{4 \cdot 2}$

Multiply $4$ by $2$ .

$a = \frac{7}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{7}{8}$

Decimal Form:

$a = 0.875$

Solve for a 1/10*(a+1)=1/6*(2-a)

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Solve for a 1/10(a+1)=1/6(2-a)