Algebra Examples

$3.\overline{3}$

Step 1

Create two equations where the only numbers to the right of the decimal place are the repeating part.

Step 2

Set up an equation with $0.\overline{3}$ .

$x = 0.\overline{3}$

Step 3

Multiply both sides of $x = 0.\overline{3}$ by $10$ to create the second equation.

$10 x = 3.\overline{3}$

Step 4

Subtract $x = 0.\overline{3}$ from $10 x = 3.\overline{3}$ to remove the repeating part.

$9 x = 3$

Step 5

Divide each term in $9 x = 3$ by $9$ and simplify.

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Step 5.1

Divide each term in $9 x = 3$ by $9$ .

$\frac{9 x}{9} = \frac{3}{9}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $9$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{3}{9}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{9}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $3$ and $9$ .

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Step 5.3.1.1

Factor $3$ out of $3$ .

$x = \frac{3 (1)}{9}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 1}{3 \cdot 3}$

Step 5.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 1}{3 \cdot 3}$

Step 5.3.1.2.3

Rewrite the expression.

$x = \frac{1}{3}$

Step 6

Write as a mixed number by taking the whole number part along with the solution to the equation.

$3 \frac{1}{3}$

Step 7

Convert $3 \frac{1}{3}$ to an improper fraction.

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Step 7.1

A mixed number is an addition of its whole and fractional parts.

$3 + \frac{1}{3}$

Step 7.2

Add $3$ and $\frac{1}{3}$ .

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Step 7.2.1

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

$3 \cdot \frac{3}{3} + \frac{1}{3}$

Step 7.2.2

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

$\frac{3 \cdot 3}{3} + \frac{1}{3}$

Step 7.2.3

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 + 1}{3}$

Step 7.2.4

Simplify the numerator.

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Step 7.2.4.1

Multiply $3$ by $3$ .

$\frac{9 + 1}{3}$

Step 7.2.4.2

Add $9$ and $1$ .

$\frac{10}{3}$