Pre-Algebra Examples

$4 \frac{1}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1

Simplify the left side.

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Step 1.1

Simplify $4 \frac{1}{2} + \frac{3}{5} x$ .

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Step 1.1.1

Convert $4 \frac{1}{2}$ to an improper fraction.

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Step 1.1.1.1

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

$4 + \frac{1}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1.1.1.2

Add $4$ and $\frac{1}{2}$ .

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Step 1.1.1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 \cdot \frac{2}{2} + \frac{1}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1.1.1.2.2

Combine $4$ and $\frac{2}{2}$ .

$\frac{4 \cdot 2}{2} + \frac{1}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1.1.1.2.3

Combine the numerators over the common denominator.

$\frac{4 \cdot 2 + 1}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1.1.1.2.4

Simplify the numerator.

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Step 1.1.1.2.4.1

Multiply $4$ by $2$ .

$\frac{8 + 1}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1.1.1.2.4.2

Add $8$ and $1$ .

$\frac{9}{2} + \frac{3}{5} x = 6 \frac{7}{10}$

Step 1.1.2

Combine $\frac{3}{5}$ and $x$ .

$\frac{9}{2} + \frac{3 x}{5} = 6 \frac{7}{10}$

Step 2

Simplify the right side.

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Step 2.1

Convert $6 \frac{7}{10}$ to an improper fraction.

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Step 2.1.1

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

$\frac{9}{2} + \frac{3 x}{5} = 6 + \frac{7}{10}$

Step 2.1.2

Add $6$ and $\frac{7}{10}$ .

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Step 2.1.2.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{9}{2} + \frac{3 x}{5} = 6 \cdot \frac{10}{10} + \frac{7}{10}$

Step 2.1.2.2

Combine $6$ and $\frac{10}{10}$ .

$\frac{9}{2} + \frac{3 x}{5} = \frac{6 \cdot 10}{10} + \frac{7}{10}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{9}{2} + \frac{3 x}{5} = \frac{6 \cdot 10 + 7}{10}$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply $6$ by $10$ .

$\frac{9}{2} + \frac{3 x}{5} = \frac{60 + 7}{10}$

Step 2.1.2.4.2

Add $60$ and $7$ .

$\frac{9}{2} + \frac{3 x}{5} = \frac{67}{10}$

Step 3

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

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Step 3.1

Subtract $\frac{9}{2}$ from both sides of the equation.

$\frac{3 x}{5} = \frac{67}{10} - \frac{9}{2}$

Step 3.2

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 x}{5} = \frac{67}{10} - \frac{9}{2} \cdot \frac{5}{5}$

Step 3.3

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

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Step 3.3.1

Multiply $\frac{9}{2}$ by $\frac{5}{5}$ .

$\frac{3 x}{5} = \frac{67}{10} - \frac{9 \cdot 5}{2 \cdot 5}$

Step 3.3.2

Multiply $2$ by $5$ .

$\frac{3 x}{5} = \frac{67}{10} - \frac{9 \cdot 5}{10}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3 x}{5} = \frac{67 - 9 \cdot 5}{10}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 9$ by $5$ .

$\frac{3 x}{5} = \frac{67 - 45}{10}$

Step 3.5.2

Subtract $45$ from $67$ .

$\frac{3 x}{5} = \frac{22}{10}$

Step 3.6

Cancel the common factor of $22$ and $10$ .

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Step 3.6.1

Factor $2$ out of $22$ .

$\frac{3 x}{5} = \frac{2 (11)}{10}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $2$ out of $10$ .

$\frac{3 x}{5} = \frac{2 \cdot 11}{2 \cdot 5}$

Step 3.6.2.2

Cancel the common factor.

$\frac{3 x}{5} = \frac{2 \cdot 11}{2 \cdot 5}$

Step 3.6.2.3

Rewrite the expression.

$\frac{3 x}{5} = \frac{11}{5}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = 11$

Step 5

Divide each term in $3 x = 11$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 11$ by $3$ .

$\frac{3 x}{3} = \frac{11}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{11}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{3}$

Decimal Form:

$x = 3.\overline{6}$

Mixed Number Form:

$x = 3 \frac{2}{3}$