Pre-Algebra Examples

$- \frac{1}{6} t - \frac{1}{12} = - \frac{1}{3} t + \frac{1}{2} \cdot \frac{5}{12} \cdot t$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{1}{3} t + \frac{1}{2} \cdot \frac{5}{12} \cdot t = - \frac{1}{6} t - \frac{1}{12}$

Step 2

Simplify $- \frac{1}{3} t + \frac{1}{2} \cdot \frac{5}{12} \cdot t$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$- \frac{t}{3} + \frac{1}{2} \cdot \frac{5}{12} \cdot t = - \frac{1}{6} t - \frac{1}{12}$

Step 2.1.2

Multiply $\frac{1}{2} \cdot \frac{5}{12}$ .

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Step 2.1.2.1

Multiply $\frac{1}{2}$ by $\frac{5}{12}$ .

$- \frac{t}{3} + \frac{5}{2 \cdot 12} \cdot t = - \frac{1}{6} t - \frac{1}{12}$

Step 2.1.2.2

Multiply $2$ by $12$ .

$- \frac{t}{3} + \frac{5}{24} \cdot t = - \frac{1}{6} t - \frac{1}{12}$

Step 2.1.3

Combine $\frac{5}{24}$ and $t$ .

$- \frac{t}{3} + \frac{5 t}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.2

To write $- \frac{t}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{t}{3} \cdot \frac{8}{8} + \frac{5 t}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.3

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{t}{3}$ by $\frac{8}{8}$ .

$- \frac{t \cdot 8}{3 \cdot 8} + \frac{5 t}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.3.2

Multiply $3$ by $8$ .

$- \frac{t \cdot 8}{24} + \frac{5 t}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- t \cdot 8 + 5 t}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $t$ out of $- t \cdot 8 + 5 t$ .

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Step 2.5.1.1

Factor $t$ out of $- t \cdot 8$ .

$\frac{t (- 1 \cdot 8) + 5 t}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.5.1.2

Factor $t$ out of $5 t$ .

$\frac{t (- 1 \cdot 8) + t \cdot 5}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.5.1.3

Factor $t$ out of $t (- 1 \cdot 8) + t \cdot 5$ .

$\frac{t (- 1 \cdot 8 + 5)}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.5.2

Multiply $- 1$ by $8$ .

$\frac{t (- 8 + 5)}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.5.3

Add $- 8$ and $5$ .

$\frac{t \cdot - 3}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.6

Reduce the expression by cancelling the common factors.

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Step 2.6.1

Cancel the common factor of $- 3$ and $24$ .

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Step 2.6.1.1

Factor $3$ out of $t \cdot - 3$ .

$\frac{3 (t \cdot - 1)}{24} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.6.1.2

Cancel the common factors.

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Step 2.6.1.2.1

Factor $3$ out of $24$ .

$\frac{3 (t \cdot - 1)}{3 (8)} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.6.1.2.2

Cancel the common factor.

$\frac{3 (t \cdot - 1)}{3 \cdot 8} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.6.1.2.3

Rewrite the expression.

$\frac{t \cdot - 1}{8} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.6.2

Simplify the expression.

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Step 2.6.2.1

Move $- 1$ to the left of $t$ .

$\frac{- 1 \cdot t}{8} = - \frac{1}{6} t - \frac{1}{12}$

Step 2.6.2.2

Move the negative in front of the fraction.

$- \frac{t}{8} = - \frac{1}{6} t - \frac{1}{12}$

Step 3

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

$- \frac{t}{8} = - \frac{t}{6} - \frac{1}{12}$

Step 4

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

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Step 4.1

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

$- \frac{t}{8} + \frac{t}{6} = - \frac{1}{12}$

Step 4.2

To write $- \frac{t}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{t}{8} \cdot \frac{3}{3} + \frac{t}{6} = - \frac{1}{12}$

Step 4.3

To write $\frac{t}{6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{t}{8} \cdot \frac{3}{3} + \frac{t}{6} \cdot \frac{4}{4} = - \frac{1}{12}$

Step 4.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{t}{8}$ by $\frac{3}{3}$ .

$- \frac{t \cdot 3}{8 \cdot 3} + \frac{t}{6} \cdot \frac{4}{4} = - \frac{1}{12}$

Step 4.4.2

Multiply $8$ by $3$ .

$- \frac{t \cdot 3}{24} + \frac{t}{6} \cdot \frac{4}{4} = - \frac{1}{12}$

Step 4.4.3

Multiply $\frac{t}{6}$ by $\frac{4}{4}$ .

$- \frac{t \cdot 3}{24} + \frac{t \cdot 4}{6 \cdot 4} = - \frac{1}{12}$

Step 4.4.4

Multiply $6$ by $4$ .

$- \frac{t \cdot 3}{24} + \frac{t \cdot 4}{24} = - \frac{1}{12}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{- t \cdot 3 + t \cdot 4}{24} = - \frac{1}{12}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Factor $t$ out of $- t \cdot 3 + t \cdot 4$ .

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Step 4.6.1.1

Factor $t$ out of $- t \cdot 3$ .

$\frac{t (- 1 \cdot 3) + t \cdot 4}{24} = - \frac{1}{12}$

Step 4.6.1.2

Factor $t$ out of $t \cdot 4$ .

$\frac{t (- 1 \cdot 3) + t (4)}{24} = - \frac{1}{12}$

Step 4.6.1.3

Factor $t$ out of $t (- 1 \cdot 3) + t (4)$ .

$\frac{t (- 1 \cdot 3 + 4)}{24} = - \frac{1}{12}$

Step 4.6.2

Multiply $- 1$ by $3$ .

$\frac{t (- 3 + 4)}{24} = - \frac{1}{12}$

Step 4.6.3

Add $- 3$ and $4$ .

$\frac{t \cdot 1}{24} = - \frac{1}{12}$

Step 4.7

Multiply $t$ by $1$ .

$\frac{t}{24} = - \frac{1}{12}$

Step 5

Multiply both sides of the equation by $24$ .

$24 \frac{t}{24} = 24 (- \frac{1}{12})$

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

Cancel the common factor of $24$ .

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Step 6.1.1.1

Cancel the common factor.

$24 \frac{t}{24} = 24 (- \frac{1}{12})$

Step 6.1.1.2

Rewrite the expression.

$t = 24 (- \frac{1}{12})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $24 (- \frac{1}{12})$ .

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Step 6.2.1.1

Cancel the common factor of $12$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$t = 24 (\frac{- 1}{12})$

Step 6.2.1.1.2

Factor $12$ out of $24$ .

$t = 12 (2) \frac{- 1}{12}$

Step 6.2.1.1.3

Cancel the common factor.

$t = 12 \cdot 2 \frac{- 1}{12}$

Step 6.2.1.1.4

Rewrite the expression.

$t = 2 \cdot - 1$

Step 6.2.1.2

Multiply $2$ by $- 1$ .

$t = - 2$