Basic Math Examples

$- \frac{1}{2} \cdot (t + 9) + 5 \frac{3}{4} \cdot t = \frac{3}{8} t$

Step 1

Simplify the left side.

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Step 1.1

Simplify $- \frac{1}{2} \cdot (t + 9) + 5 \frac{3}{4} \cdot t$ .

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Step 1.1.1

Convert $5 \frac{3}{4}$ 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.

$- \frac{1}{2} \cdot (t + 9) + (5 + \frac{3}{4}) \cdot t = \frac{3}{8} t$

Step 1.1.1.2

Add $5$ and $\frac{3}{4}$ .

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Step 1.1.1.2.1

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

$- \frac{1}{2} \cdot (t + 9) + (5 \cdot \frac{4}{4} + \frac{3}{4}) \cdot t = \frac{3}{8} t$

Step 1.1.1.2.2

Combine $5$ and $\frac{4}{4}$ .

$- \frac{1}{2} \cdot (t + 9) + (\frac{5 \cdot 4}{4} + \frac{3}{4}) \cdot t = \frac{3}{8} t$

Step 1.1.1.2.3

Combine the numerators over the common denominator.

$- \frac{1}{2} \cdot (t + 9) + \frac{5 \cdot 4 + 3}{4} \cdot t = \frac{3}{8} t$

Step 1.1.1.2.4

Simplify the numerator.

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Step 1.1.1.2.4.1

Multiply $5$ by $4$ .

$- \frac{1}{2} \cdot (t + 9) + \frac{20 + 3}{4} \cdot t = \frac{3}{8} t$

Step 1.1.1.2.4.2

Add $20$ and $3$ .

$- \frac{1}{2} \cdot (t + 9) + \frac{23}{4} \cdot t = \frac{3}{8} t$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Apply the distributive property.

$- \frac{1}{2} t - \frac{1}{2} \cdot 9 + \frac{23}{4} \cdot t = \frac{3}{8} t$

Step 1.1.2.2

Combine $t$ and $\frac{1}{2}$ .

$- \frac{t}{2} - \frac{1}{2} \cdot 9 + \frac{23}{4} \cdot t = \frac{3}{8} t$

Step 1.1.2.3

Multiply $- \frac{1}{2} \cdot 9$ .

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Step 1.1.2.3.1

Multiply $9$ by $- 1$ .

$- \frac{t}{2} - 9 (\frac{1}{2}) + \frac{23}{4} \cdot t = \frac{3}{8} t$

Step 1.1.2.3.2

Combine $- 9$ and $\frac{1}{2}$ .

$- \frac{t}{2} + \frac{- 9}{2} + \frac{23}{4} \cdot t = \frac{3}{8} t$

Step 1.1.2.4

Move the negative in front of the fraction.

$- \frac{t}{2} - \frac{9}{2} + \frac{23}{4} \cdot t = \frac{3}{8} t$

Step 1.1.2.5

Combine $\frac{23}{4}$ and $t$ .

$- \frac{t}{2} - \frac{9}{2} + \frac{23 t}{4} = \frac{3}{8} t$

Step 1.1.3

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

$- \frac{t}{2} \cdot \frac{2}{2} + \frac{23 t}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.4.1

Multiply $\frac{t}{2}$ by $\frac{2}{2}$ .

$- \frac{t \cdot 2}{2 \cdot 2} + \frac{23 t}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.4.2

Multiply $2$ by $2$ .

$- \frac{t \cdot 2}{4} + \frac{23 t}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{- t \cdot 2 + 23 t}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

Simplify the numerator.

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Step 1.1.6.1.1

Factor $t$ out of $- t \cdot 2 + 23 t$ .

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Step 1.1.6.1.1.1

Factor $t$ out of $- t \cdot 2$ .

$\frac{t (- 1 \cdot 2) + 23 t}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.6.1.1.2

Factor $t$ out of $23 t$ .

$\frac{t (- 1 \cdot 2) + t \cdot 23}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.6.1.1.3

Factor $t$ out of $t (- 1 \cdot 2) + t \cdot 23$ .

$\frac{t (- 1 \cdot 2 + 23)}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.6.1.2

Multiply $- 1$ by $2$ .

$\frac{t (- 2 + 23)}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.6.1.3

Add $- 2$ and $23$ .

$\frac{t \cdot 21}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 1.1.6.2

Move $21$ to the left of $t$ .

$\frac{21 t}{4} - \frac{9}{2} = \frac{3}{8} t$

Step 2

Simplify the right side.

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Step 2.1

Combine $\frac{3}{8}$ and $t$ .

$\frac{21 t}{4} - \frac{9}{2} = \frac{3 t}{8}$

Step 3

Multiply both sides by $8$ .

$(\frac{21 t}{4} - \frac{9}{2}) \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $(\frac{21 t}{4} - \frac{9}{2}) \cdot 8$ .

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Step 4.1.1.1

Apply the distributive property.

$\frac{21 t}{4} \cdot 8 - \frac{9}{2} \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.2

Cancel the common factor of $4$ .

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Step 4.1.1.2.1

Factor $4$ out of $8$ .

$\frac{21 t}{4} \cdot (4 (2)) - \frac{9}{2} \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{21 t}{4} \cdot (4 \cdot 2) - \frac{9}{2} \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.2.3

Rewrite the expression.

$21 t \cdot 2 - \frac{9}{2} \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.3

Multiply $2$ by $21$ .

$42 t - \frac{9}{2} \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.4

Cancel the common factor of $2$ .

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Step 4.1.1.4.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$42 t + \frac{- 9}{2} \cdot 8 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.4.2

Factor $2$ out of $8$ .

$42 t + \frac{- 9}{2} \cdot (2 (4)) = \frac{3 t}{8} \cdot 8$

Step 4.1.1.4.3

Cancel the common factor.

$42 t + \frac{- 9}{2} \cdot (2 \cdot 4) = \frac{3 t}{8} \cdot 8$

Step 4.1.1.4.4

Rewrite the expression.

$42 t - 9 \cdot 4 = \frac{3 t}{8} \cdot 8$

Step 4.1.1.5

Multiply $- 9$ by $4$ .

$42 t - 36 = \frac{3 t}{8} \cdot 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$42 t - 36 = \frac{3 t}{8} \cdot 8$

Step 4.2.1.2

Rewrite the expression.

$42 t - 36 = 3 t$

Step 5

Solve for $t$ .

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Step 5.1

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

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Step 5.1.1

Subtract $3 t$ from both sides of the equation.

$42 t - 36 - 3 t = 0$

Step 5.1.2

Subtract $3 t$ from $42 t$ .

$39 t - 36 = 0$

Step 5.2

Add $36$ to both sides of the equation.

$39 t = 36$

Step 5.3

Divide each term in $39 t = 36$ by $39$ and simplify.

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Step 5.3.1

Divide each term in $39 t = 36$ by $39$ .

$\frac{39 t}{39} = \frac{36}{39}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $39$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{39 t}{39} = \frac{36}{39}$

Step 5.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{36}{39}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $36$ and $39$ .

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Step 5.3.3.1.1

Factor $3$ out of $36$ .

$t = \frac{3 (12)}{39}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $3$ out of $39$ .

$t = \frac{3 \cdot 12}{3 \cdot 13}$

Step 5.3.3.1.2.2

Cancel the common factor.

$t = \frac{3 \cdot 12}{3 \cdot 13}$

Step 5.3.3.1.2.3

Rewrite the expression.

$t = \frac{12}{13}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{12}{13}$

Decimal Form:

$t = 0.\overline{923076}$