Calculus Examples

$\frac{3}{2} - 5 \frac{6}{3} - 2$

Step 1

Convert $5 \frac{6}{3}$ to an improper fraction.

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Step 1.1

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

$\frac{3}{2} - (5 + \frac{6}{3}) - 2$

Step 1.2

Add $5$ and $\frac{6}{3}$ .

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Step 1.2.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{2} - (5 \cdot \frac{3}{3} + \frac{6}{3}) - 2$

Step 1.2.2

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

$\frac{3}{2} - (\frac{5 \cdot 3}{3} + \frac{6}{3}) - 2$

Step 1.2.3

Combine the numerators over the common denominator.

$\frac{3}{2} - \frac{5 \cdot 3 + 6}{3} - 2$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $5$ by $3$ .

$\frac{3}{2} - \frac{15 + 6}{3} - 2$

Step 1.2.4.2

Add $15$ and $6$ .

$\frac{3}{2} - \frac{21}{3} - 2$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\frac{3}{2} \cdot \frac{3}{3} - \frac{21}{3} - 2$

Step 2.2

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\frac{3 \cdot 3}{2 \cdot 3} - \frac{21}{3} - 2$

Step 2.3

Multiply $\frac{21}{3}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 3}{2 \cdot 3} - (\frac{21}{3} \cdot \frac{2}{2}) - 2$

Step 2.4

Multiply $\frac{21}{3}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 3}{2 \cdot 3} - \frac{21 \cdot 2}{3 \cdot 2} - 2$

Step 2.5

Write $- 2$ as a fraction with denominator $1$ .

$\frac{3 \cdot 3}{2 \cdot 3} - \frac{21 \cdot 2}{3 \cdot 2} + \frac{- 2}{1}$

Step 2.6

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$\frac{3 \cdot 3}{2 \cdot 3} - \frac{21 \cdot 2}{3 \cdot 2} + \frac{- 2}{1} \cdot \frac{6}{6}$

Step 2.7

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$\frac{3 \cdot 3}{2 \cdot 3} - \frac{21 \cdot 2}{3 \cdot 2} + \frac{- 2 \cdot 6}{6}$

Step 2.8

Multiply $2$ by $3$ .

$\frac{3 \cdot 3}{6} - \frac{21 \cdot 2}{3 \cdot 2} + \frac{- 2 \cdot 6}{6}$

Step 2.9

Reorder the factors of $3 \cdot 2$ .

$\frac{3 \cdot 3}{6} - \frac{21 \cdot 2}{2 \cdot 3} + \frac{- 2 \cdot 6}{6}$

Step 2.10

Multiply $2$ by $3$ .

$\frac{3 \cdot 3}{6} - \frac{21 \cdot 2}{6} + \frac{- 2 \cdot 6}{6}$

Step 3

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 21 \cdot 2 - 2 \cdot 6}{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{9 - 21 \cdot 2 - 2 \cdot 6}{6}$

Step 4.2

Multiply $- 21$ by $2$ .

$\frac{9 - 42 - 2 \cdot 6}{6}$

Step 4.3

Multiply $- 2$ by $6$ .

$\frac{9 - 42 - 12}{6}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Subtract $42$ from $9$ .

$\frac{- 33 - 12}{6}$

Step 5.2

Subtract $12$ from $- 33$ .

$\frac{- 45}{6}$

Step 5.3

Cancel the common factor of $- 45$ and $6$ .

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Step 5.3.1

Factor $3$ out of $- 45$ .

$\frac{3 (- 15)}{6}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot - 15}{3 \cdot 2}$

Step 5.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 15}{3 \cdot 2}$

Step 5.3.2.3

Rewrite the expression.

$\frac{- 15}{2}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{15}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{15}{2}$

Decimal Form:

$- 7.5$

Mixed Number Form:

$- 7 \frac{1}{2}$