Basic Math Examples

- 3 \frac{1}{4}

$- 3 \frac{1}{4}$ ,

$- 6 \frac{1}{2}$

Step 1

Convert

$3 \frac{1}{4}$ to an improper fraction.

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Step 1.1

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

$\overline{x} = - (3 + \frac{1}{4}), - 6 \frac{1}{2}$

Step 1.2

Add

$3$ and

$\frac{1}{4}$ .

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Step 1.2.1

To write

$3$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$\overline{x} = - (3 \cdot \frac{4}{4} + \frac{1}{4}), - 6 \frac{1}{2}$

Step 1.2.2

Combine

$3$ and

$\frac{4}{4}$ .

$\overline{x} = - (\frac{3 \cdot 4}{4} + \frac{1}{4}), - 6 \frac{1}{2}$

Step 1.2.3

Combine the numerators over the common denominator.

$\overline{x} = - \frac{3 \cdot 4 + 1}{4}, - 6 \frac{1}{2}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply

$3$ by

$4$ .

$\overline{x} = - \frac{12 + 1}{4}, - 6 \frac{1}{2}$

Step 1.2.4.2

Add

$12$ and

$1$ .

$\overline{x} = - \frac{13}{4}, - 6 \frac{1}{2}$

Step 2

Convert

$6 \frac{1}{2}$ to an improper fraction.

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Step 2.1

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

$\overline{x} = - \frac{13}{4}, - (6 + \frac{1}{2})$

Step 2.2

Add

$6$ and

$\frac{1}{2}$ .

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Step 2.2.1

To write

$6$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\overline{x} = - \frac{13}{4}, - (6 \cdot \frac{2}{2} + \frac{1}{2})$

Step 2.2.2

Combine

$6$ and

$\frac{2}{2}$ .

$\overline{x} = - \frac{13}{4}, - (\frac{6 \cdot 2}{2} + \frac{1}{2})$

Step 2.2.3

Combine the numerators over the common denominator.

$\overline{x} = - \frac{13}{4}, - \frac{6 \cdot 2 + 1}{2}$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply

$6$ by

$2$ .

$\overline{x} = - \frac{13}{4}, - \frac{12 + 1}{2}$

Step 2.2.4.2

Add

$12$ and

$1$ .

$\overline{x} = - \frac{13}{4}, - \frac{13}{2}$

Step 3

The mean of a set of numbers is the sum divided by the number of terms.

$\overline{x} = \frac{- \frac{13}{4} - \frac{13}{2}}{2}$

Step 4

Simplify the numerator.

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Step 4.1

To write

$- \frac{13}{2}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\overline{x} = \frac{- \frac{13}{4} - \frac{13}{2} \cdot \frac{2}{2}}{2}$

Step 4.2

Write each expression with a common denominator of

$4$ , by multiplying each by an appropriate factor of

$1$ .

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Step 4.2.1

Multiply

$\frac{13}{2}$ by

$\frac{2}{2}$ .

$\overline{x} = \frac{- \frac{13}{4} - \frac{13 \cdot 2}{2 \cdot 2}}{2}$

Step 4.2.2

Multiply

$2$ by

$2$ .

$\overline{x} = \frac{- \frac{13}{4} - \frac{13 \cdot 2}{4}}{2}$

Step 4.3

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{- 13 - 13 \cdot 2}{4}}{2}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Multiply

$- 13$ by

$2$ .

$\overline{x} = \frac{\frac{- 13 - 26}{4}}{2}$

Step 4.4.2

Subtract

$26$ from

$- 13$ .

$\overline{x} = \frac{\frac{- 39}{4}}{2}$

Step 4.5

Move the negative in front of the fraction.

$\overline{x} = \frac{- \frac{39}{4}}{2}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\overline{x} = - \frac{39}{4} \cdot \frac{1}{2}$

Step 6

Multiply

$- \frac{39}{4} \cdot \frac{1}{2}$ .

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Step 6.1

Multiply

$\frac{1}{2}$ by

$\frac{39}{4}$ .

$\overline{x} = - \frac{39}{2 \cdot 4}$

Step 6.2

Multiply

$2$ by

$4$ .

$\overline{x} = - \frac{39}{8}$

Step 7

Divide.

$\overline{x} = - 4.875$

Step 8

The mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$\overline{x} = - 4.9$

Step 9

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

$s^{2} = \sum_{i = 1}^{n} \frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}$

Step 10

Set up the formula for variance for this set of numbers.

$s = \frac{{(- \frac{13}{4} + 4.9)}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11

Simplify the result.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

To write

$4.9$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$s = \frac{{(- \frac{13}{4} + 4.9 \cdot \frac{4}{4})}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.2

Combine

$4.9$ and

$\frac{4}{4}$ .

$s = \frac{{(- \frac{13}{4} + \frac{4.9 \cdot 4}{4})}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.3

Combine the numerators over the common denominator.

$s = \frac{{(\frac{- 13 + 4.9 \cdot 4}{4})}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.4

Simplify the numerator.

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Step 11.1.4.1

Multiply

$4.9$ by

$4$ .

$s = \frac{{(\frac{- 13 + 19.6}{4})}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.4.2

Add

$- 13$ and

$19.6$ .

$s = \frac{{(\frac{6.6}{4})}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.5

Divide

$6.6$ by

$4$ .

$s = \frac{{1.65}^{2} + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.6

Raise

$1.65$ to the power of

$2$ .

$s = \frac{2.7225 + {(- \frac{13}{2} + 4.9)}^{2}}{2 - 1}$

Step 11.1.7

To write

$4.9$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$s = \frac{2.7225 + {(- \frac{13}{2} + 4.9 \cdot \frac{2}{2})}^{2}}{2 - 1}$

Step 11.1.8

Combine

$4.9$ and

$\frac{2}{2}$ .

$s = \frac{2.7225 + {(- \frac{13}{2} + \frac{4.9 \cdot 2}{2})}^{2}}{2 - 1}$

Step 11.1.9

Combine the numerators over the common denominator.

$s = \frac{2.7225 + {(\frac{- 13 + 4.9 \cdot 2}{2})}^{2}}{2 - 1}$

Step 11.1.10

Simplify the numerator.

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Step 11.1.10.1

Multiply

$4.9$ by

$2$ .

$s = \frac{2.7225 + {(\frac{- 13 + 9.8}{2})}^{2}}{2 - 1}$

Step 11.1.10.2

Add

$- 13$ and

$9.8$ .

$s = \frac{2.7225 + {(\frac{- 3.2}{2})}^{2}}{2 - 1}$

Step 11.1.11

Divide

$- 3.2$ by

$2$ .

$s = \frac{2.7225 + {(- 1.6)}^{2}}{2 - 1}$

Step 11.1.12

Raise

$- 1.6$ to the power of

$2$ .

$s = \frac{2.7225 + 2.56}{2 - 1}$

Step 11.1.13

Add

$2.7225$ and

$2.56$ .

$s = \frac{5.2825}{2 - 1}$

Step 11.2

Simplify the expression.

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Step 11.2.1

Subtract

$1$ from

$2$ .

$s = \frac{5.2825}{1}$

Step 11.2.2

Divide

$5.2825$ by

$1$ .

$s = 5.2825$

Step 12

Approximate the result.

$s^{2} \approx 5.2825$