Algebra Examples

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- 5 \frac{1}{2} + \frac{37}{4})$

Step 1

Convert $5 \frac{1}{2}$ to an improper fraction.

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Step 1.1

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

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- (5 + \frac{1}{2}) + \frac{37}{4})$

Step 1.2

Add $5$ and $\frac{1}{2}$ .

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Step 1.2.1

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

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- (5 \cdot \frac{2}{2} + \frac{1}{2}) + \frac{37}{4})$

Step 1.2.2

Combine $5$ and $\frac{2}{2}$ .

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- (\frac{5 \cdot 2}{2} + \frac{1}{2}) + \frac{37}{4})$

Step 1.2.3

Combine the numerators over the common denominator.

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- \frac{5 \cdot 2 + 1}{2} + \frac{37}{4})$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $5$ by $2$ .

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- \frac{10 + 1}{2} + \frac{37}{4})$

Step 1.2.4.2

Add $10$ and $1$ .

$(\frac{2}{5} - \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}) (- \frac{11}{2} + \frac{37}{4})$

Step 2

Simplify each term.

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Step 2.1

Multiply the numerator and denominator of the fraction by $120$ .

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Step 2.1.1

Multiply $\frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}}$ by $\frac{120}{120}$ .

$(\frac{2}{5} - (\frac{120}{120} \cdot \frac{\frac{2}{3} + \frac{4}{5}}{1 + \frac{3}{8}})) (- \frac{11}{2} + \frac{37}{4})$

Step 2.1.2

Combine.

$(\frac{2}{5} - \frac{120 (\frac{2}{3} + \frac{4}{5})}{120 (1 + \frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.2

Apply the distributive property.

$(\frac{2}{5} - \frac{120 (\frac{2}{3}) + 120 (\frac{4}{5})}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $3$ .

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Step 2.3.1.1

Factor $3$ out of $120$ .

$(\frac{2}{5} - \frac{3 (40) \frac{2}{3} + 120 (\frac{4}{5})}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.1.2

Cancel the common factor.

$(\frac{2}{5} - \frac{3 \cdot 40 \frac{2}{3} + 120 (\frac{4}{5})}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.1.3

Rewrite the expression.

$(\frac{2}{5} - \frac{40 \cdot 2 + 120 (\frac{4}{5})}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.2

Multiply $40$ by $2$ .

$(\frac{2}{5} - \frac{80 + 120 (\frac{4}{5})}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.3

Cancel the common factor of $5$ .

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Step 2.3.3.1

Factor $5$ out of $120$ .

$(\frac{2}{5} - \frac{80 + 5 (24) \frac{4}{5}}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.3.2

Cancel the common factor.

$(\frac{2}{5} - \frac{80 + 5 \cdot 24 \frac{4}{5}}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.3.3

Rewrite the expression.

$(\frac{2}{5} - \frac{80 + 24 \cdot 4}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.4

Multiply $24$ by $4$ .

$(\frac{2}{5} - \frac{80 + 96}{120 \cdot 1 + 120 (\frac{3}{8})}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.5

Cancel the common factor of $8$ .

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Step 2.3.5.1

Factor $8$ out of $120$ .

$(\frac{2}{5} - \frac{80 + 96}{120 \cdot 1 + 8 (15) \frac{3}{8}}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.5.2

Cancel the common factor.

$(\frac{2}{5} - \frac{80 + 96}{120 \cdot 1 + 8 \cdot 15 \frac{3}{8}}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.5.3

Rewrite the expression.

$(\frac{2}{5} - \frac{80 + 96}{120 \cdot 1 + 15 \cdot 3}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.3.6

Multiply $15$ by $3$ .

$(\frac{2}{5} - \frac{80 + 96}{120 \cdot 1 + 45}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.4

Add $80$ and $96$ .

$(\frac{2}{5} - \frac{176}{120 \cdot 1 + 45}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Multiply $120$ by $1$ .

$(\frac{2}{5} - \frac{176}{120 + 45}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.5.2

Add $120$ and $45$ .

$(\frac{2}{5} - \frac{176}{165}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.6

Cancel the common factor of $176$ and $165$ .

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Step 2.6.1

Factor $11$ out of $176$ .

$(\frac{2}{5} - \frac{11 (16)}{165}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $11$ out of $165$ .

$(\frac{2}{5} - \frac{11 \cdot 16}{11 \cdot 15}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.6.2.2

Cancel the common factor.

$(\frac{2}{5} - \frac{11 \cdot 16}{11 \cdot 15}) (- \frac{11}{2} + \frac{37}{4})$

Step 2.6.2.3

Rewrite the expression.

$(\frac{2}{5} - \frac{16}{15}) (- \frac{11}{2} + \frac{37}{4})$

Step 3

To write $\frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{2}{5} \cdot \frac{3}{3} - \frac{16}{15}) (- \frac{11}{2} + \frac{37}{4})$

Step 4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{5}$ by $\frac{3}{3}$ .

$(\frac{2 \cdot 3}{5 \cdot 3} - \frac{16}{15}) (- \frac{11}{2} + \frac{37}{4})$

Step 4.2

Multiply $5$ by $3$ .

$(\frac{2 \cdot 3}{15} - \frac{16}{15}) (- \frac{11}{2} + \frac{37}{4})$

Step 5

Combine the numerators over the common denominator.

$\frac{2 \cdot 3 - 16}{15} (- \frac{11}{2} + \frac{37}{4})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $2$ by $3$ .

$\frac{6 - 16}{15} (- \frac{11}{2} + \frac{37}{4})$

Step 6.2

Subtract $16$ from $6$ .

$\frac{- 10}{15} (- \frac{11}{2} + \frac{37}{4})$

Step 7

Cancel the common factor of $- 10$ and $15$ .

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Step 7.1

Factor $5$ out of $- 10$ .

$\frac{5 (- 2)}{15} (- \frac{11}{2} + \frac{37}{4})$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $5$ out of $15$ .

$\frac{5 \cdot - 2}{5 \cdot 3} (- \frac{11}{2} + \frac{37}{4})$

Step 7.2.2

Cancel the common factor.

$\frac{5 \cdot - 2}{5 \cdot 3} (- \frac{11}{2} + \frac{37}{4})$

Step 7.2.3

Rewrite the expression.

$\frac{- 2}{3} (- \frac{11}{2} + \frac{37}{4})$

Step 8

Move the negative in front of the fraction.

$- \frac{2}{3} (- \frac{11}{2} + \frac{37}{4})$

Step 9

To write $- \frac{11}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{2}{3} (- \frac{11}{2} \cdot \frac{2}{2} + \frac{37}{4})$

Step 10

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

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Step 10.1

Multiply $\frac{11}{2}$ by $\frac{2}{2}$ .

$- \frac{2}{3} (- \frac{11 \cdot 2}{2 \cdot 2} + \frac{37}{4})$

Step 10.2

Multiply $2$ by $2$ .

$- \frac{2}{3} (- \frac{11 \cdot 2}{4} + \frac{37}{4})$

Step 11

Combine the numerators over the common denominator.

$- \frac{2}{3} \cdot \frac{- 11 \cdot 2 + 37}{4}$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 11$ by $2$ .

$- \frac{2}{3} \cdot \frac{- 22 + 37}{4}$

Step 12.2

Add $- 22$ and $37$ .

$- \frac{2}{3} \cdot \frac{15}{4}$

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{15}{4}$

Step 13.2

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{15}{4}$

Step 13.3

Factor $2$ out of $4$ .

$\frac{2 \cdot - 1}{3} \cdot \frac{15}{2 \cdot 2}$

Step 13.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{15}{2 \cdot 2}$

Step 13.5

Rewrite the expression.

$\frac{- 1}{3} \cdot \frac{15}{2}$

Step 14

Cancel the common factor of $3$ .

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Step 14.1

Factor $3$ out of $15$ .

$\frac{- 1}{3} \cdot \frac{3 (5)}{2}$

Step 14.2

Cancel the common factor.

$\frac{- 1}{3} \cdot \frac{3 \cdot 5}{2}$

Step 14.3

Rewrite the expression.

$- 1 (\frac{5}{2})$

Step 15

Rewrite $- 1 (\frac{5}{2})$ as $- (\frac{5}{2})$ .

$- \frac{5}{2}$