Finite Math Examples

$\frac{12}{22} \cdot \frac{10}{21} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $3$ .

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Step 1.1.1

Factor $3$ out of $12$ .

$\frac{3 (4)}{22} \cdot \frac{10}{21} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.1.2

Factor $3$ out of $21$ .

$\frac{3 \cdot 4}{22} \cdot \frac{10}{3 \cdot 7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.1.3

Cancel the common factor.

$\frac{3 \cdot 4}{22} \cdot \frac{10}{3 \cdot 7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.1.4

Rewrite the expression.

$\frac{4}{22} \cdot \frac{10}{7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Factor $2$ out of $22$ .

$\frac{4}{2 (11)} \cdot \frac{10}{7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.2.2

Factor $2$ out of $10$ .

$\frac{4}{2 \cdot 11} \cdot \frac{2 \cdot 5}{7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.2.3

Cancel the common factor.

$\frac{4}{2 \cdot 11} \cdot \frac{2 \cdot 5}{7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.2.4

Rewrite the expression.

$\frac{4}{11} \cdot \frac{5}{7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.3

Multiply $\frac{4}{11}$ by $\frac{5}{7}$ .

$\frac{4 \cdot 5}{11 \cdot 7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.4

Multiply $4$ by $5$ .

$\frac{20}{11 \cdot 7} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.5

Multiply $11$ by $7$ .

$\frac{20}{77} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.6

Cancel the common factor of $20$ .

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Step 1.6.1

Cancel the common factor.

$\frac{20}{77} \cdot \frac{9}{20} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.6.2

Rewrite the expression.

$\frac{1}{77} \cdot 9 \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.7

Combine $\frac{1}{77}$ and $9$ .

$\frac{9}{77} \cdot \frac{8}{19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.8

Multiply $\frac{9}{77} \cdot \frac{8}{19}$ .

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Step 1.8.1

Multiply $\frac{9}{77}$ by $\frac{8}{19}$ .

$\frac{9 \cdot 8}{77 \cdot 19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.8.2

Multiply $9$ by $8$ .

$\frac{72}{77 \cdot 19} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.8.3

Multiply $77$ by $19$ .

$\frac{72}{1463} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.9

Cancel the common factor of $18$ .

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Step 1.9.1

Factor $18$ out of $72$ .

$\frac{18 (4)}{1463} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.9.2

Cancel the common factor.

$\frac{18 \cdot 4}{1463} \cdot \frac{7}{18} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.9.3

Rewrite the expression.

$\frac{4}{1463} \cdot 7 + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.10

Cancel the common factor of $7$ .

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Step 1.10.1

Factor $7$ out of $1463$ .

$\frac{4}{7 (209)} \cdot 7 + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.10.2

Cancel the common factor.

$\frac{4}{7 \cdot 209} \cdot 7 + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.10.3

Rewrite the expression.

$\frac{4}{209} + \frac{10}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.11

Cancel the common factor of $10$ and $22$ .

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Step 1.11.1

Factor $2$ out of $10$ .

$\frac{4}{209} + \frac{2 (5)}{22} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.11.2

Cancel the common factors.

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Step 1.11.2.1

Factor $2$ out of $22$ .

$\frac{4}{209} + \frac{2 \cdot 5}{2 \cdot 11} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.11.2.2

Cancel the common factor.

$\frac{4}{209} + \frac{2 \cdot 5}{2 \cdot 11} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.11.2.3

Rewrite the expression.

$\frac{4}{209} + \frac{5}{11} \cdot \frac{9}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.12

Cancel the common factor of $9$ and $21$ .

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Step 1.12.1

Factor $3$ out of $9$ .

$\frac{4}{209} + \frac{5}{11} \cdot \frac{3 (3)}{21} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.12.2

Cancel the common factors.

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Step 1.12.2.1

Factor $3$ out of $21$ .

$\frac{4}{209} + \frac{5}{11} \cdot \frac{3 \cdot 3}{3 \cdot 7} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.12.2.2

Cancel the common factor.

$\frac{4}{209} + \frac{5}{11} \cdot \frac{3 \cdot 3}{3 \cdot 7} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.12.2.3

Rewrite the expression.

$\frac{4}{209} + \frac{5}{11} \cdot \frac{3}{7} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.13

Multiply $\frac{5}{11} \cdot \frac{3}{7}$ .

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Step 1.13.1

Multiply $\frac{5}{11}$ by $\frac{3}{7}$ .

$\frac{4}{209} + \frac{5 \cdot 3}{11 \cdot 7} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.13.2

Multiply $5$ by $3$ .

$\frac{4}{209} + \frac{15}{11 \cdot 7} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.13.3

Multiply $11$ by $7$ .

$\frac{4}{209} + \frac{15}{77} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.14

Cancel the common factor of $5$ .

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Step 1.14.1

Factor $5$ out of $15$ .

$\frac{4}{209} + \frac{5 (3)}{77} \cdot \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.14.2

Factor $5$ out of $20$ .

$\frac{4}{209} + \frac{5 \cdot 3}{77} \cdot \frac{8}{5 \cdot 4} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.14.3

Cancel the common factor.

$\frac{4}{209} + \frac{5 \cdot 3}{77} \cdot \frac{8}{5 \cdot 4} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.14.4

Rewrite the expression.

$\frac{4}{209} + \frac{3}{77} \cdot \frac{8}{4} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.15

Multiply $\frac{3}{77}$ by $\frac{8}{4}$ .

$\frac{4}{209} + \frac{3 \cdot 8}{77 \cdot 4} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.16

Multiply $3$ by $8$ .

$\frac{4}{209} + \frac{24}{77 \cdot 4} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.17

Multiply $77$ by $4$ .

$\frac{4}{209} + \frac{24}{308} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.18

Cancel the common factor of $7$ .

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Step 1.18.1

Factor $7$ out of $308$ .

$\frac{4}{209} + \frac{24}{7 (44)} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.18.2

Cancel the common factor.

$\frac{4}{209} + \frac{24}{7 \cdot 44} \cdot \frac{7}{19} \cdot \frac{6}{18}$

Step 1.18.3

Rewrite the expression.

$\frac{4}{209} + \frac{24}{44} \cdot \frac{1}{19} \cdot \frac{6}{18}$

Step 1.19

Multiply $\frac{24}{44}$ by $\frac{1}{19}$ .

$\frac{4}{209} + \frac{24}{44 \cdot 19} \cdot \frac{6}{18}$

Step 1.20

Multiply $44$ by $19$ .

$\frac{4}{209} + \frac{24}{836} \cdot \frac{6}{18}$

Step 1.21

Cancel the common factor of $6$ .

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Step 1.21.1

Factor $6$ out of $24$ .

$\frac{4}{209} + \frac{6 (4)}{836} \cdot \frac{6}{18}$

Step 1.21.2

Factor $6$ out of $18$ .

$\frac{4}{209} + \frac{6 \cdot 4}{836} \cdot \frac{6}{6 \cdot 3}$

Step 1.21.3

Cancel the common factor.

$\frac{4}{209} + \frac{6 \cdot 4}{836} \cdot \frac{6}{6 \cdot 3}$

Step 1.21.4

Rewrite the expression.

$\frac{4}{209} + \frac{4}{836} \cdot \frac{6}{3}$

Step 1.22

Cancel the common factor of $2$ .

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Step 1.22.1

Factor $2$ out of $836$ .

$\frac{4}{209} + \frac{4}{2 (418)} \cdot \frac{6}{3}$

Step 1.22.2

Factor $2$ out of $6$ .

$\frac{4}{209} + \frac{4}{2 \cdot 418} \cdot \frac{2 \cdot 3}{3}$

Step 1.22.3

Cancel the common factor.

$\frac{4}{209} + \frac{4}{2 \cdot 418} \cdot \frac{2 \cdot 3}{3}$

Step 1.22.4

Rewrite the expression.

$\frac{4}{209} + \frac{4}{418} \cdot \frac{3}{3}$

Step 1.23

Multiply $\frac{4}{418}$ by $\frac{3}{3}$ .

$\frac{4}{209} + \frac{4 \cdot 3}{418 \cdot 3}$

Step 1.24

Multiply $4$ by $3$ .

$\frac{4}{209} + \frac{12}{418 \cdot 3}$

Step 1.25

Multiply $418$ by $3$ .

$\frac{4}{209} + \frac{12}{1254}$

Step 1.26

Cancel the common factor of $12$ and $1254$ .

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Step 1.26.1

Factor $6$ out of $12$ .

$\frac{4}{209} + \frac{6 (2)}{1254}$

Step 1.26.2

Cancel the common factors.

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Step 1.26.2.1

Factor $6$ out of $1254$ .

$\frac{4}{209} + \frac{6 \cdot 2}{6 \cdot 209}$

Step 1.26.2.2

Cancel the common factor.

$\frac{4}{209} + \frac{6 \cdot 2}{6 \cdot 209}$

Step 1.26.2.3

Rewrite the expression.

$\frac{4}{209} + \frac{2}{209}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{4 + 2}{209}$

Step 2.2

Add $4$ and $2$ .

$\frac{6}{209}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{6}{209}$

Decimal Form:

$0.02870813 \dots$