Finite Math Examples

$1 - 1 + \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{9} + \frac{1}{4} - \frac{1}{16}$

Step 1

Combine the numerators over the common denominator.

$1 - 1 + \frac{1}{2} + \frac{- 1 + 1}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 2

Add $- 1$ and $1$ .

$1 - 1 + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3

Find the common denominator.

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Step 3.1

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

$\frac{1}{1} - 1 + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.2

Multiply $\frac{1}{1}$ by $\frac{144}{144}$ .

$\frac{1}{1} \cdot \frac{144}{144} - 1 + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.3

Multiply $\frac{1}{1}$ by $\frac{144}{144}$ .

$\frac{144}{144} - 1 + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.4

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

$\frac{144}{144} + \frac{- 1}{1} + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.5

Multiply $\frac{- 1}{1}$ by $\frac{144}{144}$ .

$\frac{144}{144} + \frac{- 1}{1} \cdot \frac{144}{144} + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.6

Multiply $\frac{- 1}{1}$ by $\frac{144}{144}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{1}{2} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.7

Multiply $\frac{1}{2}$ by $\frac{72}{72}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{1}{2} \cdot \frac{72}{72} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.8

Multiply $\frac{1}{2}$ by $\frac{72}{72}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0}{4} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.9

Multiply $\frac{0}{4}$ by $\frac{36}{36}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0}{4} \cdot \frac{36}{36} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.10

Multiply $\frac{0}{4}$ by $\frac{36}{36}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{1}{3} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.11

Multiply $\frac{1}{3}$ by $\frac{48}{48}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{1}{3} \cdot \frac{48}{48} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.12

Multiply $\frac{1}{3}$ by $\frac{48}{48}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{48}{3 \cdot 48} + \frac{- 1}{9} + \frac{- 1}{16}$

Step 3.13

Multiply $\frac{- 1}{9}$ by $\frac{16}{16}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{48}{3 \cdot 48} + \frac{- 1}{9} \cdot \frac{16}{16} + \frac{- 1}{16}$

Step 3.14

Multiply $\frac{- 1}{9}$ by $\frac{16}{16}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{48}{3 \cdot 48} + \frac{- 1 \cdot 16}{9 \cdot 16} + \frac{- 1}{16}$

Step 3.15

Multiply $\frac{- 1}{16}$ by $\frac{9}{9}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{48}{3 \cdot 48} + \frac{- 1 \cdot 16}{9 \cdot 16} + \frac{- 1}{16} \cdot \frac{9}{9}$

Step 3.16

Multiply $\frac{- 1}{16}$ by $\frac{9}{9}$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{2 \cdot 72} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{48}{3 \cdot 48} + \frac{- 1 \cdot 16}{9 \cdot 16} + \frac{- 1 \cdot 9}{16 \cdot 9}$

Step 3.17

Multiply $2$ by $72$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{144} + \frac{0 \cdot 36}{4 \cdot 36} + \frac{48}{3 \cdot 48} + \frac{- 1 \cdot 16}{9 \cdot 16} + \frac{- 1 \cdot 9}{16 \cdot 9}$

Step 3.18

Multiply $4$ by $36$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{144} + \frac{0 \cdot 36}{144} + \frac{48}{3 \cdot 48} + \frac{- 1 \cdot 16}{9 \cdot 16} + \frac{- 1 \cdot 9}{16 \cdot 9}$

Step 3.19

Multiply $3$ by $48$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{144} + \frac{0 \cdot 36}{144} + \frac{48}{144} + \frac{- 1 \cdot 16}{9 \cdot 16} + \frac{- 1 \cdot 9}{16 \cdot 9}$

Step 3.20

Multiply $9$ by $16$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{144} + \frac{0 \cdot 36}{144} + \frac{48}{144} + \frac{- 1 \cdot 16}{144} + \frac{- 1 \cdot 9}{16 \cdot 9}$

Step 3.21

Reorder the factors of $16 \cdot 9$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{144} + \frac{0 \cdot 36}{144} + \frac{48}{144} + \frac{- 1 \cdot 16}{144} + \frac{- 1 \cdot 9}{9 \cdot 16}$

Step 3.22

Multiply $9$ by $16$ .

$\frac{144}{144} + \frac{- 1 \cdot 144}{144} + \frac{72}{144} + \frac{0 \cdot 36}{144} + \frac{48}{144} + \frac{- 1 \cdot 16}{144} + \frac{- 1 \cdot 9}{144}$

Step 4

Combine the numerators over the common denominator.

$\frac{144 - 1 \cdot 144 + 72 + 0 \cdot 36 + 48 - 1 \cdot 16 - 1 \cdot 9}{144}$

Step 5

Simplify each term.

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Step 5.1

Multiply $- 1$ by $144$ .

$\frac{144 - 144 + 72 + 0 \cdot 36 + 48 - 1 \cdot 16 - 1 \cdot 9}{144}$

Step 5.2

Multiply $0$ by $36$ .

$\frac{144 - 144 + 72 + 0 + 48 - 1 \cdot 16 - 1 \cdot 9}{144}$

Step 5.3

Multiply $- 1$ by $16$ .

$\frac{144 - 144 + 72 + 0 + 48 - 16 - 1 \cdot 9}{144}$

Step 5.4

Multiply $- 1$ by $9$ .

$\frac{144 - 144 + 72 + 0 + 48 - 16 - 9}{144}$

Step 6

Simplify by adding and subtracting.

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Step 6.1

Subtract $144$ from $144$ .

$\frac{0 + 72 + 0 + 48 - 16 - 9}{144}$

Step 6.2

Add $0$ and $72$ .

$\frac{72 + 0 + 48 - 16 - 9}{144}$

Step 6.3

Add $72$ and $0$ .

$\frac{72 + 48 - 16 - 9}{144}$

Step 6.4

Add $72$ and $48$ .

$\frac{120 - 16 - 9}{144}$

Step 6.5

Subtract $16$ from $120$ .

$\frac{104 - 9}{144}$

Step 6.6

Subtract $9$ from $104$ .

$\frac{95}{144}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{95}{144}$

Decimal Form:

$0.6597\overline{2}$