Finite Math Examples

$8000 = \frac{200 ({(1.04)}^{n} - 1)}{0.04}$

Step 1

Rewrite the equation as $\frac{200 ({1.04}^{n} - 1)}{0.04} = 8000$ .

$\frac{200 ({1.04}^{n} - 1)}{0.04} = 8000$

Step 2

Multiply both sides of the equation by $\frac{0.04}{200}$ .

$\frac{0.04}{200} \cdot \frac{200 ({1.04}^{n} - 1)}{0.04} = \frac{0.04}{200} \cdot 8000$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{0.04}{200} \cdot \frac{200 ({1.04}^{n} - 1)}{0.04}$ .

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Step 3.1.1.1

Combine.

$\frac{0.04 (200 ({1.04}^{n} - 1))}{200 \cdot 0.04} = \frac{0.04}{200} \cdot 8000$

Step 3.1.1.2

Cancel the common factor of $0.04$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{0.04 (200 ({1.04}^{n} - 1))}{200 \cdot 0.04} = \frac{0.04}{200} \cdot 8000$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{200 ({1.04}^{n} - 1)}{200} = \frac{0.04}{200} \cdot 8000$

Step 3.1.1.3

Cancel the common factor of $200$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{200 ({1.04}^{n} - 1)}{200} = \frac{0.04}{200} \cdot 8000$

Step 3.1.1.3.2

Divide ${1.04}^{n} - 1$ by $1$ .

${1.04}^{n} - 1 = \frac{0.04}{200} \cdot 8000$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{0.04}{200} \cdot 8000$ .

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Step 3.2.1.1

Cancel the common factor of $0.04$ .

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Step 3.2.1.1.1

Cancel the common factor.

${1.04}^{n} - 1 = \frac{0.04}{200} \cdot 8000$

Step 3.2.1.1.2

Rewrite the expression.

${1.04}^{n} - 1 = \frac{1}{200} \cdot \frac{8000}{25}$

Step 3.2.1.2

Cancel the common factor of $200$ .

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Step 3.2.1.2.1

Factor $200$ out of $8000$ .

${1.04}^{n} - 1 = \frac{1}{200} \cdot \frac{200 (40)}{25}$

Step 3.2.1.2.2

Cancel the common factor.

${1.04}^{n} - 1 = \frac{1}{200} \cdot \frac{200 \cdot 40}{25}$

Step 3.2.1.2.3

Rewrite the expression.

${1.04}^{n} - 1 = \frac{40}{25}$

Step 3.2.1.3

Cancel the common factor of $40$ and $25$ .

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Step 3.2.1.3.1

Factor $5$ out of $40$ .

${1.04}^{n} - 1 = \frac{5 (8)}{25}$

Step 3.2.1.3.2

Cancel the common factors.

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Step 3.2.1.3.2.1

Factor $5$ out of $25$ .

${1.04}^{n} - 1 = \frac{5 \cdot 8}{5 \cdot 5}$

Step 3.2.1.3.2.2

Cancel the common factor.

${1.04}^{n} - 1 = \frac{5 \cdot 8}{5 \cdot 5}$

Step 3.2.1.3.2.3

Rewrite the expression.

${1.04}^{n} - 1 = \frac{8}{5}$

Step 4

Move all terms not containing $n$ to the right side of the equation.

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Step 4.1

Add $1$ to both sides of the equation.

${1.04}^{n} = \frac{8}{5} + 1$

Step 4.2

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

${1.04}^{n} = \frac{8}{5} + \frac{5}{5}$

Step 4.3

Combine the numerators over the common denominator.

${1.04}^{n} = \frac{8 + 5}{5}$

Step 4.4

Add $8$ and $5$ .

${1.04}^{n} = \frac{13}{5}$

Step 5

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({1.04}^{n}) = \ln (\frac{13}{5})$

Step 6

Expand $\ln ({1.04}^{n})$ by moving $n$ outside the logarithm.

$n \ln (1.04) = \ln (\frac{13}{5})$

Step 7

Divide each term in $n \ln (1.04) = \ln (\frac{13}{5})$ by $\ln (1.04)$ and simplify.

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Step 7.1

Divide each term in $n \ln (1.04) = \ln (\frac{13}{5})$ by $\ln (1.04)$ .

$\frac{n \ln (1.04)}{\ln (1.04)} = \frac{\ln (\frac{13}{5})}{\ln (1.04)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (1.04)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{n \ln (1.04)}{\ln (1.04)} = \frac{\ln (\frac{13}{5})}{\ln (1.04)}$

Step 7.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (\frac{13}{5})}{\ln (1.04)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (\frac{13}{5})}{\ln (1.04)}$

Decimal Form:

$n = 24.36241894 \dots$