Finite Math Examples

$120 = 7700 (\frac{0.003}{{(1.003)}^{n} - 1})$

Step 1

Rewrite the equation as $7700 (\frac{0.003}{{1.003}^{n} - 1}) = 120$ .

$7700 (\frac{0.003}{{1.003}^{n} - 1}) = 120$

Step 2

Multiply $7700 (\frac{0.003}{{1.003}^{n} - 1})$ .

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Step 2.1

Combine $7700$ and $\frac{0.003}{{1.003}^{n} - 1}$ .

$\frac{7700 \cdot 0.003}{{1.003}^{n} - 1} = 120$

Step 2.2

Multiply $7700$ by $0.003$ .

$\frac{23.1}{{1.003}^{n} - 1} = 120$

Step 3

Multiply both sides by ${1.003}^{n} - 1$ .

$\frac{23.1}{{1.003}^{n} - 1} ({1.003}^{n} - 1) = 120 ({1.003}^{n} - 1)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of ${1.003}^{n} - 1$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{23.1}{{1.003}^{n} - 1} ({1.003}^{n} - 1) = 120 ({1.003}^{n} - 1)$

Step 4.1.1.2

Rewrite the expression.

$23.1 = 120 ({1.003}^{n} - 1)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $120 ({1.003}^{n} - 1)$ .

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Step 4.2.1.1

Apply the distributive property.

$23.1 = 120 \cdot {1.003}^{n} + 120 \cdot - 1$

Step 4.2.1.2

Multiply $120$ by $- 1$ .

$23.1 = 120 \cdot {1.003}^{n} - 120$

Step 5

Solve for $n$ .

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Step 5.1

Rewrite the equation as $120 \cdot {1.003}^{n} - 120 = 23.1$ .

$120 \cdot {1.003}^{n} - 120 = 23.1$

Step 5.2

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

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Step 5.2.1

Add $120$ to both sides of the equation.

$120 \cdot {1.003}^{n} = 23.1 + 120$

Step 5.2.2

Add $23.1$ and $120$ .

$120 \cdot {1.003}^{n} = 143.1$

Step 5.3

Divide each term in $120 \cdot {1.003}^{n} = 143.1$ by $120$ and simplify.

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Step 5.3.1

Divide each term in $120 \cdot {1.003}^{n} = 143.1$ by $120$ .

$\frac{120 \cdot {1.003}^{n}}{120} = \frac{143.1}{120}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $120$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{120 \cdot {1.003}^{n}}{120} = \frac{143.1}{120}$

Step 5.3.2.1.2

Divide ${1.003}^{n}$ by $1$ .

${1.003}^{n} = \frac{143.1}{120}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $143.1$ by $120$ .

${1.003}^{n} = 1.1925$

Step 5.4

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

$\ln ({1.003}^{n}) = \ln (1.1925)$

Step 5.5

Expand $\ln ({1.003}^{n})$ by moving $n$ outside the logarithm.

$n \ln (1.003) = \ln (1.1925)$

Step 5.6

Divide each term in $n \ln (1.003) = \ln (1.1925)$ by $\ln (1.003)$ and simplify.

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Step 5.6.1

Divide each term in $n \ln (1.003) = \ln (1.1925)$ by $\ln (1.003)$ .

$\frac{n \ln (1.003)}{\ln (1.003)} = \frac{\ln (1.1925)}{\ln (1.003)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $\ln (1.003)$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{n \ln (1.003)}{\ln (1.003)} = \frac{\ln (1.1925)}{\ln (1.003)}$

Step 5.6.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (1.1925)}{\ln (1.003)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (1.1925)}{\ln (1.003)}$

Decimal Form:

$n = 58.77196328 \dots$