Basic Math Examples

$11000 = 600 (\frac{1 - {(1.015)}^{n}}{0.015})$

Step 1

Rewrite the equation as $600 (\frac{1 - {1.015}^{n}}{0.015}) = 11000$ .

$600 (\frac{1 - {1.015}^{n}}{0.015}) = 11000$

Step 2

Simplify $600 (\frac{1 - {1.015}^{n}}{0.015})$ .

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Step 2.1

Cancel the common factor of $0.015$ .

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Step 2.1.1

Factor $0.015$ out of $600$ .

$0.015 (40000) \frac{1 - {1.015}^{n}}{0.015} = 11000$

Step 2.1.2

Cancel the common factor.

$0.015 \cdot 40000 \frac{1 - {1.015}^{n}}{0.015} = 11000$

Step 2.1.3

Rewrite the expression.

$40000 (1 - {1.015}^{n}) = 11000$

Step 2.2

Apply the distributive property.

$40000 \cdot 1 + 40000 (- {1.015}^{n}) = 11000$

Step 2.3

Multiply.

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Step 2.3.1

Multiply $40000$ by $1$ .

$40000 + 40000 (- {1.015}^{n}) = 11000$

Step 2.3.2

Multiply $- 1$ by $40000$ .

$40000 - 40000 \cdot {1.015}^{n} = 11000$

Step 3

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

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Step 3.1

Subtract $40000$ from both sides of the equation.

$- 40000 \cdot {1.015}^{n} = 11000 - 40000$

Step 3.2

Subtract $40000$ from $11000$ .

$- 40000 \cdot {1.015}^{n} = - 29000$

Step 4

Divide each term in $- 40000 \cdot {1.015}^{n} = - 29000$ by $- 40000$ and simplify.

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Step 4.1

Divide each term in $- 40000 \cdot {1.015}^{n} = - 29000$ by $- 40000$ .

$\frac{- 40000 \cdot {1.015}^{n}}{- 40000} = \frac{- 29000}{- 40000}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 40000$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 40000 \cdot {1.015}^{n}}{- 40000} = \frac{- 29000}{- 40000}$

Step 4.2.1.2

Divide ${1.015}^{n}$ by $1$ .

${1.015}^{n} = \frac{- 29000}{- 40000}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 29000$ and $- 40000$ .

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Step 4.3.1.1

Factor $- 1000$ out of $- 29000$ .

${1.015}^{n} = \frac{- 1000 (29)}{- 40000}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $- 1000$ out of $- 40000$ .

${1.015}^{n} = \frac{- 1000 \cdot 29}{- 1000 \cdot 40}$

Step 4.3.1.2.2

Cancel the common factor.

${1.015}^{n} = \frac{- 1000 \cdot 29}{- 1000 \cdot 40}$

Step 4.3.1.2.3

Rewrite the expression.

${1.015}^{n} = \frac{29}{40}$

Step 5

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

$\ln ({1.015}^{n}) = \ln (\frac{29}{40})$

Step 6

Expand $\ln ({1.015}^{n})$ by moving $n$ outside the logarithm.

$n \ln (1.015) = \ln (\frac{29}{40})$

Step 7

Divide each term in $n \ln (1.015) = \ln (\frac{29}{40})$ by $\ln (1.015)$ and simplify.

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Step 7.1

Divide each term in $n \ln (1.015) = \ln (\frac{29}{40})$ by $\ln (1.015)$ .

$\frac{n \ln (1.015)}{\ln (1.015)} = \frac{\ln (\frac{29}{40})}{\ln (1.015)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (1.015)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{n \ln (1.015)}{\ln (1.015)} = \frac{\ln (\frac{29}{40})}{\ln (1.015)}$

Step 7.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (\frac{29}{40})}{\ln (1.015)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (\frac{29}{40})}{\ln (1.015)}$

Decimal Form:

$n = - 21.59930109 \dots$