Basic Math Examples

$93000 = 3200 ((\frac{{(1 + 0.04)}^{n + 1} - 1}{0.04}) - 1)$

Step 1

Rewrite the equation as $3200 (\frac{{(1 + 0.04)}^{n + 1} - 1}{0.04} - 1) = 93000$ .

$3200 (\frac{{(1 + 0.04)}^{n + 1} - 1}{0.04} - 1) = 93000$

Step 2

Add $1$ and $0.04$ .

$3200 (\frac{{1.04}^{n + 1} - 1}{0.04} - 1) = 93000$

Step 3

Divide each term in $3200 (\frac{{1.04}^{n + 1} - 1}{0.04} - 1) = 93000$ by $3200$ and simplify.

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Step 3.1

Divide each term in $3200 (\frac{{1.04}^{n + 1} - 1}{0.04} - 1) = 93000$ by $3200$ .

$\frac{3200 (\frac{{1.04}^{n + 1} - 1}{0.04} - 1)}{3200} = \frac{93000}{3200}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3200$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3200 (\frac{{1.04}^{n + 1} - 1}{0.04} - 1)}{3200} = \frac{93000}{3200}$

Step 3.2.1.2

Divide $\frac{{1.04}^{n + 1} - 1}{0.04} - 1$ by $1$ .

$\frac{{1.04}^{n + 1} - 1}{0.04} - 1 = \frac{93000}{3200}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply by $1$ .

$\frac{1 \cdot {1.04}^{n + 1} - 1}{0.04} - 1 = \frac{93000}{3200}$

Step 3.2.2.2

Rewrite $- 1$ as $1 (- 1)$ .

$\frac{1 \cdot {1.04}^{n + 1} + 1 (- 1)}{0.04} - 1 = \frac{93000}{3200}$

Step 3.2.2.3

Factor $1$ out of $1 \cdot {1.04}^{n + 1} + 1 (- 1)$ .

$\frac{1 \cdot ({1.04}^{n + 1} - 1)}{0.04} - 1 = \frac{93000}{3200}$

Step 3.2.2.4

Factor $0.04$ out of $0.04$ .

$\frac{1 \cdot ({1.04}^{n + 1} - 1)}{0.04 (1)} - 1 = \frac{93000}{3200}$

Step 3.2.2.5

Separate fractions.

$\frac{1}{0.04} \cdot \frac{{1.04}^{n + 1} - 1}{1} - 1 = \frac{93000}{3200}$

Step 3.2.2.6

Divide $1$ by $0.04$ .

$25 \frac{{1.04}^{n + 1} - 1}{1} - 1 = \frac{93000}{3200}$

Step 3.2.2.7

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

$25 ({1.04}^{n + 1} - 1) - 1 = \frac{93000}{3200}$

Step 3.2.2.8

Apply the distributive property.

$25 \cdot {1.04}^{n + 1} + 25 \cdot - 1 - 1 = \frac{93000}{3200}$

Step 3.2.2.9

Multiply $25$ by $- 1$ .

$25 \cdot {1.04}^{n + 1} - 25 - 1 = \frac{93000}{3200}$

Step 3.2.3

Subtract $1$ from $- 25$ .

$25 \cdot {1.04}^{n + 1} - 26 = \frac{93000}{3200}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $93000$ and $3200$ .

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Step 3.3.1.1

Factor $200$ out of $93000$ .

$25 \cdot {1.04}^{n + 1} - 26 = \frac{200 (465)}{3200}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $200$ out of $3200$ .

$25 \cdot {1.04}^{n + 1} - 26 = \frac{200 \cdot 465}{200 \cdot 16}$

Step 3.3.1.2.2

Cancel the common factor.

$25 \cdot {1.04}^{n + 1} - 26 = \frac{200 \cdot 465}{200 \cdot 16}$

Step 3.3.1.2.3

Rewrite the expression.

$25 \cdot {1.04}^{n + 1} - 26 = \frac{465}{16}$

Step 4

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

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Step 4.1

Add $26$ to both sides of the equation.

$25 \cdot {1.04}^{n + 1} = \frac{465}{16} + 26$

Step 4.2

To write $26$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$25 \cdot {1.04}^{n + 1} = \frac{465}{16} + 26 \cdot \frac{16}{16}$

Step 4.3

Combine $26$ and $\frac{16}{16}$ .

$25 \cdot {1.04}^{n + 1} = \frac{465}{16} + \frac{26 \cdot 16}{16}$

Step 4.4

Combine the numerators over the common denominator.

$25 \cdot {1.04}^{n + 1} = \frac{465 + 26 \cdot 16}{16}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $26$ by $16$ .

$25 \cdot {1.04}^{n + 1} = \frac{465 + 416}{16}$

Step 4.5.2

Add $465$ and $416$ .

$25 \cdot {1.04}^{n + 1} = \frac{881}{16}$

Step 5

Divide each term in $25 \cdot {1.04}^{n + 1} = \frac{881}{16}$ by $25$ and simplify.

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Step 5.1

Divide each term in $25 \cdot {1.04}^{n + 1} = \frac{881}{16}$ by $25$ .

$\frac{25 \cdot {1.04}^{n + 1}}{25} = \frac{\frac{881}{16}}{25}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $25$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{25 \cdot {1.04}^{n + 1}}{25} = \frac{\frac{881}{16}}{25}$

Step 5.2.1.2

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

${1.04}^{n + 1} = \frac{\frac{881}{16}}{25}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

${1.04}^{n + 1} = \frac{881}{16} \cdot \frac{1}{25}$

Step 5.3.2

Multiply $\frac{881}{16} \cdot \frac{1}{25}$ .

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Step 5.3.2.1

Multiply $\frac{881}{16}$ by $\frac{1}{25}$ .

${1.04}^{n + 1} = \frac{881}{16 \cdot 25}$

Step 5.3.2.2

Multiply $16$ by $25$ .

${1.04}^{n + 1} = \frac{881}{400}$

Step 6

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

$\ln ({1.04}^{n + 1}) = \ln (\frac{881}{400})$

Step 7

Expand $\ln ({1.04}^{n + 1})$ by moving $n + 1$ outside the logarithm.

$(n + 1) \ln (1.04) = \ln (\frac{881}{400})$

Step 8

Simplify the left side.

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Step 8.1

Simplify $(n + 1) \ln (1.04)$ .

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Step 8.1.1

Apply the distributive property.

$n \ln (1.04) + 1 \ln (1.04) = \ln (\frac{881}{400})$

Step 8.1.2

Multiply $\ln (1.04)$ by $1$ .

$n \ln (1.04) + \ln (1.04) = \ln (\frac{881}{400})$

Step 9

Move all the terms containing a logarithm to the left side of the equation.

$n \ln (1.04) + \ln (1.04) - \ln (\frac{881}{400}) = 0$

Step 10

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$n \ln (1.04) + \ln (\frac{1.04}{\frac{881}{400}}) = 0$

Step 11

Simplify each term.

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Step 11.1

Multiply the numerator by the reciprocal of the denominator.

$n \ln (1.04) + \ln (1.04 (\frac{400}{881})) = 0$

Step 11.2

Multiply $1.04 (\frac{400}{881})$ .

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Step 11.2.1

Combine $1.04$ and $\frac{400}{881}$ .

$n \ln (1.04) + \ln (\frac{1.04 \cdot 400}{881}) = 0$

Step 11.2.2

Multiply $1.04$ by $400$ .

$n \ln (1.04) + \ln (\frac{416}{881}) = 0$

Step 11.3

Cancel the common factor of $416$ and $881$ .

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Step 11.3.1

Rewrite $416$ as $1 (416)$ .

$n \ln (1.04) + \ln (\frac{1 (416)}{881}) = 0$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Rewrite $881$ as $1 (881)$ .

$n \ln (1.04) + \ln (\frac{1 \cdot 416}{1 \cdot 881}) = 0$

Step 11.3.2.2

Cancel the common factor.

$n \ln (1.04) + \ln (\frac{1 \cdot 416}{1 \cdot 881}) = 0$

Step 11.3.2.3

Rewrite the expression.

$n \ln (1.04) + \ln (\frac{416}{881}) = 0$

Step 12

Subtract $\ln (\frac{416}{881})$ from both sides of the equation.

$n \ln (1.04) = - \ln (\frac{416}{881})$

Step 13

Divide each term in $n \ln (1.04) = - \ln (\frac{416}{881})$ by $\ln (1.04)$ and simplify.

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Step 13.1

Divide each term in $n \ln (1.04) = - \ln (\frac{416}{881})$ by $\ln (1.04)$ .

$\frac{n \ln (1.04)}{\ln (1.04)} = \frac{- \ln (\frac{416}{881})}{\ln (1.04)}$

Step 13.2

Simplify the left side.

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Step 13.2.1

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

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Step 13.2.1.1

Cancel the common factor.

$\frac{n \ln (1.04)}{\ln (1.04)} = \frac{- \ln (\frac{416}{881})}{\ln (1.04)}$

Step 13.2.1.2

Divide $n$ by $1$ .

$n = \frac{- \ln (\frac{416}{881})}{\ln (1.04)}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (\frac{416}{881})}{\ln (1.04)}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (\frac{416}{881})}{\ln (1.04)}$

Decimal Form:

$n = 19.13204287 \dots$