Basic Math Examples

$1000 = \frac{{(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t})}{1 - {(1 + \frac{0.1}{12})}^{12}}$

Step 1

Rewrite the equation as $\frac{{(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t})}{1 - {(1 + \frac{0.1}{12})}^{12}} = 1000$ .

$\frac{{(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t})}{1 - {(1 + \frac{0.1}{12})}^{12}} = 1000$

Step 2

Multiply both sides by $1 - {(1 + \frac{0.1}{12})}^{12}$ .

$\frac{{(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t})}{1 - {(1 + \frac{0.1}{12})}^{12}} (1 - {(1 + \frac{0.1}{12})}^{12}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{{(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t})}{1 - {(1 + \frac{0.1}{12})}^{12}} (1 - {(1 + \frac{0.1}{12})}^{12})$ .

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Step 3.1.1.1

Reduce the expression by cancelling the common factors.

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Step 3.1.1.1.1

Cancel the common factor of $1 - {(1 + \frac{0.1}{12})}^{12}$ .

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Step 3.1.1.1.1.1

Cancel the common factor.

$\frac{{(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t})}{1 - {(1 + \frac{0.1}{12})}^{12}} (1 - {(1 + \frac{0.1}{12})}^{12}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.1.1.2

Rewrite the expression.

${(1 + \frac{0.1}{12})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.1.2

Simplify the expression.

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Step 3.1.1.1.2.1

Divide $0.1$ by $12$ .

${(1 + 0.008\overline{3})}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.1.2.2

Add $1$ and $0.008\overline{3}$ .

${1.008\overline{3}}^{12} (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.1.2.3

Raise $1.008\overline{3}$ to the power of $12$ .

$1.10471306 (1 - {(1 + \frac{0.1}{12})}^{12 \cdot t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.2

Simplify each term.

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Step 3.1.1.2.1

Divide $0.1$ by $12$ .

$1.10471306 (1 - {(1 + 0.008\overline{3})}^{12 \cdot t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.2.2

Add $1$ and $0.008\overline{3}$ .

$1.10471306 (1 - {1.008\overline{3}}^{12 t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.3

Simplify by multiplying through.

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Step 3.1.1.3.1

Apply the distributive property.

$1.10471306 \cdot 1 + 1.10471306 (- {1.008\overline{3}}^{12 t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.3.2

Multiply.

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Step 3.1.1.3.2.1

Multiply $1.10471306$ by $1$ .

$1.10471306 + 1.10471306 (- {1.008\overline{3}}^{12 t}) = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.1.1.3.2.2

Multiply $- 1$ by $1.10471306$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = 1000 (1 - {(1 + \frac{0.1}{12})}^{12})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $1000 (1 - {(1 + \frac{0.1}{12})}^{12})$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Divide $0.1$ by $12$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = 1000 (1 - {(1 + 0.008\overline{3})}^{12})$

Step 3.2.1.1.2

Add $1$ and $0.008\overline{3}$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = 1000 (1 - {1.008\overline{3}}^{12})$

Step 3.2.1.1.3

Raise $1.008\overline{3}$ to the power of $12$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = 1000 (1 - 1 \cdot 1.10471306)$

Step 3.2.1.1.4

Multiply $- 1$ by $1.10471306$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = 1000 (1 - 1.10471306)$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Subtract $1.10471306$ from $1$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = 1000 \cdot - 0.10471306$

Step 3.2.1.2.2

Multiply $1000$ by $- 0.10471306$ .

$1.10471306 - 1.10471306 \cdot {1.008\overline{3}}^{12 t} = - 104.71306744$

Step 4

Solve for $t$ .

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Step 4.1

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

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Step 4.1.1

Subtract $1.10471306$ from both sides of the equation.

$- 1.10471306 \cdot {1.008\overline{3}}^{12 t} = - 104.71306744 - 1.10471306$

Step 4.1.2

Subtract $1.10471306$ from $- 104.71306744$ .

$- 1.10471306 \cdot {1.008\overline{3}}^{12 t} = - 105.8177805$

Step 4.2

Divide each term in $- 1.10471306 \cdot {1.008\overline{3}}^{12 t} = - 105.8177805$ by $- 1.10471306$ and simplify.

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Step 4.2.1

Divide each term in $- 1.10471306 \cdot {1.008\overline{3}}^{12 t} = - 105.8177805$ by $- 1.10471306$ .

$\frac{- 1.10471306 \cdot {1.008\overline{3}}^{12 t}}{- 1.10471306} = \frac{- 105.8177805}{- 1.10471306}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 1.10471306$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 1.10471306 \cdot {1.008\overline{3}}^{12 t}}{- 1.10471306} = \frac{- 105.8177805}{- 1.10471306}$

Step 4.2.2.1.2

Divide ${1.008\overline{3}}^{12 t}$ by $1$ .

${1.008\overline{3}}^{12 t} = \frac{- 105.8177805}{- 1.10471306}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 105.8177805$ by $- 1.10471306$ .

${1.008\overline{3}}^{12 t} = 95.7875702$

Step 4.3

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

$\ln ({1.008\overline{3}}^{12 t}) = \ln (95.7875702)$

Step 4.4

Expand $\ln ({1.008\overline{3}}^{12 t})$ by moving $12 t$ outside the logarithm.

$12 t \ln (1.008\overline{3}) = \ln (95.7875702)$

Step 4.5

Divide each term in $12 t \ln (1.008\overline{3}) = \ln (95.7875702)$ by $12 \ln (1.008\overline{3})$ and simplify.

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Step 4.5.1

Divide each term in $12 t \ln (1.008\overline{3}) = \ln (95.7875702)$ by $12 \ln (1.008\overline{3})$ .

$\frac{12 t \ln (1.008\overline{3})}{12 \ln (1.008\overline{3})} = \frac{\ln (95.7875702)}{12 \ln (1.008\overline{3})}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $12$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{12 t \ln (1.008\overline{3})}{12 \ln (1.008\overline{3})} = \frac{\ln (95.7875702)}{12 \ln (1.008\overline{3})}$

Step 4.5.2.1.2

Rewrite the expression.

$\frac{t \ln (1.008\overline{3})}{\ln (1.008\overline{3})} = \frac{\ln (95.7875702)}{12 \ln (1.008\overline{3})}$

Step 4.5.2.2

Cancel the common factor of $\ln (1.008\overline{3})$ .

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Step 4.5.2.2.1

Cancel the common factor.

$\frac{t \ln (1.008\overline{3})}{\ln (1.008\overline{3})} = \frac{\ln (95.7875702)}{12 \ln (1.008\overline{3})}$

Step 4.5.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (95.7875702)}{12 \ln (1.008\overline{3})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (95.7875702)}{12 \ln (1.008\overline{3})}$

Decimal Form:

$t = 45.81115524 \dots$