Finite Math Examples

$5000 = \frac{100 ({(1.01)}^{n} - 1)}{0.01}$

Step 1

Multiply both sides by $0.01$ .

$5000 \cdot 0.01 = \frac{100 ({1.01}^{n} - 1)}{0.01} \cdot 0.01$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Multiply $5000$ by $0.01$ .

$50 = \frac{100 ({1.01}^{n} - 1)}{0.01} \cdot 0.01$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Simplify $\frac{100 ({1.01}^{n} - 1)}{0.01} \cdot 0.01$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor of $0.01$ .

Tap for more steps...

Step 2.2.1.1.1

Cancel the common factor.

$50 = \frac{100 ({1.01}^{n} - 1)}{0.01} \cdot 0.01$

Step 2.2.1.1.2

Rewrite the expression.

$50 = 100 ({1.01}^{n} - 1)$

Step 2.2.1.2

Apply the distributive property.

$50 = 100 \cdot {1.01}^{n} + 100 \cdot - 1$

Step 2.2.1.3

Multiply $100$ by $- 1$ .

$50 = 100 \cdot {1.01}^{n} - 100$

Step 3

Solve for $n$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $100 \cdot {1.01}^{n} - 100 = 50$ .

$100 \cdot {1.01}^{n} - 100 = 50$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Add $100$ to both sides of the equation.

$100 \cdot {1.01}^{n} = 50 + 100$

Step 3.2.2

Add $50$ and $100$ .

$100 \cdot {1.01}^{n} = 150$

Step 3.3

Divide each term in $100 \cdot {1.01}^{n} = 150$ by $100$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $100 \cdot {1.01}^{n} = 150$ by $100$ .

$\frac{100 \cdot {1.01}^{n}}{100} = \frac{150}{100}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $100$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{100 \cdot {1.01}^{n}}{100} = \frac{150}{100}$

Step 3.3.2.1.2

Divide ${1.01}^{n}$ by $1$ .

${1.01}^{n} = \frac{150}{100}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Cancel the common factor of $150$ and $100$ .

Tap for more steps...

Step 3.3.3.1.1

Factor $50$ out of $150$ .

${1.01}^{n} = \frac{50 (3)}{100}$

Step 3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.1

Factor $50$ out of $100$ .

${1.01}^{n} = \frac{50 \cdot 3}{50 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

${1.01}^{n} = \frac{50 \cdot 3}{50 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

${1.01}^{n} = \frac{3}{2}$

Step 3.4

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

$\ln ({1.01}^{n}) = \ln (\frac{3}{2})$

Step 3.5

Expand $\ln ({1.01}^{n})$ by moving $n$ outside the logarithm.

$n \ln (1.01) = \ln (\frac{3}{2})$

Step 3.6

Divide each term in $n \ln (1.01) = \ln (\frac{3}{2})$ by $\ln (1.01)$ and simplify.

Tap for more steps...

Step 3.6.1

Divide each term in $n \ln (1.01) = \ln (\frac{3}{2})$ by $\ln (1.01)$ .

$\frac{n \ln (1.01)}{\ln (1.01)} = \frac{\ln (\frac{3}{2})}{\ln (1.01)}$

Step 3.6.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.1

Cancel the common factor of $\ln (1.01)$ .

Tap for more steps...

Step 3.6.2.1.1

Cancel the common factor.

$\frac{n \ln (1.01)}{\ln (1.01)} = \frac{\ln (\frac{3}{2})}{\ln (1.01)}$

Step 3.6.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (\frac{3}{2})}{\ln (1.01)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (\frac{3}{2})}{\ln (1.01)}$

Decimal Form:

$n = 40.74890715 \dots$