Math Percentage calculation Compound interest calculation

# Compound interest calculation

After one year, you get interests on your credit. Next year, you will receive interests on these interest rates again. This is called compound interest.

$P_{new}=P_0\cdot(1+\frac{p}{100})$
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### Hint

The formula corresponds to the percentage change.

Now, you can multiply the previous result with the growth factor each time.

### Example

Lena has 200 € in her passbook with an interest rate of 5 percent. How much money does she have after one/ two/ three years?

1. #### after one year

$P_1=P_0\cdot(1+\frac{r}{100})$ $=200€\cdot(1+\frac{5}{100})$ $=200€\cdot\frac{105}{100}$ $=210€$
2. #### after two years

$P_2=P_1\cdot(1+\frac{r}{100})$ $=210€\cdot(1+\frac{5}{100})$ $=210€\cdot\frac{105}{100}$ $=220.50€$
3. #### after three years

$P_3=P_2\cdot(1+\frac{r}{100})$ $=220.50€\cdot(1+\frac{5}{100})$ $=220.50€\cdot\frac{105}{100}$ $\approx231.53€$

The following context is recognizable:

$P_1=\color{blue}{P_0\cdot(1+\frac{p}{100})}$

$P_2=\color{blue}{P_1}\cdot(1+\frac{r}{100})$ $=\color{blue}{P_0\cdot(1+\frac{r}{100})}\cdot(1+\frac{r}{100})$ $=\color{green}{P_0\cdot(1+\frac{r}{100})^2}$

$P_3=\color{green}{P_2}\cdot(1+\frac{r}{100})$ $=\color{green}{P_0\cdot(1+\frac{r}{100})^2}\cdot(1+\frac{r}{100})$ $=P_0\cdot(1+\frac{r}{100})^3$

## The compound interest formula

You use the compound interest formula so that you don’t always have to calculate all previous values. This is percentage growth.

$P_{t}=P_0\cdot(1+\frac{r}{100})^t$

### Example

Calculate Lena’s money after 20 years.

$P_{20}=P_0\cdot(1+\frac{r}{100})^{20}$ $=200€\cdot(1+\frac{5}{100})^{20}$ $=200€\cdot(\frac{105}{100})^{20}$ $\approx530.66€$