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.

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