Interest

When working with Percent, you quickly encounter the concept of Interest. 

Interest is used to describe how much a value grows over time, for example when you put money in the bank or borrow money.

 

What is interest?

The Interest is the payment you either receive or have to pay because a capital (an amount) is made available.

The interest is always stated as a percent of the capital per unit of time (often yearly).

 

Example:

If you put 1,000 euros in the bank at an annual interest rate of 5%, it means you get:

 

$$ \large \frac{5}{100} \cdot 1000 = 50\ euros $$

 

That is 50 euros in interest after one year.

 

Simple interest

Sometimes you only calculate interest on the original amount (the capital), without adding the interest to the capital. This is called simple interest.

 

The formula for simple interest is:

 

$$ \large R = K_0 \cdot r \cdot n $$

 

where \(K_0\) is the initial capital, \(r\) is the interest rate (written as a decimal), and \(n\) is the number of years.

 

Example:

If you put 1,000 euros in the bank at 5% interest for 3 years, you get:

 

$$ \large R = 1000 \cdot 0.05 \cdot 3 = 150\ euros $$

 

That is 150 euros in total interest after 3 years, so the capital is:

 

$$ \large K = 1000 + 150 = 1150\ euros $$

 

With simple interest, the capital therefore grows by the same amount each year.

 

Compound interest

If the interest is added to the capital, so that the next year you get interest on both the original amount and the previous interest, it is called compound interest.

 

After two years it looks like this:

 

$$ \large 1000 \cdot 1.05 \cdot 1.05 = 1102.50 $$

 

The extra 2.50 comes from the fact that you also received interest on the first year’s interest, equal to 5% of 50.

 

General formula for compound interest

If you know the initial capital \(K_0\), the interest rate \(r\) (written as a decimal), and the number of years \(n\), you can find the capital \(K_n\) after \(n\) years with the formula:

 

$$ \large K_n = K_0 \cdot (1+r)^n $$

 

Example:

If you put 2,000 euros in the bank at 3% interest for 4 years, you get:

 

$$ \large K_4 = 2000 \cdot (1+0.03)^4 = 2000 \cdot 1.1255 = 2251.06 $$

 

So the amount grows to about 2251 euros.

 

Summary

• Interest is a percentage of a capital.
• Compound interest means you get interest on interest.
• The general formula is \(K_n = K_0 \cdot (1+r)^n\).