When you deposit money in a savings account, you expected it to gain interest over time. The money will continue to gain interest until you decide to make a withdrawal. It is like an accumulation of interest.
In this article, we will learn more about the mathematical definition of this accumulation, named Compound interest.
Meaning of compound interest
Compound interest is the accumulation or addition of interest to a principal amount.
The idea is that the interest gained on the principal amount is reinvested, and future interest is added to the principal amount plus the earlier interest, where the principal amount is the original amount of money that was invested. This continues until the period elapses. Take a look at the compound interest graph below.
Compound interest graph
Compound interest graph
From the compound interest graph we can see that as time increases, the money also increases. This is the whole idea of compound interest.
When solving questions related to compound interest, we are actually asked to find the amount of money that is obtained or earned over a particular period of time as a result of the rate of compound interest added.
To calculate the compound interest, we should know:
- the principal or initial amount.
- the percentage rate of the compound interest on the principal amount.
- the time which is the period that the money will be withdrawn or stop gaining interest.
Calculating compound interest: Formula
There are two ways to calculate compound interest. You can calculate using a table, and you can calculate using the compound interest formula.
Using the formula of compound interest
The compound interest formula is given by:
where,
Therefore,
The result obtained from the compound interest formula is the amount of money earned or gained after interest has been added over time.
Using the compound interest table
For each year, we calculate the money to be held until the time elapses. In order to do so, we follow the following steps.
- Draw a table with two columns, one for 'amount' and another for 'rate in percentage'.
- Write the principal amount on the first row under the amount and multiply it by the percentage rate under interest.
- Add the principal amount to the interest on the second row under amount and multiply by the percentage rate on that row under interest.
- Repeat step 3 until the time elapses.
Calculating the compound interest using the table will take longer than it would if you used the compound interest formula.
In the next section, we will learn how to calculate compound interest using both methods.
Compound interest examples
Let’s take some examples using both the compound interest formula and the table.
If you deposit £4000 in a bank for three years and you are paid 4% interest per annum. How much will you have at the end of the 3 years?
Solution
Let's try to solve this problem using the table first and then we will try the formula. From the steps above, we know we have to draw a two-column table.
So, at the end of the 3 years, you will have 
.
That was quite a long process but we can get to our answer quicker using the compound interest formula. The formula is below.
From the given, we know that
We will now substitute the values in the formula.
You can see that the same answer was gotten using the table and the formula.
Let's take another example.
Jane deposits £800 in a bank paying 1% compound interest per annum. What will she have after two years? Use the table and the formula for the compound interest calculation for the following question.
Solution
First of all, state the information given:
Let's start with the table before using the formula.
| Amount | Percentage Rate - 1% |
1st year –  This is the initial money she deposited at the bank. |  This means at the end of the first year, Jane will have extra money. |
2nd year –  The second year will start with the initial amount plus the interest gained in the first year. |  This means that at the end of the second year, Jane will have  extra money. |
After 2 years, Jane will have  | The total interest earned after 2 years is  |
According to the table, by the end of 2 years, Jane will have 
.
Let's now use the formula. The compound interest formula is given by:
We will now substitute our values in the formula to get:
You can see that the same answer was obtained using the table and the formula.
Let's take one more example.
Ben takes a loan of £15000, and the bank charges him 10% compound interest per annum. If Ben does not pay off the loan in four years, how much does Ben owe the bank?
Solution
Let’s state the information given.
The amount owed after four years is the final amount and we will get it using the compound interest formula. The compound interest formula is given by:
We will substitute our values in the formula:
Ben will owe the bank 
by the end of the 4 years.
Difference between simple and compound interest
Aside from compound interest, there is also what is called simple interest.
The significant difference between simple interest and compound interest is that simple interest has to do with one-time interest on the principal amount while compound interest has to do with an accumulation of interest on the principal amount over a period of time. To find out more about simple interest, check out our article on Simple Interest.
Compound Interest - Key takeaways
- Compound interest is the accumulation or addition of interest to a principal amount.
- There are two ways to calculate compound interest. You can calculate it using a table or by using the compound interest formula.
- The compound interest formula is: .
- Aside from compound interest, there is also what is called simple interest. The significant difference between simple and compound interest is that simple interest has to do with one-time interest on the principal amount while compound interest has to do with an accumulation of interest on the principal amount over a period of time.
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extra in the first year.
extra by the end of the second year. 
extra by the end of the third year; t
This is the amount of money you will have after 3 years. 
