Growth and Compounding:
When money is invested or borrowed for more than one period, interest may be calculated in different ways. The main difference is whether the interest earned is added to the original amount before calculating the next period's interest.
This idea is called compounding.
In simple terms:
Compounding means earning interest on the interest that has already been added to the money.
Before understanding growth and compounding, let us recall some important terms.
Principal:
The principal is the original amount of money that is invested, deposited, or borrowed. It is represented by (p).
Rate of Interest:
The rate of interest is the interest charged or earned on the principal for a specified period, usually given as a percentage per year.
It is represented by (r %) per annum.
Time:
Time is the period for which the money is invested or borrowed.
It is represented by (t) and is usually measured in years.
Interest:
Interest is the additional money earned on an investment or paid on borrowed money.
There are two common ways of calculating interest:
- Simple Interest (SI)
- Compound Interest (CI)
Amount:
The amount is the total money obtained after adding the interest to the principal.
To understand growth and compounding clearly, let us look at three situations.
1. Growth with Compounding:
When compounding is done, the interest earned in one period is added to the principal. The next period's interest is then calculated on the new amount. It is also called as Compound Interest.
Therefore, the amount keeps increasing, and each period's interest can be slightly greater than the previous period's interest.
Example:
Suppose \(₹\) \(1000\) is invested at \(10 \%\) per year with annual compounding.
After 1 year:
Interest:
\(10 \%\) of \(₹ 1000\) \(=\) \(₹ 100\)
New amount:
\(₹1000 + ₹ 100 = ₹ 1100\)
So, after the first year, the amount becomes \(₹ 1100\).
After 2 year:
Now the interest is calculated on \(₹ 1100\), not on the original \(₹ 1000\).
\(10 \%\) of \(₹ 1100 = ₹ 110\)
New amount:
\(₹ 1100 + ₹ 110 = ₹ 1210\)
Notice that the second year's interest is \(₹ 110\), whereas the first year's interest was \(₹100\).
The additional \(₹ 10\) is interest earned on the interest from the first year.
This is the key idea of compounding.
Formula for finding annual compounding:
If:
- Principal \(\text{p}\)
- Rate of interest \(\text{r}\%\) per annum
- Time \(\text{t}\) years
then the amount after \(\text{n}\) years is:
\(\text{A} = \text{p} \left(1+\frac{\text{r}}{100}\right)^\text{t}\)
The compound interest is: \(\text{CI} = \text{A} - \text{P}\)
where \(\text{A}\) is the final amount.
Important!
With Compounding:
The interest earned in one period is added to the amount, and the next period's interest is calculated on the new amount
2. Growth Without Compounding:
When compounding is not done, the interest earned is not added to the principal for calculating the interest in the next period.
The interest is calculated on the original principal throughout the given period.
This is also called as simple interest.
Example:
Suppose \(₹ 1000\) is invested at \(10 \%\) per year without compounding.
Interest for 1 year:
\(10 \%\) of \(₹ 1000 = ₹ 100\)
Since the interest is not added to the principal for calculating the next year's interest, the interest remains \(₹ 100\) every year.
For 2 years:
\(\text{SI} = ₹ 100 + ₹ 100 = ₹ 200\)
Therefore, the amount after 2 years is: \(\text{A} = ₹ 1000 + ₹ 200 = ₹ 1200\)
Formula for calculating without compounding:
\(\text{SI} = \frac{\text{p} \times \text{r} \times \text{t}}{100}\)
The amount is: \(\text{A} = \text{P} + \text{SI}\)
where:
- \(\text{p}\) \(=\) Principal
- \(\text{r}\) \(=\) Rate of interest per annum
- \(\text{t}\) \(=\) Time in years
- \(\text{SI}\) \(=\) Simple Interest
- \(\text{A}\) \(=\) Amount
Important!
Without Compounding:
Interest is calculated only on the original principal, throughout the given period.
3. No Compounding:
No compounding means that interest is not added to the principal for further calculation.
In some situations, the phrase may simply mean that there is no repeated addition of interest to the principal.
For example, if a person keeps \(₹1000\) without earning any interest, the amount remains \(₹1000\).
There is no increase because no interest is being earned.
This is different from without compounding, where interest may still be earned but is calculated only on the original principal.
Important!
1. Principal is the initial amount of money invested or borrowed.
2. Interest is the additional amount earned or paid on the principal.
3. Amount is the total value after adding interest to the principal.
\(\text{A} = \text{P} + \text{Interest}\)
4. With compounding, interest earned in one period becomes part of the amount used to calculate interest in the next period.
5. Without compounding, interest is calculated on the original principal throughout the period.
6. For annual compounding, \(\text{A} = \text{p} \left(1+\frac{\text{r}}{100}\right)^\text{t}\)
7. Compound Interest: \(\text{CI} = \text{A} - \text{P}\)
8. Simple Interest: \(\text{SI} = \frac{\text{p} \times \text{r} \times \text{t}}{100}\)
9. The main difference between simple interest and compound interest is the amount on which interest is calculated in each successive period.