What will be the difference between the compound interest and simple interest on a sum of Rs. 100 at 10% per annum for 2 years?
Rs. 1
Understanding the Difference Between Compound Interest and Simple Interest
This question asks for the difference between the compound interest (CI) and the simple interest (SI) earned on a specific principal amount over a fixed period at a given annual interest rate.
Let's break down the calculation for both types of interest and then find the difference.
Given Information
- Principal Amount (P) = Rs. 100
- Rate of Interest (R) = 10% per annum
- Time Period (T or n) = 2 years
Step-by-Step Calculation of Simple Interest (SI)
Simple interest is calculated only on the original principal amount. The formula for Simple Interest is:
\(\text{SI} = \frac{P \times R \times T}{100}\)
Substituting the given values:
\(\text{SI} = \frac{100 \times 10 \times 2}{100}\)
\(\text{SI} = \frac{2000}{100}\)
\(\text{SI} = 20\)
So, the simple interest for 2 years is Rs. 20.
Step-by-Step Calculation of Compound Interest (CI)
Compound interest is calculated on the principal amount and also on the accumulated interest of previous years. The formula for the Amount (A) under Compound Interest is:
\(A = P \left(1 + \frac{R}{100}\right)^n\)
Where n is the number of years.
Substituting the given values:
\(A = 100 \left(1 + \frac{10}{100}\right)^2\)
\(A = 100 \left(1 + 0.1\right)^2\)
\(A = 100 \left(1.1\right)^2\)
\(A = 100 \times 1.21\)
\(A = 121\)
This is the total amount after 2 years, including the principal and the compound interest.
To find the Compound Interest (CI), we subtract the original principal from the amount:
\(\text{CI} = A - P\)
\(\text{CI} = 121 - 100\)
\(\text{CI} = 21\)
So, the compound interest for 2 years is Rs. 21.
Calculating the Difference (CI - SI)
The difference between the compound interest and the simple interest is:
\(\text{Difference} = \text{CI} - \text{SI}\)
\(\text{Difference} = 21 - 20\)
\(\text{Difference} = 1\)
The difference between the compound interest and the simple interest is Rs. 1.
Alternative Method: Direct Formula for 2 Years Difference
For a time period of 2 years, there is a direct formula to calculate the difference between compound interest and simple interest:
\(\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2\)
Substituting the given values:
\(\text{CI} - \text{SI} = 100 \left(\frac{10}{100}\right)^2\)
\(\text{CI} - \text{SI} = 100 \left(\frac{1}{10}\right)^2\)
\(\text{CI} - \text{SI} = 100 \times \frac{1}{100}\)
\(\text{CI} - \text{SI} = 1\)
Using the direct formula also gives the difference as Rs. 1.
| Calculation | Value |
|---|---|
| Principal (P) | Rs. 100 |
| Rate (R) | 10% |
| Time (T) | 2 years |
| Simple Interest (SI) | Rs. 20 |
| Compound Interest (CI) | Rs. 21 |
| Difference (CI - SI) | Rs. 1 |
Final Answer
The difference between the compound interest and simple interest on a sum of Rs. 100 at 10% per annum for 2 years is Rs. 1.
Revision Table: Compound vs Simple Interest
| Feature | Simple Interest (SI) | Compound Interest (CI) |
|---|---|---|
| Calculation Basis | Always on the original principal amount. | On the principal amount plus accumulated interest from previous periods. |
| Interest Earned | Linear growth. Same amount of interest is added each period. | Exponential growth. Interest earned increases each period. |
| Formula (for 1 year) | \(\frac{P \times R \times 1}{100}\) | Same as SI for the first year. |
| Formula (for T years) | \(\frac{P \times R \times T}{100}\) | \(P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]\) |
| Growth Type | Arithmetic Progression | Geometric Progression |
| Difference CI - SI | Zero for 1 year. Positive for > 1 year (assuming R > 0). | Zero for 1 year. Positive for > 1 year (assuming R > 0). |
Additional Information on Interest Calculations
Understanding the difference between simple and compound interest is fundamental in finance and mathematics. Compound interest is often referred to as 'interest on interest'. This is why it grows faster than simple interest over time, especially for longer periods and higher interest rates.
- Compounding Frequency: Compound interest can be calculated annually, semi-annually, quarterly, monthly, or even daily. The more frequent the compounding, the higher the effective annual rate and thus the higher the compound interest. The formula \(A = P \left(1 + \frac{R/k}{100}\right)^{nk}\) is used, where k is the number of times interest is compounded per year.
- Rule of 72: A quick way to estimate how long it takes for an investment to double at a fixed annual rate (compounded annually) is to divide 72 by the interest rate. For example, at 10% compound interest, it would take approximately \(72 / 10 = 7.2\) years for the principal to double. This rule applies only to compound interest.
- Applications: Simple interest is commonly used in short-term loans and basic calculations. Compound interest is used in most savings accounts, investments, mortgages, and loans over longer periods.
The difference between CI and SI highlights the power of compounding, where earning interest on previously earned interest significantly boosts the total return over time compared to earning interest only on the initial principal.
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