The difference between compound interest and simple interest on x at 15% per annum for 2 years is 9. What is the value of x?
400
Understanding Compound Interest vs Simple Interest Difference
The question asks us to find the principal amount, denoted as 'x', given the difference between the compound interest (CI) and simple interest (SI) earned over a period of 2 years at a specific annual interest rate. The difference is stated to be 9, and the interest rate is 15% per annum.
Key Concepts: Simple Interest and Compound Interest
- Simple Interest (SI): Calculated only on the initial principal amount. The interest earned each year is the same. The formula for SI is \( \text{SI} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \).
- Compound Interest (CI): Calculated on the initial principal and also on the accumulated interest from previous periods. The interest earned each year grows because it's added to the principal for the next calculation. The formula for the Amount (Principal + CI) is \( \text{Amount} = \text{Principal} \left( 1 + \frac{\text{Rate}}{100} \right)^{\text{Time}} \). Then, CI = Amount - Principal.
Calculating Simple Interest for 2 Years
Given: Principal = \(x\), Rate = 15%, Time = 2 years.
Using the SI formula:
\( \text{SI} = \frac{x \times 15 \times 2}{100} \)
\( \text{SI} = \frac{30x}{100} \)
\( \text{SI} = \frac{3x}{10} \)
Calculating Compound Interest for 2 Years
Given: Principal = \(x\), Rate = 15%, Time = 2 years.
First, calculate the Amount (A) after 2 years:
\( A = x \left( 1 + \frac{15}{100} \right)^2 \)
\( A = x \left( 1 + \frac{3}{20} \right)^2 \)
\( A = x \left( \frac{20+3}{20} \right)^2 \)
\( A = x \left( \frac{23}{20} \right)^2 \)
\( A = x \left( \frac{529}{400} \right) \)
Now, calculate the CI:
\( \text{CI} = A - \text{Principal} \)
\( \text{CI} = x \left( \frac{529}{400} \right) - x \)
\( \text{CI} = x \left( \frac{529}{400} - 1 \right) \)
\( \text{CI} = x \left( \frac{529 - 400}{400} \right) \)
\( \text{CI} = x \left( \frac{129}{400} \right) \)
Using the Difference Formula for CI and SI for 2 Years
For a period of 2 years, there is a direct formula to find the difference between Compound Interest and Simple Interest:
\( \text{CI} - \text{SI} = P \left( \frac{R}{100} \right)^2 \)
Where P is the Principal, and R is the annual Rate.
We are given that the difference is 9, the Principal is \(x\), and the Rate is 15%. Substitute these values into the formula:
\( 9 = x \left( \frac{15}{100} \right)^2 \)
\( 9 = x \left( \frac{3}{20} \right)^2 \)
\( 9 = x \left( \frac{9}{400} \right) \)
Solving for the Principal (x)
Now, we solve the equation \( 9 = x \left( \frac{9}{400} \right) \) for \(x\).
To isolate \(x\), multiply both sides of the equation by \( \frac{400}{9} \):
\( 9 \times \frac{400}{9} = x \times \frac{9}{400} \times \frac{400}{9} \)
\( 400 = x \)
So, the value of \(x\) is 400.
Verification
Let's verify the result with \(x=400\) at 15% for 2 years.
SI = \( \frac{400 \times 15 \times 2}{100} = \frac{12000}{100} = 120 \)
Amount after 2 years (CI): \( 400 \left( 1 + \frac{15}{100} \right)^2 = 400 \left( \frac{23}{20} \right)^2 = 400 \times \frac{529}{400} = 529 \)
CI = Amount - Principal = \( 529 - 400 = 129 \)
CI - SI difference = \( 129 - 120 = 9 \)
The difference is indeed 9, which matches the information given in the question. Thus, our calculated value for \(x\) is correct.
Conclusion
The value of \(x\) for which the difference between compound interest and simple interest at 15% per annum for 2 years is 9 is 400.
| Term | Value |
|---|---|
| Principal (x) | ? |
| Rate (R) | 15% p.a. |
| Time (T) | 2 years |
| CI - SI Difference | 9 |
Revision Table: Compound Interest and Simple Interest Basics
| Feature | Simple Interest (SI) | Compound Interest (CI) |
|---|---|---|
| Interest Calculation Basis | Only on original principal | On principal + accumulated interest |
| Interest Amount Annually | Constant | Increases over time |
| Formula (Amount) | \( P(1 + \frac{RT}{100}) \) | \( P(1 + \frac{R}{100})^T \) |
| Formula (Interest) | \( \frac{PRT}{100} \) | \( P((1 + \frac{R}{100})^T - 1) \) |
| Growth Type | Linear | Exponential |
Additional Information: CI and SI Difference
The difference between CI and SI grows with time and rate. For a time period of more than one year, CI is always greater than SI when the rate is positive.
- For T = 1 year, CI = SI.
- For T = 2 years, \( \text{CI} - \text{SI} = P \left( \frac{R}{100} \right)^2 \). This formula is very useful for quick calculations when the time is exactly 2 years.
- For T = 3 years, \( \text{CI} - \text{SI} = P \left[ \left( 1 + \frac{R}{100} \right)^3 - 1 - \frac{3R}{100} \right] = P \frac{R^2}{100^2} \left( \frac{R}{100} + 3 \right) \).
These difference formulas highlight how the extra interest in compound interest arises from earning interest on previously accumulated interest.
Important Questions from Simple and Compound Both
What is the compound interest (in Rs.) at the rate of 10%, compounded annually, for 3 years on the principal which in 8 years at the rate of 12% per annum gives Rs. 4,800 as simple interest?
A certain sum amounts to Rs. 15,500 in 2 years at 12% p.a. simple interest. If the same sum is compounded half-yearly at 10% per annum for \(1 \frac{1}{2}\) years, what will be the amount received?
The difference in compound interest on a certain sum at 10% p.a. for one year, when the interest is compounded half-yearly and yearly, is Rs. 88.80. What is the simple interest on the same sum for \(1\frac{2}{3}\) years at the same rate?
A sum of Rs. 7500 amounts to Rs. 9075 at 10% p.a, interest being compounded yearly in a certain time. The simple interest (in Rs.) on the same sum for the same time and the same rate is:
A certain sum amounts to Rs.291600 in 2 years and to Rs.314928 in 3 years on compound interest compounded annually. How much will be the simple interest (in Rs.) on Rs.40000 at the same rate for 2 years?
