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?
6400
Understanding the Compound Interest and Simple Interest Problem
This question asks us to first find the annual interest rate based on given compound interest amounts over consecutive years. Once we determine this rate, we need to calculate the simple interest on a different principal amount for a specific time period using the same rate.
We are given:
- Amount after 2 years on compound interest: Rs. 291600
- Amount after 3 years on compound interest: Rs. 314928
- Compounding is annual.
We need to find the simple interest on Rs. 40000 at the same rate for 2 years.
Finding the Annual Interest Rate
When interest is compounded annually, the amount after $n$ years becomes the principal for calculating interest in the $(n+1)$-th year. The difference between the amount after 3 years and the amount after 2 years is the compound interest earned during the 3rd year.
Interest earned in the 3rd year = Amount after 3 years - Amount after 2 years
Interest earned in the 3rd year = $\text{Rs. } 314928 - \text{Rs. } 291600$
Interest earned in the 3rd year = $\text{Rs. } 23328$
This interest of Rs. 23328 is earned on the amount at the end of the 2nd year, which is Rs. 291600, over a period of 1 year. We can use the simple interest formula for this one-year period to find the rate, as compound interest for one year is the same as simple interest on the principal at the beginning of that year.
Let the annual interest rate be $R\%$.
Simple Interest = $\frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}$
Here, Principal = Amount after 2 years = Rs. 291600, Interest = Rs. 23328, Time = 1 year.
So, $23328 = \frac{291600 \times R \times 1}{100}$
$23328 = 2916 \times R$
To find $R$, we divide the interest by the principal (after 2 years) divided by 100:
$R = \frac{23328}{2916}$
Let's perform the division:
| Calculation | Value |
|---|---|
| Interest for 3rd year | 23328 |
| Amount at end of 2nd year | 291600 |
| Rate $R = (23328 / 291600) \times 100$ | $(23328 / 2916) \times 1$ |
| Calculated Rate $R$ | 8% |
So, the annual interest rate is 8%.
Calculating Simple Interest
Now we need to calculate the simple interest on a principal of Rs. 40000 at the rate of 8% per annum for 2 years. The simple interest formula is:
Simple Interest (SI) = $\frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}$
Here, Principal = Rs. 40000, Rate = 8%, Time = 2 years.
SI = $\frac{40000 \times 8 \times 2}{100}$
SI = $\frac{40000 \times 16}{100}$
SI = $400 \times 16$
SI = $6400$
Final Simple Interest Value
The simple interest on Rs. 40000 at 8% per annum for 2 years is Rs. 6400.
Revision Table: Key Formulas
| Concept | Formula |
|---|---|
| Simple Interest (SI) | $SI = \frac{P \times R \times T}{100}$ |
| Compound Amount (A) | $A = P \left(1 + \frac{R}{100}\right)^T$ |
| Compound Interest (CI) | $CI = A - P$ or $CI = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right]$ |
| Rate from consecutive Compound Amounts | $R = \left(\frac{A_{T+1}}{A_T} - 1\right) \times 100$ |
Where: P = Principal, R = Rate of Interest per annum, T = Time in years, $A_T$ = Amount after T years, $A_{T+1}$ = Amount after T+1 years.
Additional Information: Compound vs. Simple Interest
Simple Interest: Interest is calculated only on the initial principal amount for the entire duration. The interest earned does not get added back to the principal to earn further interest.
Compound Interest: Interest is calculated on the initial principal as well as on the accumulated interest from previous periods. The interest earned is added to the principal, and subsequent interest is calculated on this new, larger principal. This leads to faster growth in the amount compared to simple interest, especially over longer periods.
In this problem, we used the compound interest amounts to find the rate and then applied that rate to calculate simple interest, demonstrating the distinction between the two concepts.
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:
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?
