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?
1,655
Understanding the Problem
The question asks us to calculate the compound interest (CI) for a specific principal amount over 3 years at a 10% annual rate. However, the principal is not given directly. Instead, we are told that this principal earns Rs. 4,800 as simple interest (SI) over 8 years at a 12% annual rate. Therefore, the first step is to find this principal amount using the simple interest information, and then use this principal to calculate the compound interest.
Step 1: Finding the Principal using Simple Interest
We are given the simple interest earned, the rate of interest, and the time period. The formula for simple interest is:
\(\text{SI} = \frac{P \times R \times T}{100}\)
Where:
- \(\text{SI}\) = Simple Interest
- \(P\) = Principal amount
- \(R\) = Rate of interest per annum
- \(T\) = Time period in years
From the question, we have:
- \(\text{SI} = \text{Rs. } 4,800\)
- \(R = 12\%\) per annum
- \(T = 8\) years
We need to find the Principal (\(P\)). Let's plug the given values into the formula:
\(4800 = \frac{P \times 12 \times 8}{100}\)
Now, let's solve for \(P\):
\(4800 \times 100 = P \times (12 \times 8)\)
\(480000 = P \times 96\)
\(P = \frac{480000}{96}\)
\(P = 5000\)
So, the principal amount is Rs. 5,000.
Step 2: Calculating Compound Interest
Now that we have the principal amount (Rs. 5,000), we can calculate the compound interest. We are given:
- Principal (\(P\)) = Rs. 5,000
- Rate of interest (\(R\)) = 10% per annum
- Time period (\(T\)) = 3 years
- Compounding: Annually
The formula for the amount (\(A\)) under compound interest compounded annually is:
\(A = P \left(1 + \frac{R}{100}\right)^T\)
Let's plug in the values:
\(A = 5000 \left(1 + \frac{10}{100}\right)^3\)
\(A = 5000 \left(1 + 0.1\right)^3\)
\(A = 5000 \left(1.1\right)^3\)
\(A = 5000 \times 1.331\)
\(A = 6655\)
The amount after 3 years is Rs. 6,655.
The compound interest (\(\text{CI}\)) is the difference between the amount and the principal:
\(\text{CI} = A - P\)
\(\text{CI} = 6655 - 5000\)
\(\text{CI} = 1655\)
Final Answer
The compound interest at the rate of 10%, compounded annually, for 3 years on the principal of Rs. 5,000 is Rs. 1,655.
Revision Table: Simple vs. Compound Interest
| Feature | Simple Interest (SI) | Compound Interest (CI) |
|---|---|---|
| Calculation Basis | Only on the original principal amount. | On the principal amount plus accumulated interest from previous periods. |
| Interest Growth | Linear growth. | Exponential growth (interest earns interest). |
| Formula (Annual) | \(\text{SI} = \frac{P \times R \times T}{100}\) | \(A = P \left(1 + \frac{R}{100}\right)^T\) \(\text{CI} = A - P\) |
| Earnings Over Time | Constant amount of interest earned each period. | Increasing amount of interest earned each period. |
Additional Information: Interest Calculations
Interest is a key concept in finance, representing the cost of borrowing money or the return on an investment. Simple interest and compound interest are the two primary ways to calculate interest.
Simple Interest: Simple interest is the easiest to calculate. It is always based on the initial principal amount. This type of interest is often used for short-term loans or basic calculations.
Compound Interest: Compound interest is more complex because the interest earned in each period is added to the principal, and the next period's interest is calculated on this new, larger amount. This effect is often referred to as "interest on interest" and leads to faster growth of money over time, especially over longer periods. The frequency of compounding (annually, semi-annually, quarterly, monthly, daily) significantly impacts the total compound interest earned. More frequent compounding results in higher interest.
Understanding the difference between simple and compound interest is crucial for evaluating loans, investments, and savings plans. Compound interest is often seen in savings accounts, fixed deposits, and most loan products like mortgages and personal loans.
Important Questions from Simple and Compound Both
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?
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?
