If C is the compound interest on Rs. 10,000 for one year at 4% per annum when compounded quarterly, then which one of the following is correct ?
Calculating Compound Interest Compounded Quarterly
The question asks us to find the range of the compound interest (C) earned on a principal amount of Rs. 10,000 for one year at an annual interest rate of 4%, where the interest is compounded quarterly.
Let's break down the given information:
- Principal amount (P) = Rs. 10,000
- Annual interest rate (R) = 4% per annum
- Time period (T) = 1 year
- Compounding frequency = Quarterly (n = 4 times a year)
When interest is compounded quarterly, the interest rate per compounding period is the annual rate divided by the number of quarters in a year, and the number of compounding periods is the number of years multiplied by the number of quarters per year.
- Interest rate per period \(r = \frac{R}{n} = \frac{4\%}{4} = 1\%\) or \(0.01\)
- Number of periods \(N = n \times T = 4 \times 1 = 4\)
The formula for the compound amount (A) after \(N\) periods with principal \(P\) and rate \(r\) per period is:
\(A = P (1 + r)^N\)
Substituting the values:
\(A = 10000 \left(1 + \frac{4/100}{4}\right)^{4 \times 1}\)
\(A = 10000 (1 + 0.01)^4\)
\(A = 10000 (1.01)^4\)
Now, let's calculate \((1.01)^4\):
\(1.01^2 = 1.01 \times 1.01 = 1.0201\)
\(1.01^4 = (1.01^2)^2 = (1.0201)^2\)
To calculate \((1.0201)^2\):
\(1.0201 \times 1.0201 = 1.04060401\)
So, the compound amount is:
\(A = 10000 \times 1.04060401\)
\(A = 10406.0401\)
The compound interest (C) is the difference between the compound amount (A) and the principal amount (P):
\(C = A - P\)
\(C = 10406.0401 - 10000\)
\(C = 406.0401\)
Now, we need to compare this calculated compound interest (C = 406.0401) with the given options:
Let's evaluate each option:
- Option 1: \(C < \text{Rs. } 100\)
Is \(406.0401 < 100\)? No. - Option 2: \(\text{Rs. } 100 < C < \text{Rs. } 200\)
Is \(100 < 406.0401 < 200\)? No. - Option 3: \(\text{Rs. } 200 < C < \text{Rs. } 400\)
Is \(200 < 406.0401 < 400\)? No. - Option 4: \(C > \text{Rs. } 400\)
Is \(406.0401 > 400\)? Yes.
Based on the calculation, the compound interest is approximately Rs. 406.04, which is greater than Rs. 400. Therefore, the correct option is \(C > \text{Rs. } 400\).
Revision Table: Compound Interest Concepts
| Term | Definition | Formula (Annual Compounding) |
|---|---|---|
| Principal (P) | The initial amount of money. | - |
| Rate (R) | The annual interest rate (as a percentage). | - |
| Time (T) | The duration for which the money is borrowed or invested (in years). | - |
| Compound Amount (A) | The total amount including principal and accumulated interest after a certain period. | \(A = P(1 + R/100)^T\) |
| Compound Interest (C) | The interest earned, calculated on the principal amount and also on the accumulated interest of previous periods. | \(C = A - P\) or \(C = P((1 + R/100)^T - 1)\) |
Additional Information: Quarterly Compounding vs. Annual Compounding
When interest is compounded more frequently than annually, such as quarterly, the effective annual rate is higher than the nominal annual rate. This is because interest earned in each period is added to the principal, and subsequent interest is calculated on this larger amount.
In this problem:
- Nominal annual rate = 4%
- Rate per quarter = 1%
Let's compare the compound interest with simple interest for the same period and rate to see the difference compounding makes.
Simple Interest (SI) = \(\frac{P \times R \times T}{100}\)
SI = \(\frac{10000 \times 4 \times 1}{100} = \frac{40000}{100} = \text{Rs. } 400\)
Compound Interest (Quarterly) = Rs. 406.0401
As expected, the compound interest is slightly higher than the simple interest for the same principal, rate, and time period, demonstrating the effect of earning interest on interest, especially with more frequent compounding like quarterly compounding.
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