Find the compound interest (Compounded annually) on Rs. 2,000 for \(2\frac{1}{2}\) years at 10% per annum.
Rs. 541
Calculating Compound Interest for Fractional Years
This problem asks us to calculate the compound interest on a principal amount of Rs. 2,000 for a period of \(2\frac{1}{2}\) years at an annual interest rate of 10%, compounded annually. Compound interest for a fractional period requires calculating the interest for the whole number of years and then the simple interest on the accumulated amount for the remaining fractional part.
Understanding the Given Information
- Principal amount (P) = Rs. 2,000
- Time (n) = \(2\frac{1}{2}\) years
- Rate of interest (R) = 10% per annum
- Compounding frequency: Annually
Formula for Amount with Fractional Time Period
When the time period is in the form of \(a\frac{b}{c}\) years, where 'a' is the whole number of years and \(\frac{b}{c}\) is the fractional part, and the interest is compounded annually, the amount (A) can be calculated using the formula:
\[A = P \left(1 + \frac{R}{100}\right)^a \left(1 + \frac{\frac{b}{c}R}{100}\right)\]
Where:
- P = Principal
- R = Annual interest rate
- a = Whole number of years
- \(\frac{b}{c}\) = Fractional part of a year
Step-by-Step Compound Interest Calculation
Here, P = 2000, a = 2 years, \(\frac{b}{c} = \frac{1}{2}\) year, and R = 10%.
Step 1: Calculate the Amount after the Whole Number of Years (2 Years)
We can first calculate the amount after 2 full years. This is the standard compound interest calculation for a whole number of years.
Amount after 2 years \(A_2 = P \left(1 + \frac{R}{100}\right)^2\)
\[A_2 = 2000 \left(1 + \frac{10}{100}\right)^2\]
\[A_2 = 2000 \left(1 + 0.1\right)^2\]
\[A_2 = 2000 \left(1.1\right)^2\]
\[A_2 = 2000 \times 1.21\]
\[A_2 = 2420\]
So, after 2 full years, the amount is Rs. 2,420.
Step 2: Calculate Interest for the Fractional Part (Remaining \(\frac{1}{2}\) Year)
Now, we need to calculate the interest for the remaining \(\frac{1}{2}\) year. This interest is calculated on the amount accumulated at the end of the whole number of years (Rs. 2,420) using the simple interest method. The rate for this period is the annual rate multiplied by the fraction of the year.
Rate for \(\frac{1}{2}\) year = \(R \times \frac{b}{c} = 10\% \times \frac{1}{2} = 5\%\)
Interest for \(\frac{1}{2}\) year = Simple Interest on Rs. 2,420 at 5% per annum for 1 year (or at 10% p.a. for \(\frac{1}{2}\) year).
Using the formula for Simple Interest \(SI = \frac{P \times R \times T}{100}\), where P is the amount at the start of the period, R is the annual rate, and T is the time in years:
\[\text{Interest for } \frac{1}{2} \text{ year} = \frac{2420 \times 10 \times \frac{1}{2}}{100}\]
\[\text{Interest for } \frac{1}{2} \text{ year} = \frac{2420 \times 5}{100}\]
\[\text{Interest for } \frac{1}{2} \text{ year} = \frac{12100}{100}\]
\[\text{Interest for } \frac{1}{2} \text{ year} = 121\]
The interest for the remaining half year is Rs. 121.
Step 3: Calculate the Total Amount after \(2\frac{1}{2}\) Years
The total amount at the end of \(2\frac{1}{2}\) years is the sum of the amount after 2 years and the interest for the next \(\frac{1}{2}\) year.
Total Amount \(A = A_2 + \text{Interest for } \frac{1}{2} \text{ year}\)
\[A = 2420 + 121\]
\[A = 2541\]
The total amount after \(2\frac{1}{2}\) years is Rs. 2,541.
Step 4: Calculate the Compound Interest
Compound Interest (CI) is the total amount minus the original principal.
\[CI = \text{Total Amount} - \text{Principal}\]
\[CI = 2541 - 2000\]
\[CI = 541\]
The compound interest is Rs. 541.
Summary Table
| Item | Value |
|---|---|
| Principal (P) | Rs. 2,000 |
| Time (n) | \(2\frac{1}{2}\) years |
| Rate (R) | 10% p.a. |
| Amount after 2 years | Rs. 2,420 |
| Interest for next \(\frac{1}{2}\) year | Rs. 121 |
| Total Amount after \(2\frac{1}{2}\) years | Rs. 2,541 |
| Compound Interest | Rs. 541 |
Final Answer
The compound interest on Rs. 2,000 for \(2\frac{1}{2}\) years at 10% per annum, compounded annually, is Rs. 541.
Revision Table: Compound Interest Key Concepts
| Concept | Description |
|---|---|
| Compound Interest (CI) | Interest calculated on the initial principal and also on the accumulated interest of previous periods. |
| Principal (P) | The initial amount of money borrowed or invested. |
| Amount (A) | The total sum of principal and compound interest after a certain period. \(A = P + CI\). |
| Rate (R) | The percentage of interest charged or earned per period (usually per year). |
| Time (n) | The duration for which the money is invested or borrowed. |
| Compounding Frequency | How often the interest is added to the principal (e.g., annually, half-yearly, quarterly). |
Additional Information: Handling Fractional Time in CI
When calculating compound interest for a period that includes a fraction of a year, and compounding is annual:
- The full number of years are compounded normally.
- For the remaining fractional part of the year, the interest is calculated as simple interest on the amount accumulated at the end of the last full year.
- The rate for the simple interest calculation is the annual rate multiplied by the fraction of the year. For example, for \(\frac{1}{2}\) year at 10% p.a., the rate used for simple interest calculation on the accumulated amount is \(10\% \times \frac{1}{2} = 5\%\).
This method ensures that interest earned during the full periods is compounded, while the interest for the partial period is treated as a simple accrual on the amount at the beginning of that partial period.
Important Questions from Compound Interest
- A sum of money triples itself at a certain rate of compound interest in 11 years. In how many years will it amount to 27 times of itself?
A sum of ₹10,000 amounts to ₹12,321 in 2 years at a certain rate of interest per annum, compounded annually. The rate of interest per annum is:
- If the interest earned during the 2nd year on a certain sum is ₹2,352, and the rate of interest is 5% per annum compounded annually, then the sum is:
- A sum of money triples itself at a certain rate of compound interest in 17 years. In how many years will it amount to 81 times of itself?
If the interest earned during the $2^{nd}$ year on a certain sum is ₹3,342, and the rate of interest is 20% per annum compounded annually, then the sum is:
