Find the compound interest on Rs. 2000 at 5% per annum, compounded yearly for `2(1)/(2)` years ?

Answer

To find the compound interest on Rs. 2000 at 5% per annum, compounded yearly for \(2 \frac{1}{2}\) years, we will follow these steps:

### Step 1: Understand the formula for compound interest
The formula for the amount \(A\) after \(t\) years with principal \(P\) and rate \(r\) is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \(A\) = Amount after time \(t\)
- \(P\) = Principal amount (initial investment)
- \(r\) = Rate of interest per annum
- \(t\) = Time in years

### Step 2: Identify the values from the question
From the question:
- Principal \(P = 2000\)
- Rate \(r = 5\%\)
- Time \(t = 2 \frac{1}{2} = 2.5\) years

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ A = 2000 \left(1 + \frac{5}{100}\right)^{2.5} \]

### Step 4: Simplify the expression inside the parentheses
Calculate \(1 + \frac{5}{100}\):
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]

### Step 5: Calculate \(A\) using the exponent
Now we need to calculate:
\[ A = 2000 \times (1.05)^{2.5} \]
To calculate \((1.05)^{2.5}\), we can break it down:
\[ (1.05)^{2.5} = (1.05^2)^{1.25} = 1.1025^{1.25} \]
Using a calculator, we find:
\[ (1.05)^{2.5} \approx 1.128 \]
Thus:
\[ A \approx 2000 \times 1.128 = 2256 \]

### Step 6: Calculate the compound interest
The compound interest \(CI\) is given by:
\[ CI = A - P \]
Substituting the values:
\[ CI = 2256 - 2000 = 256 \]

### Final Answer
The compound interest on Rs. 2000 at 5% per annum, compounded yearly for \(2 \frac{1}{2}\) years is approximately Rs. 256.

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