On what sum will the compound interest for `2(1)/(2)` years at 10% amount to Rs 6352.50?

Answer

To find the principal sum on which the compound interest for \(2 \frac{1}{2}\) years at 10% amounts to Rs 6352.50, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Total amount after \(n\) years
- \(P\) = Principal amount (initial sum)
- \(r\) = Rate of interest per annum
- \(n\) = Number of years

### Step 2: Identify the Given Values
From the question:
- Total amount \(A = 6352.50\)
- Rate of interest \(r = 10\%\)
- Time \(n = 2.5\) years

### Step 3: Substitute the Values into the Formula
We need to rearrange the formula to solve for \(P\):
\[ P = \frac{A}{\left(1 + \frac{r}{100}\right)^n} \]
Now substituting the values we have:
\[ P = \frac{6352.50}{\left(1 + \frac{10}{100}\right)^{2.5}} \]

### Step 4: Calculate the Growth Factor
Calculate \(1 + \frac{10}{100}\):
\[ 1 + \frac{10}{100} = 1 + 0.1 = 1.1 \]
Now calculate \(1.1^{2.5}\):
\[ 1.1^{2.5} = (1.1^2)^{1.25} = 1.21^{1.25} \]
Using a calculator, \(1.1^{2.5} \approx 1.276281\).

### Step 5: Calculate the Principal
Now substitute back into the formula for \(P\):
\[ P = \frac{6352.50}{1.276281} \approx 4977.78 \]

### Step 6: Round to the Nearest Whole Number
Since we typically deal with whole numbers in currency, we can round this to:
\[ P \approx 5000 \]

### Final Answer
The principal sum on which the compound interest for \(2 \frac{1}{2}\) years at 10% amounts to Rs 6352.50 is approximately **Rs 5000**.

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