A sum of money is lent out at compound interest rate of 20 % per annum for 2 years. It would fetch Rs 482 more if interest is compounded half-yearly. Find the sum. Solution: Suppose the sum is Rs P.

Answer

To solve the problem, we need to find the principal amount (P) that is lent out at a compound interest rate of 20% per annum for 2 years. We also know that if the interest is compounded half-yearly, it would yield Rs 482 more than when compounded annually.

Let's break down the solution step by step:

### Step 1: Calculate the amount when compounded annually
The formula for the amount (A) when compounded annually is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest (20%)
- \( T \) = Time in years (2)

Substituting the values:
\[ A = P \left(1 + \frac{20}{100}\right)^2 \]
\[ A = P \left(1 + 0.2\right)^2 \]
\[ A = P \left(1.2\right)^2 \]
\[ A = P \times 1.44 \]
Thus, the amount when compounded annually is:
\[ A_1 = 1.44P \]

### Step 2: Calculate the amount when compounded half-yearly
When compounded half-yearly, the rate becomes half (10%) and the time doubles (4 years):
\[ A = P \left(1 + \frac{R}{100}\right)^{2T} \]
Substituting the values:
\[ A = P \left(1 + \frac{10}{100}\right)^{4} \]
\[ A = P \left(1 + 0.1\right)^{4} \]
\[ A = P \left(1.1\right)^{4} \]
Calculating \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
Thus, the amount when compounded half-yearly is:
\[ A_2 = 1.4641P \]

### Step 3: Set up the equation based on the difference
According to the problem, the amount when compounded half-yearly is Rs 482 more than the amount when compounded annually:
\[ A_2 = A_1 + 482 \]
Substituting the amounts we found:
\[ 1.4641P = 1.44P + 482 \]

### Step 4: Solve for P
Rearranging the equation:
\[ 1.4641P - 1.44P = 482 \]
\[ 0.0241P = 482 \]
Now, divide both sides by 0.0241:
\[ P = \frac{482}{0.0241} \]
Calculating the value:
\[ P \approx 20000 \]

### Conclusion
The principal amount (P) is Rs 20,000.

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