An amount was lent for one year at the rate of `18%` per annum compounding annually. Had the compounding been done half yearly, the interest would have increased by 324. What was the amount (in Rs.) lent?

Options

A. A) 32000
B. B) 36000
C. C) 40000
D. D) 72000

Answer

To solve the problem step by step, we will first calculate the compound interest for both annual and half-yearly compounding, and then find the principal amount lent.

### Step 1: Calculate the Compound Interest for Annual Compounding
Given:
- Rate of interest (R) = 18% per annum
- Time (T) = 1 year

The formula for compound interest (CI) when compounded annually is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Where \(P\) is the principal amount.

So, for annual compounding:
\[ CI_A = P \left(1 + \frac{18}{100}\right)^1 - P = P(1.18) - P = 0.18P \]

### Step 2: Calculate the Compound Interest for Half-Yearly Compounding
When compounding is done half-yearly, the rate for each half-year is:
\[ \text{Half-yearly rate} = \frac{18}{2} = 9\% \]
The time for half-yearly compounding for 1 year is 2 periods.

Using the compound interest formula:
\[ CI_B = P \left(1 + \frac{9}{100}\right)^2 - P \]
Calculating this:
\[ CI_B = P \left(1.09\right)^2 - P = P(1.1881) - P = 0.1881P \]

### Step 3: Find the Increase in Interest
According to the problem, the increase in interest when switching from annual to half-yearly compounding is given as Rs. 324. Therefore, we can set up the equation:
\[ CI_B - CI_A = 324 \]
Substituting the values we found:
\[ 0.1881P - 0.18P = 324 \]
This simplifies to:
\[ 0.0081P = 324 \]

### Step 4: Solve for Principal Amount (P)
To find \(P\), we can rearrange the equation:
\[ P = \frac{324}{0.0081} \]
Calculating this gives:
\[ P = 40000 \]

Thus, the amount lent is Rs. 40,000.

### Final Answer
The amount lent was **Rs. 40,000**.

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