A sum of `Rs. 15000` is lent at compound Interest (compounded annually) at an interest rate of `20%` per annum. If the interest is compounded half yearly, then how much more interest in Rs.) will be obtained in one year?

Options

A. 225
B. 150
C. 75
D. 300

Answer

To solve the problem step by step, we will calculate the compound interest for both annual compounding and half-yearly compounding, and then find the difference in interest earned.

### Step 1: Calculate the compound interest when compounded annually.
- **Principal (P)** = Rs. 15,000
- **Rate (R)** = 20% per annum
- **Time (T)** = 1 year

The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Substituting the values:
\[ A = 15000 \left(1 + \frac{20}{100}\right)^1 \]
\[ A = 15000 \left(1 + 0.20\right)^1 \]
\[ A = 15000 \times 1.20 \]
\[ A = 18000 \]

Now, calculate the compound interest (CI):
\[ CI = A - P \]
\[ CI = 18000 - 15000 \]
\[ CI = 3000 \]

### Step 2: Calculate the compound interest when compounded half-yearly.
- **Principal (P)** = Rs. 15,000
- **Rate (R)** = 20% per annum, so for half-yearly compounding, the rate will be half:
- Half-yearly rate = 10% (which is 20% / 2)
- **Time (T)** = 2 half-years (1 year = 2 half-years)

Using the same formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = 15000 \left(1 + \frac{10}{100}\right)^2 \]
\[ A = 15000 \left(1 + 0.10\right)^2 \]
\[ A = 15000 \times (1.10)^2 \]
\[ A = 15000 \times 1.21 \]
\[ A = 18150 \]

Now, calculate the compound interest (CI):
\[ CI = A - P \]
\[ CI = 18150 - 15000 \]
\[ CI = 3150 \]

### Step 3: Find the difference in interest earned.
Now we will find how much more interest is earned when compounded half-yearly compared to annually:
\[ \text{Difference} = CI_{\text{half-yearly}} - CI_{\text{annually}} \]
\[ \text{Difference} = 3150 - 3000 \]
\[ \text{Difference} = 150 \]

### Final Answer:
The additional interest earned when compounded half-yearly compared to annually is **Rs. 150**.

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