The different between the compound interest on a sum of ₹ 8,000 for 1 year at the rate of 10% per annum, interest compounded yearly and half yearly is:

Options

A. ₹40
B. ₹10
C. ₹30
D. ₹20

Answer

To find the difference between the compound interest on a sum of ₹ 8,000 for 1 year at the rate of 10% per annum compounded yearly and half-yearly, we can follow these steps:

### Step 1: Calculate Compound Interest when compounded yearly
- **Principal (P)** = ₹ 8,000
- **Rate (R)** = 10% per annum
- **Time (T)** = 1 year

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

Substituting the values:
\[ CI_{yearly} = 8000 \left(1 + \frac{10}{100}\right)^1 - 8000 \]
\[ = 8000 \left(1 + 0.10\right) - 8000 \]
\[ = 8000 \times 1.10 - 8000 \]
\[ = 8800 - 8000 = 800 \]

### Step 2: Calculate Compound Interest when compounded half-yearly
- **Principal (P)** = ₹ 8,000
- **Rate (R)** = 10% per annum (which is halved for half-yearly compounding, so 5%)
- **Time (T)** = 1 year (which is 2 half-year periods)

Using the same formula for compound interest:
\[ CI_{half-yearly} = P \left(1 + \frac{R/2}{100}\right)^{2T} - P \]
Substituting the values:
\[ CI_{half-yearly} = 8000 \left(1 + \frac{5}{100}\right)^{2} - 8000 \]
\[ = 8000 \left(1 + 0.05\right)^{2} - 8000 \]
\[ = 8000 \left(1.05\right)^{2} - 8000 \]
Calculating \( (1.05)^{2} \):
\[ (1.05)^{2} = 1.1025 \]
Thus,
\[ CI_{half-yearly} = 8000 \times 1.1025 - 8000 \]
\[ = 8820 - 8000 = 820 \]

### Step 3: Find the difference between the two compound interests
Now, we need to find the difference between the compound interest calculated for yearly and half-yearly compounding:
\[ \text{Difference} = CI_{half-yearly} - CI_{yearly} \]
\[ = 820 - 800 = 20 \]

### Final Answer
The difference between the compound interest on ₹ 8,000 for 1 year at the rate of 10% per annum, compounded yearly and half-yearly is ₹ 20.

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