The difference of compounds interest on Rs. 800 for 1 year at 20% per annum when compounded half yearly and quarterly is :

Answer

To solve the problem of finding the difference in compound interest on Rs. 800 for 1 year at 20% per annum when compounded half-yearly and quarterly, we will follow these steps:

### Step 1: Identify the principal amount and the rate of interest
- Principal (P) = Rs. 800
- Annual Rate of Interest (R) = 20%

### Step 2: Calculate the effective rate and number of compounding periods for half-yearly compounding
- For half-yearly compounding, the rate per half year = 20% / 2 = 10%
- Number of half-year periods in 1 year = 2

### Step 3: Calculate the compound interest for half-yearly compounding
Using the formula for compound interest:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal
- \( r \) = Rate of interest per period
- \( n \) = Number of periods

For half-yearly compounding:
\[ A_{half-yearly} = 800 \left(1 + \frac{10}{100}\right)^2 \]
\[ A_{half-yearly} = 800 \left(1 + 0.10\right)^2 \]
\[ A_{half-yearly} = 800 \left(1.10\right)^2 \]
\[ A_{half-yearly} = 800 \times 1.21 = 968 \]

Now, calculate the compound interest:
\[ CI_{half-yearly} = A_{half-yearly} - P = 968 - 800 = 168 \]

### Step 4: Calculate the effective rate and number of compounding periods for quarterly compounding
- For quarterly compounding, the rate per quarter = 20% / 4 = 5%
- Number of quarterly periods in 1 year = 4

### Step 5: Calculate the compound interest for quarterly compounding
For quarterly compounding:
\[ A_{quarterly} = 800 \left(1 + \frac{5}{100}\right)^4 \]
\[ A_{quarterly} = 800 \left(1 + 0.05\right)^4 \]
\[ A_{quarterly} = 800 \left(1.05\right)^4 \]
\[ A_{quarterly} = 800 \times 1.21550625 \approx 972.40 \]

Now, calculate the compound interest:
\[ CI_{quarterly} = A_{quarterly} - P = 972.40 - 800 = 172.40 \]

### Step 6: Find the difference in compound interest
Now, we find the difference between the two compound interests:
\[ \text{Difference} = CI_{quarterly} - CI_{half-yearly} \]
\[ \text{Difference} = 172.40 - 168 = 4.40 \]

### Final Answer
The difference in compound interest when compounded half-yearly and quarterly is **Rs. 4.40**.

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