What is the difference (in Rs.) between the compound interest on Rs. 5000 for 1 year at `4%` per annum compounded yearly and half yearly?

Options

A. 2
B. 3
C. 4
D. 8

Answer

To find the difference between the compound interest on Rs. 5000 for 1 year at 4% per annum compounded yearly and half-yearly, we will follow these steps:

### Step 1: Calculate Compound Interest when Compounded Yearly
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = the amount after time \( n \)
- \( P \) = principal amount (initial investment)
- \( r \) = rate of interest per annum
- \( n \) = number of years

For yearly compounding:
- \( P = 5000 \)
- \( r = 4\% \)
- \( n = 1 \)

Substituting the values:
\[ A = 5000 \left(1 + \frac{4}{100}\right)^1 \]
\[ A = 5000 \left(1 + 0.04\right) \]
\[ A = 5000 \times 1.04 \]
\[ A = 5200 \]

Now, the compound interest (CI) for yearly compounding is:
\[ CI_{yearly} = A - P = 5200 - 5000 = 200 \]

### Step 2: Calculate Compound Interest when Compounded Half-Yearly
For half-yearly compounding:
- The rate per half year = \( \frac{4}{2} = 2\% \)
- The number of compounding periods in a year = 2

Using the same formula:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where \( n \) is now 2 for half-yearly compounding.

Substituting the values:
\[ A = 5000 \left(1 + \frac{2}{100}\right)^2 \]
\[ A = 5000 \left(1 + 0.02\right)^2 \]
\[ A = 5000 \left(1.02\right)^2 \]
\[ A = 5000 \times 1.0404 \]
\[ A = 5202 \]

Now, the compound interest (CI) for half-yearly compounding is:
\[ CI_{half-yearly} = A - P = 5202 - 5000 = 202 \]

### Step 3: Calculate the Difference in Compound Interest
Now, we find the difference between the two compound interests:
\[ \text{Difference} = CI_{half-yearly} - CI_{yearly} \]
\[ \text{Difference} = 202 - 200 = 2 \]

### Final Answer:
The difference in compound interest between yearly and half-yearly compounding is Rs. 2.

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