What is the difference (in ₹) between the com pound interests on ₹ 12,500 for 1 year at 8% per annum compounded yearly and halfyearly?

Options

A. A. 16
B. B. 25
C. C. 20
D. C. 40

Answer

To find the difference in compound interest on ₹12,500 for 1 year at an interest rate of 8% per annum compounded yearly and half-yearly, we will follow these steps:

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

For yearly compounding:
- \(P = 12,500\)
- \(r = 8\%\)
- \(n = 1\)

Substituting the values:
\[ A = 12500 \left(1 + \frac{8}{100}\right)^1 \]
\[ A = 12500 \left(1 + 0.08\right) \]
\[ A = 12500 \times 1.08 = 13500 \]

Now, the compound interest (CI) for yearly compounding is:
\[ CI_{yearly} = A - P = 13500 - 12500 = 1000 \]

### Step 2: Calculate Compound Interest Compounded Half-Yearly
For half-yearly compounding, the formula remains the same, but the rate and time period change:
- The rate per half-year = \(\frac{r}{2} = \frac{8}{2} = 4\%\)
- The number of compounding periods in 1 year = 2

Substituting the values:
\[ A = 12500 \left(1 + \frac{4}{100}\right)^2 \]
\[ A = 12500 \left(1 + 0.04\right)^2 \]
\[ A = 12500 \left(1.04\right)^2 \]
Calculating \(1.04^2\):
\[ 1.04^2 = 1.0816 \]
Now substituting back:
\[ A = 12500 \times 1.0816 = 13520 \]

Now, the compound interest (CI) for half-yearly compounding is:
\[ CI_{half-yearly} = A - P = 13520 - 12500 = 1020 \]

### Step 3: Calculate the Difference in Compound Interests
Now, we find the difference between the two compound interests:
\[ \text{Difference} = CI_{half-yearly} - CI_{yearly} = 1020 - 1000 = 20 \]

### Final Answer
The difference in compound interests is ₹20.

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