The difference between simple interest and compound on Rs.900 for one year at `10%` per annum reckoned half-yearly is:

Options

A. Rs.3
B. Rs.2.25
C. Rs.4.5
D. Rs.4

Answer

To find the difference between simple interest (SI) and compound interest (CI) on Rs. 900 for one year at a rate of 10% per annum, compounded half-yearly, we can follow these steps:

### Step 1: Identify the Given Values
- Principal (P) = Rs. 900
- Rate (R) = 10% per annum
- Time (T) = 1 year

### Step 2: Adjust the Rate and Time for Half-Yearly Compounding
Since the interest is compounded half-yearly:
- Half-yearly rate = \( \frac{10\%}{2} = 5\% \)
- Time in half-years = \( 1 \text{ year} \times 2 = 2 \text{ half-years} \)

### Step 3: Calculate Simple Interest (SI)
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{900 \times 10 \times 1}{100} = \frac{9000}{100} = 90 \]

### Step 4: Calculate Compound Interest (CI)
The formula for compound interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^n - P \]
Where \( n \) is the number of compounding periods. Here, \( n = 2 \) (for 2 half-years).
Substituting the values:
\[ CI = 900 \left(1 + \frac{5}{100}\right)^2 - 900 \]
Calculating further:
\[ CI = 900 \left(1 + 0.05\right)^2 - 900 = 900 \left(1.05\right)^2 - 900 \]
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
Now substituting back:
\[ CI = 900 \times 1.1025 - 900 = 991.25 - 900 = 91.25 \]

### Step 5: Calculate the Difference Between CI and SI
Now, we find the difference:
\[ \text{Difference} = CI - SI = 91.25 - 90 = 1.25 \]

### Final Answer
The difference between simple interest and compound interest on Rs. 900 for one year at 10% per annum, compounded half-yearly, is Rs. 1.25.

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