Simple interest on a certain sum of money at 9% is rupees 450 in 2 years . Find the compound interest. On the same sum , at the same rate for 1 year, if the interest is reckoned half yearly .

Answer

To solve the problem step by step, we will first calculate the principal amount using the simple interest formula, and then we will calculate the compound interest for one year when the interest is compounded half-yearly.

### Step 1: Calculate the Principal Amount
We know the formula for Simple Interest (SI):
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal Amount
- \( R \) = Rate of Interest (in percentage)
- \( T \) = Time (in years)

From the problem:
- \( SI = 450 \)
- \( R = 9\% \)
- \( T = 2 \) years

Substituting the values into the formula:
\[ 450 = \frac{P \times 9 \times 2}{100} \]

### Step 2: Rearranging the Equation
To find \( P \), we can rearrange the equation:
\[ 450 = \frac{18P}{100} \]
Multiplying both sides by 100:
\[ 45000 = 18P \]
Now, divide both sides by 18:
\[ P = \frac{45000}{18} = 2500 \]

### Step 3: Calculate the Compound Interest
Now that we have the principal amount, we can calculate the compound interest for 1 year with half-yearly compounding.

The formula for the amount \( A \) when interest is compounded half-yearly is:
\[ A = P \left(1 + \frac{R}{2 \times 100}\right)^{2n} \]
Where:
- \( n \) = number of years

For our case:
- \( P = 2500 \)
- \( R = 9\% \)
- \( n = 1 \)

Substituting the values into the formula:
\[ A = 2500 \left(1 + \frac{9}{2 \times 100}\right)^{2 \times 1} \]
\[ A = 2500 \left(1 + \frac{9}{200}\right)^{2} \]
\[ A = 2500 \left(1 + 0.045\right)^{2} \]
\[ A = 2500 \left(1.045\right)^{2} \]

### Step 4: Calculate \( (1.045)^{2} \)
Calculating \( (1.045)^{2} \):
\[ (1.045)^{2} = 1.092025 \]

### Step 5: Calculate the Amount
Now substituting back into the amount formula:
\[ A = 2500 \times 1.092025 = 2730.0625 \]

### Step 6: Calculate Compound Interest
Now, we can find the compound interest \( CI \):
\[ CI = A - P \]
\[ CI = 2730.0625 - 2500 = 230.0625 \]

### Final Answer
The compound interest on the same sum at the same rate for 1 year, when the interest is reckoned half-yearly, is approximately \( \text{Rs } 230.06 \).