Simple interest on a sum of money for 2 years at 4% growth rate is Rs 450. Find compound interest on the same sum and at the same rate for 1 year, if the interest is reckoned half yearly.

Answer

To solve the problem step by step, we will follow the outlined process to find the compound interest on the given sum.

### Step 1: Identify the given values
- Simple Interest (SI) = Rs 450
- Time (T) = 2 years
- Rate (R) = 4%

### Step 2: Use the formula for Simple Interest to find the Principal (P)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Rearranging the formula to solve for Principal (P):
\[ P = \frac{SI \times 100}{R \times T} \]

### Step 3: Substitute the known values into the formula
Substituting the known values into the rearranged formula:
\[ P = \frac{450 \times 100}{4 \times 2} \]

### Step 4: Calculate the Principal (P)
Calculating the denominator:
\[ 4 \times 2 = 8 \]
Now substituting back:
\[ P = \frac{45000}{8} = 5625 \]
Thus, the Principal (P) is Rs 5625.

### Step 5: Calculate the Compound Interest for 1 year at 4% half-yearly
Since the interest is compounded half-yearly, we will use the formula:
\[ A = P \left(1 + \frac{R}{2 \times 100}\right)^{2T} \]
Where:
- \(T = 1\) year
- \(R = 4\%\)

### Step 6: Substitute the values into the compound interest formula
Substituting the values:
\[ A = 5625 \left(1 + \frac{4}{2 \times 100}\right)^{2 \times 1} \]
Calculating \(\frac{4}{2 \times 100}\):
\[ \frac{4}{200} = 0.02 \]
Now substituting back:
\[ A = 5625 \left(1 + 0.02\right)^{2} \]
\[ A = 5625 \left(1.02\right)^{2} \]

### Step 7: Calculate \((1.02)^{2}\)
Calculating \((1.02)^{2}\):
\[ (1.02)^{2} = 1.0404 \]
Now substituting back:
\[ A = 5625 \times 1.0404 \]

### Step 8: Calculate the Amount (A)
Calculating the amount:
\[ A = 5850.75 \]

### Step 9: Calculate the Compound Interest (CI)
The formula for Compound Interest is:
\[ CI = A - P \]
Substituting the values:
\[ CI = 5850.75 - 5625 \]
Calculating:
\[ CI = 225.75 \]

### Final Answer
The compound interest on the same sum at the same rate for 1 year, if the interest is reckoned half-yearly, is Rs 225.75.

---