Simple interest on a certain sum of money for 4 years at 4% per annum exceeds the compound interest on the same sum for 3 years at 5 percent per annum by Rs 228. Find the sum.

Answer

To solve the problem step by step, we will follow the mathematical principles of simple interest and compound interest.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the principal amount (the sum of money) be \( P \).

2. **Calculate Simple Interest (SI):**
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Here, \( R = 4\% \) and \( T = 4 \) years.
So, the simple interest for 4 years at 4% per annum is:
\[ SI = \frac{P \times 4 \times 4}{100} = \frac{16P}{100} = \frac{4P}{25} \]

3. **Calculate Compound Interest (CI):**
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Here, \( R = 5\% \) and \( T = 3 \) years.
Thus, the amount after 3 years is:
\[ A = P \left(1 + \frac{5}{100}\right)^3 = P \left(\frac{105}{100}\right)^3 = P \left(\frac{21}{20}\right)^3 \]
Now, calculating \( \left(\frac{21}{20}\right)^3 \):
\[ \left(\frac{21}{20}\right)^3 = \frac{21^3}{20^3} = \frac{9261}{8000} \]
Therefore, the amount \( A \) is:
\[ A = P \times \frac{9261}{8000} \]
The compound interest is then:
\[ CI = A - P = P \times \frac{9261}{8000} - P = P \left(\frac{9261 - 8000}{8000}\right) = P \left(\frac{1261}{8000}\right) \]

4. **Set Up the Equation:**
According to the problem, the simple interest exceeds the compound interest by Rs 228:
\[ SI - CI = 228 \]
Substituting the values we calculated:
\[ \frac{4P}{25} - P \left(\frac{1261}{8000}\right) = 228 \]

5. **Finding a Common Denominator:**
The common denominator for 25 and 8000 is 8000. We rewrite the equation:
\[ \frac{4P \times 320}{8000} - \frac{1261P}{8000} = 228 \]
This simplifies to:
\[ \frac{1280P - 1261P}{8000} = 228 \]
Which further simplifies to:
\[ \frac{19P}{8000} = 228 \]

6. **Solve for \( P \):**
Multiply both sides by 8000:
\[ 19P = 228 \times 8000 \]
\[ 19P = 1824000 \]
Now, divide by 19:
\[ P = \frac{1824000}{19} = 96000 \]

7. **Conclusion:**
The sum of money is \( \text{Rs } 96000 \).