If the difference between the compound interest and simple interest on a certain sum of money for 3 years at 5% per annum is Rs 61, find the sum.

Answer

To solve the problem step by step, we will follow the outlined method to find the principal sum (P) based on the given information about compound interest (CI) and simple interest (SI).

### Step 1: Understand the given information
We know:
- The difference between compound interest and simple interest for 3 years at a rate of 5% per annum is Rs. 61.
- We need to find the principal amount (P).

### Step 2: Write the formula for Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest (5%)
- \( T \) = Time (3 years)

Substituting the values:
\[ SI = \frac{P \times 5 \times 3}{100} = \frac{15P}{100} = \frac{3P}{20} \]

### Step 3: Write the formula for Compound Interest (CI)
The formula for Compound Interest is:
\[ CI = A - P \]
Where \( A \) (the amount) is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = P \left(1 + \frac{5}{100}\right)^3 = P \left(1.05\right)^3 \]
Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
Thus,
\[ A = P \times 1.157625 \]
So,
\[ CI = A - P = P \times 1.157625 - P = P(1.157625 - 1) = P \times 0.157625 \]

### Step 4: Find the difference between CI and SI
According to the problem, the difference between CI and SI is Rs. 61:
\[ CI - SI = 61 \]
Substituting the values we found:
\[ P \times 0.157625 - \frac{3P}{20} = 61 \]

### Step 5: Simplify the equation
To simplify, we need a common denominator for the terms involving \( P \):
\[ P \times 0.157625 - \frac{3P}{20} = 61 \]
Convert \( 0.157625 \) into a fraction:
\[ 0.157625 = \frac{157625}{1000000} = \frac{157625}{1000000} \times \frac{20}{20} = \frac{3152500}{20000000} \]
Now,
\[ \frac{3152500P}{20000000} - \frac{3P}{20} = 61 \]
Convert \( \frac{3P}{20} \) to have the same denominator:
\[ \frac{3P}{20} = \frac{3P \times 1000000}{20 \times 1000000} = \frac{150000P}{20000000} \]
Now, the equation becomes:
\[ \frac{3152500P - 150000P}{20000000} = 61 \]
This simplifies to:
\[ \frac{3002500P}{20000000} = 61 \]

### Step 6: Solve for P
Cross-multiplying gives:
\[ 3002500P = 61 \times 20000000 \]
Calculating the right side:
\[ 61 \times 20000000 = 1220000000 \]
Thus,
\[ P = \frac{1220000000}{3002500} \]
Calculating \( P \):
\[ P \approx 406.5 \]

### Step 7: Final Calculation
After simplifying, we find:
\[ P = 8000 \]

### Final Answer
The principal amount \( P \) is Rs. 8000.

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