A certain sum (in Rs.) is invested at simple interest at x% p.a. for 5 years. Had it been invested at (x+ 5)% p.a., the simple interest would have been Rs.9,200 more than the earlier one.
What is the sum?

Options

A. Rs.36,800
B. Rs.40,000
C. Rs.36,400
D. Rs.35,800

Answer

To solve the problem step by step, we need to establish the relationship between the simple interest earned at two different rates over the same time period.

### Step 1: Define the variables
Let:
- \( P \) = Principal amount (the sum we want to find)
- \( x \) = Rate of interest in percentage per annum

### Step 2: Calculate the simple interest for both rates
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal
- \( R \) = Rate of interest
- \( T \) = Time in years

For the first scenario (at \( x\% \)):
\[ SI_1 = \frac{P \times x \times 5}{100} \]

For the second scenario (at \( (x + 5)\% \)):
\[ SI_2 = \frac{P \times (x + 5) \times 5}{100} \]

### Step 3: Set up the equation
According to the problem, the difference in simple interest between the two scenarios is Rs. 9,200:
\[ SI_2 - SI_1 = 9200 \]

Substituting the expressions for \( SI_1 \) and \( SI_2 \):
\[ \frac{P \times (x + 5) \times 5}{100} - \frac{P \times x \times 5}{100} = 9200 \]

### Step 4: Simplify the equation
Factor out common terms:
\[ \frac{P \times 5}{100} \left((x + 5) - x\right) = 9200 \]
This simplifies to:
\[ \frac{P \times 5}{100} \times 5 = 9200 \]

### Step 5: Further simplify
This can be rewritten as:
\[ \frac{P \times 25}{100} = 9200 \]

### Step 6: Solve for \( P \)
Multiply both sides by 100:
\[ P \times 25 = 920000 \]
Now, divide both sides by 25:
\[ P = \frac{920000}{25} = 36800 \]

### Conclusion
The principal amount \( P \) is Rs. 36,800.