A certain sum (in ₹)is invested at simple interest at y% per annum for 4 years. Had it been invested at (y + 4) % per annum at simple interest, it would have fetched ₹4,452 more as interest. What is the sum?

Options

A. ₹24,400
B. ₹31,800
C. ₹30,400
D. ₹27,825

Answer

To solve the problem step by step, we will use the formula for simple interest and set up the equations based on the information given in the question.

### Step 1: Define the Variables
Let the principal sum (the amount invested) be \( X \) (in ₹).

### Step 2: Write the Simple Interest Formula
The formula for simple interest is given by:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest per annum, and \( T \) is the time in years.

### Step 3: Calculate Simple Interest for Both Rates
1. For the first investment at \( y\% \) for 4 years:
\[ \text{SI}_1 = \frac{X \times y \times 4}{100} \]

2. For the second investment at \( (y + 4)\% \) for 4 years:
\[ \text{SI}_2 = \frac{X \times (y + 4) \times 4}{100} \]

### Step 4: Set Up the Equation Based on the Given Information
According to the problem, the difference in interest between the two investments is ₹4,452:
\[ \text{SI}_2 - \text{SI}_1 = 4452 \]

Substituting the expressions for \( \text{SI}_1 \) and \( \text{SI}_2 \):
\[ \frac{X \times (y + 4) \times 4}{100} - \frac{X \times y \times 4}{100} = 4452 \]

### Step 5: Simplify the Equation
Factoring out the common terms:
\[ \frac{4X}{100} \left((y + 4) - y\right) = 4452 \]
This simplifies to:
\[ \frac{4X}{100} \times 4 = 4452 \]

### Step 6: Solve for \( X \)
Now, simplifying further:
\[ \frac{16X}{100} = 4452 \]
Multiplying both sides by 100:
\[ 16X = 445200 \]
Now, divide by 16:
\[ X = \frac{445200}{16} = 27825 \]

### Conclusion
The sum invested is ₹27,825.