A certain sum (in Rs) is invested at simple interest at y% per annum for 3% years. Had it been invested at (y + 4)% per annum at simple interst, it would have fetched Rs 4,452 more as interest. What is the sum

Options

A. Rs 42,400
B. Rs 31,800
C. Rs 30,400
D. Rs 42,800

Answer

To solve the problem step by step, we will follow the information given in the question and use the formula for simple interest.

### Step 1: Define the variables
Let the principal sum be Rs. \( X \) and the rate of interest be \( y\% \) per annum. The time period is given as \( 3\frac{1}{2} \) years, which can be converted to an improper fraction:
\[ 3\frac{1}{2} = \frac{7}{2} \text{ years} \]

### Step 2: Write the formula for simple interest
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

### Step 3: Calculate the simple interest for both rates
1. **For the rate \( y\% \)**:
\[ SI_1 = \frac{X \times y \times \frac{7}{2}}{100} \]

2. **For the rate \( (y + 4)\% \)**:
\[ SI_2 = \frac{X \times (y + 4) \times \frac{7}{2}}{100} \]

### Step 4: Set up the equation based on the difference in interest
According to the problem, the difference in interest earned at the two rates is Rs. 4452:
\[ SI_2 - SI_1 = 4452 \]
Substituting the expressions for \( SI_1 \) and \( SI_2 \):
\[ \frac{X \times (y + 4) \times \frac{7}{2}}{100} - \frac{X \times y \times \frac{7}{2}}{100} = 4452 \]

### Step 5: Simplify the equation
Factor out the common terms:
\[ \frac{X \times \frac{7}{2}}{100} \left( (y + 4) - y \right) = 4452 \]
This simplifies to:
\[ \frac{X \times \frac{7}{2}}{100} \times 4 = 4452 \]

### Step 6: Solve for \( X \)
Now, simplify the equation:
\[ \frac{X \times 28}{200} = 4452 \]
Multiply both sides by 200:
\[ X \times 28 = 4452 \times 200 \]
Calculate \( 4452 \times 200 \):
\[ 4452 \times 200 = 890400 \]
Now divide by 28 to solve for \( X \):
\[ X = \frac{890400}{28} \]
Calculating this gives:
\[ X = 31800 \]

### Final Answer
The sum invested is Rs. \( 31,800 \).

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