A sum was invested at simple interset at x% p.a for `2(1)/(2)` years. Had it been invested at `(x + 3)`% for the same time,it would have fetched Rs585 more . The simple interset on the same sum for `4(2)/(3)` year at 14% p.a is .

Options

A. Rs 5,096
B. Rs 5, 460
C. Rs 4,732
D. Rs 4, 900

Answer

To solve the problem step by step, we need to determine the principal amount (P) based on the information provided and then calculate the simple interest for a different time period and interest rate.

### Step 1: Set up the equations based on the information given
We know that the simple interest (SI) formula is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

From the problem, we have two scenarios:
1. The first investment at \( x\% \) for \( 2\frac{1}{2} \) years (which is \( \frac{5}{2} \) years).
2. The second investment at \( (x + 3)\% \) for the same time period, which yields Rs 585 more than the first investment.

### Step 2: Write the equation for the first scenario
For the first scenario:
\[ \text{SI}_1 = \frac{P \times x \times \frac{5}{2}}{100} = \frac{5Px}{200} \]

### Step 3: Write the equation for the second scenario
For the second scenario:
\[ \text{SI}_2 = \frac{P \times (x + 3) \times \frac{5}{2}}{100} = \frac{5P(x + 3)}{200} \]

### Step 4: Set up the equation based on the difference in interest
According to the problem:
\[ \text{SI}_2 - \text{SI}_1 = 585 \]
Substituting the expressions from Steps 2 and 3:
\[ \frac{5P(x + 3)}{200} - \frac{5Px}{200} = 585 \]

### Step 5: Simplify the equation
Combining the two fractions:
\[ \frac{5P((x + 3) - x)}{200} = 585 \]
This simplifies to:
\[ \frac{5P \cdot 3}{200} = 585 \]
\[ \frac{15P}{200} = 585 \]

### Step 6: Solve for P
Multiplying both sides by 200:
\[ 15P = 585 \times 200 \]
\[ 15P = 117000 \]
Now, divide by 15:
\[ P = \frac{117000}{15} = 7800 \]

### Step 7: Calculate the simple interest for the new scenario
Now we need to find the simple interest for \( P = 7800 \), \( R = 14\% \), and \( T = 4\frac{2}{3} \) years (which is \( \frac{14}{3} \) years).

Using the SI formula:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ \text{SI} = \frac{7800 \times 14 \times \frac{14}{3}}{100} \]

### Step 8: Calculate the simple interest
Calculating the above expression:
\[ \text{SI} = \frac{7800 \times 14 \times 14}{300} \]
Calculating \( 7800 \times 14 = 109200 \):
\[ \text{SI} = \frac{109200 \times 14}{300} \]
Calculating \( 109200 \times 14 = 1528800 \):
\[ \text{SI} = \frac{1528800}{300} = 5096 \]

Thus, the simple interest for \( 4\frac{2}{3} \) years at 14% p.a. is **Rs 5096**.