If the simple interest on a certain sum for `2(2)/(3)` years at `15%` p.a. is Rs. 514.80 less than the simple interest on the same sum for `4(1)/(4)` years at `12%` p.a. than the sum is

Options

A. a) Rs. 4,680
B. b) Rs. 4,784
C. c) Rs. 4,580
D. d) Rs. 4, 860

Answer

To solve the problem step by step, we will break down the information given and use the formula for simple interest.

### Step 1: Understand the Simple Interest Formula
The formula for simple interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount (the sum of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Calculate Simple Interest for the First Case
For the first case:
- Time \( T_1 = 2 \frac{2}{3} = \frac{8}{3} \) years
- Rate \( R_1 = 15\% \)

Using the formula:
\[ SI_1 = \frac{P \times 15 \times \frac{8}{3}}{100} \]
\[ SI_1 = \frac{P \times 15 \times 8}{300} = \frac{120P}{300} = \frac{2P}{5} \]

### Step 3: Calculate Simple Interest for the Second Case
For the second case:
- Time \( T_2 = 4 \frac{1}{4} = \frac{17}{4} \) years
- Rate \( R_2 = 12\% \)

Using the formula:
\[ SI_2 = \frac{P \times 12 \times \frac{17}{4}}{100} \]
\[ SI_2 = \frac{P \times 12 \times 17}{400} = \frac{204P}{400} = \frac{51P}{100} \]

### Step 4: Set Up the Equation Based on the Given Condition
According to the problem, the simple interest for the first case is Rs. 514.80 less than the simple interest for the second case:
\[ SI_1 + 514.80 = SI_2 \]
Substituting the expressions we found:
\[ \frac{2P}{5} + 514.80 = \frac{51P}{100} \]

### Step 5: Clear the Fractions
To eliminate the fractions, multiply the entire equation by 100:
\[ 100 \left(\frac{2P}{5}\right) + 100 \times 514.80 = 51P \]
\[ 20P + 51480 = 51P \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 51P - 20P = 51480 \]
\[ 31P = 51480 \]

### Step 7: Solve for Principal \( P \)
Now, divide both sides by 31:
\[ P = \frac{51480}{31} \]
Calculating this gives:
\[ P = 1660 \]

### Step 8: Final Calculation
Thus, the principal amount is:
\[ P = 1660 \]

### Conclusion
The sum is Rs. 1660.