A sum of 7500 amounts to 8748 after 2 years at a certain rate per cent per annum compounded annually. What will be the simple interest (in) on the same sum for 3 years at double the earlier rate ?

Options

A. 3600
B. 3750
C. 2400
D. 2800

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We need to find the simple interest on a principal amount of 7500 for 3 years at double the rate of interest, which we first need to determine from the given compound interest scenario.

### Step 2: Identify the Given Values
- Principal (P) = 7500
- Amount (A) after 2 years = 8748
- Time (n) = 2 years

### Step 3: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the known values:
\[ 8748 = 7500 \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Rearranging the Equation
We can rearrange the equation to isolate the term involving R:
\[ \frac{8748}{7500} = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Simplifying the Left Side
Calculating the left side:
\[ \frac{8748}{7500} = 1.1664 \]

### Step 6: Taking the Square Root
Now we take the square root of both sides:
\[ 1 + \frac{R}{100} = \sqrt{1.1664} \]

Calculating the square root:
\[ 1 + \frac{R}{100} = 1.08 \]

### Step 7: Isolating R
Now, we isolate R:
\[ \frac{R}{100} = 1.08 - 1 = 0.08 \]
\[ R = 0.08 \times 100 = 8\% \]

### Step 8: Doubling the Rate
Now, we double the rate:
\[ \text{New Rate} = 2 \times 8\% = 16\% \]

### Step 9: Calculate Simple Interest
Now we calculate the simple interest for 3 years at the new rate:
\[ \text{Simple Interest (SI)} = \frac{P \times R \times T}{100} \]
Where:
- P = 7500
- R = 16
- T = 3

Substituting the values:
\[ SI = \frac{7500 \times 16 \times 3}{100} \]

### Step 10: Simplifying the Calculation
Calculating the simple interest:
\[ SI = \frac{7500 \times 48}{100} = 7500 \times 0.48 = 3600 \]

### Final Answer
The simple interest on the same sum for 3 years at double the earlier rate is **3600**.

---