The simple interest on a certain sum for 4 years is 28 % of the sum .What will be the simple interest on a sum of Rs. 2,880 for `3(3)/(4)` years at double the earlier rate?

Options

A. Rs. 1,296
B. Rs. 1,528
C. Rs. 1,620
D. Rs. 1,512

Answer

To solve the problem step by step, we need to find the simple interest on a sum of Rs. 2,880 for \(3\frac{3}{4}\) years at double the earlier rate.

### Step 1: Determine the original rate of interest
The problem states that the simple interest for 4 years is 28% of the principal sum.

Let the principal sum be \(P\).

The simple interest (SI) can be calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \(R\) is the rate of interest and \(T\) is the time in years.

From the problem, we know:
\[ SI = 0.28P \quad \text{(since 28% of P)} \]
and \(T = 4\) years.

Substituting the values into the SI formula:
\[ 0.28P = \frac{P \times R \times 4}{100} \]

### Step 2: Solve for the original rate \(R\)
We can simplify the equation:
\[ 0.28 = \frac{R \times 4}{100} \]
Multiplying both sides by 100:
\[ 28 = 4R \]
Now, divide by 4:
\[ R = 7\% \]

### Step 3: Determine the new rate of interest
The problem states that we need to find the simple interest at double the earlier rate. Therefore, the new rate \(R'\) is:
\[ R' = 2 \times R = 2 \times 7\% = 14\% \]

### Step 4: Calculate the time in years
The time given is \(3\frac{3}{4}\) years, which can be converted to an improper fraction:
\[ 3\frac{3}{4} = \frac{15}{4} \text{ years} \]

### Step 5: Calculate the simple interest on Rs. 2,880
Now, we can use the SI formula again with the new principal \(P = 2880\), new rate \(R' = 14\%\), and time \(T = \frac{15}{4}\):
\[ SI = \frac{2880 \times 14 \times \frac{15}{4}}{100} \]

### Step 6: Simplify the calculation
Calculating step by step:
1. Calculate \(2880 \times 14\):
\[ 2880 \times 14 = 40320 \]
2. Multiply by \(\frac{15}{4}\):
\[ 40320 \times \frac{15}{4} = 100800 \]
3. Finally, divide by 100:
\[ SI = \frac{100800}{100} = 1008 \]

Thus, the simple interest on the sum of Rs. 2,880 for \(3\frac{3}{4}\) years at double the earlier rate is **Rs. 1008**.