A sum amounts to Rs 6,050 in 2 years and to Rs 6, 655 in 3 years at a certain rate percentage p.a. when the interest is compounded yearly. What is the simple interest on a sum of Rs 6,000 at the same rate for `5(3)/(4)` years ?

Options

A. Rs 3,450
B. Rs 3,150
C. Rs 3,300
D. Rs 3,200

Answer

To solve the problem step by step, we will first determine the rate of interest using the compound interest information provided and then calculate the simple interest on a sum of Rs 6,000 for the specified time period.

### Step 1: Determine the Rate of Interest

We know:
- Amount after 2 years (A1) = Rs 6,050
- Amount after 3 years (A2) = Rs 6,655

Using the compound interest formula:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]

For the first amount:
\[ 6050 = P \left(1 + \frac{r}{100}\right)^2 \quad \text{(1)} \]

For the second amount:
\[ 6655 = P \left(1 + \frac{r}{100}\right)^3 \quad \text{(2)} \]

### Step 2: Form a Ratio

Dividing equation (2) by equation (1):
\[ \frac{6655}{6050} = \frac{P \left(1 + \frac{r}{100}\right)^3}{P \left(1 + \frac{r}{100}\right)^2} \]
This simplifies to:
\[ \frac{6655}{6050} = 1 + \frac{r}{100} \]

### Step 3: Calculate the Left Side

Calculating the left side:
\[ \frac{6655 - 6050}{6050} = \frac{605}{6050} = \frac{1}{10} \]

Thus, we have:
\[ 1 + \frac{r}{100} = 1.1 \]

### Step 4: Solve for r

From the equation:
\[ 1 + \frac{r}{100} = 1.1 \]
Subtracting 1 from both sides gives:
\[ \frac{r}{100} = 0.1 \]
Multiplying by 100:
\[ r = 10\% \]

### Step 5: Calculate Simple Interest

Now, we need to calculate the simple interest on Rs 6,000 at the same rate (10%) for \(5 \frac{3}{4}\) years.

Convert \(5 \frac{3}{4}\) years to an improper fraction:
\[ 5 \frac{3}{4} = \frac{23}{4} \text{ years} \]

### Step 6: Use the Simple Interest Formula

The formula for simple interest (SI) is:
\[ SI = \frac{P \times r \times t}{100} \]
Where:
- \(P = 6000\)
- \(r = 10\)
- \(t = \frac{23}{4}\)

Substituting the values:
\[ SI = \frac{6000 \times 10 \times \frac{23}{4}}{100} \]
\[ SI = \frac{6000 \times 10 \times 23}{400} \]
\[ SI = \frac{1380000}{400} = 3450 \]

### Final Answer

The simple interest on a sum of Rs 6,000 at the same rate for \(5 \frac{3}{4}\) years is Rs 3,450.

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