A sum amounts to Rs. 6,050 in years and to Rs. 6655 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 and then calculate the simple interest on the given sum.

### Step 1: Understanding the Problem
We know that a certain sum amounts to Rs. 6,050 in 2 years and Rs. 6,655 in 3 years when compounded yearly. We need to find the simple interest on Rs. 6,000 at the same rate for \(5 \frac{3}{4}\) years.

### Step 2: Setting Up the Equations
Let the principal amount be \(P\) and the rate of interest be \(R\% \) per annum.

From the problem, we have:
1. Amount after 2 years:
\[ A_2 = P \left(1 + \frac{R}{100}\right)^2 = 6050 \]
2. Amount after 3 years:
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 = 6655 \]

### Step 3: Dividing the Two Equations
To eliminate \(P\), we can divide the second equation by the first:
\[ \frac{A_3}{A_2} = \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)^2} \]
This simplifies to:
\[ \frac{6655}{6050} = 1 + \frac{R}{100} \]

### Step 4: Calculating the Rate of Interest
Now, we calculate:
\[ \frac{6655}{6050} = 1 + \frac{R}{100} \]
Calculating the left side:
\[ \frac{6655}{6050} \approx 1.1 \]
Thus,
\[ 1.1 = 1 + \frac{R}{100} \]
Subtracting 1 from both sides gives:
\[ 0.1 = \frac{R}{100} \]
Multiplying by 100:
\[ R = 10\% \]

### Step 5: Finding Simple Interest
Now that we have the rate, we can find the simple interest on Rs. 6,000 for \(5 \frac{3}{4}\) years.

First, convert \(5 \frac{3}{4}\) years into an improper fraction:
\[ 5 \frac{3}{4} = \frac{23}{4} \text{ years} \]

The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{6000 \times 10 \times \frac{23}{4}}{100} \]

### Step 6: Calculating the Simple Interest
Calculating the above expression:
\[ SI = \frac{6000 \times 10 \times 23}{400} \]
\[ SI = \frac{1380000}{400} = 3450 \]

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

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