A sum of Rs. 400 becomes Rs 448 at simple interest in 2 years. In how many years will the sum of Rs. 550 amounts to Rs. 682 at the same rate ?

Options

A. 2
B. 3
C. 3.5
D. 4

Answer

To solve the problem step by step, we need to first determine the rate of interest from the first part of the question and then use that rate to find the time required for the second part.

### Step 1: Calculate the Simple Interest for the first case
The formula for Simple Interest (SI) is:
\[ \text{SI} = \text{Principal} \times \text{Rate} \times \text{Time} \]

From the question:
- Principal (P) = Rs. 400
- Amount (A) = Rs. 448
- Time (T) = 2 years

First, we calculate the Simple Interest:
\[ \text{SI} = \text{Amount} - \text{Principal} \]
\[ \text{SI} = 448 - 400 = 48 \]

### Step 2: Use the Simple Interest to find the Rate
Now we can use the SI to find the rate of interest (R):
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 48 = \frac{400 \times R \times 2}{100} \]

Now, simplify and solve for R:
\[ 48 = \frac{800R}{100} \]
\[ 48 = 8R \]
\[ R = \frac{48}{8} = 6\% \]

### Step 3: Calculate the time for the second case
Now we need to find out how many years it will take for Rs. 550 to amount to Rs. 682 at the same rate of interest (6%).

From the question:
- Principal (P) = Rs. 550
- Amount (A) = Rs. 682

First, calculate the Simple Interest for this case:
\[ \text{SI} = \text{Amount} - \text{Principal} \]
\[ \text{SI} = 682 - 550 = 132 \]

Now, we can use the SI to find the time (T):
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 132 = \frac{550 \times 6 \times T}{100} \]

Now, simplify and solve for T:
\[ 132 = \frac{3300T}{100} \]
\[ 132 = 33T \]
\[ T = \frac{132}{33} = 4 \]

### Final Answer
The sum of Rs. 550 will amount to Rs. 682 in **4 years**.

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