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, we need to follow these steps:

### Step 1: Calculate the Simple Interest Rate
We know that the formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

From the question, we have:
- Principal amount \( P = 400 \)
- Amount after 2 years = \( 448 \)
- Time \( T = 2 \)

First, we calculate the Simple Interest earned in 2 years:
\[ SI = Amount - Principal = 448 - 400 = 48 \]

Now, we can use the SI formula to find the rate \( R \):
\[ 48 = \frac{400 \times R \times 2}{100} \]
\[ 48 = \frac{800R}{100} \]
\[ 48 = 8R \]
\[ R = \frac{48}{8} = 6\% \]

### Step 2: Calculate the Time for Rs. 550 to Amount to Rs. 682
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%).

Using the same formula for Simple Interest:
\[ SI = Amount - Principal = 682 - 550 = 132 \]

Now, substituting the values into the SI formula:
\[ 132 = \frac{550 \times 6 \times T}{100} \]
\[ 132 = \frac{3300T}{100} \]
\[ 132 = 33T \]
\[ T = \frac{132}{33} = 4 \]

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

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