A sum of Rs 450 amounts to Rs 495 at simple interest in 2 years. In what time will the sum of Rs 820 amount to ` 943 at the same rate ?

Answer

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

### Step 1: Calculate the Rate of Interest

We know the following from the question:
- Principal (P) = Rs 450
- Amount (A) = Rs 495
- Time (T) = 2 years

Using the formula for Simple Interest (SI):
\[ SI = A - P \]
Substituting the values:
\[ SI = 495 - 450 = 45 \]

Now, we can use the formula for Simple Interest to find the rate (R):
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 45 = \frac{450 \times R \times 2}{100} \]

### Step 2: Solve for R

Rearranging the equation to solve for R:
\[ 45 = \frac{900R}{100} \]
\[ 45 = 9R \]
Dividing both sides by 9:
\[ R = 5\% \]

### Step 3: Use the Rate to Find Time for the Second Case

Now we will use the rate we found to calculate the time for the second part of the question:
- Principal (P) = Rs 820
- Amount (A) = Rs 943
- Rate (R) = 5%

Using the formula for Simple Interest again:
\[ SI = A - P \]
Substituting the values:
\[ SI = 943 - 820 = 123 \]

Now we can use the formula for Simple Interest to find the time (T):
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 123 = \frac{820 \times 5 \times T}{100} \]

### Step 4: Solve for T

Rearranging the equation to solve for T:
\[ 123 = \frac{4100T}{100} \]
\[ 123 = 41T \]
Dividing both sides by 41:
\[ T = \frac{123}{41} \]
Calculating the value:
\[ T = 3 \]

### Conclusion

The time required for the sum of Rs 820 to amount to Rs 943 at the same rate is **3 years**.