A certain sum in certain time becomes X 500 at the rate of 8% per annum SI and the same sum amounts to X 200 at the rate of 2% per annum SI in the same duration. Find the sum and time.

Answer

To solve the problem step-by-step, we will use the formulas for Simple Interest and the relationships between the amounts, principal, rate, and time.

### Given:
- Amount A1 = 500 at the rate of R1 = 8% per annum
- Amount A2 = 200 at the rate of R2 = 2% per annum
- Time (T) is the same for both cases.

### Step 1: Use the formula for Simple Interest to find the Principal (P).

The formula for the amount in Simple Interest is given by:
\[ A = P + SI \]
Where:
\[ SI = \frac{P \times R \times T}{100} \]

From the two cases, we can express the amounts as:
1. For A1:
\[ A1 = P + \frac{P \times R1 \times T}{100} \]
\[ 500 = P + \frac{P \times 8 \times T}{100} \]

2. For A2:
\[ A2 = P + \frac{P \times R2 \times T}{100} \]
\[ 200 = P + \frac{P \times 2 \times T}{100} \]

### Step 2: Rearranging the equations.

From the first equation:
\[ 500 = P \left(1 + \frac{8T}{100}\right) \]
From the second equation:
\[ 200 = P \left(1 + \frac{2T}{100}\right) \]

### Step 3: Express P from both equations.

From the first equation:
\[ P = \frac{500}{1 + \frac{8T}{100}} \]
From the second equation:
\[ P = \frac{200}{1 + \frac{2T}{100}} \]

### Step 4: Set the two expressions for P equal to each other.

\[ \frac{500}{1 + \frac{8T}{100}} = \frac{200}{1 + \frac{2T}{100}} \]

### Step 5: Cross-multiply to eliminate the fractions.

\[ 500 \left(1 + \frac{2T}{100}\right) = 200 \left(1 + \frac{8T}{100}\right) \]

### Step 6: Expand both sides.

\[ 500 + \frac{1000T}{100} = 200 + \frac{1600T}{100} \]

### Step 7: Simplify the equation.

\[ 500 + 10T = 200 + 16T \]

### Step 8: Rearranging to isolate T.

\[ 500 - 200 = 16T - 10T \]
\[ 300 = 6T \]

### Step 9: Solve for T.

\[ T = \frac{300}{6} = 50 \text{ years} \]

### Step 10: Substitute T back to find P.

Using the value of T in either expression for P:
\[ P = \frac{500}{1 + \frac{8 \times 50}{100}} = \frac{500}{1 + 4} = \frac{500}{5} = 100 \]

### Final Answers:
- Principal (P) = 100
- Time (T) = 50 years