The difference between the simple interest obtained by investing an amount X rupees at 8% per annum for one year and the simple interest obtained by investing X plus 1400 rupees at 8% per annum for two years is 240 rupees. Find the value of X.

Introduction / Context:
This problem involves comparing two simple interest amounts that arise from slightly different investments and then using their difference to determine the original principal X. Both investments use the same rate of 8 percent per annum, but one uses X for one year and the other uses X plus 1400 for two years. The question states that the difference between these two interest amounts is 240 rupees, and we must find X. This is a linear equation in one variable based on the simple interest formula.

Given Data / Assumptions:
First investment: principal is X rupees, rate is 8 percent per annum, time is 1 year.

Second investment: principal is X + 1400 rupees, rate is 8 percent per annum, time is 2 years.
Difference between the interest from the second and the first investment is 240 rupees.
Interest is simple interest in both cases with the same annual rate.

Concept / Approach:
We use the simple interest formula I = P * r * t / 100 for both cases. Let I1 be the interest from X for one year and I2 be the interest from X plus 1400 for two years. Because the rate is the same for both, the only differences are principal and time. The given relationship is I2 - I1 = 240. By substituting the expressions for I1 and I2 and simplifying, we get a linear equation in X that can be solved easily.

Step-by-Step Solution:
For the first investment, simple interest I1 = X * 8 * 1 / 100 = 0.08X.For the second investment, simple interest I2 = (X + 1400) * 8 * 2 / 100.Simplify I2 to I2 = (X + 1400) * 16 / 100 = 0.16(X + 1400).The difference between the interests is I2 - I1 = 240 rupees.Write the equation: 0.16(X + 1400) - 0.08X = 240.Expand the first term: 0.16X + 0.16 * 1400 - 0.08X = 240.Compute 0.16 * 1400 = 224, so we have 0.16X + 224 - 0.08X = 240.Combine like terms for X to get 0.08X + 224 = 240.Rearrange to 0.08X = 240 - 224 = 16.Solve for X: X = 16 / 0.08 = 200 rupees.Therefore the value of X is Rs. 200.

Verification / Alternative check:
For X = 200, compute I1 and I2. Simple interest I1 for the first investment is 200 * 8 / 100 = 16 rupees. Second principal is X + 1400 = 1600 rupees, and I2 for two years is 1600 * 8 * 2 / 100 = 1600 * 16 / 100 = 256 rupees. The difference is 256 - 16 = 240 rupees, which matches the problem statement. This confirms that X = 200 is the correct solution.

Why Other Options Are Wrong:
If X were 100 rupees, the interest difference would be much smaller than 240 rupees.
X = 400 or X = 300 would produce larger or smaller differences that do not equal 240 rupees when interest is computed correctly.
X = 250 also fails to satisfy the difference equation. Only X = 200 gives the exact difference of 240 rupees.

Common Pitfalls:
Students sometimes reverse the order of subtraction and write I1 - I2 instead of I2 - I1, leading to a negative difference. Another issue is mishandling the multiplication by 16 when calculating the second interest. Small arithmetic mistakes in decimal operations, such as miscomputing 0.16 * 1400, also cause incorrect answers. Careful equation setup and stepwise simplification helps avoid these errors.

Final Answer:
The value of X that satisfies the given simple interest difference condition is Rs. 200.