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. Using the simple interest formula, find the value of X, that is, the original smaller amount.

Introduction / Context:
This problem compares simple interest earned on two different principals over different time periods, both at the same rate of interest. The interest on a smaller amount X for one year is compared with interest on a larger amount (X + 1400) for two years. The difference between these two simple interests is given as Rs 240. The task is to translate this relationship into an algebraic equation using the simple interest formula and then solve for X. Such questions build comfort with forming equations involving unknowns from verbal descriptions.

Given Data / Assumptions:

• Rate of interest for both cases, R = 8% per annum.
• Case 1: Principal = X rupees, time T1 = 1 year.
• Case 2: Principal = X + 1400 rupees, time T2 = 2 years.
• Difference between simple interest in Case 2 and Case 1 is Rs 240.
• Interest is calculated using simple interest only.

Concept / Approach:
For simple interest, the formula is:
SI = (P * R * T) / 100 We compute the interest for each case in terms of X. Let SI1 be the interest in the first case and SI2 be the interest in the second case. The problem states that the difference between these two interests is Rs 240. Since the second investment has larger principal and time, SI2 is greater than SI1, so we set SI2 − SI1 = 240. Substituting the expressions for SI1 and SI2 leads to a linear equation in X that can be solved easily.

Step-by-Step Solution:
Rate R = 8% per annum. Case 1: Principal = X, time T1 = 1 year. SI1 = (X * 8 * 1) / 100 = 0.08X. Case 2: Principal = X + 1400, time T2 = 2 years. SI2 = ((X + 1400) * 8 * 2) / 100. Simplify: SI2 = (X + 1400) * 16 / 100 = 0.16(X + 1400). Given that SI2 − SI1 = 240. So 0.16(X + 1400) − 0.08X = 240. Expand: 0.16X + 0.16 * 1400 − 0.08X = 240. 0.16X − 0.08X + 224 = 240. 0.08X + 224 = 240. 0.08X = 240 − 224 = 16. X = 16 / 0.08 = 200. Therefore, the required value of X is Rs 200.

Verification / Alternative check:
Compute the two interests with X = 200. For Case 1, principal = 200, so SI1 = (200 * 8 * 1) / 100 = 16. For Case 2, principal = 200 + 1400 = 1600, time = 2 years, so SI2 = (1600 * 8 * 2) / 100 = (1600 * 16) / 100 = 256. The difference SI2 − SI1 = 256 − 16 = 240, which matches the given condition. This confirms that X = Rs 200 is correct.

Why Other Options Are Wrong:
If X were Rs 100, SI1 would be 8, and SI2 would be (1500 * 16) / 100 = 240, giving a difference of 232, not 240.
If X were Rs 400, SI1 would be 32, and SI2 would be (1800 * 16) / 100 = 288, giving a difference of 256.
If X were Rs 300, or Rs 250, similar calculations show that the difference between SI2 and SI1 does not equal 240. Only X = Rs 200 satisfies the exact relationship.

Common Pitfalls:
A common error is to set up the equation as SI1 − SI2 = 240 instead of SI2 − SI1 = 240, which can lead to negative values and confusion. Another mistake is to forget that the time in the second case is 2 years and use T = 1 year for both cases. Some students also mishandle the decimal 0.08 and 0.16 calculations. Careful substitution and stepwise simplification reduce the risk of such errors.

Final Answer:
The value of the original amount X is Rs 200.