In a business, investor B invests an amount for 15 months, while investor A invests Rs. 1,200 more than B and keeps it invested for 4 months longer than B (that is, 19 months). The total profit is Rs. 1,240, and B's share of profit is Rs. 280 less than A's share. What amount (in rupees) did A invest?

Introduction / Context:
This is a partnership and profit-sharing problem. The partners invest different amounts for different periods, and their profit shares are proportional to the product of capital and time. We are given the relationship between the profit shares of A and B and the fact that A invests more money for a longer duration. From the total profit and the difference in shares, we must work backwards to determine A's invested capital. Problems of this type are standard in aptitude tests, especially in the section on simple partnerships.

Given Data / Assumptions:

• B invests some amount, say B rupees, for 15 months.
• A invests (B + 1,200) rupees for 19 months (which is 4 months more than B).
• Total profit from the business = Rs. 1,240.
• B's profit share is Rs. 280 less than A's profit share.
• Profits are shared in the ratio of (capital * time) for each partner.

Concept / Approach:
In simple partnership problems, the profit ratio equals the ratio of investments multiplied by the time for which the money remains invested. Let A's and B's profit shares be Pa and Pb respectively. We know that Pa + Pb = 1,240 and Pa − Pb = 280. From these two equations, we find Pa and Pb. Once we know the ratio Pa:Pb, we equate it to the ratio of (capital * time) for A and B. From there, we can solve for the unknown capital of B and hence of A.

Step-by-Step Solution:
Let A's profit share be Pa and B's profit share be Pb.Given: Pa + Pb = 1240 and Pa − Pb = 280.Add the equations: (Pa + Pb) + (Pa − Pb) = 1240 + 280 ⇒ 2Pa = 1520 ⇒ Pa = 760.Then Pb = 1240 − 760 = 480.So the profit ratio A : B = Pa : Pb = 760 : 480.Simplify this ratio by dividing by 40: 760/40 : 480/40 = 19 : 12.Now express the ratio of capital * time.Let B's capital be b; then A's capital = b + 1200.B's investment product = b * 15; A's investment product = (b + 1200) * 19.Thus, (b + 1200) * 19 : b * 15 = 19 : 12.This implies (b + 1200) * 19 / (b * 15) = 19 / 12.Cancel 19 from both sides: (b + 1200) / (15b) = 1 / 12.Cross multiply: 12(b + 1200) = 15b ⇒ 12b + 14400 = 15b ⇒ 3b = 14400 ⇒ b = 4800.Hence A's capital = b + 1200 = 4800 + 1200 = Rs. 6,000.

Verification / Alternative check:
A's investment product = 6000 * 19 = 114,000.B's investment product = 4800 * 15 = 72,000.Ratio = 114000 : 72000 = 19 : 12, which matches the profit ratio.Total profit = 1,240, and splitting in 19 : 12 gives A = 760, B = 480, with a difference of 280, consistent with the problem.

Why Other Options Are Wrong:
If A's capital were 7,000, then the ratio of capital * time would not simplify to 19 : 12.Similarly, 5,000 or 8,000 fail to satisfy both the total profit and the difference of 280 between profits.Only Rs. 6,000 fits all given conditions simultaneously.

Common Pitfalls:
Some learners mix up the given difference in profit with the difference in investments, which are not the same.Another mistake is to use 4 months instead of 19 months for A's time without adding it to B's 15 months.It is also easy to mishandle the ratio simplification and lose the factor of 19, leading to incorrect calculations.

Final Answer:
The amount invested by A in the business is Rs. 6,000.