Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

Q.Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

Let the total profit be \(x\) rupees. ∴ The amount to be equally divided = 50% of \(x\) \(= x × \frac{50}{100} = \frac{x}{2}\) rupees And the amount to be divided in the ratio of capital = \(x - \frac{x}{2} = \frac{x}{2}\) rupees From the equally divided portion, each friend receives: \(\frac{\frac{x}{2}}{2} = \frac{x}{4}\) rupees Now, the ratio of their investments = ₹40,000 : ₹50,000 \(= 4 : 5 = \frac{4}{9} : \frac{5}{9}\) [∵ 4 + 5 = 9] ∴ From the capital-based portion of \(\frac{x}{2}\) rupees: First friend receives \(= \frac{4}{9} × \frac{x}{2} = \frac{4x}{18}\) rupees Second friend receives \(= \frac{5}{9} × \frac{x}{2} = \frac{5x}{18}\) rupees ∴ Total profit share of the first friend = \(\frac{x}{4} + \frac{4x}{18} = \frac{9x + 8x}{36} = \frac{17x}{36}\) rupees Total profit share of the second friend = \(\frac{x}{4} + \frac{5x}{18} = \frac{9x + 10x}{36} = \frac{19x}{36}\) rupees According to the condition: \(\frac{19x}{36} - \frac{17x}{36} = 800\) i.e., \(\frac{2x}{36} = 800\) i.e., \(x = \frac{800 × 36}{2} = 14400\) ∴ Total profit = ₹14,400 First friend’s share = \(\frac{17 × 14400}{36} = ₹6800\) So, the first friend’s profit share is ₹6800.