Three friends, 'X', 'Y', and 'Z', decided to start a business together. They made investments of ₹6,000, ₹8,000, and ₹5,000, respectively. The time for which 'X', 'Y', and 'Z' kept their money invested is 't' months, (t - 3) months, and (t + 3) months, respectively. During the period for which 'X' had his investment, he also worked as a manager for which he receive a monthly salary of ₹3,000 from the gross profit. At the end of the business venture, the amount received by 'X' was ₹25,000 more than that received by 'Z'. The total profit was ₹81,000. Find the profit share of Y.

### Concept & Partnership Ratios In partnership problems, profit is distributed based on the "Investment $\times$ Time" ratio. Additionally, when a working partner receives a salary, it must be deducted from the total profit before distributing the remainder according to the ratio. $$ \text{Profit Ratio} = (I_1 \times T_1) : (I_2 \times T_2) : (I_3 \times T_3) $$ ### Step-by-Step Solution 1. **Find Profit Sharing Ratios:** Investments: $I_X = 6000$, $I_Y = 8000$, $I_Z = 5000$ Time: $T_X = t$, $T_Y = t - 3$, $T_Z = t + 3$ Ratio of Profit Share (X : Y : Z) = $6000t : 8000(t - 3) : 5000(t + 3) = 6t : 8(t - 3) : 5(t + 3)$ Total units of ratio = $6t + 8t - 24 + 5t + 15 = 19t - 9$ 2. **Calculate Distributable Profit:** X receives a salary of ₹3000 per month for $t$ months = $3000t$. Remaining profit to distribute = $81000 - 3000t$. 3. **Formulate the Equation:** Total amount X received = Salary + Profit Share = $3000t + \frac{6t}{19t - 9} \times (81000 - 3000t)$ Total amount Z received = Profit Share = $\frac{5(t + 3)}{19t - 9} \times (81000 - 3000t)$ Given: X's total - Z's total = 25000 $3000t + \frac{6t - 5(t + 3)}{19t - 9} \times 3000(27 - t) = 25000$ $3t + \frac{t - 15}{19t - 9} \times 3(27 - t) = 25$ 4. **Solve for $t$:** $3t(19t - 9) + 3(27t - t^2 - 405 + 15t) = 25(19t - 9)$ $57t^2 - 27t + 3(-t^2 + 42t - 405) = 475t - 225$ $57t^2 - 27t - 3t^2 + 126t - 1215 = 475t - 225$ $54t^2 + 99t - 1215 = 475t - 225$ $54t^2 - 376t - 990 = 0$ Dividing by 2 gives: $27t^2 - 188t - 495 = 0$ Using the quadratic formula: $t = \frac{188 \pm \sqrt{35344 + 53460}}{54} = \frac{188 \pm \sqrt{88804}}{54} = \frac{188 \pm 298}{54}$ $t = \frac{486}{54} = 9$ (Ignoring the negative root) 5. **Calculate Y's Share:** With $t = 9$, total remaining profit = $81000 - 3000(9) = 54000$. Ratio X : Y : Z = $6(9) : 8(6) : 5(12) = 54 : 48 : 60 = 9 : 8 : 10$. Y's share = $\frac{8}{9 + 8 + 10} \times 54000 = \frac{8}{27} \times 54000 = 8 \times 2000 = 16000$. ### Exam Strategy & Shortcut To avoid large numbers when solving quadratics in partnership questions, divide equations by common factors early (like dividing the whole equation by 1000). Also, if you suspect an integer root for time, you can plug in values like 6, 9, 12 into $3t + \dots = 25$ to save time over expanding the full quadratic. ### Common Pitfall Forgetting to subtract the manager's salary from the gross profit before splitting the remaining funds into the investment ratios is the most common reason answers go off-track. ### Final Answer Therefore, the correct answer is **₹16,000**.