P, Q and R invested ₹ 45000, ₹ 70000 and ₹ 90000 respectively to start a business. At the end of 2 years, they earned a profit of ₹ 164000. What will be Q's share in the profit?

### Concept & Equal Time Partnership In a partnership where all members invest their capital for an identical duration, the profit is distributed precisely in the ratio of their initial investments. $$ \text{Profit Ratio} = \text{Investment}_1 : \text{Investment}_2 : \text{Investment}_3 $$ ### Step-by-Step Solution * **Given:** * Investments: P = ₹ $45000$, Q = ₹ $70000$, R = ₹ $90000$ * Total Profit = ₹ $164000$ * Time = $2$ years (Constant for all, thus negligible for ratio) * **Calculation/Deduction:** 1. Find the ratio of their investments: $$P : Q : R = 45000 : 70000 : 90000$$ 2. Simplify by canceling out three zeros from each term: $$P : Q : R = 45 : 70 : 90$$ 3. Further simplify by dividing all terms by their highest common factor, $5$: $$P : Q : R = 9 : 14 : 18$$ 4. Sum the parts of the ratio to find the total denominator: $$\text{Total parts} = 9 + 14 + 18 = 41$$ 5. Calculate Q's share (which is $14$ parts out of $41$): $$\text{Q's Share} = 164000 \times \left(\frac{14}{41}\right)$$ 6. Simplify and solve: $$\frac{164000}{41} = 4000$$ $$\text{Q's Share} = 4000 \times 14 = 56000$$ ### Exam Strategy & Shortcut Recognize that $41 \times 4 = 164$. Once you get the total parts as $41$, the division step becomes mental math: $164000 / 41 = 4000$. Q represents $14$ units, so $14 \times 4000 = 56000$. ### Common Pitfall A time-consuming mistake is multiplying each investment by $2$ (for the years) before finding the ratio. Since the time is the same for all investors, it will eventually cancel out. Save time by ignoring uniform time periods immediately. ### Final Answer Therefore, the correct answer is **₹ 56000**.