Gautam started a business with a sum of ₹ 60000. Jatin joined him 8 months later with a sum of ₹ 35000. At what respective ratio will the two share the profit after two years?

### Concept & Active Investment Duration For partnerships spanning multiple years, carefully convert the total duration into months. Calculate the active duration for late joiners by subtracting their joining delay from the total duration. $$ \text{Profit Ratio} = (C_1 \times T_1) : (C_2 \times T_2) $$ ### Step-by-Step Solution * **Given:** * Total duration = $2$ years = $24$ months. * Gautam's capital = ₹ $60000$ (invested for the full $24$ months). * Jatin's capital = ₹ $35000$ (joined $8$ months later). * **Calculation/Deduction:** 1. Determine Jatin's active time: $$\text{Jatin's time} = 24 - 8 = 16 \text{ months}$$ 2. Set up the profit ratio based on Capital $\times$ Time: $$\text{Gautam} : \text{Jatin} = (60000 \times 24) : (35000 \times 16)$$ 3. Simplify by dividing by $1000$ (drop three zeros): $$\text{Ratio} = (60 \times 24) : (35 \times 16)$$ 4. Factor and simplify further. Divide by $8$: $$\text{Ratio} = (60 \times 3) : (35 \times 2)$$ $$\text{Ratio} = 180 : 70$$ 5. Divide by $10$: $$\text{Ratio} = 18 : 7$$ ### Exam Strategy & Shortcut To quickly simplify $(60 \times 24) : (35 \times 16)$, look for common denominators across both sides of the ratio. $24$ and $16$ are divisible by $8$ (leaving $3$ and $2$). $60$ and $35$ are divisible by $5$ (leaving $12$ and $7$). So the ratio simplifies directly to $(12 \times 3) : (7 \times 2) \implies 36 : 14 \implies 18 : 7$. ### Common Pitfall Students often miss that the total duration is "two years" and mistakenly calculate using a 12-month base, which completely skews the time ratio. Always verify the total timeline. ### Final Answer Therefore, the correct answer is **18 : 7**.