A and B enter into a partnership with ₹ 50000 and ₹ 60000 respectively. C joins them after $x$ months, contributing ₹ 70000 and B leaves $x$ months before the end of the year. If they share the profit in the ratio of 20 : 18 : 21, then the value of $x$ is

### Concept & Formula Partnership profits are shared in proportion to the product of investment amounts and time periods. $$Profit Ratio = (C_A \times T_A) : (C_B \times T_B) : (C_C \times T_C)$$ ### Step-by-Step Solution 1. A invested ₹ 50000 for the full 12 months. (Equivalent capital = $50 \times 12 = 600$ units of 1000) 2. B invested ₹ 60000 but left $x$ months before the end. So B's time period is $(12 - x)$ months. (Equivalent capital = $60(12 - x)$) 3. C joined after $x$ months, contributing ₹ 70000. So C's time period is also $(12 - x)$ months. (Equivalent capital = $70(12 - x)$) 4. The ratio of their profits is given as 20 : 18 : 21. 5. Equating A's and B's shares: $\frac{600}{60(12 - x)} = \frac{20}{18}$ 6. Simplify the equation: $\frac{10}{12 - x} = \frac{10}{9}$ 7. By comparing the denominators (since numerators are equal), we get: $12 - x = 9$ 8. $x = 12 - 9 = 3$. 9. Verify with C's share: C's equivalent capital = $70(12 - 3) = 70 \times 9 = 630$. Checking ratio A : C $\Rightarrow 600 : 630 \Rightarrow 60 : 63 \Rightarrow 20 : 21$. This perfectly matches the given ratio. ### Exam Strategy & Shortcut Just look at A and B. A's capital is 50k for 12 months = 600k. The ratio gives A 20 units. So 1 unit = 30k. B gets 18 units, so B's capital-time product is $18 \times 30k = 540k$. Since B invests 60k, time is $\frac{540}{60} = 9$ months. B left $x$ months early, so $12 - 9 = 3$. ### Common Pitfall Mishandling the variables: B leaves $x$ months *before* the end (time = $12 - x$), C joins *after* $x$ months (time = $12 - x$). It's easy to make a sign error and use $x$ as the time period itself. ### Final Answer Therefore, the correct answer is **3**.