Sonia started a business investing ₹ 60000. After 6 months Vivek joined him with an amount of ₹ 140000. After 1 year Kirti also joined them with ₹ 120000. After 2 years the business yielded a total profit of ₹ 450000. What is the share of Vivek in the profit?

### Concept & Ratio of Equivalent Capitals Profits are distributed in proportion to the equivalent capital of each partner, which is the product of their investment and the time period it remained in the business. $$ \text{Equivalent Capital} = \text{Investment} \times \text{Time} $$ ### Step-by-Step Solution * **Given:** * Total duration = $2 \text{ years} = 24 \text{ months}$ * Sonia's investment = $₹ 60000$ for $24 \text{ months}$ * Vivek's investment = $₹ 140000$ for $(24 - 6) = 18 \text{ months}$ * Kirti's investment = $₹ 120000$ for $(24 - 12) = 12 \text{ months}$ * Total Profit = $₹ 450000$ * **Calculate Profit Ratio:** * $\text{Ratio} = (60000 \times 24) : (140000 \times 18) : (120000 \times 12)$ * Cancel out $10000$ from each: * $\text{Ratio} = (6 \times 24) : (14 \times 18) : (12 \times 12)$ * $\text{Ratio} = 144 : 252 : 144$ * Divide by $12$: * $\text{Ratio} = 12 : 21 : 12$ * Divide by $3$: * $\text{Ratio} = 4 : 7 : 4$ * **Calculate Vivek's Share:** * Sum of ratio parts = $4 + 7 + 4 = 15$ * $\text{Vivek's Share} = \left(\frac{7}{15}\right) \times 450000$ * $\text{Vivek's Share} = 7 \times 30000 = ₹ 210000$ ### Exam Strategy & Shortcut Quickly simplify the investment numbers by removing the common four zeros at the very beginning ($6 : 14 : 12$). Then simplify the time periods if they share a common factor (24, 18, 12 share a factor of 6). Investments: $6 : 14 : 12$ -> $3 : 7 : 6$ Times: $4 : 3 : 2$ (divided by 6) Ratio = $(3 \times 4) : (7 \times 3) : (6 \times 2) = 12 : 21 : 12 = 4 : 7 : 4$. Much faster! ### Common Pitfall Misreading "After 1 year" for Kirti as meaning she invested for 1 year from that point. "After 1 year" means 12 months after the start, so out of the 24 months, her money was only active for the remaining 12 months. ### Final Answer Therefore, the correct answer is **₹ 210000**.