In a partnership, A invests $\frac{1}{6}$ of the capital for $\frac{1}{6}$ of the time, B invests $\frac{1}{3}$ of the capital for $\frac{1}{3}$ of the time and C, the rest of the capital for the whole time. Out of a profit of ₹ 4600, B's share is (M.B.A., 2004)

### Concept & Fractional Investment Proportions When investments and time periods are given as fractions of a whole, multiply the capital fraction by the time fraction for each partner to establish the unsimplified profit ratio, then clear the denominators. ### Step-by-Step Solution * Let total capital be $1$ unit and total time be $1$ unit. * **A's Investment Profile:** * Capital = $\frac{1}{6}$ * Time = $\frac{1}{6}$ * Product = $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$ * **B's Investment Profile:** * Capital = $\frac{1}{3}$ * Time = $\frac{1}{3}$ * Product = $\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$ * **C's Investment Profile:** * Capital = Rest of the capital = $1 - (\frac{1}{6} + \frac{1}{3}) = 1 - (\frac{1}{6} + \frac{2}{6}) = 1 - \frac{3}{6} = \frac{1}{2}$ * Time = Whole time = $1$ * Product = $\frac{1}{2} \times 1 = \frac{1}{2}$ * **Profit Ratio:** * $A : B : C = \frac{1}{36} : \frac{1}{9} : \frac{1}{2}$ * Multiply by LCM (36) to clear fractions: * $1 : 4 : 18$ * **B's Share:** * Total ratio parts = $1 + 4 + 18 = 23$ * Total profit = ₹ 4600 * B's share = $\frac{4}{23} \times 4600 = 4 \times 200 = 800$ ### Exam Strategy & Shortcut To avoid fractions entirely, assume a total capital and total time that are easily divisible by the denominators (6 and 3). Let Total Capital = 6 and Total Time = 6. A: $1 \times 1 = 1$. B: $2 \times 2 = 4$. C: remaining capital (3) $\times$ whole time (6) = 18. The ratio $1 : 4 : 18$ is found instantly. ### Common Pitfall Incorrectly calculating C's remaining capital by forgetting to subtract both A's and B's shares from the total whole (1). ### Final Answer Therefore, the correct answer is **₹ 800**.