Three partners A, B, C start a business. Twice A's capital is equal to thrice B's capital and B's capital is four times C's capital. Out of a total profit of ₹ 16500 at the end of the year, B's share is :

### Concept & Ratio Unification When relationships between capitals are given as separate equations, convert them into a unified continuous ratio ($A : B : C$) before distributing the profit. ### Step-by-Step Solution * **Given Equations:** 1. $2A = 3B \implies \frac{A}{B} = \frac{3}{2} \implies A : B = 3 : 2$ 2. $B = 4C \implies \frac{B}{C} = \frac{4}{1} \implies B : C = 4 : 1$ * **Unifying the Ratios:** * To combine $A : B$ and $B : C$, make the value of $B$ the same in both ratios. * Multiply the first ratio ($3 : 2$) by 2 to make $B = 4$: * $A : B = 6 : 4$ * Now, $A : B = 6 : 4$ and $B : C = 4 : 1$. * The combined ratio is $A : B : C = 6 : 4 : 1$. * **Distributing the Profit:** * Total ratio parts = $6 + 4 + 1 = 11$ parts. * Total profit = ₹ 16500. * Value of 1 part = $\frac{16500}{11} = \text{₹ } 1500$. * B's share = 4 parts. * B's share = $4 \times 1500 = \text{₹ } 6000$. ### Exam Strategy & Shortcut You can also use a common variable. Let $C = x$. Then $B = 4x$. Since $2A = 3B$, $2A = 3(4x) = 12x$, so $A = 6x$. The ratio $A : B : C$ is instantly $6x : 4x : x$, which simplifies to $6 : 4 : 1$. This avoids ratio manipulation entirely. ### Common Pitfall Incorrectly combining ratios without finding a common multiple for the bridging term (B in this case). ### Final Answer Therefore, the correct answer is **₹ 6000**.