Three friends had dinner at a restaurant. When the bill was received, Amita paid $\frac{2}{3}$ as much as Veena paid and Veena paid $\frac{1}{2}$ as much as Tanya paid. What fraction of the bill did Veena pay?

### Concept & Strategy Express each person's relative contribution as a single ratio chain, similar to multi-variable component sharing. Let $A$ = Amita, $V$ = Veena, $T$ = Tanya. $$A = \frac{2}{3}V \implies \frac{A}{V} = \frac{2}{3}$$ $$V = \frac{1}{2}T \implies \frac{V}{T} = \frac{1}{2}$$ ### Step-by-Step Solution * **Unify the component ratios:** We need $V$ to be identical in both ratios. The common value for 3 and 1 is 3. Multiply the second ratio ($V:T = 1:2$) by 3: $$V : T = 3 : 6$$ Now join them: $$A : V : T = 2 : 3 : 6$$ * **Calculate the total bill parts:** $$\text{Total Parts} = 2 + 3 + 6 = 11 \text{ parts}$$ * **Determine Veena's fraction of the total:** $$\text{Veena's Fraction} = \frac{\text{Veena's Parts}}{\text{Total Parts}} = \frac{3}{11}$$ ### Exam Strategy & Shortcut Assume a smart concrete value for the final independent variable, Tanya. Let Tanya pay ₹ 6 (the LCM of the denominators 3 and 2). * Tanya = ₹ 6 * Veena = $\frac{1}{2} \times 6 =$ ₹ 3 * Amita = $\frac{2}{3} \times 3 =$ ₹ 2 * Total bill = $6 + 3 + 2 =$ ₹ 11 * Veena's portion = $\frac{3}{11}$ This entirely avoids algebraic variables. ### Common Pitfall Summing the initial denominators directly without adjusting them to form a cohesive, unified ratio expression. ### Final Answer **Therefore, the correct answer is \frac{3}{11}.**