Question

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

• A. $\tfrac{1}{3}$
• B. $\tfrac{3}{11}$
• C. $\tfrac{12}{31}$
• D. $\tfrac{5}{8}$

Hint:

Choose one person’s amount as a variable and express all others in that variable; the required fraction is then \(\dfrac\textthat amount\texttotal\). Common factors cancel cleanly.

Answer 1

The Correct Option is B

Step 1: Translate the relations into algebra.
Let Veena’s share be $V$. Then Amit paid $A=\tfrac{2}{3}V$.
Veena paid half as much as Tanya, i.e., $V=\tfrac{1}{2}T ⇒ T=2V$.
Step 2: Write the total bill in terms of $V$.
Total $B = A+V+T = \tfrac{2}{3}V + V + 2V
= \Big(\tfrac{2}{3}+1+2\Big)V
= \tfrac{11}{3}V$.
Step 3: Fraction of the bill paid by Veena.
\[ \text{Veena's fraction}=\frac{V}{B}=\frac{V}{\tfrac{11}{3}V} = \frac{3}{11}. \] \[ \boxed{\tfrac{3}{11}} \]