Question

Three friends had a 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?

• A. \(\frac{1}{3}\)
• B. \(\frac{3}{11}\)
• C. \(\frac{12}{31}\)
• D. \(\frac{5}{8}\)

Answer 1

The Correct Option is B

Let's determine what fraction of the bill Veena paid by setting up equations based on the given conditions. Let \( T \) be the amount Tanya paid. According to the problem, Veena paid \(\frac{1}{2}\) as much as Tanya:

\( V = \frac{1}{2}T \)

Additionally, we know Amita paid \(\frac{2}{3}\) as much as Veena:

\( A = \frac{2}{3}V \)

The total bill \( B \) is the sum of what each person paid:

\( B = A + V + T \)

Substitute for \( A \) and \( V \):

\( B = \frac{2}{3}\left(\frac{1}{2}T\right) + \frac{1}{2}T + T \)

Simplify:

\( B = \frac{1}{3}T + \frac{1}{2}T + T \)

Find a common denominator (6) and combine:

\( B = \frac{2}{6}T + \frac{3}{6}T + \frac{6}{6}T \)

\( B = \frac{11}{6}T \)

We need the fraction of the bill that Veena paid \( \frac{V}{B} \):

\( \frac{V}{B} = \frac{\frac{1}{2}T}{\frac{11}{6}T} \)

Cancel \( T \) and simplify:

\( \frac{V}{B} = \frac{1}{2} \times \frac{6}{11} = \frac{3}{11} \)

Thus, Veena paid \(\frac{3}{11}\) of the entire bill.