Question

One third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. One third of the passengers, in the buses stopping at City C, continue to City B, while the rest alight at City C. All the buses have equal capacity and always start full from City A. What proportion of the passengers going to City B from City A travel by a bus stopping at City C?

• A. \( \frac{1}{7} \)
• B. \( \frac{1}{9} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{7}{9} \)
• E. \( \frac{4}{9} \)

Answer 1

The Correct Option is A

Suppose the total number of buses is \( 3 \), and each bus has capacity \( N \).

Step 1: Distribution of buses

• \(\tfrac{1}{3}\) of buses stop at City C \(\Rightarrow 1\) bus stops at C.
• The remaining \(\tfrac{2}{3}\) go non-stop to City B \(\Rightarrow 2\) buses go directly.

Step 2: Passengers on buses stopping at City C

In the bus stopping at City C: \[ \tfrac{1}{3} \text{ of passengers continue to B} = \tfrac{1}{3}N \] \[ \tfrac{2}{3}N \text{ passengers get down at C.} \]

Step 3: Passengers on non-stop buses

Each of the 2 non-stop buses goes directly to B with all \( N \) passengers. So total = \( 2N \).

Step 4: Total passengers reaching B

From bus stopping at C = \( \tfrac{1}{3}N \) From 2 direct buses = \( 2N \) 
Therefore, total passengers reaching B: \[ 2N + \tfrac{1}{3}N = \tfrac{7}{3}N \]

Step 5: Proportion

Proportion of passengers going to B via bus stopping at C: \[ \frac{\tfrac{1}{3}N}{\tfrac{7}{3}N} = \frac{1}{7} \]

Final Answer:

\[ \boxed{\tfrac{1}{7}} \]