A question from IMAT

Question

"Two companies have just started a round-the-clock air taxi service between Rome and Milan. They use the same flight path and fly at constant speeds at different altitudes. Planes owned by the company Alpha-Air take off from Rome every 10 minutes and take 90 minutes to reach Milan. Planes owned by the company Beta-Air take off every 5 minutes and take 60 minutes to reach Milan. Captain Johnston, who flies for Beta-Air, takes off from Rome 5 minutes after the previous Alpha-Air flight has departed.

How many Alpha-Air planes (flying from Rome to Milan) will Captain Johnston have passed as he lands in Milan? "

This is a question from IMAT. The solution for this question is $2$. However, I think it should be $3$. My argument is as below.

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Consider the moment when Captain Johnston takes off. I call the Alpha planes that have taken off for 5,15,25,35 minutes $a_1,a_2,a_3,a_4$. $60$ minutes later, Captain Johnston lands in Milan, while $a_1,a_2,a_3$ have only taken off for $65,75,85$ minutes, hence have not arrived and were passed by. Meanwhile, $a_4$ have taken $95>90$ minutes, hence it has arrived and cannot be passed by.

In conclusion, the answer is $3$.

What is wrong with my argument? Or simply the solution was wrong?

Answer 1

I agree with you that $3$ looks correct

Let's suppose that Captain Johnston takes off at 10:05 and lands at 11:05

Meanwhile the Alpha-Air timetable includes

takeoff  landing
 09:30    11:00 
 09:40    11:10 
 09:50    11:20 
 10:00    11:30
 10:10    11:40 

So Captain Johnston passes the middle three of these in that he takes off later and lands earlier