IMAT 2013 Q25 [Taxi Service]

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 Beta­Air 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?

A. 0
B. 3
C. 2
D. 1
E. 4

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Simple steps to solve word problems:

  • Underline key information
  • Determine what they are trying to ask, and what you will need to solve it
    • Eliminate any non-essential information
  • Draw a picture, graph, or equation
    • In moments of high stress like exam taking, always work with the paper they give you to avoid careless mistakes.
  • Solve.

The first thing we want to do is read the question and figure out what they are asking. They want to know how many Alpha-Air flights Captain Johnson will pass on his way to Milan. He is flying to Milan, so we need to find out how many planes he will overtake, meaning he will start after but then pass by (because he has a faster plane). Now, rereading the question, we can take out the important information.

Alpha Air
Leave every 10 min
90 min flight

BetaAir
Leave every 5 min
60 min flight

Captain Johnson leaves 5 min after the AlphaAir flight.

Now, our goal is to see how many planes he passes, so lets first make an arbitrary flight time and then compare the two flight companies using this timing. Let’s say that Captain Johnson leaves at 1:05. This time was chosen arbitrarily, but it makes it easier when you use 5 minutes after an hour because he leaves 5 min after the AlphaAir.

Next, we can figure out that since he has a 1 hour flight, he will land at 2:05, so he will pass every flight that leaves before him but lands after 2:05 but leaves earlier than him. He is faster than the planes that leave after him so he cannot pass them. He is also cannot pass planes that leave before him and land before him, so this is how we determined this range.

Knowing this, we can make a table of AlphaAir planes to see which ones leave before him but arrive after. In this solution, there is one interval on each side to show that he cannot pass him. The 00:30 flight leaves before and lands before so it cannot be included, and the 1:10 leaves after and lands after, so it aswell is not included in our final count.

\begin{array} {|r|r|}\hline Takeoff & Landing \\ \hline 00:30 & 2:00 \\ \hline 00:40 & 2:10 \\ \hline 00:50 & 2:20 \\ \hline 1:00 & 2:30 \\ \hline 1:10 & 2:40 \\ \hline \end{array}

As you can see, the middle 3 rows fit our criteria.

\fcolorbox{red}{grey!30}{Therefore B is correct, as there are $3$ planes Captain Johnson can pass.}
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