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Choose an Elevator
Problem Statement
Two elevators leave the nth floor at 2:00 P.M. The faster elevator takes one minute between floors and the slower elevator takes two minutes between floors. The first elevator to reach a floor must stop for three minutes to take on passengers. If both elevators arrive at a floor at the exact same time, they become confused and do not stop for passengers.

If the final stop for the elevator is the lobby (1st floor), then describe n if the faster elevator arrives at the lobby first. Describe n if the slower elevator arrives at the lobby first.

(Source: Adapted from Mathematics Teaching in the Middle School, Apr 1994).

Problem setup

Here, I am not finding out how many floors are in this building.  I am being asked to determine a pattern for the floor total if the faster elevator comes in first or if the slower elevator comes in first.

Plans to Solve/Investigate the Problem

Initially, I used pencil and paper to write down the possible arrival times.  This proved to be rather arduous, and I returned to my first love...Excel!

Investigation/Exploration of the Problem

After much time and effort, I realized that I had overlooked an important part of the problem.  The first elevator to reach a floor must stop for three minutes to take on passengers.   Below is a part of the initial and erroneous spreadsheet:

 Floor Faster Slower N 0.083333 0.083333 N-1 0.084028 1 0.084722 N-2 0.086806 0.086111 N-3 0.0875 1 0.089583 N-4 0.090278 1 0.090972 N-5 0.093056 1 0.094444 N-6 0.095833 0.095833 N-7 0.096528 1 0.099306 N-8 0.099306 1 0.102778 N-9 0.102083 1 0.10625 N-10 0.104861 1 0.109722 N-11 0.107639 1 0.113194 N-12 0.110417 1 0.116667 N-13 0.113194 1 0.118056 N-14 0.115972 1 0.119444 N-15 0.11875 1 0.122917 N-16 0.119444 1 0.126389 N-17 0.120139 1 0.129861 N-18 0.122917 1 0.133333 N-19 0.125694 1 0.136806

Here, I had not taken into account the first elevator situation.  I proceeded to insert an if statement that checks to see which time was lowest in an effort to determine which elevator reached each floor first.  The floor with the earlier arrival has a 1 to the right of its time.

Please note that the time was changed to a value using a formula in excel.

This determination told me where to add the time for travel between floors only and where to include time for passenger pick up and drop off.  The resultant and hopefully correct spreadsheet sample is below.

 Floor Faster Slower N 0.083333 0.083333 N-1 0.084028 1 0.084722 N-2 0.086806 0.086111 1 N-3 0.0875 1 0.089583 N-4 0.090278 1 0.090972 N-5 0.093056 0.092361 1 N-6 0.09375 1 0.095833 N-7 0.096528 1 0.097222 N-8 0.099306 0.098611 1

The floors where the slower cab arrives first are highlighted in green (the color of the season I might add).

Now the ultimate question was what is true about the number of floors in the building if the slower car reaches the lobby first.  Take a look at each place where the floors are shaded green.  If these places were the lobby of the building, then the place would have 3, 6, 9, 12,... floors (a multiple of 3).  If the faster cab reaches the lobby first, the number of floors could be anything EXCEPT a multiple of 3.

Elevators

Extensions of the Problem

What would the case be if there were 3 elevators, the middle one takes 1.5 minutes to travel between floors?

Author & Contact
Michelle Houston

Link(s) to resources, references, lesson plans, and/or other materials