Intermath | Workshop Support
Write-up

 

Title
 Turning Red

Problem Statement
 Two traffic lights turn red together at exactly 5:00 p.m. One light is on a 36-second cycle from red back to red, and the other is on a 48-second cycle. At what time will they again turn red together? Give the exact time including seconds.

Problem setup
At 5:00 p.m. both lights are red.  You are given the amount of time for each light to turn back to red.  Periodically the lights will turn red simultaneously.  You are to discover the time of the first occasion they both turn red simultaneously.

 

Plans to Solve/Investigate the Problem
This problem is a Least Common Multiple word problem.  The solution will lie with the least common multiple of 36 and 48.  We shall use a table to determine the times each will turn red, and look for the first time common to both lights.  This is known as the List the Multiples method of finding Least Common Multiple. 

 

Investigation/Exploration of the Problem

The use of a table enables us to list and see the times each turns red.  We need merely compare the times to see the first occasion they both turn red simultaneously.  The elapsed time and actually time are shaded in gray.

 

Traffic Light Red Lights

1

2

3

4

5

6

7

8

9

10

Traffic Light One Time Elapsed Between Red Lights

0

0:00:36

0:01:12

0:01:48

0:02:24

0:03:00

0:03:36

0:04:12

0:04:48

0:05:24

Traffic Light One Red Light Times

5:00:00 PM

5:00:36 PM

5:01:12 PM

5:01:48 PM

5:02:24 PM

5:03:00 PM

5:03:36 PM

5:04:12 PM

5:04:48 PM

5:05:24 PM

Traffic Light Two Time Elapsed Between Red Lights

0

0:00:48

0:01:36

0:02:24

0:03:12

0:04:00

0:04:48

0:05:36

0:06:24

0:07:12

Traffic Light Two Red Light Times

5:00:00 PM

5:00:48 PM

5:01:36 PM

5:02:24 PM

5:03:12 PM

5:04:00 PM

5:04:48 PM

5:05:36 PM

5:06:24 PM

5:07:12 PM

 

Both lights will turn red simultaneously for the first time at 5:02:24.  The 36-second light will achieve this on itís fourth instance of turning red, and the 48-second light on itís third instance of turning red. 

 

As you can see, the next time both lights will turn red is at 5:04:48.  The 36-second light will achieve this on itís eighth instance of turning red, and the 48-second light on itís sixth instance of turning red.  Every two minutes and twenty-four seconds the two lights will turn simultaneously. 

 

We have solved this problem using time.  However, the same problem might be more easily solved in running seconds.  Two minutes and twenty-four seconds = 144 running seconds.  The least common multiple of 36 and 48 is 144.  Every 144 seconds the lights will turn red simultaneously.  144, 288, and 432 are the first three multiples of 36 and 48, and mark the first three occasions the lights will turn red simultaneously.


Extensions of the Problem
Instead of extensions, students might find this problem easier to solve given a little practice in such problems not including having to deal with the element of time.  Teachers could practice with students solving word problems such as: hot dogs come in packs of 10 and hot dog rolls come in packs of 6.  What is the smallest number of each one could buy and have the same amount of hot dogs as rolls?


Author & Contact

Pat Devane

Memorial Middle School

Conyers, GA

pjdevane@rockdale.k12.ga.us