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On or Off?

Date: 05/26/2001 at 23:38:06
From: Lisa Williams
Subject: Maths word problems

Our school has an automatic sprinkler system for watering the hockey 
field overnight. The sprinkler system first switches on at 10:00 p.m. 
and remains on for a full number of minutes. Then it switches off for 
three times as many minutes as it is on. It switches off again, on 
again, and so on throughout the night, repeating the cycle with the 
same time lengths for each on and off period.

Just a few seconds before 10:11 p.m., the security guard notices that 
the sprinkler system is off; just a few seconds after 1:03 a.m., he 
notices that it is on. If it is on again at exactly 2:15 a.m., will it 
be on or off at 3:30 a.m.?

I have tried different numbers of minutes, but nothing works. Please 
show how to work this out.

Date: 05/27/2001 at 16:50:31
From: Doctor Jaffee
Subject: Re: Maths word problems

Hi Lisa,

Here is how I solved this problem. Since the sprinkler is off for 
three times as long as it is on, during a 24-hour period the sprinkler 
will be on for exactly 6 hours and off for 18 hours. Since 6 hours is 
360 minutes, the amount of time the sprinkler is on during a cycle 
must be a factor of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, ..., 180, 
or 360.

But we know that at 10:11 p.m. the sprinkler is off, so that 
eliminates all the numbers greater than 10.

A few seconds after 1:03 a.m. the sprinkler is on. 1:03 a.m. is 183 
minutes after 10:00 p.m. So, let's check out our possible answers one 
at a time. If the sprinkler is on for 1 minute and off for 3 minutes, 
then we could say that one cycle of the sprinkler is 4 minutes. If we 
divide 183 by 4 we get 45 remainder 3, which means that the sprinkler 
has gone through 45 complete cycles and is in the 3rd minute of its 
current cycle. But during the third minute of a four minute cycle the 
sprinkler is off, so we can eliminate "1 minute on" as a possibility. 
On the other hand, if the sprinkler is on for 5 minutes and off for 15 
minutes, one cycle lasts 20 minutes. If you divide 183 by 20 you get 9 
with a remainder of 3 which means that it has completed 9 cycles and 
is in the 3rd minute of its current cycle. Since the first 5 minutes 
of the cycle have the water on, we can conclude that the sprinkler 
will be on at 1:03 a.m.

Now 2:15 a.m. is 255 minutes after 10:00 p.m. So, divide 255 by 20 and 
we get 12 remainder 15 which means that the sprinkler has completed 12 
cycles and is in the 15th minute of its current cycle. So, the water 
should be turned off. We can conclude, then, that we haven't found our 
solution yet.

If you go through the numbers 1, 2, 3, 4, 5, 6, 8, 9, 10 one at a 
time, however, you will find one that will satisfy all the conditions 
of the problem.

Give it a try and if you want to check your answer with me, write 
back. If you are having difficulties, let me know and show me what you 
have done so far, and I'll try to help you some more.

Good luck.

- Doctor Jaffee, The Math Forum   
Associated Topics:
Middle School Puzzles
Middle School Word Problems

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