Grandfather Clock and 7-Second Chime

Date: 06/06/2002 at 11:06:29
From: Cindy Harris
Subject: Word problem

If it takes a grandfather clock 7 seconds to chime 7 o'clock, how long 
will it take the same clock to chime 10 o'clock?

(The answer is not 10 seconds.)

Date: 06/06/2002 at 14:11:00
From: Doctor Ian
Subject: Re: Word problem

Hi Cindy,

A good picture can help a lot:

   c  r
  [ ]___[ ]___[ ]___[ ]___[ ]___[ ]___[ ]

  \_____________________________________/
              7 seconds

From the time the first chime starts to the time the last one ends is 
7 seconds. Let's say that each chime lasts c seconds, and there is a 
rest of r seconds between chimes. Add up the chimes and the rests, and 
we get 7 seconds:

   7c + 6r = 7

If we extend this for another three chimes, we get three more 
rests, so

   10c + 9r = ?

But we have three unknowns and only two equations. Let's make an 
assumption: A chime takes no time at all, i.e., c=0. Then

   7*0 + 6r = 7

         6r = 7

          r = 7/6

So

          ? = 9r

            = 9(7/6)

            = 3(7/2)

            = 21/2

            = 10.5 seconds

But for fun, let's assume that each chime takes half a second to 
finish. Then we have

  7(1/2) + 6r = 7

     7/2 + 6r = 7

           6r = 7 - 7/2

           6r = 7/2

            r = 7/12

Now for 10 chimes, we get

          ? = 10(1/2) + 9(7/12)

            = 5 + 3(7/4)

            = 5 + 21/4

            = 5 + 5.25

            = 10.25 seconds

So the answer appears to depend on the duration of each chime.

Can we do this more generally?  Let's look at that first equation 
again.  We know that

    7c + 6r = 7

         6r = 7 - 7c

         6r = 7(1 - c)

          r = (7/6)(1 - c)

We can substitute this into the second equation:

          ? = 10c + 9r

            = 10c + 9[(7/6)(1 - c)]

            = 10c + 10.5(1 - c)

            = 10c + 10.5 - 10.5c

            = 10.5 + 10c - 10.5c

            = 10.5 - 0.5c

Let's check the cases we already know:

   c = 0       

         10.5 - 0.5*0 = 10.5 - 0

                      = 10.5                (Okay)

   c = 1/2   

         10.5 - 0.5*(1/2) = 10.5 - 0.25

                          = 10.25           (Okay)

So if you want to get technical, you can't really answer the question 
without knowing how long each chime lasts.  If you assume that each 
chime is instantaneous, then what's actually going on is that you're 
spreading (N-1) rests over N seconds; so the time needed for (N+K) 
chimes is the time needed for (N+K-1) rests:

   N   seconds 
  ------------ * (N+K-1) rests
 (N-1) rests

If 7 chimes take 7 seconds, 10 chimes will take

  7 seconds
  --------- * 9 rests = 63/6 seconds = 10.5 seconds
  6 rests

If 5 chimes take 5 seconds, 10 chimes will take

  5 seconds
  --------- * 9 rests = 45/4 seconds = 11.25 seconds
  4 rests

Does that make sense?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/