Clock Hands and Hours Worked

Date: 01/29/2003 at 09:08:16
From: Brady
Subject: Hours Worked Word Problem

A man arrives at work between 8:00 and 9:00 a.m. at exactly the 
moment the minute hand and hour hand of the clock point in the same 
direction. He takes a quick break for lunch from exactly noon to the 
next time the clock hands point in opposite directions. He leaves work 
between 4:00 and 5:00 p.m. at exactly the moment the clock hands point 
opposite one another. 

How many hours did he work that day? 

Date: 01/29/2003 at 11:44:33
From: Doctor Greenie
Subject: Re: Hours Worked Word Problem

Hi, Brady -

Let me sketch out a method for solving this problem the slow way and 
then give you the quick and easy answer....

Solving this problem the slow way, you need to find the exact times 
when...

(a) between 8:00 and 9:00 the hands point in the same direction;
(b) between 12:00 and 1:00 they point in opposite directions; and
(c) between 4:00 and 5:00 they point in opposite directions

Let's look at how we might go about finding the exact time (a).  We 
use the fact that the minute hand moves 12 times as fast as the hour 
hand (because the minute hand goes around 12 times each time the hour 
hand goes around once).

  Let x = # degrees the hour hand travels between 8:00 and
           the time when the hands are together
  Then 240+x = # degrees the minute hand travels between 8:00 and
           that time (the minute hand has to travel 240 degrees
           farther in that time, because at 8:00 it is pointing
           at "12" whereas the hour hand is pointing at "8")

  So (since the minute hand travels 12 times as far as the hour hand)

     240+x = 12(x)
     240 = 11x
     x = 240/11

  Since the hour hand travels 30 degrees in 1 hour, (240/11) degrees
  represents (8/11) of an hour.

So the exact time when the two hands are together between 8:00 and 
9:00 is at the time "8 and 8/11 hours."  (Our final answer will be 
much easier to express if we leave this result in this awkward form, 
rather than going through the quite awkward process of converting 
this time to hours, minutes, seconds, and fractions of seconds.)

We can use the same type of process to determine the exact times (b) 
and (c).  I leave that exercise to you, because there is a quicker 
way to the answer to your problem.

To solve the problem the quick way, we need to recognize that

(a) in each 12-hour period, there are 11 times when the hands point 
    in exactly the same direction; and
(b) those 11 times are equally spaced over the 12-hour time period.

So the amount of time between successive times when the hands point in 
the same direction is 12/11 hours, or 1 1/11 hours. Since the hands 
point in the same direction at 12:00, the times when the hands point 
in the same direction are at the following "hours":

  12  0/11
   1  1/11
   2  2/11
   3  3/11
   4  4/11
   5  5/11
   6  6/11
   7  7/11
   8  8/11
   9  9/11
  10 10/11

Similar reasoning, along with the fact that the hands point in 
opposite directions at 6:00, give us the following "hours" at which 
the hands point in opposite directions:

   6  0/11
   7  1/11
   8  2/11
   9  3/11
  10  4/11
  11  5/11
  12  6/11
   1  7/11
   2  8/11
   3  9/11
   4 10/11

Using these results, we see that the man

(1) started work at "8 8/11 hours" and stopped at noon ("12 hours"),
    thus working for 3 3/11 hours; and
(2) resumed work at "12 6/11 hours" and stopped for the day at
    "4 10/11 hours", thus working for 4 4/11 hours

So the total time the man worked was 7 7/11 hours.

I hope all this helps.  Please write back if you have any further 
questions about any of this.

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

Date: 01/30/2003 at 09:46:03
From: Brady
Subject: Hours Worked Word Problem

How can I find the answer to the precice fraction of a second?

Date: 01/30/2003 at 13:48:12
From: Doctor Greenie
Subject: Re: Hours Worked Word Problem

Hi, Brady -

You have to convert the 7 7/11 hours to hours, minutes, seconds, and 
fractions of seconds

There are two basic ways to do that. You of course can leave the 7 
whole hours alone and work on converting the 7/11 hours. One approach 
is to change the hour to 3600 seconds and find 7/11 of 3600 seconds, 
and then convert that number of seconds back to minutes and seconds.  
The alternate approach is to convert the hour to 60 minutes and find 
7/11 of the 60 minutes and then convert the fraction of a minute 
remaining to seconds. It appears to me the first approach is less 
complicated.

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