Counting Seconds

Date: 12/13/2002 at 18:29:29
From: James Yoder
Subject: Word problem about time (seconds)

Ali counted to 10, speaking a number every 5 seconds. How long did the 
complete count take?  

I thought the answer was 50, but my teacher says it's 45, because you 
start by saying "One" and going from there. So, if that were the 
case, wouldn't the answer be 46?

Help!

Date: 12/13/2002 at 23:39:05
From: Doctor Peterson
Subject: Re: Word problem about time (seconds)

Hi, James.

The trouble here is that we don't know how long it takes to SAY one 
number, from opening the mouth to closing it. So your teacher is 
assuming that it takes no time at all, which is pretty close to 
reality (try saying "ten" and timing it!):

   "1    2    3    4    5    6    7    8    9   10"
    +----+----+----+----+----+----+----+----+----+
     \___/\___/\___/\___/\___/\___/\___/\___/\___/
       5    5    5    5    5    5    5    5    5  = 9*5 = 45 sec

Your answer of 50 assumes that Ali not only speaks one number every 5 
seconds, but that it takes the full five seconds to say it:

   "   1    2    3    4    5    6    7    8    9   10  "
     ==== ==== ==== ==== ==== ==== ==== ==== ==== ====
    +----+----+----+----+----+----+----+----+----+----+
     \___/\___/\___/\___/\___/\___/\___/\___/\___/\___/
       5    5    5    5    5    5    5    5    5    5 = 10*5 = 50 sec

Your new proposal seems to be that it takes one second to say each 
number:

   " 1    2    3    4    5    6    7    8    9   10 "
     =    =    =    =    =    =    =    =    =    =
    +----+----+----+----+----+----+----+----+----++
     \___/\___/\___/\___/\___/\___/\___/\___/\___/|
       5    5    5    5    5    5    5    5    5  1 = 9*5+1 = 46 sec

But there's really no reason to assume that. The best thing we can do 
is to say it will take 45 seconds PLUS however long it takes to 
say "ten".

If you have any further questions, feel free to write back.

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