Elapsed Time and the Grouping Factor

Date: 03/08/2003 at 15:02:46
From: Marylyn Wills
Subject: Elapsed time

How do I explain why you can't just subtract the two times given in a 
word problem and get the time elapsed?

For instance... Carl studies from 5:25 to 5:45. How long does he 
study? The answer is simply 20 minutes, after subtracting the 2. BUT 
if he studies from 3:48 p.m. to 7:05 p.m., if you subtract the two, I 
get 3:57.  

I know you have to convert something to 60 minutes, but WHY?
 
Thank you!

Date: 03/09/2003 at 03:46:34
From: Doctor Ian
Subject: Re: Elapsed time

Hi Marylyn,

The easiest way to find elapsed times is to add, not subtract.

 If Carl studies from 3:48 to 7:05, then 

  he studies for 12 minutes from 3:48 to 4:00

  he studies 3 hours from 4:00 to 7:00

  he studies 5 minutes from 7:00 to 7:05

for a total of 

  12 minutes + 3 hours + 5 minutes = 3 hours + 17 minutes

This is how I'd do it.  It's very much like making change. 

But if you want to use subtraction, you have to be careful about
borrowing. 

Let's look at a typical subtraction problem:

   45
 - 17
 ----

Here, we can't subtract 7 from 5, so we have to borrow from the next
column over.  So we end up with 

   3 | 15
 - 1 |  7
 ----|---
   1 |  8

and then we're okay. But what happened here? We made use of the fact
that a digit in the tens column is worth 10 times as much as the same
digit in the ones column. That is, we traded one 10 for ten 1's.   

Now let's look at a subtraction of times, e.g., 

   7:05 
 - 3:48
 ------

Now the 'normal' rules of borrowing no longer work. A 1 in the 
leftmost column is worth 1 minute; a 1 in the second column is worth
10 minutes; but a 1 in the third column is NOT worth 100 minutes. 
It's only worth 60 minutes. So borrowing from hours would look like

   7:05             6:65
 - 3:48    --->   - 3:48
 ------           ------

To get a better feel for what's going on, suppose we want to do the
following subtraction:

    3 gallons, 2 quarts,  4 ounces
  - 1 gallon,  3 quarts, 11 ounces
  --------------------------------

We start with the smallest units, and find that we can't subtract 11
from 4, so we have to borrow some ounces by converting a quart.  A
quart is 32 ounces, so now we have

    3 gallons, 1 quart,  36 ounces
  - 1 gallon,  3 quarts, 11 ounces
  --------------------------------

Now we have a similar problem with quarts.  A gallon is 4 quarts, so
we have

    2 gallons, 5 quart,  36 ounces
  - 1 gallon,  3 quarts, 11 ounces
  --------------------------------

and now we can proceed smoothly:

    2 gallons, 5 quart,  36 ounces
  - 1 gallon,  3 quarts, 11 ounces
  --------------------------------
    1 gallon,  2 quarts, 25 ounces

The point here is that in the general case, the way borrowing works
depends on how things are grouped, because borrowing is the inverse of
grouping.  

In the decimal system, the grouping factor is always the same - you
group 10 of something into the next largest thing, 

  10 ones     -> 1 ten
  10 tens     -> 1 hundred
  10 hundreds -> 1 thousand

and so on.  But in other systems, the grouping might be just about
anything at all:

   8 ounces  -> 1 cup
   2 cups    -> 1 pint
   2 pints   -> 1 quart
   4 quarts  -> 1 gallon

  60 seconds -> 1 minute
  60 minutes -> 1 hour
  24 hours   -> 1 day
  365 days   -> 1 year

  12 inches  -> 1 foot
  3 feet     -> 1 yard
  1760 yards -> 1 mile

So the moral of the story is that when you borrow, it pays to keep
track of what kind of thing you're borrowing.

Does this make sense?  

I hope this helps.  Write back if you'd like to talk more about this,
or anything else.   

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