Adding and Subtracting Units of Time

Date: 12/11/98 at 10:40:25
From: Shellie
Subject: Adding and subtracting units of time

How do you solve subtracting units of time? For example, how do you do:

     15 hours  52 minutes
   - 12 hours  85 minutes 
   ----------------------

I thought that the answer would be 3 hours and 7 minutes because you 
can't do 52 - 85, so I thought you would carry, and that would make it 
12 - 5. Since you left the 5 as 4, you have to carry, and the 4 will 
be a 14, and 14 - 8 is 6. Finally, you have 4 - 2 = 2. So that's how 
you get 2 hours 67 mins, but then you have to regroup. Since 
60 mins = 1 hour, you add another hour to 2 to get 3 with 7 mins left 
over. My teacher said that my answer was wrong. How do you do it the 
right way?

Date: 12/11/98 at 12:45:50
From: Doctor Rick
Subject: Re: Adding and subtracting units of time

Hi, Shellie. 

First, I'm wondering why one of the times in your example is 
12 hr 85 min because this could be better written as 13 hr 25 min 
(replacing 60 minutes by 1 hour). But I'll do the example as you have 
it so you can see what is different.

You are correct that you can use borrowing (you use the word "carry" 
but I use that word only when I'm adding). But times aren't in base 10, 
so you have to think carefully about what borrowing means. Times are in 
base 60 (which is what the ancient Babylonians used for all their 
math).

Let's not go digit by digit, but look at the minutes as a whole. You 
are correct that you can't subtract 85 from 52, so you must borrow. 
How do we borrow? We go to the Hours' house and ask for an hour. Then 
we change it into minutes: 60 minutes, to be precise. Now we have 
52 + 60 = 112 minutes (and 15 - 1 = 14 hours). We can subtract 
112 - 85 = 27 (this much is in base 10, so you can borrow in the 
normal way). Moving on to the hours, we just do the subtraction: 
14 - 12 = 2. The answer is 2 hr 27 min.

Do you see what you did wrong? When you borrowed an hour, you used the 
wrong exchange rate - you changed the hour into 100 minutes (10 tens 
of minutes) instead of 60. That's why you ended up with 40 minutes too 
many.

I hope this helps you. One nice thing about adding and subtracting 
times is that it can give you a better understanding of what carrying  
and borrowing is all about. Just remember this:

 - Borrowing and carrying in regular arithmetic mean exchanging 10 
   ones for a ten (or 10 tens for a hundred, etc.).

 - Borrowing and carrying in time arithmetic mean exchanging 60 
   minutes for an hour.

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