Ways to Add and Divide More Easily

Date: 11/17/2008 at 22:08:12
From: Sofia
Subject: Word Problem

Hi!  Here is my question...Tianna has 2398 candies. Marina has 5752.
They both have 37 friends in total. They want to split their candy in
to 37 piles so each friend can get some candy. Can you tell me the
quickest and easiest way to figure this out?

I know that I have to add 2398 plus 5752. Then I have to find out 
how many times 37 goes into that number. But it takes me a really long
time! I really dislike that! I want a good example so I know how to do
division.

Date: 11/18/2008 at 08:07:14
From: Doctor Ian
Subject: Re: Word Problem

Hi Sofia,

That sounds about right. 

One thing to consider is that this is the sort of problem you'd
normally do with a calculator.  You're probably being asked to do it
without one, but that's just to prove that you CAN.  Once everyone
knows you CAN do it, you won't have to anymore.  Weird, but that's how
it is.  

A second thing to realize is that this is kind of a silly problem,
since they're going to have to count the candies out into 37 piles
anyway, so it's kind of like dealing cards from a deck.  They can just
do it, and each friend gets what he gets.  Knowing how many candies
will be in each pile doesn't change anything.  It's a pointless
calculation.

Okay?

Anyway, you want to add 2398 and 5752, and then divide by 37.  That
addition looks kind of ugly,   
   
    2398
  + 5752
  ------

but you can make it nicer by taking some away from one number and
adding it to the other.  For example, I can take 2 from the bottom
number and add it to the top number to get

    2400
  + 5750
  ------

Do you see why the sum doesn't change?  I could do the same sort of
thing again, taking 600 from the bottom number and adding it to the
top number to get

    3000
  + 5150
  ------

And now the addition is so easy I can do it in my head:

    3000
  + 5150
  ------
    8150

So far, so good?  Now, I want to divide this by 37.  The first thing
I'd do is estimate the answer, so I'll be able to check my actual
answer later.  What do I mean by that?  Well, 8150 is close to 8000,
and 37 is close to 40, and 8000 divided by 40 will be about 200.  So
my answer had better be close to 200, or I'll know I have a problem.

What about the actual division?  Probably you've learned how to do
long division, or you wouldn't be given this kind of problem.  So you
can go ahead and do that,
 
           ?
       _____
   37 ) 8150

I probably wouldn't.  I'd use multiplication instead:

  37 * 200 = 7400   <-  This gets me close.
  37 *  20 =  740   <-  This gets me closer. 
       ---   ----
       220   8140

and that's as close as I'm going to get.  So '220 remainder something'
is going to be my answer. 

Do you see what I did there?  I just kept adding multiples of 37,
until all my multiples add up to something close to my target number.
To see why that works, it's important to always keep in mind that 

       c
      __
   a ) b      means the same thing as       a * c = b

That is, division and multiplication are just two ways of expressing
the same information.  Whenever you have a multiplication, you can
change it to a division if that will make your life easier; or you can
change a division to a multiplication.  It's all the same.  

Does this make sense?

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

Date: 11/18/2008 at 21:24:10
From: Sofia
Subject: Thank you (Word Problem)

Thank you so much Doctor Ian!  That helps me a ton!  I think this site
really rocks!