Determining Which Number to Divide by Which

Date: 10/04/2004 at 08:49:07
From: Dominick
Subject: figuring out which is the divisor

Julia plans to use a recipe that yields 15 pounds of fudge.  She 
intends to wrap the fudge she makes in 3/4 pound boxes and give one 
box to each of her friends and relatives for gifts.  How many gifts 
will she have?

I don't know how to tell if I calculate 15 divided by 3/4 or if it's
3/4 divided by 15.  I get it confused sometimes.

15 divided by 3/4                      3/4 divided by 15
15/1 divided by 3/4                    3/4 divided by 15/1
15/1 times 4/3                         3/4 times 1/15
5/1 times 4/1                          1/4 times 1/5
20 answer                              1/20 answer

Date: 10/04/2004 at 10:06:07
From: Doctor Peterson
Subject: Re: figuring out which is the divisor

Hi, Dominick.

If you look at what you are doing in your two cases, you will see that
the answers are reciprocals of one another.  But that's not the main
issue here.

How can you decide what to divide by what?  One way is to think about 
what division means.  You want some number of boxes, 3/4 pound each, 
which will be 15 pounds all together.  So you want to find the number 
by which you can multiply 3/4, to give 15:

  x * 3/4 = 15

The opposite of this multiplication is division BY the known factor:

  x = 15 / 3/4

(I'm using "*" to mean multiplication and "/" to mean division.)

Thinking in this way makes it clear what you divide by what: to find 
one factor, you divide the product by the other factor.

Along the same lines, once you get an answer you can check it.  What 
is the total weight of 20 3/4-pound boxes?  It is 20 times 3/4, which 
does give 15.  Also, just look at the answers.  Does it make sense 
that 1/20 of a box should weigh 15 pounds?  No; 1/20 times 3/4 pound 
gives 3/80 of a pound!  That sort of check should be a regular part of 
your problem-solving method.

Another thing that can help is to change the numbers in the problem to 
avoid fractions, which often makes the answer more obvious.  If you 
wanted to divide 15 pounds into 3-pound packages, how many will you 
get?  This time you can picture making packages until you reach a 
total of 15 pounds, and you can see quickly that you can find that 
answer by dividing 15 by 3.  Do the same with the real numbers.

Finally, often you can use units to help.  You have

  15 pounds per batch
  3/4 pounds per box
  X boxes per batch

That is, you want to find how many 3/4-pound boxes you can make from a 
15-pound batch.  If you divided the 3/4 by the 15, and kept the units 
with the calculation, you would get

  3/4 pounds   15 pounds   3/4 pounds    1 batch
  ---------- / --------- = ---------- * --------- = 1/20 batch/box
    1 box       1 batch      1 box      15 pounds

So your answer of 1/20 is not the number of boxes in a batch, but the 
number of batches in a box--that is, each box is 1/20 of the batch. 
And therefore the whole batch is 20 boxes.  If you do the calculation 
the right way, you'll see the answer come out to 20 boxes/batch.  If 
you don't quite follow what I'm doing here, see this page:

  Unit Conversions, Unit Cancellation     
http://mathforum.org/library/drmath/sets/select/dm_unit_convert.html 

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

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