To Multiply or Divide? That Is the Question.

Date: 09/02/2003 at 21:26:36
From: Simona
Subject: how to recognize multiplication as opposed to division 

There are 36 grapefruits in 1 crate.  How many grapefruits are in 495 
crates?

I don't know whether to multiply or divide. 

Date: 09/02/2003 at 21:51:21
From: Doctor Rick
Subject: Re: how to recognize multiplication as opposed to division 

Hi, Simona.

One question that may help you tell whether to use multiplication or 
division is this:

Could you solve the problem using addition?

Each crate holds 36 grapefruits. How many grapefruits are in 495 
crates? Well, you can just start counting: 1 to 36 in the first 
crate, then you continue with number 37 through 72 in the second 
crate, and so on.

Then you realize that addition will speed up the counting. You add 
36 + 36 to get 72 in two crates, then add another 36 to get 108 in 
three crates, and so on.

Before you've finished the 495 additions, you realize that 
MULTIPLICATION will speed up the addition! Putting down 36 495 times 
and adding them is the same as multiplying 36 times 495.

Now, what if the problem had been this: There are 17,856 grapefruits 
in crates. Each crate holds 36 grapefruits. How many crates are there?

This time addition can't do the trick. You can start adding 36+36+36 
and so on, but you don't know how many crates to add up! Instead, the 
problem is to DIVIDE 17,856 grapefruits into groups of 36 each; each 
group can then go into a crate. How many groups do you get? Divide 
17,856 by 36 to find out.

You can also solve this second problem by setting it up like the 
first problem, as a multiplication. To do this, you must pretend that 
you know the number of crates. This is the algebraic method. Since 
you don't really know the number, you give it a name; I'll call it n. 
Then to find the number of grapefruits, you multiply the number of 
crates (n) by the number of grapefuits per crate (36). But you know 
that the total number is 17,856. In other words:

  n * 36 = 17,856

You have probably learned about multiplication fact families, for 
instance 

  2 * 3 = 6
  3 * 2 = 6
  6 / 2 = 3
  6 / 3 = 2

My little equation is a member of its own fact family:

       n * 36 = 17,856
      36 *  n = 17,856
  17,856 /  n = 36
  17,856 / 36 = n

Take another look at that last one! It tells us how to find the 
number n: divide 17,856 by 36. 

You see that both problems can be set up as multiplication problems. 
The difference is which number in the problem is unknown. If it's the 
product, then you find it by multiplying. If it's one of the 
multipliers, you find it by dividing.

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