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### 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/
```
Associated Topics:
Elementary Division
Elementary Multiplication
Elementary Word Problems
Middle School Division
Middle School Equations
Middle School Word Problems

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