Multiplying Fractions

Date: 1/09/98 at 11:07:37
From: Mike
Subject: Multiplying fractions

I can't figure this out:

  __ x 20 = 16

This is 7th grade math.

Date: 1/9/98 at 12:44:04 (EST)
From: Dr. Sonya
Subject: Re: Multiplying fractions

Dear Mike,

First let me put your problem into words. You want to find some number 
so that when you multiply it by 20 you get 16. This type of problem is 
called solving an equation, and you'll see a lot more like it when you 
study algebra.

An equation is an expression with an equals sign in the middle, and 
two equal things, one on either side of it. For instance, 4 = 4 is an 
equation. 

Equations get more interesting when you don't know what one side is.  
That's the kind of equation you have. On one side you have 16, and on 
the other you have a mystery number times 20. I'm going to call this 
mystery number n, and then I can rewrite your equation:

   n x 20 = 16

Our job is to find out what n is.

Before I launch into finding n, let me show you another example. What 
if I had the equation:

   m + 3 = 5  ?

In this equation, m is our unknown number, and I want to find out what 
it is. So what number plus three equals 5?  Think, think...




no answer yet...




Did you say 2?  That's right!  

   2 + 3 = 5

Now here's another way to do this. Write the equation again:

   m + 3 = 5

Because both sides are equal, if I do the same thing to both sides, 
the equation will stay equal, won't it? Make sure you believe this.  

Now, if I subtract 3 from both sides of the equation, look what 
happens:

         m + 3 = 5
     m + 3 - 3 = 5 - 3

Now, 3 - 3 = 0, and 5 - 3 = 2, so we can write:

         m + 0 = 2, or
             m = 2

This is the same answer that we got before. You have just solved an 
equation.  Let's try this with the equation:

  m + 6 = 8

Subtract 6 from both sides. Did you get m = 2?  Good job.
 
Now, for an equation like yours, where there's a multiplication 
instead of an addition, again, we want to do something to both sides 
of the equation. Let's say we have the equation:

   3m = 6  (3m means 3 x m)

If you know your multiplication tables, you already know that m = 2.  
But I'm going to show you how to do it the algebra way. What can I do 
to both sides so get m all by itself?  How about dividing both sides 
by 3?

       3m = 6
     3m/3 = 6/3

Now, 3m/3 = m, and 6/3 = 2, right?  So,

  m = 6/3 = 2, and this is the same answer we had before.

Lets try it with one that's a little more complicated:

       4m = 6

Just as in the last one, we'll divide both sides by 4 and see what 
we get.

       4m = 6
     4m/4 = 6/4

Now, 4m/4 = m, but 6/4 = ...Oh no, a fraction!  All right, don't 
panic.  m can be equal to a fraction. There was never a rule that said 
it had to be a whole number.  6/4 = 1 1/2, right?  So:

        m = 6/4 = 1 1/2

One way to see if we are right is to plug the vaule of m we got back 
into the equation.  If:

         4m = 6 and
          m = 1 1/2, then
  4 (1 1/2) = 6 should be true.
      1 1/2 = 3/2, and 4 x 3/2 = 6  Ta da!

Now try this method with your equation:

   20n = 16  

Note that sometimes it takes more than one step to find out what the 
unknown variable is.  Can you solve the equation:

    4m + 3 = 11   ?

Good luck with 7th grade.

-Doctor Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/