What is 4/5 x 10?

Date: 10 Feb 1995 15:19:52 -0500
From: Joy Sakoguchi
Subject: Michael's queston

Dr. Math,

What is the answer to this fraction?
Here the problem is 4/5 x10.  Please explain. 

Date: 15 Feb 1995 20:05:35 GMT
From: Dr. Math
Subject: Re: Michael's queston

Hello there!

When I multiply fractions, I like to write _both_ of the numbers as
fractions.  So I would take your problem and notice that 10 = 10/1, so we
have

  4            4   10
  - x 10 =     - x --
  5            5    1


Then we have to remember how to multiply two fractions.  When we multiply
two fractions together, we multiply everything in the top together, and
then everything in the bottom together.  Then we take what we get and put
it in a new fraction.  So we get 

  4   10    4 x 10
  - x -- =  ------  
  5    1    5 x  1

This, technically, is our answer, but we can make it simpler in a couple
of ways.  First, notice that 5 x 1 = 5.  So we can write the fraction as 

  4 x 10    4 x 10
  ------ =  ------
   5 x 1      5

Now we do the tricky part.  Notice that 10 = 2 x 5.  So let's replace 10
in the top by 2 x 5, to get

  4 x 10    4 x 2 x 5
  ------ =  ---------
    5           5


Now when we have the same thing in the top as we do in the bottom, we can
cancel them both out, but (this is very important) we can only do it when
the top and bottom each have multiplication separating the parts, not
addition or subtraction or anything else!!  So if the top had been 4 + 2 +
5, we wouldn't have been able to cancel anything.  In this case though,
we're fine.  So we get

  4 x 2 x 5   4 x 2
  --------- = ----- = 4 x 2 = 8.
       5        1

So the answer to our problem is 8.  This may seem like a lot of work, but
if you do it enough, you'll get really good at it, and you'll start doing 
it in your head.  To let you know what you have to look forward to when you 
get good at these, when I looked at this problem, I knew the answer in 
less than a second!

I hope this helps you.

-Ken "Dr." Math