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Multiplying Two 2-Digit Numbers


Date: 03/25/97 at 19:20:44
From: Rachel
Subject: Multiplying Two Numbers with Two Digits
 
How do you multiply two numbers with two digits each -- 2 by 2 
multiplication? 

For example, how do you do twenty eight times twenty four?


Date: 03/26/97 at 01:34:11
From: Doctor Mike
Subject: Re: Multiplying Two Numbers with Two Digits

Hello Rachel.

   First I'll just do your sample problem, and then we can talk about 
why you do it that way.
   
       2 8 
       2 4 
     ------
     1 1 2 
     5 6   
     ------
     6 7 2 
  
If what you wanted was to see this worked out, then we are done.
Some people like to write a 0 to the right of the 56, like 560.
   
If you want to understand better why you do it this way, let's
look at two related, but simpler, "1 by 2" problems for review.  
    
       2 8             2 8 
         4               2 
     ------           -----
     1 1 2             5 6 
  
In the left one you start with the "units" or "ones" place, which
has an 8 in it, and multiply that by 4 to get 32.  Here's the really
important thing: 32 = 30 + 2 which is 3 tens plus 2 ones.  All you
write down in the ones column of the answer is the 2.  That makes
sense, doesn't it?  At this point, REMEMBER THE 3 TENS , okay?
  
The next thing is to look in the tens place to see what is there.
You find a 2 there, but that doesn't mean plain 2, it really means
2 tens, since the twenty from 28 is 2 times ten.  If you multiply
that 2 tens by 4 you get 8 tens.  But wait!  Is that all there is?
No, you promised to REMEMBER the 3 tens from before.  Now we have
8 tens plus 3 more tens, making 11 tens, so that is what we write
down.  That's how we get the answer 112.
   
In the right one you start with the 8 in the ones place, multiply
it by 2 to get 16, which is really a 10 plus a 6.  You write down
the 6 in the ones position in the answer, and REMEMBER THE 10.  

Next, multiply the 2 in the tens position of 28 by the multiplier
2 to get 4 tens.  But wait!  Remember the 10 from the first step,
so 4 tens plus another 10 is 5 tens.  Now you write 5 in the tens
position of the answer, and we are done.    
  
Now, I'll explain the diagonal thing. The answer to "28 times 20" is 
the same as the answer to "28 times 2" but multiplied by 10 (which 
adds a zero on the end).  Since 28*2 is 56, we know that 28*20 must be 
560.  We write it down like : 
  
      2 8 
      2 0 
     -----
    5 6 0 
   
We are about to the end of your example now.  You understand that
28*4=112.  You understand that 28*2=56.  You also understand the
related fact that 28*20=560.  Now, the whole 2-by-2 problem just
amounts to doing the 28*4=112 and the 28*20=560 in the same spot,
together.  To see this, let's go back to the full problem : 
  
       2 8 
       2 4 
     ------
     1 1 2 
     5 6 0 
     ------
     6 7 2 
  
Notice I wrote it with the 560 the way I said above that some people
like to do.  When you do a 2-by-2 multiplication problem you are
doing 2 different 1-by-2 problems in the same place, and then
adding those answers together to get the grand total.  
  
To summarize, if you want to multiply 28 by 24, you  ......
  
  1. Multiply 28 by 4 , which is 112 , 
  2. Multiply 28 by 20 , which is 560 , which some people
     like to write down as 56 shifted to the left one place ,
  3. Add the results from 1. and 2. 
  
I know that this is a lot of difficult reading.  I am glad you asked 
this question, and I encourage you to try more than once to follow 
what I have sent to you. Perhaps you can even print it out and
show it to an adult in your family, and ask them to help you
go through some similar problems.  It might even be that the
adult (or maybe a teen) never quite figured out why this works,
and would enjoy learning the real Math Secret along with you. 
   
So, I hope this helps, then.   Good luck.   Enjoy.  

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Multiplication

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