Multiplying by 9, Step by Step

Date: 11/05/2001 at 23:52:33
From: Sara
Subject: Math multiplying

I need help with 

240,531 x 9

Date: 11/06/2001 at 08:58:48
From: Doctor Peterson
Subject: Re: Math multiplying

Hi, Sara.

      240,531
    x       9
    ---------

The basic idea is that you are multiplying every part of the number by 
9; the answer will be 

   2 hundred-thousands times 9, plus 
   4 ten-thousands times 9, plus 
   5 hundreds times 9, plus 
   thirty times 9, plus 
   1 times 9. 

Our usual method (in fact, any method we use) is just an orderly way 
to keep track of that without having to think too much.

You start at the right, multiplying 9 times 1. This gives 9, so you 
can write that down:

      240,531
    x       9
    ---------
            9

Now you multiply 9 times 3 and get 27. What do you do with that? Well, 
since it is really 3 tens, or thirty, that you are multiplying by 9, 
the 27 really means 27 TENS. But we can only write one digit in the 
tens place of our answer, so we have to "carry" the 2 to the next 
column. This really means that we are thinking of 27 tens as 2 
hundreds and 7 tens:

          2
      240,531
    x       9
    ---------
           79

I wrote the ones digit, the 7, in the tens place of the answer (right 
under the 3 we just multiplied), and put the 2 above the hundreds 
place.

Now we multiply 9 times 5, which gives 45; but we have to add the 
extra 2 we saved from the tens. That gives us 47, and again we write 
down the 7 directly under the 5 we are multiplying, and "carry" the 4 
over to the next column to the left:

        4 2
      240,531
    x       9
    ---------
          779

We keep doing the same thing, until we end up with

      3 4 2
      240,531
    x       9
    ---------
    2,164,779

Notice that in the leftmost place I got 9 x 2 = 18, plus 3 is 21. I 
didn't have to write the 2 as a carry, but just put it immediately in 
the next place in the answer, since there was nothing else to add it 
to.

Now here's something worth thinking about, to help you remember when 
to use the "carries." What we've just done is really nothing more than 
a short way of writing the sum of 9 copies of 240,531:

      3 4 2
      240,531
      240,531
      240,531
      240,531
      240,531
      240,531
      240,531
      240,531
    + 240,531
    ---------
    2,164,779

For example, when you add up the hundreds column, you add together 
nine 5's to get 45, and also add on the "carry" from the tens column 
(2) to get 47. You write down the 7 and "carry" the 4. So what we did 
in multiplication is exactly the same as in addition. If you remember 
that, you will not make the mistake of adding the 2 before you 
multiply, or something like that.

Feel free to write back if you need any more help.

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