Multiplying Decimal Numbers and Placing the Decimal Point

Date: 09/20/2007 at 10:38:39
From: tel
Subject: Multiply numbers with fractions.


When I multiply something like 24.5 x 36.7, I get the decimal point in
the wrong place.

   24.5
   36.7
--------
  171.5
 1470.0
 7350.0
-------
 8991.5

answer is 899.15 - get decimal point wrong ....

Date: 09/20/2007 at 11:50:17
From: Doctor Ian
Subject: Re: Multiply numbers with fractions.

Hi Tel,

I like to work with integers, so I usually change decimals to
fractions when I want to multiply or divide:

    2.57 * 2.63

    257   263   257 * 263
  = --- * --- = ---------
    100   100    10,000

Now I can do the multiplication on top without worrying about decimal
points:

    257
  * 263
  -----
    771      <-  3   * 257
  15420      <-  60  * 257
  51400      <-  200 * 257
  -----
  67591

to get

  67,591
  ------
  10,000

And that means the same thing as

  6.7591

As a quick check, note that I've got

  2 point something * 2 point something

so the answer should be a little bigger than 4.  This can be a good
way to make sure you didn't slide a decimal point too far, or not far
enough!  If I end up with something like 67, I can see right away
that's too big; similarly, something like 0.67 would be too small. 

Anyway, the moral is that being able to go back and forth between
decimal and fraction notations is a big advantage, both in being able
to explain WHY various decimal operations work the way they do, and in
being able to AVOID lots of complicated operations entirely, by
working with integers. 

Does that make sense?  Try your problem this way, and please let me
know how it works out for you.

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

Date: 09/20/2007 at 12:04:17
From: tel
Subject: Thank you (Multiply numbers with fractions.)

Got it - a good way to simplify.  Thanks.  I am sure I used to do it
with the decimal point and putting a zero, then 2 zero's - etc... or
am I making it up?

Regards T

Date: 09/20/2007 at 12:17:16
From: Doctor Ian
Subject: Re: Thank you (Multiply numbers with fractions.)

Hi Tel,

The usual way of doing it is to 

  1. Count up the total number of decimal places.

  2. Forget about them, and multiply as if you just have integers.

  3. 'Put back' the number of decimal places you found in step (1).

Do you see why it amounts to the same thing? 

  We're doing step (1) when we convert to fraction notation 
  and multiply the denominators.  

  We're doing step (2) when we multiply the numerators.

  We're doing step (3) when we convert back to decimal notation. 

Once you know what's going on, it's quicker to do the multiplication
without explicitly constructing the fractions.  But the fraction
method is always there as a backup.  

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

Date: 09/20/2007 at 12:19:51
From: tel
Subject: Thank you (Multiply numbers with fractions.)

Even clearer - many thanks indeed!