Multiplication Trick

Date: 02/08/98 at 21:49:38
From: Rosemarie Lee
Subject: Why is the answer right?

I am a fourth grader and I have been doing multiplication and 
division. While playing with multplication I discovered that when I 
multiply 25x25 or 35x35 or 45x45 all the way up to 95x95 I can do the 
answer in my head by multiplying the first digit times the next 
highest number - like in 25 x 25 I go 2x3 is 6 and 25x 25 is 625; or 
45x 45 is 4x5 is 20 so the answer is 2025.

I am thinking about three-digit numbers like 225x225 because I think 
they should work too. I need to think about it some more. Does this 
work with numbers ending in 25 because 25 multiplies itself, or why 
does it work?  

I did my answers on a calculator and I know they are right. I would 
like to know. Can you help me find out? 

Thanks you for helping me. 
Douglas

Date: 02/10/98 at 11:15:00
From: Doctor Pete
Subject: Re: Why is the answer right?

Hi,

First of all, let me say that you're very observant! What you're 
saying is quite true: For example,

   35x35 = 1225.

But strictly speaking, this is because

   35x35 = (40 - 5) x (30 + 5)
         = 40 x (30 + 5) - 5 x (30 + 5)
         = (40x30 + 40x5) - (30x5 + 25)
         = (4x3)x100 + (40-30)x5 - 25
         = 12x100 + 50 - 25
         = 12x100 + 25
         = 1225.

And so you can see how this might be generalized to the square of any 
number that ends in 5; for example,

   115x115 = (11x12)x100 + 25 = 13225.

For example, let's pick the number

   3251165

and see what its square might be. We have (skipping a few steps)

   3251165x3251165 = (3251170 - 5) x (3251160 + 5)
                   = (325117x325116)x100 + (3251170 - 3251160)x5 - 25
                   = (325117x325116)x100 + 10x5 - 25
                   = (325117x325116)x100 + 25.

(Now, I'm going to cheat a bit and just say that 325117x325116 = 
105700738572, since the point isn't really about finding the square, 
but that you could pick any number and see that it still works.) So we 
see that we are always left with some multiple of 100, times 25. You 
chop off the units' 5 of the original number, and take product of this 
number with the number 1 greater.  When you become familiar with 
algebra, it will become more clear why this is always the case.

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