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#1 2010-08-12 05:41:39

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

multiply further apart of 2 numbers

Here's a round-about way
of multiplying two numbers
whereby we spread the numbers
further apart.  This may have
greater purpose other than amusement. 
What purpose that is, is up to you.

67 times 56 is the problem.
Okay, add 3 to 67 for 70.
Then subtract 3 from 56 for 53.
Now multiply 70 times 53 in head if can.
3710 is 70 times 53.
Now because we spread 67 and 56 apart,
then 3710 is smaller than the answer we want.
This is an important point.
Because if we had made the 2 numbers go
closer together, then the product (times'ing them)
would have been bigger than the real answer
we are looking for.
So now we have to find out how far off
we are from the right answer, and add that onto
the 3710.  To do this, we must subtract two
funny things.
A.) 67 - 56 = 11
       11 is 5 + 6, so 5 times 6 = 30
B.) 70 - 53 = 17
       17 is 8 + 9, so 8 times 9 = 72
             (I chose 5 and 6 for 11 because it is close to half of 11)
          (Also 8 and 9 are close to half of 17.   This is necessary.)
Now take the B or 72 and subtract A or 30 from it.
72 - 30 = 42
That is the funny number we have to add to 3710.
3710 + 42 = 3752, and that is the answer to the original problem.
So 56 times 67 should be 3752.


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#2 2010-08-12 05:45:36

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: multiply further apart of 2 numbers

And if you had chosen to make the two numbers go
closer together, then A might be larger than B, and
simply subtract the smaller one from the larger one,
but know that the products of the spreaded numbers
get bigger if go closer to center, and smaller if go
further from center, so you know to add or subtract
the funny number.  Have fun.


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#3 2010-08-12 06:04:06

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: multiply further apart of 2 numbers

Here's the 2 x 7 example from the other thread done this way.
It is almost the same, involves a zero term that does nothing.
Because the 5 - 4 = 1 and 1 = 0 + 1 (closest to average of 1/2)
and 0 times 1 is 0.  I'll do it out below...

7 times 2 is the problem.

We'll go closer together.  6 times 3 is closer together.
And then again closer, we'll choose 5 times 4.
Note that 5 + 4 is the same as 7 + 2.

Now 5 times 4 is 20 and this should be larger than 7 times 2
because we went closer together with the 2 numbers.

Now,
A.) 7 - 2 = 5 and 5 = 2 + 3 and 2 times 3 = 6, remember 6.
B.) 5 - 4 = 1 and 1 = 0 + 1 and 0 times 1 = 0, remember 0.

Subtract the smaller one from the bigger one.
A - B = 6 - 0 = 6.

Take this 6 and remove it from the 20 of the 5 times 4.

20 - 6 = 14

and this is the answer to 7 times 2.

So it still works this way. 

This way is a little more general than the previous thread
called "multiply near the center of 2 numbers".

You could spread or compress your numbers apart or
together any distance you like I suppose, and it
should work out fine, according to my calculations.
I developed this method by playing around with numbers
for a few days and comparing how far apart different
products were from each other. 
Like 25x25  and 24x26 and 23x27 and 22x28.
Also I compared odd and even and even/even and odd/odd
cases to come up with this method.
To get the most out of this, it is
probably encouraged to try to develop
a method all your own by yourself too.
Then you will be more interested in it.
smile


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