Algebra/Maths

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Question
Trying to work out:

8.75squared - 1.25 squared

I can do it by just multiplying out each side but wanted to know if there us a simpler formula. No Calculator allowed.

Answer
The problem is 8.75-1.25.  There are three ways to do it without a calculator.


1) As can be remembered from the equations in algebra, a-b=(a+b)(a-b).
Since a=8.75 and b=1.25, it can be seen that a+b = 10 and a-b = 7.5.
Thus, that answer is 10*7.5 = 75.


2) Note that if we have (n - m/4) where n and m are integers,
the answer is n - nm/2 + m/16 (Well, n and m don't have to be integers, but it sure
makes for easy calculations if they are).

Here, we have n=9 and m=1, so the answer is 81 - 9/2 + 1/16.
Since 9/2 = 4.5 and 1/16 = .5/8 = .25/4 = .125/2 = .0625,
that gives 81 - 4.5 + 0.0625 = 76.5625.

It is known that 1.25 is 5/4, and the square of 5/4 is 25/16.
Now 25/16 = 1 9/16 = 1 + 8/16 + 1/16 = 1 + 1/2 + 1/16 = 1 + 0.5 + 0.0625 = 1.5625.

Thus, if we take 76.5625 - 1.5625, the decimals all cancel and the result is 75.


3) One last way to look at the problem it to note 8.75 - 1.25 = (35/4) - (5/4).

The 4 (4=16)can be factored out, leaving (35 - 5)/16.
Now 35 = 30 + 10*30 + 25 = 1,225 and 5=25, so the difference is 1,200.
Since 12 is 3/4 of 16, it can be seen that 1200 = (3/4)1,600.
Using this, we get (3/4)1,600/16.  Now 1,600/16 = 100, so we have 3/4 of 100,
which is known to be 75.


Note that (1) is probably the easiest way to do it, but I did it in my head by using (2) at first.  After I thought of (2), I then realized the numbers could be expressed as fourths and the fractions combined, giving (3).  Once I had done (2) and (3), I then realized (1) could be done, and I believe this is the easiest method to use.

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