Mental Math Techniques

Question

What are some interesting mental math techniques that you know?

Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number minus $25$ and put that in the first two digits, and then add the square of $50$ minus the number: $$(\text{number}-25)\times100+(50-\text{number})^2$$ For example, to do $47^2$ we have $47-25=22$ for the first two digits and $(50-47)^2=9$ for the last two so we get $47^2=2209$.

Bonus points if you include justification! For this trick, $$100(n-25)+(50-n)^2=100n-2500+2500-100n+n^2=n^2$$

Answer 1

Makes multiplication of multi-digit numbers easier. The above is the following problem:

   14759
x 365421

This is how they teach multiplication in Japan. You may be thinking, you draw this in your mind? No, there's a shortcut for this method.

Take for instance:

  21
x 32

You can draw it to get the answer. But the drawing is basically giving you a simpler way of solving it. This is how you solve it:

http://sketchtoy.com/60373071

Here's a little tougher one that I did mentally:

http://sketchtoy.com/60373153

Makes mental multiplication of multi-digit numbers easier.

Answer 2

Here is a trick to multiply two numbers between 10 and 19 together, say $10 + x$ and $10 + y$: Compute $10 + x + y$, put a zero at the end (multiply by 10), and add $x\cdot y$. Thus $(10+x)(10+y) = 10\cdot(10 + x + y) + xy$.

Easy with algebra. I learned this from my mother who had only an 8th grade education and no algebra.

Answer 3

Of course, we have the classic trick $$9\times n=(10\times n)-n$$ which works because of distributivity. This can be generalized as follows: $$99n=100n-n$$ so for example $$99\cdot 54=5400-54=5346$$ It also helps simplify generic calculations. E.g., $$17\cdot 8=17\cdot 10-17\cdot 2=170-34=136$$