When I was a child I had my own mental tricks for getting
the solutions to math problems more quickly. Using them meant
that I didn't "show my work" in math class, which annoyed
many of the teachers, and lowered my grades. I had the correct
solutions, but I was simply using different algorithms, ones
which I had a hard time expressing on paper.

In my mind, for example, 97 x 16 became 100 x 16 (1600) minus
3 x 16 (48). This was easier, and thinking this way became almost
automatic, so I might just write down 1552 even though I couldn't
explain very well how I arrived at the answer. Teachers called
that a problem, but I have noticed that many years later such
math shortcuts are being sold in seminars and books.

You can create your own math shortcuts, and the following
may give you some ideas on how to do that. Otherwise, you can
try any of the shortcuts and algorithms you read about and adopt
the ones that seem to work best for you. Our minds work in slightly
different ways after all.

Let's look at an example. Suppose you want to multiply 68
x 6. I immediately think "60 x 6 = 360 and 8 x 6 = 48, and
360 + 48 is 408." This is one way to quickly arrive at a
solution without pen and paper. On paper it would likely be expressed
like this: (60 x 6) + (8 x 6) = 408.

Here is another shortcut: See it as (70 x 6) - (2 x 6). The
accompanying "internal dialog" might go like this:
"70 x 6 = 420, but that is two "sixes" too many,
so take away two sixes (12) and I have 408." There is often
more than one way, and you can use whichever math shortcut is
easier for you.

By the way, if the problem was 68 x 9, my mind would immediately
focus on the 9, because it is close to 10, and multiplying by
10 is easy. 68 x 10 = 680. Then I just have subtract the extra
68 to arrive at the solution of 612. Look for the numbers that
are close to 10, 50, 100, 200 or 1000, and you'll find the easier
way to do the math, especially if you are trying to do it in
your head.

Here's a video that demonstrates
a math shortcut...

Percentages are trickier to do as mental math, but there are
ways. Suppose you want to figure what the 4.6% sales tax will
amount to on your $29 book. A quick way to estimate it is to
take 10%, or $2.90, cut that in half to arrive at 5%, or $1.45,
and then just guess at around $1.35, because you know 4.6% is
a little less than 5%. You could also think of 5% as a 20th of
the price - if this is easier - and then round that figure down
a bit.

Want a more precise solution? 1% of $29 is easy to arrive
at (.29), so multiply that by 4 to arrive at $1.16. (You might
think of this as (4 x 30) - 4.) Now add .6% to that. For that,
think 6 x 29 = 174, and put the decimal in the right place: .174.
Add that .18 (rounded up as the store will likely do) to the
1.16 and you have $1.34 in sale's tax, pretty close to our quick
estimate. This isn't as difficult as it might seem once you practice
a bit.

Many of these simple methods do require a basic understanding
of math. For example, you should be able to immediately place
the decimal in the right place in the above example. If you understand
the basics, you know that .6% has to be less than 1%, or .29,
so it can't be 1.74, and .0174 will immediately appear as too
much less than 1%.

Another example: if you don't understand that 123 multiplied
by 199 is just adding 123 to itself 199 times - that multiplication
is just another way to do addition - you will have problems with
these math shortcuts. In that case, you may want to simply use
the easiest math shortcut of all - a calculator.

Oh and the solution to that last one is 24,477. And yes, I
did do that in my head, so let me know if it is incorrect. Here
is the mental dialog: 100 x 123 is 12,300 (just add two 0s),
so 200 x 123 is 24,600 (just double the 12,300). Subtract the
extra 123 from that (which actually goes like this when I slow
my brain down to watch: 24,600 - 200 = 24,400, then add back
the extra 77 I subtracted) and you have 24,477.

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Fun With Figures

Brilliant mental math short cuts that will amaze everyone!