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# Mental computations and trains

This thread is born from a remark Jaybarti made in a recent discussion:

If you took the complete timeline of life on earth and put it to a 24 hour clock, we probably only came into existence in the last minute on that 24 hour clock.

How do you people deal with complex computations, when you don't have a calculator at hand (or accept the challenge of not using one)? It seems everyone have their own tricks and shortcuts, and I'm curious about yours. I'm not a mental computation specialist, although I'd rate myself above average, and I just love smart math quizzes.

Finding the correct solution to this 'clock' case is actually rather trivial (no real trick involved). Here's how I did it (reasoning broken down to its 'atomic' components - note there aren't any complex computations):

Assuming life is 3.5G years old, and Homo Sapiens emerged 200,000 years ago -

3.5G = 3.5 * 10^9
200,000 = 2 * 10^5
3.5 / 2 = 1.75
9 - 5 = 4 (comparing the powers)
1.75 * 10^4 = 17500

I.e., the life history of Homo Sapiens is 1/17500th of the history of life on Earth.

Now, an hour is 60 * 60 * 24 seconds = 86400 seconds. Actually I didn't remember the number and had to mentally compute it as -

3600 * 24 =

36.. * 2.
36.. * .4
---------
72...
144..
---------
864..

Now to the final result -

17500/86400 = 17.5/86.4 (I find it easier to compare and approximate 2-digit numbers)

17.5 * 5 = 35 + 35 + 17.5 = 70 + 17.5 = 87.5 which is superior to 86.4 (the actual ratio is 1/4.937...)

Thus my reply to Jaybarti: make that minute less than 5 seconds.

OK, this one was pretty straightforward and dull. Here's a much more interesting example: I once was in a train with two college students doing their math homework. They had only one pocket calculator to share. One of them asked:
"9 to the power of 4?"
"Wait a minute, have to finish with that first..."

My brain instantly clicked:

9^4 = (9^2)^2 = 81^2 = (80 + 1)^2
(a + b)^2 = a^2 + b^2 + 2ab
Thus (80 + 1)^2 = 80^2 + 1^2 + 2*80*1

64..
...1
.160
----
6561

It took about the same time to compute and give the result, than it took them to check using the calculator. These two guys were much better at math than I ever was, but until I explained the whole thing, they thought they were sitting close to a math genius. It was sort of a shock for them to realize I needed only a pretty basic formula, 8*8, 9*9, and a simple addition (without carry digits). Actually I was just as stunned myself.

Any similar trick or story to share? If not, math quizzes that are best solved with outside-the-box thinking?

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### Replies to This Discussion

I am like you in that I use mental arithmetricks (I know it is not a real word) to caluculate stuff in my head.

My example is a little more basic, but it is how I calculate 15% for tip...

Take any number or bill lets say \$75

15%=10%+5% so... 10% of 75 is 7.5 + 3.25 (half of 7.5) = 10.75 at this point depending on service I round up or down (mostly up).

I do a lot of theses style things when I am doing math, breaking it down in to easily calculable segments.

Another example is a trivia questions that asked what the numbers 1-10 added up to...

1+2+3+4+5+6+7+8+9+10=??

My short hand is making 10's
(1+9)+(2+8)+(3+7)+(4+6)+5+10=55

I impressed my friends during that round with my supposed math skills.

I suspect that most people who are good at math have an ability to break complex calculations down into easy repeatable patterns.
Other cheat, the average of your pairs is 5.5, there are 10 of them. 5.5*10=55
Don't forget to think about the numbers as going from 0 to 9 instead of 1 to 10.
I did this in fourth grade but deprogrammed myself since my teacher would have been a totalitarian about it and bitch at me for not being in line with her idea of how mental computation must work. I later saw it on Unscrewed (R.I.P.) on TechTV (R.I.P.)--unless this guy just friggin' rehearsed adding and multiplying the numbers given on the show.

I don't remember how it worked to be so friggin awesome, but everything just fucking harmonized--and if it didn't, it was just a few steps to make it do so. I think it had something to do with how far from 9 each digit was and adding from place values. I decay.
Simple method to compute squares of numbers ending with '5', like 35 or 195 -

1. remove the last digit.
2. multiply the remaining number with (itself + 1)
3. append '25' to the result.

Example:
35 (1)> 3 (2)> 3 * 4 = 12 (3)> 1225

195 (1)> 19 (2)> 19 * 20 = 380 (3)> 38025

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