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
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
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?