same as 50x is better done by multiplying by 100 and dividing by 2

then the trick for squares is one below x one above +1
- ie 998x1000 +1= 998,001

explanation : where B is your number (B-1)(B+1) = (B x B) - 1

49 x 49 = 48 x 50 + 1= 48 x 100/2 +1 = 4800/2 + 1 = 2401

That was Melvin Bragg's history of science series - In Our Time talking about mathematical patterns
The Fibonacci Sequence - the numbers in nature

we use these short cuts all the time but nobody teaches them

using a trick of rounding to the nearest ten
27 x 27= 1 + 26 x 28 = 1 + (26 x 30) - (2 x 26) = 1 +780 - 52 = 729

to divide by 50 - divide by 100 and double it

to divide by 25 - divide by 100 and quad it

to multiply by 25 - multiply by 100 and divide by 4

to divide by 33 - divide by 100 and triple it - about right

Vedic maths page says
= to get square of a 5 number: left part =multiply the first part by itself plus 1 and right part is 25
e.g. 85 x 85 = so left part is 8 x 9 =72 so total is 7225

Or expressed : to get square of B+5
25+ (B+1)x 100B = 25+ 100BxB +100B

explanation - first do the unit then the tens so : (5B +25) then B.B + 5B = BB + 10B + 25 = B(10+B) + 25 e.g. 35.35 = 3x 4 x 100 +25 = 1225

seems to me if you don't have a calculator, you can look for patterns
e.g 1 . 13x - i the last part will always be 9
- 169, 529, 1089, 1849 see the first part is z6 x z 16 x 1, 26 x 2, 36 x 3 so 63 x 63= 66 x 6 = 396 so 3969
e.g 2 . 14x - i the last part will always be 16
- 196, 576, 1156,1936 see the first part is 18, 56, 114, 192 18 x 1, 28 x 2, 38 x 3, 48 x 4 so 64x 64= 68 x 6 +1 6 = 4080 + 16 = 4096

suddenly there a general rule
to get qr x qr : (q2r) x q x 10 + rxr
- test with 86 = 92 x 8 x 10 + 36= 7360+36 = 7396
or 57 = 64 x 5 x10 + 49 = 3249 79 : 88 x 7 x 10 = (560+56) x 10 + 81 = 6160 + 81 = 6241

from first principles (10q+ r) x (10q+ r) = 100q.q +20r.q + r.r or 10q(10q +2r) + r.r
79 : 4900 + 20.63 + 81 = 4900 +1260 + 81 - 6241

So now as long as the digits are easy then maths is very easy
21 : 20(22)+ 1 =

www.hinduism.co.za/vedic.htm, http://members.aol.com/REBissell/SpeedMath.html, http://www.slideshare.net/simonesu/amazing-math-trick/

From BBC More or Less - Qn You go for a walk you walk 2 miles an hour cos it's uphill. Returning you walk at 3mph what's the average ?

Now I would have thought you just average the 2, but the thing is you spend more time walking at the slower speed.

go @ 2miles an hour back @ 3 x miles takes x/2 hours back takes x miles takes x/3
- av = D/T = 2x/ (x/2 + x/3)
- then mulyiply by 6/6 so = 12/(3+2) = 2.4

If you think about it what you are doing is averaging the inverse of the speeds.
----- (1/3 + 1/2)/2 = 5/12 : av speed= 2.4

e.g.2 drive at 30 K walk at 6 = (1/30 + 1/6)/2 =(36/180)/2=1/10 = av speed

or if 30 Km then time t= 1+5, and d= 60, so av speed = 60/6 = 10

A small change in percent can be a huge change in real numbers
- a 100g bowl of cherries is 99% water
- overnight water evaporates
- so it's now 98% water
-what's the weight now ?

non water is 1g = 2% so 100%= 50g
I thinks that’s amazing ! I would not have thought the 1% change in concentration would mean a 50% drop in weight.
- once we are at 10g then the mix is still 90% water

So if you were mining gold
- mine 1 has a 99% impurity means to to get a 1Kg of gold you have to remove 99Kg of spoil
- mine 2 has a 98% impurity means to to get a 1Kg of gold you have to remove 49 Kg of spoil
- mine 3 has a 90% impurity means to to get a 1Kg of gold you have to remove 9 Kg of spoil

Wow that 1% difference in impurity makes a huge cost saving of course cos it has double the gold purity from 1% to 2%
likewise in cleaning the waste water the amount of work needed to get a 99% pure solution compared to a 98% is huge.

Doesn't this principle apply to risk ?
- If you are sure something might happen like i. 50% risk, then money is well spent, but if the risk is small say ii. 2% the amount of money it costs to reduce the risk to 1% is huge. But of course you can say we will save double the lives. Circular thinking ? no not quite i-you save 48 lives in 100 ii you save 1 extra life.

So $10bn spent reducing CO2 by 1% is a waste of money. If you could reduce it 50% that would be worth it, though I wouldn't spend $100bn to do this. But bear in mind the recirculation effect of government spending a fair amount of money comes back in tax etc, that's why wars aren't as expensive as they seem.