Ok, this might be a little basic for some guys here, but I'm interested to hear about some of the shortcuts that can be used to do more 'complex' equations. For example, let's say you want 10% of a price. It's not that hard, you just move the decimal once place forward to get 10% of a number. This can be taken further. If you want 5%, just put the decimal forward and halve the result. If you want 20%, move the decimal forward and double the result. Very basic. I was hoping that maybe some of you guys might know handy little shortcuts like that one....
When I want to mentally do something like 21*8 I do (20 + 1)*8 = 160 + 8 = 168 Or something like 232*50 = (200 + 30 + 2)*5*10 = (2000 + 300 + 20)*5 = 10000 + 1500 + 100 = 11600 After doing it for a while, you get better at it. That is something I started to do when I got too lazy to pull out the calculator.
Ohh, that's a good one. A good, simple one that I learnt was from one of my old math teachers. Let's say you have the multiplication 12 * 24. Instead of working it out of paper, all you have to do is use your multiplication tables. 12 * 12 is 144. Double that, and you get 288. That's the answer Please Register or Log in to view the hidden image!
How would you work out 84 squared in your head? Or any number for that matter? Here is how I do it. Step 1. 80^2 = 6400 (Think 8^2 * 10^2 = 6400 Or commit to memory the table below) Step 2. 4^2 = 16 Step 3. 8 * 4 * 2 = 64 (Multiply all numbers that appear in the calculation. Simple! That's just 8 * 4 * 2!!) That's it! Line the numbers up in your head like so... 6400 **16 *64* ------ 7056 Voila! Try 92^2 90^2 = 8100 2^2 = 4 9*2*2 = 36 8100 ***4 *36* 8464 Bingo! Table 1. 10^2 = 100 20^2 = 400 30^2 = 900 40^2 = 1600 50^2 = 2500 60^2 = 3600 70^2 = 4900 80^2 = 6400 90^2 = 8100
Once you get good at calculating squares then calculating square roots becomes very easy. Find sqrt(1089) We know that 30^2 = 900 and 40^2 = 1600 so sqrt(1089) must be between 30 and 40. Compare the number (1089) with 900 and 1600. It is very close to 900 so the solution we are looking for will be either 31, 32, 33, or 34. Now comes the trick, look at the last digit. It is 9. sqrt(9) = 3 so the answer is 33. Find sqrt(5041) Straight away you should recognise that 5041 is close to 4900. Also it ends in a 1 so that could mean either 1^2 or 9^2. Since it it certainly not 79 it must be 71. Find sqrt(7396) 1. 6400 < r^2 < 8100 80 < r < 90 2. Last digit in question is six so we are looking for either 86 or 84. 3. Since r is closer to 8100 than 6400, r = 86 Find sqrt(2209) 1. 1600 < r^2 < 2500 40 < r < 50 2. Last digit in question is 9 so we are looking for either 43 or 47. 3. Obvious choice is 47. When you get really good you can work these out in about 2 seconds and surprise your friends!
The method for squaring numbers can be altered slightly for numbers between 100 and 999. Find 156^2 1. 56^2 = 3136 (by the above method) 2. 56 + 156 = 212 3. 24336 (By adding 21200 + 3136) Find 188^2 1. 88^2 = 7744 (by above method) 2. 88 + 188 = 276 3. 35344 Someone asks you what is 190^2. Your thought pattern should be... 1. 90^2 = 8100 2. 90 + 190 = 280 3. 36100 Too simple! What about 176^2? 1. 76^2 = 4900 + 36 + 7*6*2*10 = 4936 + 840 = 5776 2. 76 + 176 = 252 3. 25200 + 5776 = 30976 When you get good at these you should be able to do them in about 10 seconds or better. With these types step 1 is usually the hardest.
I always like tricks for factoring such as if the sum of the digits of a number equal three the number is divisble by three. Also, if the sum of the odd digits subtracted from the sum of the even digits is divisible by 11 or by 0 the number is divisible by 11.(This works as a general rule, but it may limited in some ways). Cheers!
I nice little method I found for solving the forced,damped harmonic oscillator with some initial conditions is to use Laplace transforms. All you then proceed to do is some algebra to solve the entire problem, it is quite easy.
You used Laplace Transforms for such an elementary problem? You should be ashamed of yourself man, I thought Electrical Engineers were the only ones who used Laplace Transforms to solve such simple questions, and I never believed that Physics students would do such things. (For anyone who can't tell, pretty much everything past this point is a joke.) What's next? Using a table instead of doing the inverse Laplace Transforms by hand because you don't feel like computing the contour integral? Using Mathematica to solve all of your integrals and differential equations for you? Changing Majors to English? Developing a fondness for the works of Nietzsche?!?!
Square root of 84? Hmm.. try this.. y = sqrt(x) dy/dx = (1/2)/sqrt(x) dy = 0.5*1/sqrt(x)*dx so dx = 3 sqrt(84) is about 0.5*1/(sqrt(81))*3 + sqrt(81) = 9.16666... sqrt(84) is actually about 9.1675151
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Also, to extend the 11 trick, it can also work with other double digits like 22,33,44,etc... All you do for let's say 22 * 13 would be add up the digits 1+3=4, then double it so it's 8 which would be your middle digit, then to get the other digit you do 3*2 = 6 so the answer would be 2 8 6. Here's just one of the great vedic math tricks for adding fractions. Say you wanted to add 2/3 and 4/7: Code: 2 4 (3*4)+(7*2) --- + --- = -------------- = 26/21 3 7 3*7 First you multiply horizontally 3 and 7 to get 21. Then you cross multiply and add 3*4 + 2*7 = 26. And that's it, it's 26/21. Works for subtraction too (only you subtract). In all math tricks you do the same thing, only it's faster for the mind.
I always use this simple trick when doing a simple two-number multiplication Example: 39x45=? (39+45)/2=42 |39-42|=|45-42|=3 39x45=(42x42)-(3^2)--> I have found this trick in elementary school by accident 39x45=1764-9=1755 Proof: 39x45=(40-1)x45--> simple distribution law =1800-45=1755 I've been using this technique for long, easily put, almost everytime.
During my school years (less my universitary years),not so many years ago,I used to make many arithmetical calculations in mind.I lost this habit...deh computers... An easy method to multiply numbers in mind (my father has taught me once it) is: Let's say that we have to multiply numbers having 2 digits for example 97x86.The result of a multiplication of numbers having 2 digits can have the most 4 digits (for 100x100=10000).Let the result be ABCD. 97x 86 --------- ABCD The steps involved to find the result (also enough easy to do them mentally,for some people at least) are: 1. 7x6=42 ---> D=2 and we have a carry R1=4 2. (9x6)+(7x8)+R1=54+56+4=114 ---> C=4;R2=11 3. (9x8)+R2=72+11=83 ---> B=3 and A=8 The number seeked is 8342.The method is not really new being a variation of the usual way to make multiplications but is easier to use for for mental calculations especially for numbers having 2 or 3 digits (as I've already said at least for some people who have a good enough short time memory). It can be generalized for a higher number of digits,I will present only an example with 3 digit numbers which is still in the reach of normal people (sometimes!),beyond this one need to have a fantastic memory... 784x 529 ---------- ABCDEF The result can have at most 6 digits (1000x1000=1,000,000 has 7 digits) The steps involved: 1. 4x9=36 ---> F=6 and a carry R1=3 2. (4x2)+(8x9)+R1=8+72+3=83 ---> E=3 and R2=8 3. (4x5)+(7x9)+(8x2)+R2=20+63+16+8=107 ---> D=7 and R3=10 4. (7x2)+(8x5)+R3=14+40+10=64 ---> C=4 and R4=6 5. (7x5)+6=35+6=41 ---> B=1 and A=4 The seeked number is therefore 414736 The method does not seem easy but in reality is not so,once you learn the principles and get some practice it becomes much more friendly...and reliable (OK,in the majority of times).Has this method been of help for me?I'd argue that yes,even in the exterme case that I could remember only the time when I was in the army...I was doing such types of mental calculations during the long boring nights when I was on guard... Please Register or Log in to view the hidden image! Another easy trick I know from my father is how to square numbers that end in 5 Let say 95x95: 95x 95 ---- ABCD 1. (5x5)=25 ---> we have always C=2 and D=5 2.To find A and B we must multiply 9 (here) with the next natural number that is 10;(9x10)=90 ---> B=0 and A=9 The number seeked is 9025. If we have for exaqmple 105x105 ---> the last two digits are 25 and the first are given by the multiplication 10x11=110.The seeked number is 11025.
Wow, you're patient enough to invent those (so I talked to all of you)? Okay, back to thread: I also remember this particular principle everytime I'm dealing with multiplication: 4=5-1 6=5+1 9=10-1 After that, distributionlaw works wonders....
Not at my school! We look for a common multiple... Thanks metachristi. That is far easier way of adding fraction than the way I do it!
There's another nice thing to remember when dealing with division by seven, particularly if we are into decimal (not fraction) result: This order of number: 142857 Here's the fun: 7n / 7 = n 7n+1 / 7 = n.142857... 7n+2 / 7 = n.285714... 7n+3 / 7 = n.428571... 7n+4 / 7 = n.571428... 7n+5 / 7 = n.714285... 7n+6 / 7 = n.857142... funny, isn't it?
