Numbers - Any help would be appreciated
Hi guys,
can someone please explain an easy way to work out:
1) 99 squared
2) the square root of 100,000?
A former uni peer briefly explained it in some pictorial manner where you form a mental square in your head and it seemed really simple - so much so that I have completely forgotten everything he told me about it.
Any help would be greatly appreciated.
Thanks!
Well 99 squared is easy, it is 199 less than 100 squared, or 9801.
And the only way I can think of square root of 100,000 is to think of the square root of 10, and then multiply your answer by 100, since 10 is 10,000 times less then 100,000.
So you you know 3.2 is too much and 3.1 is too little (simple math) so you get around 3.15-6, I'm sure someone has a better way to solve this.
99 squared can be broken up into:
99 1 from 100 multiplied by 99 1 from 100
this you get
99 1 99 1
1 x 1 = 01 so your last two numbers are 01 99 - 1 = 98 so first two are 98
so your answer is 9801
This works for all numbers close to 100, i.e
97 x 95
97 3 95 5
Last two numbers are 15 (3x5) first two numbers are 92 (97-5)
answer is 9215
uh... 99^2= 99*99
so just do 99*100=9900 and subtract 99=9801
Thats another quick way to go about it, but the method coffeebateman presented works in particulary well for numbers 80-100. This would be my strategy if I were to be presented something like 83 x 93 in test/interview, takes about 10-15 secs mentally.
For squaring numbers: x^2 = (x+a)(x-a) + (a^2) choose a convinient (x+a) or (x-a) such that it is a multiple of 10 or 100... example: 88^2 = (100)(76) + 144 = 7744
For square roots: I use the algorithm here to calculate roots quickly. Need to scroll down to the section title "Finding square roots using an algorithm".
This would be my approach:
For 99 ^2
(100-1)^2 = 100^2 -200 + 1 = 10000-199 = 9801
For sqrt of 100,000
I would factor the 100,000 as simple as possible .
i.e Sqrt 100000 = sqrt (100 x 1000) = sqrt (100 x 100 x10) =sqrt (100) x sqrt (100) x sqrt (10) = 10 x 10 x sqrt(10)
Now to find sqrt of 10, well you now 3^2 is 9 and 4^2 is 16. So number is b/w 3 and 4. My trick here is to divide the closest square to 10 by 10. That is 10 divide by 3 is 3 and 1/3 (0.33). Now to find an approximate answer to sqrt of 10 you would:
(3 + 3.33)/2 = 6.33/2 = 3.166 = 3.17
Hence sqrt of 100000 = 10 x 10 x 3.17 = 100 x 3.17 = 317 (approximately)
Mind you this can be done in around 20 seconds in your head if you have enough practice.
Square roots of 1-10 and squares of 1-30 or so are useful things to remember by heart
Hey I got an off topic question for some of those whom have experience with quant/trading interviews,
For questions such as 78 x 54 , 99^2 etc...how long does the interviewer generally expect for you to spend on the question mentally. Like for 78 x 54, do they expect you to do it in like 10 secs?
Let's see...78*54 you should be able to do pretty quickly by breaking it down into 4 simple multiplies and 3 simple add operations, so I would say 10-15seconds sounds fair. 99^2 using the trick I talked about earlier, probably 2-3seconds.
They key in such problems, however, is to not think about time. This will cause you to get pressurized, and inevitably make mistakes. Just be calm, and no matter how much time it ends up taking, be thorough in your calculation. Also, say out loud the calculations you're doing in your head...Interviewers like hearing your thoughts.
Anyways, If you're really into metal math, you probably should give this a read: http://www.scribd.com/doc/4108845/Trachtenberg-speed-system-of-math
While fast mental math isn't required for most jobs, I tend to use it as a benchmark of intelligence. It's not about just memorizing multiplication shortcuts, its about being comfortable with numbers and, indirectly, "logic."
I would expect 78*54 to be 10-15 seconds like unqwertyfied said. The way I look at it:
7850 = 3900 480= 320 320 - 8 = 312 312 + 3900 = 4212
Thanks for the insight guys.
Yeah I have looked at the Trachtenberg system but I think his methods are more practical for larger numbers. For relatively simpler stuff I have been using:
http://www.scribd.com/doc/14078410/Secrets-of-Mental-Math
as we as some vedic maths. At the moment I am just trying to familiarize myself with the most effective strategy because in alot of the barrier tests such as Optiver's 80 q's in 8 minutes you haven't got the time you really think through numbers, you have hit the ground running.
What's your take on Vedic math? Have you actually found it helpful?
To an extent, Yes. I think it is very effective for multiplication because the method is relatively simple and can be applied for a range of numbers.But if your dealing with big numbers, best to see the Tractenberg system.
There are definitely some very nifty tricks with vedic, not necessarily logical but if you can learn it you will be amazed at how quick you can calculate some problems. I myself have read Benjamins book on mental maths and skimmed through some vedic maths and I have used a combination of both because I feel both have their strengths and weakness'. But ultimately it comes down to the person and what works for them!
For vedic maths try this book;
http://books.google.com.au/books?id=SPHX0rdxhV0C&pg=PA6&dq=vedic+math+s…
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