Ug... This test just proved to me how slow at these tests. I only got to #34, but I only missed one. So that means 161 for me on that one. Anyway, I have some questions.
Thanks for the solution to the area problem. I thought the answer was 3/2 when I was taking the test; I seriously divided 216 by 144, but thought that it couldnt be right because the answer wasnt a whole number.
As for the foiling technique, I was wondering if it gets that much more complicated as the numbers get larger. I mean, I love that way of doing it because you can actually multiply number in your mind, but I'm not sure how general Loifing can be. Say you get 123 *153. What do you do then? or 27 * 84?
for 123*153: (120+3)(150+3): 120*150 = 18000 (12*15 = 180 should be automatic)
since ones digits are the same, for the outside/inside you can combine 120 and 150 to do 270*3 = 810, so its now 18810 + 3*3 = 18819. It looks difficult at first... but practice is really the key with these
if 27*84 seems hard to foil, you could double/half it to 54*42 to make it a bit more manageable. Again, with foiling practice, 27*84 should be routine
See on those area conversion problems, you know the conversion factors, and you know the number given, so most of the times you can approximate because the answer won't be that nasty.
e.g: 6912 cu.in = _____ cu.ft, you know 1 cu.ft=1728cu.in, then what is 69/17 close to, 4. So the answer is 4. Plus, if you double 1728, you get a mnemonic number -- 3456. So it's even easier to remember 2 cu.ft = 3456 cu.in.
Also, milage problems are important. It is good to have an idea of how many feet 1/3, 1/4, 1/5, 1/6, 1/8 mile is. You may see questions like 1 2/3 mile = 8800 ft.
Foiling may involve a few more steps sometimes, but it's never hard. Like 123*153:
123=100+23 and 153=100+53: the 100s'd be 100+(23+53)=176 but don't write it down, foil 23*53 first: 1000 and 3*(20+50)=210 and 3by3=9. So from 1219 you carry 12 to the 176 which give the final answer of 18819.
Or(after a few seconds, I thought of...):
123=120+3 and 153=150+3: the 100s'd be 12*15=180, and 3*(120+150)=810, then 3*3=9, therefore 18819.
Please don't look at it and be discouraged by the number of steps you see. I promise you your brain can do it <6sec. If you can't, practice until you can.
FOILing is hardcore math skill. After some practicing, I arrived at the phase that my fastest column is the 3rd column because there's in fact no as many computations since there's a shortcut to all the problems. 1st and 2nd columns have some hardcore calculations that test your basics, and the line drawn between good NSers and worse NSers are whether they can pass the bottleneck. We have to know how to FOIL fast.
27*84: honestly, my first reaction is 27*(80+4) which is probably long, but after I study the problem a few more seconds, I have a quicker way: You know that 27*4=108, double that you get 216=27*8, so 27*84=2160+108=2268. Then after a few more second, I found another approach, likely just as fast: 27*(74+10)=27*37*2+270=2*999+270=2270-2=2268.
This is what separates us from Aaron and Sam and Chris. I bet they don't need those few seconds I took to figure out what I just figured out -- their number SENSE enables them to go to the fastest method straight away.
You guys are thinking way too much into this. The easy way is to foil and that's that. You should be able to foil two by two's, two by three's, and three by three's with relative ease. The only way to do it is to PRACTICE! That's what I did and now it takes me just a matter of seconds to do even the most complicated looking numbers. There are some that are easier if you double and half, but mostly only if you can double something to make a multiple of 10 such as 35 x 48 = 70 x 24 = 24 x 7 then add a 0 to the end. and there are other special cercimstances but if you don't know what they are foiling is always an option. and remember PRACTICE!
Thank you very much, that's the message I've been trying to deliver.
In many cases though, foiling is not just foiling. Look at the 123*153, there are 2 ways to foil it, but obviously the second way is better. However, it's only so if you remember 12*15=180(thus no time wasted on that step). And I didn't deliberately memorize it, they come after you've done enough problems.
That's it (or at least how I do it, going on 7 years I don't think I'll change). It's not too hard, just keep practicing with different numbers and different sizes, 2x3 or 3x4, you'll get better and quicker,and practice is the only way. Some people think the only way to get better at NS is to practice test but you actually have to understand what is going on and disect the tests to find quicker ways to do it. It helps to discuss with teammates, friends, co-competitors whomever to see how they do it, even if you don't change your way of doing it at least it might bring it a specific example that you might see later. I don't know, that's just what my team has done and we have been pretty successful with that, if any of us had questions we weren't afraid to ask.
Well, I like your way of foiling because it can be done without paper. That, in my opinion, is the most powerful kind of Number Sense that a competitor can develop.
The easy way is to foil and that's that. You should be able to foil two by two's, two by three's, and three by three's with relative ease. The only way to do it is to PRACTICE! That's what I did and now it takes me just a matter of seconds to do even the most complicated looking numbers. There are some that are easier if you double and half, but mostly only if you can double something to make a multiple of 10 such as 35 x 48 = 70 x 24 = 24 x 7 then add a 0 to the end.
and there are other special cercimstances but if you don't know what they are foiling is always an option. and remember PRACTICE!
