With most of calculations nowadays done with the aid of a calculator (or another computer tool, such as excel), there is little surprise in the fact that student’s ability to calculate by heart has deteriorated immensely during the last two decades. Many claim that studying the basic (e.g. the Table of Multiplication) is, in fact, unnecessary waste of time… Well, there are also those, who say that teaching writing should be abandoned as well – since the majority of communication is done with the aid of the computerized keyboard – or some sort of cell phone touch screen / sms-friendly keypad layout. Luckily, those groups are still a minority – as the basics should never be neglected. Ignoring the principles of studying will ruin the most important thing of all – UNDERSTANDING.
One of the aspects of standardized tests (such as SAT, TOEFFL, GMAT etc.) is the necessity to make all the calculations involved either by heart or with the aid of pen and paper. This requirement has raised several questions over the years – why is this so important? After all, there is little chance that we won’t have some sort of a computer at our disposal when required to make certain calculation.
This is certainly true. But it is also true that all the calculations required at the standardized tests can be easily performed without any “computerized aid”. Moreover, with certain training (and not a big amount of it) – you will actually get the result in your head faster than typing the numbers on the keyboard. All you need is to memorize the table of multiplication (which is, in fact, a very basic skill) and learn some basic techniques (those that students call “tricks”). Here is an example:
There are in fact several ways to get the result very quickly.
a) 19×11 is 19×10 + 19×1, which is 190+19 = 209.
b) 19×11 is 11×19, which is 11×10 + 11×9, which is 110+99 = 209. Even quicker is:
c) 19×11 is 11×19, which is 11×20 – 11×1, which is 220-11 = 209.
d) 19×11 is (15+4) x (15-4), which is 152-42, which is 225-16 = 209.
All those are very basic techniques that we actually use in our everyday life. What we have to look for in calculating is the round numbers, which are very easy to operate with. This and basic of Binomial Theorem will help you to solve most multiplications relatively fats.
Let’s remember the part of the binomial theorem that is of importance to us.
The general Nth grade is not really important for now, so we should remember the three basic formulas:
(a+b)2 = a2+2ab+b2
(a – b)2 = a2-2ab+b2
a2 − b2 = (a + b)(a − b)
You should be familiar with them, but your usage, probably, was limited to one direction (the one shown). However, as you know, in GMAT, GRE and such you are expected to think differently. The questions posed are different than those you are accustomed to. Thus, reverse direction is something that should help you.
Another example: How much is 562?
Well, 562 is (60-4)2, which is 602+42-2x60x4, which is 3600+16-480 = 3136
Another way is: 562 is (55+1)2, which is 552 + 12 + 2x55x1 = 3025+1+110 = 3136. Did you type it on your calculator yet?
But wait – what about this 552 result. How exactly one should get the 3025 immediately?
Well, there is a little secret, regarding the squares of the number that end with 5. To get this, you should hide the 5, multiply the remainder by the following number and write 25 in the end. For example:
952 = [9×10]25 = 9025.
752 = [7×8]25 = 5625
Now let’s learn another small trick
1452 = [14×15]25 = [14×10+14×5]25.
Of course, when multiplying by fifteen, we can multiply by ten and then add half of the result to itself:
[14×10+14×5]25 = [14×10+14×10/2]25 = [140+70]25 = 21025.
Let’s get back to the 562 question.
There is another way:
562 is approximately the same as 60×52, right? Well, approximately is not good enough (although APPROXIMATION can do us a lot of good during the standardized test, as it might eliminate some obscure answers.). However, according to the formulas above:
60×52 = (56=4)x(55-4), which is, of course, 562-42. This means that
60×52 = 562-16. Thus, 562 is 60×52+16 = 3120+16 = 3136.
This trick is especially “powerful” when round numbers are near:
1032 = 106×100+9 = 10609
982 = 96×100+4 = 9604 (or (100-2)2 = 1002+22-100x2x2 = 10000+4-400 = 9604)
So, in fact, calculating by heart can be done pretty fast. When preparing for a standardized test (or even just for fun) – try to make everyday computations without using your Excel – and you will see your abilities improve immensely. Not only you are now more ready for the exam – I am sure you brain is working slightly better now.