Be quick to make both mental calculations and calculations using just pen and paper is very useful when you take a math exam and also exams of other scientific subjects, especially if timing is important, such as in GCSEs and in tests in general. In fact:

- there are some specific exercises in Mathematics GCSEs that require you to estimate some values without a calculator;
- even in tests in which calculators are allowed, you can save time per- forming part of the calculations in mind (yes! thinking is often quicker than typing a number, especially in mathematical tests);
- to have at least a rough idea of the result before displaying it on the screen of your
calculator screen acts as a rough check to avoid typing errors.An anecdote tells that in
primary school, after the young Gauss misbehaved, his teacher asked him to sum all the
integer numbers from 1 to 100, thinking to keep him busy for a long time, but he
surprised him giving the correct answer within seconds. Gauss’s presumed method was to
realize that the pairwise addition of terms from opposite ends of the list of numbers
yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, … and so
on, for a total sum of 50·101 = 5050. Now we know how to deal with arithmetic
progressions, but in those days no. Note that to perform that sum with a calculator you
have to press 292 buttons, which would take 2 minutes and a half, if you typed 2 buttons
per second without making mistakes, that is a very optimistic assumption.Another example
(no anecdote this time) is the following: “If you want to calculate 312 − 292, instead
of calculate the two squares, it is much simpler if you use the equivalence 312 −292 =
(31+29)·(31−29) = 60·2 = 120. If you have a well-trained mind, you will spend less time
doing so rather than typing 10 buttons in your calculator!
Here I list some of the most famous tricks with examples to show their efficiency. (Do not have to read them all at once)

1. To add a number to another one, or to subtract a number from another one, sometimes it is easier to round a number to the nearest ten, then add or subtract to the result the complement to nearest ten of the rounded number.

Ex:

25 + 9 = 25 + (10 − 1) = (25 + 10) − 1 = 35 − 1 = 34

191 − 37 = 191 − (40 −
3) = (191 − 40) + 3 = 151 + 3 = 154 29 + 15 = (30 − 1) + 15 = (30 + 15) − 1 = 45 − 1
= 44

- To multiply a long number by 2 or 3 or even larger one-digit numbers, first
try to calculate the product digit by digit; if it does not work, i.e. you
obtain a two-digit numbers (except if it is on the first from the left), you
can consider block of digits. It works if the products of each block have
the same number of digits of the original block.Ex:

3241 · 2 = (6)(4)(8)(2) = 6482

721 · 3 = (21)(6)(3) = 2163

31520·2 = (3)(2)(10)(4)(0) it is not ok. Better: 31520·2 = (3)(15)(20)· 2 = (6)(30)(40) = 63040

1107 · 7 = (1)(1)(07) · 7 = (7)(7)(49) = 7749 - To multiply or divide a number by 2 or 3 or even larger one-digit numbers,
you can write the first number as a sum or difference of other well chosen
numbers, then perform the two multiplications or divisions, and sum or
subtract their results.Ex:

201/3 = (180 + 21)/3 = (180/3) + (21/3) = 60 + 7 = 67 201/3 = (210 − 9)/3 = (210/3) − (9/3) = 70 − 3 = 67 - To divide a long number by 2 or 3 or even larger one-digit numbers, try to divide that number in blocks. Add zeroes on the left of the result of a block, except the first one on the left, to have results with the same number of digit of the original blocks.1527201/3 = (15)(27)(201)/3 = (5)(09)(067) = 509067
- To multiply a number by 5, first multiply that number by 10, then divide it
by 2.

Ex:

24 · 5 = 24 · 10/2 = 240/2 = 12037 · 5 = 37 · 10/2 = 370/2 = 185 - To multiply a number by 9, first multiply that number by 10, then subtract
the original number from this result.

Ex:

468 · 9 = 4680 − 468 = 4212 - To multiply a number by 11, first multiply that number by 10, then add the original number to this result.2

Ex:

124 · 11 = 1240 + 124 = 1364

- To multiply or divide a number by 4, multiply or divide that number by 2
twice.Ex:

314 · 4 = (314 · 2) · 2 = 628 · 2 = 1256 - To multiply a number by 3, first multiply that number by 2, then add the
original number to this result.

Ex:

314 · 3 = (314 · 2) + 314 = 628 + 314 = 942 - To multiply a number by 25, first multiply that number by 100, then divide
it by 4.Ex:

48 · 25 = (48 · 100)/4 = 4800/4 = 1200 - To divide a number by 25, first multiply that number by 4, then divide it by
100.Ex:

31/25 = (31 · 4)/100 = 124/100 = 1.24

Note that often you should use more than one trick in a calculation.
Those are just a few of the possible tricks; you can also invent new tricks when
you need. At first glance, maybe you do not see great advantages in using them,
but I am sure you will change idea after a bit of practice. You can become
confident with certain particular operations if you associate numbers with
something else, for example you know that a football match is made up two times
of 45 minutes each, for a total of 90 minutes, that there are 24 hours in a day
and 12 is midday, etc.

To conclude, I want to show you two complete reasonings to solve two exer- cises from GCSEs. They are quite long and technical to be read as if they were tales. I suggest you to take pen and paper, then read slowly, stopping from time to time when you do not understand a passage and to try to re- peat the passages without reading them. Important: Do not memorize as if it were a poem!

How to estimate

207 · 148 49

3

Firstly note that:

148 ∼ 150 = 3 49 50

Butif50·3=150,then49·3=(50−1)·3=150−3=147. So: 148=147+ 1 =3+ 1

Again:

49 49 49 49

207 · 148

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