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
 

2. 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

 

3. 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

 

4. 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

 

5. 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

 

6. 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

 

7. 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

8. 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

 

9. 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

 

10. 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

 

11. 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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