Although mental calculation is not a life-saving skill (and is crucial in only very few professions), it is one of those skills we definitely need from time to time throughout our lives. In addition, it keeps us mentally fit, so it is well worth investing some time to learn the basic tricks and practice them wherever we can.

For that reason, I recently bought the following book (written by the presenter in this interesting video ^[1]):

It is a good book that is well written and contains useful techniques for mental calculation. As with most books, the core ideas can be listed on a page or two (reading the entire book isn’t necessary), and that’s what I’d like to do with this article.

The book also contains more sophisticated calculations (e.g., multiplying two five-digit numbers), but these skills usually are not required in everyday life.

Addition

For me, the main eye-opener in this book is the advice to add numbers “from left to right” instead of the other way around, as I was taught in school.

For example:

   723 
+ 259 

I used to add the last two digits first (9 + 3 = 12), remembering the “1” (12) and add it to the next level of digits (5 + 2 + 1). Now I know it is much easier to start from the left: add (7 + 2) first and work your way toward the right (the result is 982). With this technique, you also have the advantage of knowing quickly in what range the final number will be (7 + 2 is nine hundred….).

Try it yourself:

        274              329             538             732              321  
    + 125           + 428          + 417          + 321          + 972 
 

Subtraction

Subtraction, too, is much easier to calculate from left to right (use this as your default approach).

For example:

  47
- 23

… is 24 (4 – 2 = 2; 7 – 3 = 4).

However, it becomes tricky if the number you have to subtract from is smaller than the number you are subtracting.

For example:

  87
- 38

Here it would be easier to apply the following approach: round the number you are subtracting (38) to the nearest multiple of 10, which in this case is 40. Subtract this number from 87, which is 47 (easy). Then add 2 to this number (because you’ve rounded 38 to 40, with a difference of two) and you get the final result of 49.

Try it yourself:

  45           44           84           33           334
- 23        - 34         – 49         – 22        - 233 

Multiplication

For multiplication, you must know the multiplication table by heart:

As with addition and subtraction, you also multiply from left to right. For example:

  42
x   8

You first calculate [4 x 8] = 32. Because the “4” is in the tens position, you have 320. Then you multiply [2 times 8] = 16 and add it to 320. The result is 336.

As described above, in some cases it makes sense to use a rounding technique, especially if the number is not far from a multiple of 10. For example:

  78
x   9

You could use the standard approach (“addition-technique”) by first calculating 7 x 9 = 63, which would be 630, and then calculating 8 x 9 = 72 and adding these two numbers, getting the final result of 630 + 72 = 702. However, in this case it may be easier to round 9 to 10. That is, multiply 78 by 10 (=780) and then subtract (1 x 78), getting the same result of 702. When in doubt, however, use the addition technique, because selecting the technique to use may take longer than simply performing the calculation via the addition technique.

Now it’s your turn: 

   23           35           55           66           89
x    4        x   8         x   4         x  6          x  9

Division

How would you calculate the following problem without using paper?

 182
/     8

The easiest way seems to be the following. First multiply the latter number (8) by a multiple of 10 and see how close you get to the number above. So in the example above, I first calculate 8*10 and get 80. This is too low, so I multiply 8 by 20 instead and get 160, and then I already know the result will be in the 20-30 range (this can be calculated very quickly). The remainder is 22 (182-160); applying the same approach as before, the 8 “fits” twice in the 22, so I know the result will have to be 22… and something. The last part can be expressed as a fraction, with the remainder (22-16 = 6) as the numerator and the “8” as the denominator. In this case, the fraction is 6/8 or ¾. So the final result is 22 ¾.

The same principle applies for dividing a number by a two-digit number (e.g., 624 / 12), although of course it is more complex. Sometimes it may help to further simplify the calculations by breaking down the latter number into components. That is, you can split up the 12 into 6 and 2 and calculate 624 / 6 first (= 104), and then divide it by 2, getting the final result of 52.

Try it out: 

    34         454        122        442        984
  /   4         /    9        /   3        /  12       /   20

Other Advice in the Book

The book contains some other useful advice:

• If you need to calculate 25% of something, it is easier to halve the number twice. For example, 25% of $30 is calculated as 30/2 = 15, and 15/2 = 7.5.
• In most cases, an approximation is fine and we don’t need the exact result. So instead of calculating 8,261 + 4,754, round both numbers (e.g., to 8,000 and 5,000) and add them.
• You can do the same thing to estimate the price of your purchases in the supermarket. Round the prices for individual items to the next half-dollar (i.e., $1.20 becomes $1.00, $4.71 becomes $4.50, etc.) and add those numbers instead. Most of the time your result will be very close to the real sum.

I hope these techniques will prove useful to you!

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