Rapid Mental Calculations - 3

To square a number ending in 5 (e.g. 85), multiply the number of tens (8) by itself plus one (8 • 9 = 72), and attach 25 (in this case it we obtain 7225).

More examples: let’s calculate 252. We multiply the number of tens 2 • (2 + 1) = 2 • 3 = 6, attach to it 25 and get 625. Let’s calculate 452. 4 • (4 + 1) = 4 • 5 = 20, we obtain 2025. Let’s calculate 1452. 14 • 15 = 210, we get 21025. This method follows from (10x + 5)2 = 100x2 + 100x + 25 = 100x (x + 1) + 25.

We will apply this method to decimals ending in 5: 8.52 = 72.25, 14.52 = 210.25, 0.352 = 0.1225, and so on. As can be seen, the fractional number is first multiplied by 10, 100 and so on (or comma is transferred to the right by the required number of digits) to make it an integer, then that number is squared, then the result is divided by 102, 1002 and so on (or comma is transferred to the left by twice the number of digits, by which it was transferred to the right previously). Since 0.5 = 1/2, 0.25 = 1/4, 0.125 = 1/8, 0.0625 = 1/16 and so on, the described technique can be used also for the squaring of numbers ending by fractions 1/2, 1/4, 1/8, 1/16 and so on.

When mentally squaring a number, it is often convenient to use the formula (a + b)2 = a2 + b2 + 2ab or formula (a-b)2 = a2 + b2-2ab. For example: 412 = 402 + 1 + 2 • 40 = 1601 + 80 = 1681, 692 = 702 + 1-2 • 70 = 4901-140 = 4761, 362 = 352 + 1 + 2 • 35 = 1226 + 70 = 1296. This method is particularly convenient for numbers ending in 1, 4, 6 and 9, as they are easily represented in the form (a + 1)2, or (a-1)2.

Suppose you want to mentally multiply 52 • 48. Imagine these factors presented in the form of (50 + 2) (50-2) and apply the formula contained in the title: (50 + 2) (50-2) = 502-22 = 2500-4 = 2496.

You can act in a similar way whenever one factor is conveniently expressed as the sum of two numbers, and the other one as the difference between the same numbers: 69 • 71 = (70-1) (70 + 1) = 702-12 = 4900-1 = 4899, 33 • 27 = (30 + 3) (30-3) = 302-32 = 900-9 = 891, 53 • 57 = (55-2) (55 + 2) = 552-22 = 3025-4 = 3021, 84 • 86 = (85-1) (85 + 1) = 852-12 = 7225-1 = 7224.

It is convenient to use the aforementioned method for calculations of the following kind: 7.5 • 6.5 = (7 + 0.5) (7-0.5) = 72-0.52 = 49-0.25 = 48.75, 11.75 • 12.25 = (12-0.25) (12 + 0.25) = 122-0.252= 144-0.0625 = 143.9375.

It is useful to remember that 37 • 3 = 111. With this in mind, it is easy to mentally multiply 37 by 6, 9, 12 and so on: 37 • 6 = 37 • (3 • 2) = 111 • 2 = 222, 37 • 9 = 37 • (3 • 3) = 111 • 3 = 333, 37 • 12 = 37 • (3 • 4) = 111 • 4 = 444, 37 • 15 = 37 • (3 • 5) = 111 • 5 = 555, and so on.

It is also useful to remember: 7 • 11 • 13 = 1001. With this in mind, it is easy to perform oral multiplying the following kind: 77 • 13 = (7 • 11) • 13 = 1001, 77 • 26 = (7 • 11) • (2 • 13) = 2002, 77 • 39 = (7 • 11 ) • (3 • 13) = 3003, and so on; 91 • 11 = (7 • 13) • 11 = 1001, 91 • 22 = (7 • 13) • (2 • 11) = 2002, 91 • 33 = (7 • 13) • (3 • 11) = 3003 and so on; 143 • 7 = (11 • 13) • 7 = 1001, 143 • 14 = (11 • 13) • (2 • 7) = 2002, 143 • 21 = (11 • 13) • (3 • 7) = 3003 and so on.


The first article of the series: Rapid Mental Calculations: Part1

Previous article of the series: Rapid Mental Calculations: Part2

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