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May 23, 2012

Two digit mental multiplication tips

Filed under: mathematics — ckrao @ 12:58 pm

Here I have put down all the methods I know to multiply two-digit numbers, many of which can be used to do so mentally. Some are easier than others but all take at least some practice to master.

Firstly, most of the answers have four digits, and the calculations often involve adding or subtracting from these. One way I do this is to imagine the four-digit number as two two-digit numbers. For example I say the number 8723 as “eighty-seven twenty-three”. Then if I want to add say 315 to this, I add 3 to 87 and 15 to 23. If one is comfortable with two-digit addition or subtraction, then this makes things easier.

When subtracting it is beneficial to know the “complement” of a number n, meaning 10^d - n where d is the number of digits of n. (While not a formal term, it is used for convenience here.) For example, the complement of 34 is 66 and the complement of 330 is 670. This is simply 9 minus each digit, or 10 minus the last non-zero digit and any zero-digits following. Then a-b is equivalent to adding a -10^d and the complement of b. This is helpful if there is carry. For example, to do 1234 - 456, find the complement of 56 as 44, then do 12 – 4 – 1 (the one represents carry) for the hundreds digit and 34 + 44 = 78 for the remainder, to arrive at 778.

If doubling a three- or four-digit number it helps to treat the number as convenient groups of digits. For example to double 4567, I would double 67 (134), then add the carry of 1 to twice 45 to obtain 9134.

Finally, if I use the notation a | b, it means either 100a + b or 10a + b depending on the context (hopefully clear). For example a square of a number 50 + a has the form (25 + a) | a^2, while a multiple of 11 has the form a | (a + b) | b.

Squaring two digit numbers

  • Numbers ending in 0: (10a)^2 = a^2 | 00
    e.g. 40^2 = 1600
  • Numbers ending in 5: (10a + 5)^2 = 100a(a+1) + 25 = a(a+1) | 25
    e.g. 45^2 = (4 \times 5)|25 = 2025
  • Numbers close to 50: (50+a)^2 = (25 + a) | a^2
    e.g. 47^2 = (25 - 3) | 3^2 = 2209
  • Numbers close to 25: (25 + a)^2 = (6 + a/2) | a^2+25
    e.g. 28^2 = (6 + 3/2) | (3^2 + 25) = 784
  • Numbers close to 75: (75 + a)^2 = (56 + 3a/2) | a^2 + 25
    e.g. 67^2 = (56 - 3\times 8/2) | (8^2 + 25) = 4489
  • Numbers ending in 1 or 9: (10a \pm 1)^2 = a^2 \pm 20a + 1
    e.g. 61^2 = 3600 + 120 + 1 = 3721
  • Any number that looks tough: use a^2 = (a-b)(a+b) + b^2 for b that makes a-b or a+b easy to work with (usually the number rounded to a multiple of 5).
    e.g. 63^2 = 60 \times 66 + 3^2 = 3969
  • Alternative: use (a+b)^2 = a^2 + 2ab + b^2 where b is small and a^2 is easy to find (a multiple of 5).
    e.g. 37^2 = (35 + 2)^2 = 1225 + 140 + 4 = 1369

High school method (distributive law)

  • (10a + b) x (10c + d) = (10a + b)d + 10(10a + b)c = 100ac + 10bc + 10ad + bd
    e.g. 34 x 87 = 2400 + 240 + 210 + 28 = 2958

This is more or less what is taught in many schools, but the products  34 x 7 and 34 x 80 are usually written one above the other with their sum written below.

The difficulty with this method for mental calculation is keeping track of so many products in one’s head. This can be simplified by use of the following variants.

Butterfly or criss-cross method

  • (10a + b)(10c +d) = 100ac + 10(ad + bc) + bd

Here I find it easiest to work out ad + bc (the criss-cross part) first, multiply by 10, add this to bd, then add to 100ac at the end. It helps to interpret the two numbers as four digits in this method.

e.g. to find 86 x 67 I work out 6 x 6 = 36 and 8 x 7 = 56. Adding these gives 92. Then add 10 times this (920) to  6 x 7 to obtain 962. Finally add this to 8 x 6 x 100 = 4800 to end with 5762.

  • Special case: if a = c and b + d = 10, this becomes a(a + 1) | bd
    e.g. 47 x 43 = 2021

I find the next three methods to be among the most common I use.

Work out a similar, simpler product, then adjust for the difference

  • Use a(b+c) = ab + ac where ab and ac are easier to find than a(b+c).
    e.g.  35 x 33  = 35 x 35 – 35 x 2 = 1225 – 70 = 1155

One may often choose b to be a round number and c to be small in magnitude.

This method works especially well when one of the numbers is close to a multiple of 10 (especially 20 or 50).

Anchor method

This term comes from [1] and it makes use of the following identity.

  • (a + b)(a + c) = a(a + b +c) + bc

Here we choose a to be the anchor (a nice number), decrease one number by b, increase the other by b, then add bc, where c is the difference between the number to be increased and the anchor.

This works beautifully when the two numbers are close together. For example 54 x 52 = 50 x 56 + 2 x 4 = 2808. If the anchor is between the two numbers, the bc term is negative.

e.g. 82 x 77 = 80 x 79 – 2 x 3 = 6400 – 80 – 6 = 6314.

Using difference of perfect squares

  • ab = \left(\frac{a + b}{2}\right)^2 - \left(\frac{a - b}{2}\right)^2

This is especially nice if one is comfortable with evaluating squares (see above) and the two numbers differ by an even number, better still that difference is small or if they add to a multiple of 10.

e.g. 33 x 47 = 40 x 40 – 7 x 7 = 1600 – 49 = 1551.

32 x 38 = 35 x 35 – 3 x 3 = 1225 – 9 = 1216.

Note that this is a special case of the anchor method, where c = -b

If the two numbers differ by an odd number, we can make a slight adjustment to the above identity.

  • ab = \left(\frac{a + b - 1}{2}\right)\left(\frac{a + b +1}{2}\right) - \left(\frac{a - b - 1}{2}\right)\left(\frac{a - b + 1}{2}\right)

e.g. 34 x 47 = 40 x 41 – 6 x 7 = 1640 – 42 = 1598

Multiples of 5

  • To multiply by 5, halve it, then multiply by 10.
  • To multiply by 10, append a 0.
  • To multiply by 15, first multiply by 10 then add half the result.
  • To multiply by 20, double it then append a 0.
  • To multiply by 50, halve it, then multiply by 100.
  • To multiply by 25, take a quarter of it, then multiply by 100.
  • To multiply by 75, subtract a quarter off it, then multiply by 100.
  • Also potentially helpful is the identity (10x+5)(10y+5) = 100xy + 50(x+y) + 25 = 100(xy + (x+y)/2) + 25, but it is often easier to use difference of squares
    e.g. 45 x 75 = (4×7 + (4+7)/2) | 25 = 3350 + 25 = 3375 or 45 x 75 = 60^2 – 15^2 = 3600 – 225 = 3375.

Convenient products

Most of the following products are close to multiples of 100 and hence may prove to be useful stepping stones when multiplying.

12 x 9 = 108
13 x 8 = 104
15 x 7 = 105
17 x 6 = 102, 17 x 47 = 899, 17 x 53 = 901
21 x 19 = 399
23 x 9 = 207, 23 x 13 = 299
29 x 7 = 203
31 x 13 = 403
33 x 3 = 99
34 x 3 = 102
37 x 3 = 111, 37 x 19 = 703, 37 x 27 = 999
38 x 8 = 304
39 x 23 = 897
41 x 17 = 697, 41 x 22 = 902
43 x 7 = 301, 43 x 31 = 1333
45 x 9 = 405
47 x 17 = 799
53 x 17 = 901
54 x 13 = 702
56 x 9 = 504
59 x 17 = 1003
63 x 8 = 504
64 x 11 = 704
67 x 3 = 201
68 x 3 = 204
69 x 29 = 2001
72 x 7 = 504
73 x 11 = 803
76 x 4 = 304
77 x 13 = 1001
78 x 9 = 702
84 x 6 = 504
87 x 23 = 2001
88 x 8 = 704
89 x 9 = 801
91 x 11 = 1001
93 x 43 = 3999

Use of multiples

It is often helpful to break the numbers up into their factors and regroup them, or perform single-digit multiplication.

For example, multiplying an even number by a multiple of 5 is easy because 5x \times 2y = 10xy.

e.g. 15 x 16 = 3 x 5 x 2 x 8 = 3 x 10 x 8 = 240

Here are a couple of other examples:

42 x 63 = 2 x 21 x 21 x 3 = 6 x 441 = 6 x 450 – 6 x 9 = 2700 – 54 = 2646

38 x 21 = 2 x 19 x 21 = 2 x 399 = 798

77 x 39 = 7 x 11 x 13 x 3 = 1001 x 3 = 3003

These last two examples use the “convenient products” shown above.

How about multiplying primes? If they end in 1 or 9, one can multiply by the closest multiple of 10 and adjust.

e.g. 19 x 53 = 20 x 53 – 53 = 1060 – 53 = 1007

If they end in 3 or 7, one can often resort to difference of square methods.

e.g. 37 x 83 = 60^2 – 23^2 = 3600 – 529 = 3071

A few more specific tricks:

  • 13x: use 13 x 8 = 104
    e.g. 13 x 28 = 13 x 8 x 3.5 = 104 x 3.5 = 412 + 52 = 464
  • 17x: use 17 x 6 = 102
    e.g. 17 x 43 = 17 x 42 + 17 = 17 x 6 x 7 + 17 = 102 x 7 + 17 = 714 + 17 = 731
  • 23x: multiply by 25, then subtract double the number
    e.g. 23 x 38 = 25 x 38 – 2 x 38 = 950 – 76 = 874
  • 37x: use 37 x 3 = 111
    e.g. 37 x 59 = 37 x 60 – 37 = 37 x 3 x 20 – 37 = 111 x 20 – 37 = 2220 – 37 = 2183
  • 43x: use 43 x 7 = 301
    e.g. 43 x 66 = 43 x 7 x 9 + 43 x 3 = 301 x 9 + 129 = 2709 + 129 = 2838
  • 47x, 53x: adjust from 50x
    e.g. 53 x 64 = 50 x 64 + 3 x 64 = 3200 + 192 = 3392
  • 67x: use 67 x 3 = 201
    e.g. 67 x 78 = 67 x 3 x 26 = 201 x 26 = 5226
  • 73x: use 73 x 7 = 511
    e.g. 73 x 43 = 73 x 42 + 73 = 73 x 7 x 6 + 73 = 511 x 6 + 73 = 3066 + 73 = 3139
  • 83x: use 83 x 6 = 498
    e.g. 83 x 18 = 83 x 6 x 3 = 498 x 3 = 1500 – 6 = 1494
  • 97x: use (100 – 3)x
    e.g. 97 x 28 = 100 x 28 – 3 x 28 = 2800 – 84 = 2716

Other products

  • To multiply by any multiple of 9, multiply first by its nearest multiple of 10, then subtract 10% of that number – i.e. 9x = 10x – x
    e.g. 27 x 53 = 30 x 53 – 3 x 53 = 1590 – 159 = 1431
  • To multiply by 11, add adjacent digits, taking carry into account (ab x 11 = a | a+b | b)
    e.g. 45 x 11 = 495 since 9 = 4 + 5, 46 x 11 = 506 since 10 = 4 + 6
  • To multiply by 24 or 26, multiply by 25 first, then adjust
    e.g. 26 x 64 = 25 x 64 + 64 = 1664
    Multiplying 26 by a multiple of 4 turns out to be very easy since the last two digits are always the number itself!
  • To multiply by 33, take a third of the number, multiply by 100 (ignoring anything after the decimal), then subtract the first two digits.
    e.g. 33 x 59 = 1966 – 19 = 1947, 33 x 87 = 2900 – 29 = 2871
  • To multiply by 49 or 51, multiply by 50 first, then adjust
    e.g. 51 x 66 = 50 x 66 + 66 = 3366
    Multiplying 51 by a multiple of 2 is very easy since the last two digits are the number itself and the first two digits are half that number!
  • If both numbers are close to 100, use (100 – a)(100 – b) = (100 – a – b) | ab
    e.g. 98 x 97 = 9556
  • If both numbers are close to 50, use (50 + a)(50 + b) = (25 + (a + b)/2) | ab
    e.g. 53 x 54 = 2850 + 12 = 2862
  • If one number is close to 50, the other close to 100, use (50 + a)(100 – b) = (50 + a – b/2) | -ab
    e.g. 48  x 97 = (50 – 2 – 3/2) | 06 = 4656
  • If the numbers have the same unit digit and their tens digit adds to 10, then (10a + b)(10c + b) = ac + b | b^2 when a + c = 10
    e.g. 36 x 76 = (21 + 6) | 36 = 2736

Multiplying by smallish numbers

  • Both numbers having tens digit 1: (10 + b)(10  + d) = 10(10 + b + d) + bd
    e.g. 14 x 13 = 10(14 + 3) + 4 x 3 = 182
  • One of numbers having tens digit 1: (10 + b)(10c + d) = 10(bc + 10c + d) + bd
    e.g. 13 x 64 = 10(3 x 6 + 64) + 3 x 4 = 832
  • One of numbers having tens digit 2: (20 + b)(10c + d) = 10(bc + 2(10c + d))  + bd (i.e. double the larger number)
    e.g. 23 x 64 = 10(3 x 6 + 128) + 3 x 4 = 1472

A number times its mirror reflection

  • If we wish to multiply (10a + b) by (10b + a) their product can be written as 101ab + 10(a^2 + b^2).

It is easy to multiply a one or two digit number by 101 making this method appealing.

e.g. to find 47 x 74 we first find 4^2 + 7^2 = 65. Multiply this by 10 to get 650. We then add this to 2828 (101 x 4 x 7) to arrive at 3478.

Another example: 68 x 86 = 4848 + 100 x 10 = 5848. Easy!

This method only becomes tricky where there is a possibility of carry between the tens and hundreds place. For example,

98 x 89 = 7272 + 1450 = 8722.

This carry occurs in the following cases:  25 x 52, 58 x 85, 88 x 88, 49 x 94, 69 x 96, 59 x 95, 89 x 98, 99 x 99. In most instances one of the digits is 8 or 9.

In these cases it may be easier to resort to other methods described above.

e.g. 98 x 89 = (89 – 2)100 + 2 x 11 = 8922, 58 x 85 = 29 x 170 = (510 – 17)10 = 4930

One could try similarly expanding (10a + b)(10b - a) = 99ab + 10(b^2 - a^2), but this is a harder form to recognise (e.g. 37 x 67 has this form).

 

Finally, there are methods that may require a combination of the above.
e.g. 58 \times 27 = 2 \times 29 \times 27 = 2 \times \left(28^2 - 1\right) = 2 \times 783 =1566

The beauty of two-digit multiplication is that there are many ways in which each problem can be attempted, giving one ways to check the answer. Don’t forget to do a quick sanity check of your answer by using rounded estimates and by checking if the last digit is correct.

References

[1] Ron Doerfler, Dead Reckonings » Lightning Calculators II: The Methods

[2]  Bill Handley “Speed Mathematics: Secret Skills for Quick Calculation”, John Wiley & Sons, 2012.

[3] A. Benjamin and M. Shermer, “Secrets of Mental Math: the Mathemagician’s guide to Lightning Calculation and Amazing Math Tricks“, Three Rivers Press, 2006.

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