Master List of Mental Calculation Techniques

You rightly noticed that this is not the fastest way in this case
In case 82 * 44 the best way is 82 * 11 * 4 or 44 * 80 + 44 * 2
But these are only examples of these methods, not the best solutions

Then, can you give the best example of this method?

Because, I still can’t understand why I have to do this so steps.

It’s very difficult to find an example. I guess this method is really useless.
What about 68 * 46?
Is 68 * 46 = 68 * (45 + 1) = 34 * 90 + 68 the fastest way?
I guess it could be half a second faster than addition/substraction methods

It will again , applying only for even numbers, right??

I am doing by this way, I don’t know this is faster or not but here it is -
68 × 46 (50-4)

3400 - 272
3128

Note : in this I have not too worry about even numbers and odd numbers.

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Since so many steps are taken, why not directly calculate it?

Flou, If you don’t mind, would you like to give some examples.
I like to read that.

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There are many creative ways to do small multiplications, but I also often calculate with obvious methods rather than complicated shortcuts.
If you calculate fast enough, it doesn’t make sense to think about the method, but sometimes there may be a slightly faster way that is more creative.
On the other hand, the brute force can be much slower than the smarter methods.

68 * 46 (50-4) is the obvious way
68 * 46 = 68 * (45 + 1) = 34 * 90 + 68 more creative and complicated way but maybe a bit faster

Another example
27 * 69 = (27 * 23) * 3 = (20 * 30 + 7 * 3) * 3 complicated way
27 * 69 = 69 * 9 * 3 or 27 * 69 = 27 * (70 - 1) obvious methods, the fastest way in this case
27 * 69 = 27 * 60 + 27 * 9 brute force, it’s the slowest way

I’m not sure what you consider brute force, but I don’t think criss cross is necessarily slower here. In any case, if you do want a shortcut…


68*{\color{gray}46}=68*({\color{red}5}0-{\color{red}5}+1)

…seems the most obvious and least complicated to me, because {\color{red}5}={10\over2}; so all you have to do is take {68\over2}=34 and then add the appropriate zeroes to 50 and 5, respectively. After that you just subtract and add 68.

Obviously, the same principle applies to 44 as well as 56 and 54; however, I’d personally only use it for 45 or 55 because by the time you’ve performed the third step you’re probably already done doing criss cross anyway.

It’s a judgement call… taking half of a number isn’t complicated, especially when the number is not odd as is the case here. The next step from 50 to 45 reuses the previously “calculated” result with the decimal point shifted. Lastly, just adding the original number isn’t a big deal either, but… do whatever you’re comfortable with. This way seems to required the least “actual calculations” to get the result.

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So, why do you consider the first one complicated? Does it have to do with seeing that {69\over3}=23 or with the resulting calculation of 27 * 23? I mean you wrote yourself, in the section about squares…

…and the same principle applies when multiplying if the tens digit is the same and the unit digits add up to 10, so not just 5+5 for squares but also 6+4, 7+3, etc… so one more than 2 is 3 for \color{blue}6\_ on the left-hand-side and then \color{blue}\_21 from multiplying the unit digits. All you’re left with then is \color{blue}621*3


So the idea is that 9=(10-1) and you just add a 0 to 69 and subtract 69 from 690? Doesn’t seem better or worse than the one above. And, just like above you now have to do \color{blue}621*3. I’d say both are the same… but I guess it’s easier to see 69=(70-1) rather than 23={69\over3}.


Alternatively, if you’re comfortable with your 2-digit squares… just solve either 27*73 or 69*31 via difference of squares and then adjust the result accordingly by subtracting either 27*4 or 69*4 depending on which of the two approach you took to get 50^2-\delta^2.

Even easier to further simplify 69*31 to 70*30 and then add the missing 69 and subtract the extra 30 to get to 69*31=2,100+69-30=2,139 . Lastly, subtract 4*69=276 so basically just 2,139-276=1,863. This last one is the fastest if done right…

Basically, use 30 instead of 27 and 70 instead of 69 to get to 2,100 (good start). Rather than adding the missing 69 just subtract three more (very smart) because you adjusted your 27 to be 30 and finally subtract the extra 30. It’s the same as above just not bothering with adding 69 because we know that we got four extra ones of those anyways from adjusting the 27 initially… untimely, it’s kinda this in your head…

27*69 = (70*30)-(3*70+3*1)-30=1,863

^that middle but is just how I do 693 in my head… and to be perfectly honest, I subtract the 30 form the last step before I even add the 3*

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I prefer this too, 69 (70 - 1) instead of 23 × 3 (well it’s my personal choice)

How much time it takes you to do this in your head??

I dunno… not long, it’s definitely the fastest way compared to the others mentioned… let’s do it together.

  1. Adjust 27 to be 30 and 69 to be 70
  2. 3x7 from the multiplication tables that you know
  3. add two zeros

…I think we’re still under 1 second…

  1. know that 69+1=70 (not really a step)
  2. recall that you adjusted 27 to be 30 (not really a step)

…does that add to the time at all?

  1. subtract 7x3 (see step 4 and 2) but add a zero first
  2. subtract the 30 from the other adjusted number
  3. add 3

…maybe under 1 second or maybe under 2? Try it yourself… all you do is \color{red}7*3=21… the first time with two zeroes… 2,100… then with one zero… 210… 2,100 - 200 = 1,900… another 10 is then 1,890. Take off 30 and add 3… you got 1,863… done.

I mean the only two numbers I see here are \color{blue}21 and \color{blue}3… ignoring that you have to shift the decimal points… but really, it’s just…

\color{blue}2,100-210-30+3

It’s harder to spot that 69 / 3 = 23 than 69 = 70 - 1

That’s why I factor 69 as 23 * 3 to have the multiplication 23 * 27, which can be done this way

Exactly

I prefer this way
27 * 73 = 27 * (100 - 27) = 2700 - 27 ^ 2 = 1971
69 * 31 = 30 * 70 + 39 = 2139

I guess between 1 and 1.5 seconds (without typing answer) for me

Is that “between 1 and 1.5 seconds” for the intermediate step of either 27x73 or 69x31… OR do you mean actually getting the answer to the final step 27x69 for the original problem?

between 1 and 1.5 seconds using my standard method 27 * 69 = 27 * 70 - 27 * 1
I think that’s faster than 27 * 73 - 27 * 4

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@Rajadodve786 I’m so sorry for the late reply, It seems that I missed your post by accident…

Umm… As Bjoern mentioned above, some general method like cross method are not necessarily slower
for task like 68*46.

In this case I think the direct one,68x46 = 68x40+68x6 are not tougher considering the effort spend to apply the shortcuts.

Thanks for your patience.