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 canât understand why I have to do this so steps.

Itâs 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 donât know this is faster or not but here it is -

68 Ă 46 (50-4)

3400 - 272

3128

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

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

Flou, If you donât mind, would you like to give some examples.

I like to read that.

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 doesnât 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, itâs the slowest way

Iâm not sure what you consider *brute force*, but I donât think criss cross is necessarily slower here. In any case, if you do want a shortcutâŚ

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

âŚseems the * most obvious* and

*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*

**least complicated****add 68**.

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

Itâs a judgement callâŚ taking half of a number isnât complicated, especially when the number is not odd as is the case here. The next step from 50 to 45 reuses the previously âcalculatedâ result with the decimal point shifted. Lastly, just adding the original number isnât a big deal either, butâŚ do whatever youâre comfortable with. This way seems to required the least âactual calculationsâ to get the result.

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 squaresâŚ

âŚand 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**, etcâŚ 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 youâre 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? Doesnât seem better or worse than the one above. And, just like above you now have to do \color{blue}621*3. Iâd say both are the sameâŚ but I guess itâs easier to see 69=(70-1) rather than 23={69\over3}.

Alternatively, if youâre comfortable with your 2-digit squaresâŚ 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 rightâŚ

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. Itâs 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 initiallyâŚ untimely, itâs kinda this in your headâŚ

*^that middle but is just how I do 69*3 in my headâŚ and to be perfectly honest, I subtract the 30 form the last step before I even add the 3*

I prefer this too, 69 (70 - 1) instead of 23 Ă 3 (well itâs my personal choice)

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

I dunnoâŚ not long, itâs definitely the fastest way compared to the others mentionedâŚ letâs do it together.

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

âŚI think weâre still under 1 secondâŚ

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

âŚdoes that add to the time at all?

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

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

I mean the only two numbers I see here are \color{blue}21 and \color{blue}3âŚ ignoring that you have to shift the decimal pointsâŚ but really, itâs justâŚ

Itâs harder to spot that 69 / 3 = 23 than 69 = 70 - 1

Thatâs 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 *âbetween 1 and 1.5 secondsâ* for the **intermediate** step of either 27x73 or 69x31âŚ 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 thatâs faster than 27 * 73 - 27 * 4

@Rajadodve786 Iâm so sorry for the late reply, It seems that I missed your post by accidentâŚ

Flou, If you donât mind, would you like to give some examples.

I like to read that.

UmmâŚ As Bjoern mentioned above, some general method like cross method are not necessarily slower

for task like 68*46.

68 Ă 46 (50-4)

3400 - 272

3128

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.