Splitting multiplications and divisions into pieces for mental calculations

[Yora]

I was just wondering if this is something that people do a lot or an individual quirk that I have. In school I've always been told that my ways of calculating things are somewhat unusual, but I always find it much simpler than using the more elaborate "tidy" ways that are being taught.

For example, if I want to divide 2018 by four, then I split it up into (2000/4)+(18/4). Or when I want to calculate 36*32 I break it up into (30*30)+(30*2)+(6*30)+(6*2). I even use such simplifications when doing most calculations on paper without a calculator because I somehow never can remember the formulas that are being taught. Is that common?

[Brother Oni]

For example, if I want to divide 2018 by four, then I split it up into (2000/4)+(18/4). Or when I want to calculate 36*32 I break it up into (30*30)+(30*2)+(6*30)+(6*2). I even use such simplifications when doing most calculations on paper without a calculator because I somehow never can remember the formulas that are being taught. Is that common?

I'd do the first one the same as you, although since I know 2000/4 is 50, I'd do the 18/4 bit then just add 50.

The second one I'd break it up into smaller steps, then smaller steps if required:

36*32
(36*30)+(36*2)
(((30*3=90)+(3*6=18)=108) then add the zero to multiply by 10)+72
(1080)+72
(80+70=150) then add the 1000 and the 2
1152

Edit: Oh, you meant on paper, in which case for multiplication, I'd just follow the layout and work it accordingly, carrying the numbers as appropriate. I never learnt how to do long division the formal way, so I just use short division all the way through.

[erikun]

Factors is another way of doing it. 36=3x12 and 32=8x4, so:
36*32
12*8*4*3
96*4*3
(100-4)*4*3
(400-16)*3
1200-(16*3)
1200-(4*4*3)
1200-(4*12)
1200-48
1152

Just working it out one line at a time, without checking work. I suppose I could also fully factor things out and then work from there. 36=3x3x2x2, 32=2^5, and so I'd just start with 9 and continue doubling it seven times. Or figure out 2^7 (128) and triple it twice.

But yes, for larger numbers, I try to break them down into a rather easy to work with number and one, or possibly two, smaller ones which are easier to handle in my head. 1997+65 is just 2000-3+65 = 2000+62 = 2062.

[FreddyNoNose]

I was just wondering if this is something that people do a lot or an individual quirk that I have. In school I've always been told that my ways of calculating things are somewhat unusual, but I always find it much simpler than using the more elaborate "tidy" ways that are being taught.

For example, if I want to divide 2018 by four, then I split it up into (2000/4)+(18/4). Or when I want to calculate 36*32 I break it up into (30*30)+(30*2)+(6*30)+(6*2). I even use such simplifications when doing most calculations on paper without a calculator because I somehow never can remember the formulas that are being taught. Is that common?

As to the general population any math seems to be something they don't think about.

What are you doing, certainly seems natural to me.

IIRC, this was a way that was taught in my grade school as one way to solve the problem.

It makes me think of this story: http://mathcentral.uregina.ca/qq/dat...02.06/jo1.html

[jayem]

I was just wondering if this is something that people do a lot or an individual quirk that I have. In school I've always been told that my ways of calculating things are somewhat unusual, but I always find it much simpler than using the more elaborate "tidy" ways that are being taught.

For example, if I want to divide 2018 by four, then I split it up into (2000/4)+(18/4). Or when I want to calculate 36*32 I break it up into (30*30)+(30*2)+(6*30)+(6*2). I even use such simplifications when doing most calculations on paper without a calculator because I somehow never can remember the formulas that are being taught. Is that common?

I always do quadratics by completing the square, because:
a) It's a lot more intuitive than the formula
b) you have to add a few extra lines, but you get to keep the terms the same so you save a double substitution.
c) you don't have to prove the formula's conditions are met

Also the way you are doing them is not that dissimilar to the main approach. The multiplication one was actively taught here (UK) when my younger brother was at school,
The way I did it you basically went 36*32 = [36*3*10 + 36*2 = 6*3*10+3*3*10*10+ 36*2]=918+72 It's just that that was in a column and you didn't write the middle steps.

Being able to pick the right splitting up is a skill, though. Which is why you're taught the way that often saves paper and works effectively on most things (plus being standardized, you can compare with a friend to see where a mistake occurred).

I quite like the pseudo-egyption way of doing multiplication/division (where you double/halve on side).

[halfeye]

Multiplication is multiple addition and Division is multiple subtraction, this is obvious, but a lot of the time we don't remember it.

[Yora]

I never have any luck with formulas. That was my bane with physics class in the final years. Where I was really good in math was geometry and statistics. Because there you're dealing with actual things and I can work out my own paths to figure it out with just additions and subtractions.

[factotum]

If I'm calculating something in my head then I'll often use shortcuts, yes--for example, if I'm subtracting £3.97 from £12.53, I'll subtract £4 and then add 3p to what's left to get £8.56. If I was doing that on paper I'd do it the "proper" way, though, it's just easier to follow when you have a specific method for doing these things.

[Knaight]

These are fairly standard mental math shortcuts among people who are decent at math. I don't tend to use them on paper a great deal, mostly because the intersection of "on paper" and "no calculator" is pretty minimal for me - or at least if I'm using these I'm not doing it somewhere where I'd actually show interim steps.

Using your examples - 2018/4 breaks down into 500 +18/4, which is 504 +2/4, which is 504.5. Some of that process is essentially subconscious, and it's certainly faster than standard long division. For 32*36, the same (30+2)(30+6) process holds, although the exact order it gets split into is going to be a bit different. 900+240+12, 1152, done. Again, this is less clunky than standard multiplication.

I'd do the first one the same as you, although since I know 2000/4 is 50, I'd do the 18/4 bit then just add 50.

Is it though? Is it really?

[Peelee]

That's how I figure out how much to tip in restaurants. Take the total, move the decimal over, then add half that, usually rounding that half up a bit because I'm lazy and it's easier. Super easy, never need to pull out my calculator.

[Lvl 2 Expert]

All of these are the right way to do it. There have since forever been ways to take the whole numbers and get an answer straight from there, typically involving paper or an abacus or a calculator or excel, but those are all shortcuts. It's fine to use them, and they're typically faster then the things described here when working with large numbers, but they take away from your understanding of what's happening and thereby your ability to check your own work. It's never wrong to do things in a way that makes you actually think about it.

[Kato]

As has been said, I think those are pretty standard techniques among people who actually still do these things without calculator for whatever reason.. They are what most techniques to do it on paper are based on.

There are loads of similar things...
I like dividing by ten and multiplying by two instead of dividing by five, or the other way around for multiplying. (maybe that's just me, because it feels faster)
Or if I need to divide by any even number I'll always divide by two first and see from there. (division by twelve is a nuisance but division by two, two and three is usually easy)
Obviously you just round until you get easy numbers if you want just an estimate, e.g. 358.7:13 is a stupid question but 360:12=30 and 350:14=175:7=25 so you can get a decent idea of what you're looking for.

[factotum]

Obviously you just round until you get easy numbers if you want just an estimate, e.g. 358.7:13 is a stupid question but 360:12=30 and 350:14=175:7=25 so you can get a decent idea of what you're looking for.

That's a whole 'nother thing there--using easier numbers to get an estimate of what you want is actually a defined mathematics technique, whereas "divide by 100 and multiply by 4 rather than divide by 25" is something purely up to the individual.

[Jay R]

Yes, of course. There are lots of ways to handle arithmetic problems.

For instance, 32 and 36 are so close that I solve 32 * 36 by difference of squares. Since 32 = 34 - 2, and 36 = 34 + 2, then

32 * 36 = 34² - 2² = 1156 - 4 = 1152.

[Rockphed]

Yes, of course. There are lots of ways to handle arithmetic problems.

For instance, 32 and 36 are so close that I solve 32 * 36 by difference of squares. Since 32 = 34 - 2, and 36 = 34 + 2, then

32 * 36 = 34² - 2² = 1156 - 4 = 1152.

This implies that people know what 34² is. My method would be to multiply 40 * 30, add 2*30, then subtract 4 * 32.

[factotum]

This implies that people know what 34² is. My method would be to multiply 40 * 30, add 2*30, then subtract 4 * 32.

That seems more complicated than just doing 30 * 36 + 2 * 36, though!

Also, it's possible he has a bunch of squares pre-memorised, the same way people memorise their times tables.

[Rockphed]

That seems more complicated than just doing 30 * 36 + 2 * 36, though!

Also, it's possible he has a bunch of squares pre-memorised, the same way people memorise their times tables.

In practice it ended up being 40 * 32 - 4 * 32. The hard part was subtracting 128 from 1280, not getting the 128 and 1280 (which helpfully turn out to be decimal multiples of eachother).

[Knaight]

This implies that people know what 34² is. My method would be to multiply 40 * 30, add 2*30, then subtract 4 * 32.

This sort of thing is why shortcuts vary - most people know squares up to 10, some people go well beyond that (a similar thing applies to memorized powers of two), and the quick shortcuts are generally built around what's memorized. The core matter is that the sort of mathematical familiarity that lets you do this is broadly helpful, and frequently works better than the standard algorithms.

[Jay R]

This sort of thing is why shortcuts vary - most people know squares up to 10, some people go well beyond that (a similar thing applies to memorized powers of two), and the quick shortcuts are generally built around what's memorized. The core matter is that the sort of mathematical familiarity that lets you do this is broadly helpful, and frequently works better than the standard algorithms.

Yup. Also, people learn what they need for their tools to work. I know many more squares numbers than most people because I use them to solve problems quickly.

[Eurus]

More or less what I do, yeah. Especially for, say, figuring out an 18% tip or something. Breaking it down into units of 10% and 1% helps.

I don't necessarily know if it's faster, but it feels like it's easier to keep track of when I need to do some quick math in my head.

[Peelee]

More or less what I do, yeah. Especially for, say, figuring out an 18% tip or something. Breaking it down into units of 10% and 1% helps.

I don't necessarily know if it's faster, but it feels like it's easier to keep track of when I need to do some quick math in my head.

I rarely need to bother with 18% tip. 15 for standard, 10 for subpar, 20 for stellar. Easy peasy.

[Bucky]

Real nerds do it like this:
36 * 32 = 32 * 32 + 4 * 32
= 2^5 * 2^5 + 2^5 * 2^2
= 2^10 + 2^7
= 1024 + 128
= 1152.

[veti]

I rarely need to bother with 18% tip. 15 for standard, 10 for subpar, 20 for stellar. Easy peasy.

Am I the only one who adds "and then round to the nearest convenient unit" anyway? So in practice, 15 is likely to become "something between 14 and 18 (tend to round up), depending what the numbers are"?

[Peelee]

Am I the only one who adds "and then round to the nearest convenient unit" anyway? So in practice, 15 is likely to become "something between 14 and 18 (tend to round up), depending what the numbers are"?

True enough. My whole "move the decimal and then add half" is always rounded to the easiest number to deal with.

[Jay R]

More or less what I do, yeah. Especially for, say, figuring out an 18% tip or something. Breaking it down into units of 10% and 1% helps.

I don't necessarily know if it's faster, but it feels like it's easier to keep track of when I need to do some quick math in my head.

Take one fifth of the number. That's 20%. Then subtract the same number one decimal point over. That's 18%.

[Peelee]

Take one fifth of the number. That's 20%. Then subtract the same number one decimal point over. That's 18%.

One fifth of, say, $37 is not something I can think of right off the bat. But 2 x 3.7 = 7ish I can.

[Knaight]

One fifth of, say, $37 is not something I can think of right off the bat. But 2 x 3.7 = 7ish I can.

Then as a second step you know that 7.4*.1 is .74, and 7.4-.74 is 6.66, getting you to the 18%. I find that a bit easier than dividing by 5, though it's pretty marginal (30/5=6, 7/5=1.4, thus 7.4) and the methods converge at the second step.

I just use 20% anyways though.

[Peelee]

Then as a second step you know that 7.4*.1 is .74, and 7.4-.74 is 6.66, getting you to the 18%. I find that a bit easier than dividing by 5, though it's pretty marginal (30/5=6, 7/5=1.4, thus 7.4) and the methods converge at the second step.

I just use 20% anyways though.

Yeah, but that's too much work to get to a stupid tipping percent to start with, though. As much as I hate tipping as a system, the 10/15/20% as the poor/average/superb rates doesn't ever need to change, because as a percentage it automatically scales with inflation. There's no reason to move to a 15/18/20% for poor/average/superb, not to mention it makes the mental math immediately more difficult for no good reason (difficult, in this case, being incredibly relative. I'm lazy).