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2018-03-25, 01:11 PM (ISO 8601)
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- Apr 2009
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Splitting multiplications and divisions into pieces for mental calculations
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?We are not standing on the shoulders of giants, but on very tall tower of other dwarves.
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2018-03-25, 01:28 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.Last edited by Brother Oni; 2018-03-25 at 01:32 PM.
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2018-03-25, 01:46 PM (ISO 8601)
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- Jun 2008
Re: Splitting multiplications and divisions into pieces for mental calculations
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.SpoilerThank you to zimmerwald1915 for the Gustave avatar.
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2018-03-25, 01:55 PM (ISO 8601)
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- Nov 2016
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.htmlLast edited by FreddyNoNose; 2018-03-25 at 02:32 PM.
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2018-03-25, 02:11 PM (ISO 8601)
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- Sep 2016
Re: Splitting multiplications and divisions into pieces for mental calculations
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).
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2018-03-25, 02:36 PM (ISO 8601)
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- Jun 2013
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- Bristol, UK
Re: Splitting multiplications and divisions into pieces for mental calculations
Multiplication is multiple addition and Division is multiple subtraction, this is obvious, but a lot of the time we don't remember it.
The end of what Son? The story? There is no end. There's just the point where the storytellers stop talking.
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2018-03-25, 02:53 PM (ISO 8601)
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- Apr 2009
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.
We are not standing on the shoulders of giants, but on very tall tower of other dwarves.
Spriggan's Den Heroic Fantasy Roleplaying
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2018-03-25, 03:59 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.
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2018-03-25, 11:37 PM (ISO 8601)
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- Aug 2008
Re: Splitting multiplications and divisions into pieces for mental calculations
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.
Is it though? Is it really?
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2018-03-25, 11:41 PM (ISO 8601)
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- Dec 2009
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- Birmingham, AL
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.
Last edited by Peelee; 2018-03-26 at 01:32 PM.
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2018-03-26, 12:57 AM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.
Last edited by Lvl 2 Expert; 2018-03-26 at 08:24 AM.
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2018-03-26, 07:40 AM (ISO 8601)
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- Apr 2008
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.
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2018-03-26, 10:06 AM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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2018-03-28, 08:53 PM (ISO 8601)
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- Oct 2010
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- Dallas, TX
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.
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2018-04-06, 05:20 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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2018-04-07, 12:36 AM (ISO 8601)
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2018-04-08, 12:37 AM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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2018-04-08, 03:03 AM (ISO 8601)
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- Aug 2008
Re: Splitting multiplications and divisions into pieces for mental calculations
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.
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2018-04-08, 01:13 PM (ISO 8601)
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2018-04-11, 03:05 PM (ISO 8601)
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- Apr 2008
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Re: Splitting multiplications and divisions into pieces for mental calculations
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.Avatar by araveugnitsuga.
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2018-04-11, 07:00 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
Cuthalion's art is the prettiest art of all the art. Like my avatar.
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2018-04-12, 04:15 PM (ISO 8601)
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- Nov 2008
Re: Splitting multiplications and divisions into pieces for mental calculations
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.The gnomes once had many mines, but now they have gnome ore.
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2018-04-13, 03:17 PM (ISO 8601)
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- May 2009
Re: Splitting multiplications and divisions into pieces for mental calculations
"None of us likes to be hated, none of us likes to be shunned. A natural result of these conditions is, that we consciously or unconsciously pay more attention to tuning our opinions to our neighbor’s pitch and preserving his approval than we do to examining the opinions searchingly and seeing to it that they are right and sound." - Mark Twain
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2018-04-13, 05:00 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
Last edited by Peelee; 2018-04-13 at 05:03 PM.
Cuthalion's art is the prettiest art of all the art. Like my avatar.
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2018-04-13, 10:19 PM (ISO 8601)
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2018-04-13, 11:28 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
Cuthalion's art is the prettiest art of all the art. Like my avatar.
Number of times Roland St. Jude has sworn revenge upon me: 2
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2018-04-14, 02:26 AM (ISO 8601)
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- Aug 2008
Re: Splitting multiplications and divisions into pieces for mental calculations
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.I would really like to see a game made by Obryn, Kurald Galain, and Knaight from these forums.
I'm not joking one bit. I would buy the hell out of that. -- ChubbyRain
Current Design Project: Legacy, a game of masters and apprentices for two players and a GM.
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2018-04-14, 12:01 PM (ISO 8601)
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Re: Splitting multiplications and divisions into pieces for mental calculations
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).
Cuthalion's art is the prettiest art of all the art. Like my avatar.
Number of times Roland St. Jude has sworn revenge upon me: 2