(Assuming that you do, of course.)
If someone gives you a figure like 4573 and asks you to divide it by 53, what is your mental working?
Mine is: 53 into 457 is 8, and 8 times 53 is 424, so 4240 subtracted from 4573 is 240 from 573 is 300 plus 33: 53 into 333 is 6, 6 times 53 is 318, 33 minus 18 is 15. The answer is 86 remainder 15.
Which I just checked on Google Calculator, in case I had misplaced a digit somewhere, which is the chief problem with doing mental arithmetic. For a figure that size it would be easier if I had somewhere to write down "80", or ask someone to remember it for me, but I can do it in my head without that, though with the uncomfortable feeling that I want to be able to check it somewhere. For larger figures I would definitely want to have a scrap of paper to write it down on, though the basic technique is good for knowing what the answer should be.
The odd thing is - this came up in the course of a conversation, and I realised that I do not recall ever being told how to calculate figures this way. The technique just arrived, sometime when I was a teenager, as a consequence - I assume - of being taught my times-tables and addition and subtraction until they were coming out my ears, and then discovering that in fact there were realworld situations where it was useful to be able to do long division in my head.
In other news, can you spell schadenfreude? Heh.
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No-one ever taught me long division either, weirdly, though otherwise my mathematical education (up to A-level) seemed pretty comprehensive, but unlike you I didn't have the wherewithal to figure it out for myself, so basically I don't really try. Someone else is usually on hand who can come up with the answer quicker than I could. If I had to I'd multiply 53 by various things till it came close to 4573 and try and narrow it down that way. But it'd probably take a while.
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Oh no, I was taught how to do long division at school: it was just by a completely different method (it requires pen and paper) than the method I use when I have to do long division in my head.
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I was taught subtraction by columns and long multiplication at primary school over and over again until I was so bored I deliberately got thrown out of the classroom so I could go to the library, but never long division, a friend at secondary school taught me it because everybody else knew it and I didn't. Now, I don't remember how to do either of those techniques on paper, I do the sums another way in my head, but I can remember just about how to do long division given a piece of paper.
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Yeah: the methods I was taught at primary school all required pencil and paper, but we were strongly encouraged to memorise the multiplication tables from 1x1 to 12x12...
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I dare say I'd do it differently at different times, but today I went:
1000/50 = 20. Multiply that by 4 = 80.
(I mean that's a quick thing, I'm not saying it took me ages to work out 50 into 4000, but I'm aware that I didn't go straight to 4000, the first thing that I 'knew' was 50s into 1000 if you know what I mean?)
80 x 3 = 240.
So 80 x 53 = 4240, which leaves 333.
And then it was like you said, 50 into 300 = 6. 3 x 6 = 18
And 33-18 = 15 so it's 86r15.
I really normally wouldn't bother, though. I'd either find pen and paper or wait for other people to work it out. Oh the laziness.
1000/50 = 20. Multiply that by 4 = 80.
(I mean that's a quick thing, I'm not saying it took me ages to work out 50 into 4000, but I'm aware that I didn't go straight to 4000, the first thing that I 'knew' was 50s into 1000 if you know what I mean?)
80 x 3 = 240.
So 80 x 53 = 4240, which leaves 333.
And then it was like you said, 50 into 300 = 6. 3 x 6 = 18
And 33-18 = 15 so it's 86r15.
I really normally wouldn't bother, though. I'd either find pen and paper or wait for other people to work it out. Oh the laziness.
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I do 50 into 4000 as 5 into 40 multiply by 10, but other than that, it all looks very like my mental "working".
I really normally wouldn't bother, though. I'd either find pen and paper or wait for other people to work it out. Oh the laziness.
Oh me too. Working it out in my head is Hard Work.
I really normally wouldn't bother, though. I'd either find pen and paper or wait for other people to work it out. Oh the laziness.
Oh me too. Working it out in my head is Hard Work.
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If I were asked to do that division in my head, I would think "It's close enough to 50 divided into halfway between 4550 and 4600, which is halfway between 91 and 92, so it's very roughly 91.5. If you want it better than that, don't ask me to do it in my head."
If you made me do it in my head, I'd do it the same way you did: mental long division. I was certainly taught how to do long division in school (both a full version and a shorter one in which the subtraction isn't written out, just done mentally--easier to explain on paper). My stepchildren apparently were not, being apparently expected instead to approximate and guesstimate until they reached the right answer, which I find appalling. Their mom did so too, and taught them long division at home.
But I can spell schadenfreude.
If you made me do it in my head, I'd do it the same way you did: mental long division. I was certainly taught how to do long division in school (both a full version and a shorter one in which the subtraction isn't written out, just done mentally--easier to explain on paper). My stepchildren apparently were not, being apparently expected instead to approximate and guesstimate until they reached the right answer, which I find appalling. Their mom did so too, and taught them long division at home.
But I can spell schadenfreude.
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My stepchildren apparently were not, being apparently expected instead to approximate and guesstimate until they reached the right answer, which I find appalling.
Well, I start out with an approximation - I know 53 will "almost fit" into a number greater than 4000 more than 80 times. The first moment of mental calculation is always occupied with figuring out what the nearest "fit" will be.
What this question was about, though, was it occurring to me that while I use the same techniques to add, subtract, and multiply in my head as I do on paper, when I do long-division in my head I use an entirely different method - which in effect involves as much if not more approximation, multiplication, and addition/subtraction as it does division. It's as if the latter three functions are somehow easier for my brain - whereas a "goesinta" operation is difficult.
I remember reading once that the modern technique of long division was only invented/discovered in the 10th or 11th century (I forget where or how I read this, so it may be one of those factoids that is essentially untrue even if a detail is true). This seems odd, because presumably any sizable military unit would have had to be able to do long-division just out of pure practicality; "We have a ration of 100 sacks of flour to last us for the next eight days, and we have 1142 soldiers, of whom 203 are entitled to double-rations: how much flour can we issue as a daily ration until the next delivery of flour?"
Well, I start out with an approximation - I know 53 will "almost fit" into a number greater than 4000 more than 80 times. The first moment of mental calculation is always occupied with figuring out what the nearest "fit" will be.
What this question was about, though, was it occurring to me that while I use the same techniques to add, subtract, and multiply in my head as I do on paper, when I do long-division in my head I use an entirely different method - which in effect involves as much if not more approximation, multiplication, and addition/subtraction as it does division. It's as if the latter three functions are somehow easier for my brain - whereas a "goesinta" operation is difficult.
I remember reading once that the modern technique of long division was only invented/discovered in the 10th or 11th century (I forget where or how I read this, so it may be one of those factoids that is essentially untrue even if a detail is true). This seems odd, because presumably any sizable military unit would have had to be able to do long-division just out of pure practicality; "We have a ration of 100 sacks of flour to last us for the next eight days, and we have 1142 soldiers, of whom 203 are entitled to double-rations: how much flour can we issue as a daily ration until the next delivery of flour?"
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