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I thought I was pretty good at doing math in my head until yesterday I saw someone do $17.4/4$ in their head, without writing anything down and it took them less than 20 seconds.

What do you do to do this in your head? Even with the answer of 4.35 I could not work backwards to the right answer. I thought maybe if you break the problem up into $16/4=4$ which leaves $1.4$ so to get 1.4 from 4 you have to just get a fraction of 4. Half of four is too much, 1.3 of four is too small. So now I know that the answer is somewhere between $1/2$ and $ 1/3$ but from here it seems like it would get even more complicated. Would I just want to determine that a tenth of four is .4 and then add that to the final answer? I have no idea and this seems to be too much to do in my head.

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17.4/4 = (16+1+0.4)/4=4+0.25+0.1=4.35; that can take barely a couple seconds mentally. –  anon Jan 31 '12 at 15:55
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@Jordan: while it is true that some math people can be arrogant and condescending, I object to your use of their behavior as an excuse to condemn all math people. –  Qiaochu Yuan Jan 31 '12 at 16:08
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@Jordan: I understand that, but that's no reason to insult the group of people you're trying to solicit help from. –  Qiaochu Yuan Jan 31 '12 at 16:38
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I never said you "shouldn't have any problems with it." I sincerely regret if I've hurt your feelings, and while I do believe your observation that math skills and teaching skills do not go hand-in-hand is correct and even applies to me often, and that I could have spent more effort here to understand and explicate my own thought process in evaluating basic arithmetic operations, I sense you are simply jaded and projecting things into my mental state that are just not there. I feel your fixation on these dramatic narratives is discouraging to others. –  anon Jan 31 '12 at 17:03
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I would add (pun intended?) that skill in arithmetic is neither necessary nor sufficient for skill in mathematics. There are arithmetic savants who never do anything more than calculate, and there is the story of Grothendieck offering 57 as an example of a prime number. –  Austin Mohr Jan 31 '12 at 18:40
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7 Answers

up vote 10 down vote accepted

As you said, notice that 17.4 = 16 + 1.4, and the only hard bit is to calculate 1.4 / 4

To divide 1.4 by 4, use the fact that dividing by 4 is the same as dividing by 2 twice. If you divide by 1.4 by 2 once, you get 0.7. If you divide by 2 again, you get 0.35, so the answer is 4.35

It might help, when dealing with decimals, to multiply them by a power of 10 in your head before doing the division. For example, when calculating 1.4 / 2 I mentally convert that to 14 / 2 (which is 7) and then divide by 10 again to get 0.7

Now to do 0.7 / 2, I multiply by 10 to get 7 / 2 (which is 3.5) and then divide by 10 to get 0.35

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Maybe: Half of of 17.4 is 8.7. Half of 8.7 is 4.35? (${17.4\over 4}={1\over2}\cdot{1\over2}\cdot17.4$.)

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Thank you, this makes the most sense to me but I am not sure why anyone would go this route right away. It seems easy to manage all the numbers. I said 20 seconds but this guy probably did it in under 5 seconds. –  user138246 Jan 31 '12 at 16:00
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@Jordan A lot of mental arithmetic is memorizing large numbers of patterns to simplify your computations and being able to quickly recognize which ones are appropriate to use for a specific problem. –  Dan Neely Jan 31 '12 at 21:15
    
+1 This is the way I would do it, too. And under five seconds. There's nothing to it. 174/2=87 et cetera. –  Jyrki Lahtonen Jan 31 '12 at 22:10
    
This is also how I did it. Half of 17.4 is half of 17 plus half of .4; half of 17 is half of sixteen, plus a half; sp 17.4/2 = 8.5 + .2 = 8.7. Lather, rinse, repeat. –  katrielalex Feb 2 '12 at 17:57
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For the purposes of mental math what you could do is split 17.4 or similar numbers into pieces that are much easier to work with. Here I would have gone with 17.4=16+1+0.4, because dividing 16 or the decimal 0.4 by 4 is easy and the middle leftover term of '1' is also easy (0.25). The easiest way to get better at this is pure practice where you should attempt to recognize and remember mental things like "what kind of numbers can I divide by 4 really easily in my head?" so you can split larger numbers accordingly with greater ease.

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Figuring out the cost per ounce of items you buy at a grocery store is a good way to practice. A curious finding is that sometimes the larger box has a lower cost per ounce and sometimes the smaller box has a lower cost per ounce. –  Jay Jan 31 '12 at 20:02
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A fast intuitive approach I would use in supermarkets is to ignore the decimals for the moment.

4 goes into 17.4 about 4 times=16. The leftover is 17.4-16=1.4

4 goes into 14 about 3 times=12. The leftover is 14 -12 = 2

4 goes into 20 exactly 5 times=20. No leftover.

If you store those original times in your head, they are 4, 3, and 5. Considering the decimal part, you know it can't be 435 or 43.5 because they are too big. The bulk of the value went into the first divisor, which was the 4 value. Using that as the decimal break, 4.35 sounds about right and it is.

What slows us down (at least me, judging from the other comments) is that we don't ordinarily think in terms of fractions and decimals. Thinking about the problem in whole numbers makes processing much easier and faster in our minds.

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This is what I do on a problem like this. I do it the same way I'd do it on paper, remembering that I have 3, 2, 5 - and then I go back and put the decimal point in the right place later. –  dannysauer Jan 31 '12 at 20:11
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Since money was the origin of this problem, and I'm from the U.S., here is a suggestion. After you identified 16 as something to separate out, you have $\$$1.40. That's one dollar plus 40 cents. A quarter of each of those: one quarter (25 cent-piece) and one dime. So you have your 4 from 16, your quarter, and your dime: $\$$4.35

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uh... It's mostly memory. "normal" (I hate that word but it is sufficient) people do arithmetic by doing things like carries ("carry the 1") or borrows, like in division.

These methods are stupid. They are inefficient and rely on paper unless you have a weirdly good memory. In short, you can't really do that in your head very well. And even if you can, eventually your brain will melt from all the unnecessary processing. I used to do really poorly in math as a little kid because i REFUSED to do this (and therefore either didn't show my work or showed it happening in a way the teachers weren't prepared to see) and even though my answers were right.

The key is in understanding what is happening to numbers when you combine them. Think of this example.. We'll take a really easy one that you probably CAN do in your head. Say you want to multiply something by 5. Sure, ok you can count by 5s. But isn't it easier to just multiply that thing by TEN and then take half of that answer? Because 5 is half of 10.

This doesn't work for every number but there are tricks for almost all of them.

For example, if you wanna multiply 77 x 99, that's the same as 77 x 100 - 77 How do i know this? Think of multiples as "sets of things". 77 x 99 is just another way of saying "I have 99 groups, each containing 77 numbers." It's just arithmetic! (on speed..) If i have 100 groups with 77 numbers in each one, the answer is easy. 7700. But we want to take away ONE SET (because we want 99 sets, not 100) So we take away one set of 77.

Uh.. somebody can explain this better but this is how you do it.

By that logic, If i want to multiply 77 x 98, I can also say 77 x 100 - (77 + 77) If i want to multiply 77 x 97, i can also say 77 x 100 - (77 + 77 + 77) and so on.

(And there are also tricks within tricks. 77 + 77 + 77, or 77 x 3, can also have a 3 trick applied to it.)

I'm guessing there are tricks you can use for each number if you google them. I can't really take a ton of time to list them all but i bet anything they ARE out there.

(17 is a freak number and there aren't many "easy" ways of dealing with multiples of 17. Don't let 17 get you down. That guy is a douche. )

There are even MORE (vastly more) efficient tricks than this but this will get you started in thinking this way. As you get used to working around outdated and BAD solutions you were taught in school, you'll find more efficient ways. Some of them are harder to explain and sound ridiculously complex. But it's really about retraining your brain. I STILL DON'T UNDERSTAND how people use carries and borrows and if i do it this way, i often make stupid mistakes. You can't do these well if your handwriting is illegible. Mine is.

The truth is that anybody can do arithmetic in their heads but if your memory is good, you'll be better at it than those who don't have great memories. The most important thing you can know? FORGET SCHOOL. Forget those teachers who hated math as much as they made you guys hate it. They taught you bad methods because THEY were taught bad methods. Sure, I had good methods but i like numbers and find them friendlier than people :) So i more or less ignored things that didn't make sense and instead focused on finding ways that DID.

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The approach I have found most useful is to consider the places from left to right one at a time until you get a zero remainder, reach your desired accuracy, or hit a repeat:

For 17.4/4:

Place       Calculation                   Action

Tens:       17.4 / 40 = 0 rem 17.4    --> no tens (normally you skip this step)

Ones:       17.4 / 4 = 4 rem 1.4      --> say "4"

                                      --> say "point"

tenths:     1.4 / .4 = 3 rem 2        --> say "3"

hundredths: .20 / .04 = 5 rem 0       --> say "5"

                                      --> stop

Notes:

  • You only need to remember the divisor and remainder after each step. You can even forget the previous digits of the solution since you've already said them and they don't affect the remaining calculations!
  • I find it nice to shift the decimal place in my head so the divisions are consistent. For example, I'd normally think of .2/.04 as 20/4 in the steps above.
  • If you get good at this method, you should be able to say the solution as you calculate it at close to normal talking speed for this and similar problems.
  • Try 3/7 for practice :)
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