Doing simple math in your head 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.
 A: 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.
A: 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.
A: 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 :)

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