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.

Solution 1:

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

Solution 2:

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

Solution 3:

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.

Solution 4:

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.

Solution 5:

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 :)