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How to divide numbers that has decimal results mentally

  1. Nov 27, 2011 #1


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    Like what the title says, How can I do it?

    For example,

    1/3 = 0.333

    43/3 = 14.3333

    I've seen people do it mentally and I find it very astonishing. I want to learn how.
  2. jcsd
  3. Nov 27, 2011 #2
    I am sure different people do it differently..here are various techniques I use

    First of all, you need to have good imagination and visualize your numbers up in mid "air" inside your head. You may need to have at least one "shelf" for temporary storage as you calculate the next number to add to the previous result.

    You need to factorize (separate) your numerator into thousands, hundreds, tens, ones...AND keep a running sum of the divisions of the thousands, hundreds, tens, ones...

    It helps to know a few fractions in advance...back in high school, during shop, we learned to use the caliper and the teacher made us memorize the typical fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 and the corresponding multiples between 0 and 1 inch...since then, I became pretty good at impressing people when coming up with results with 3 or 4 decimal place accuracy.

    Another thing is to be good with percentages...even if you can't get 3 decimal place accuracy, 1 or 2 will do just fine.

    For sales with 20% off, you need to skip calculating the discount and THEN subtract that from the price....instead, take 100-20=80 and go straight and multiply the price times 80...again, separate the hundreds, from the tens, and one...multiply each times 80% (0.8) and then add them all back up.

    Or, if the percent is not a nice multiple of 10%, but instead it is something line 35%...you need to separate this one too...100-35= 65% = 60% + 5% ...take the 60% of the price, plus 5% of the price ...this is often easier than trying to calculate 65% of the price in one shot.

    Hope this helps.
  4. Nov 27, 2011 #3


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    Science Advisor

    there are certain tricks:

    a repeating 1-digit cycle is that digit over 9:

    .7777777.... = 7/9

    a repeating 2-digit cycle is those 2 digits over 99:

    .16161616..... = 16/99

    if the "top" you get is divisible by 3 or 9, you can "simplifiy":

    .48484848.... = 48/99 = 16/33

    decimals that terminate, have powers of 2 and 5 in the denominator:

    .125 = 1/8

    .04 = 1/25

    fractions involving 6ths (when reduced) will have one "funny digit" and then repeat:

    .166666.... = 1/6
    .833333.... = 5/6

    this is because 1/6 = (1/2)(1/3), the "odd digit in the pattern" comes from the 1/2 part, the repeating comes from the 1/3 part, or you could think of it as:

    1/6 = 1/10 + (1/10)(2/3) = 3/30 + 2/30 = 5/30

    there is a similar pattern with 12ths, "two funny digits" and then a repeat:

    .41666666.... = 5/12

    fractions involving odd primes > 5 are things most people can't do in their head very fast, because the cycles of repetition can be very long. for 9/17, i have to use a calculator.

    fractions involving only powers of 2 can be found this way:

    1/4 = (1/2)(1/2). suppose we know 1/2 = .5

    to find 1/2 of .5, we "multiply by 5, and move right one decimal place".

    5x5 = 25, so we get .25

    so to find 3/8, let's say, we know that 3/4 = .75, we "multiply by 5"

    (75 x 5 = 350 + 25 = 375), and move over 1 decimal place, to get .375, done.

    the reason this works is because 1/2 = (1/10)(5), and multiplying by 1/10 just "shifts the decimal point".
  5. Nov 27, 2011 #4


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    Thanks for the tips, I'm going to review these replies and help improve myself.

    If you have anything else to add please do so. :D
  6. Nov 29, 2011 #5
    you can see the pattern if you play with the numbers many times..
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