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Fractured Memories

Some fractions are a part of everyday life; dimes, quarters, nickels, hours, minutes, seconds, etc. These are relatively easy to manage mainly because we deal with them so often. Everyone just “knows” that 1/2 is 0.5, and 1/4 is .25, and 1/10 is 0.1; we’ve had it ingrained in us through massive amounts of repetition. I go one step further; I can usually estimate the decimal equivalent of just about any fraction that comes up in my life. Super useful? Maybe not, but it has good show-off value, and I think it’s fun!


Joe commented below and suggested using 0.[3] to denote a 0. followed by an endlessly repeating 3. I thought it was a good idea, so I changed the text below to use it. Hope it’s not confusing.

Learning the first 12 fractions can make it super-easy to do division in your head and produce answers down to the 10ths or even 1000ths quickly and easily. Let’s take a look:

Denominator Values Tips
1/1 1 This is the easy one, put here for completeness. It could be beneficial to remember that any non-zero number N over N = N/N = 1.
1/2 1/2 = 0.5 Sure, it’s simple, but it’s useful when trying to compute 1/20th, etc.
1/3 1/3 = 0.[333]
2/3 = 0.[666]
That’s 0.[333], where the 3s never stop, also called 0.3 repeating. And yes, 0.9 repeating does equal 1!
1/4 1/4 = 0.25
2/4 = 0.5
3/4 = 0.75
Here’s the first one where just memorizing keeps you from having to reduce 2/4 = 1/2.
1/5 1/5 = 0.2
2/5 = 0.4
3/5 = 0.6
4/5 = 0.8
N/5 = 0.(2*N). Note you’re really multiplying N by 2, then dividing by 10 (which just moves the decimal): 3/5 = (3*2)/10 = 0.6!
1/6 1/6 = 0.1[6]
2/6 = 0.[3]
3/6 = 0.5
4/6 = 0.[6]
5/6 = 0.8[3]
Ok, this one’s not so simple. It helps to realize that 0.[3] / 2 = 0.1[6], and go from there. Having 3/6 = 0.5 in the middle can help too, since 5/6 = 3/6 + 2/6 = 0.5 + 0.[3] = 0.8[3], see?
1/7 1/7 = 0.[142857]
2/7 = 0.[285714]
3/7 = 0.[428571]
4/7 = 0.[571428]
5/7 = 0.[714285]
6/7 = 0.[857142]
This is by far my favorite fraction. Note that in all cases, all six digits repeat, so 1/7 = 0.142857142857… Also note that the same six digits appear in the same order for all 6 fractions, you just start with a different digit. I use the fact that 14 is half 28 is half (just about) 57 to help remember the digits, too. This is the impressive one, guys. Someone asks, “what’s 1/7th of 100?” and you say “14.2857” instantly. Nice.
1/8 1/8 = 0.125
2/8 = 0.25
3/8 = 0.375
4/8 = 0.5
5/8 = 0.625
6/8 = 0.75
7/8 = 0.875
Seems like a lot to know, but most are easily computable from knowing 1/8 and reducing the rest. 5/8 = 4/8 + 1/8 = 0.5 + 0.125 = 0.625
1/9 1/9 = 0.[1]
2/9 = 0.[2]

7/9 = 0.[7]
8/9 = 0.[8]
Just take the numerator and repeat it over and over. And again, 9/9 = 0.[9] = 1. Also of note, any number N (up to 99) over 99 0.[N] too, but use both digits, so 5/99 = 0.[05], 63/99 = 0.[63], etc. This continues for 999, 9999, etc.
1/10 1/10 = 0.1
2/10 = 0.2

8/10 = 0.8
9/10 = 0.9
These are pretty self-evident. You’re dividing by 10, so just slide the decimal place.
1/11 1/11 = 0.[09]
2/11 = 0.[18]
3/11 = 0.[27]
4/11 = 0.[36]
5/11 = 0.[45]
6/11 = 0.[54]
7/11 = 0.[63]
8/11 = 0.[72]
9/11 = 0.[81]
10/11 = 0.[90]
See what’s happening? N/11 = 0.[N*9] repeating, with both digits repeating (Note, 1*9 = 09 in this case). This becomes obvious when you think that 11/11 must equal 0.[9], so dividing that by 11 must divide each of those 99s in the decimal by 11 as well: 0.[9] / 11 = 0.[09].
1/12 1/12 = 0.08[3]
2/12 = 0.1[6]
3/12 = 0.25
4/12 = 0.[3]
5/12 = 0.41[6]
6/12 = 0.5
7/12 = 0.58[3]
8/12 = 0.[6]
9/12 = 0.75
10/12 = 0.8[3]
11/12 = 0.91[6]
I must admit, I don’t really have these memorized. I know that 1/12 = 0.08[3] and work from there. 7/12 = 6/12 + 1/12 = 0.5 + 0.08[3] = 0.58[3], etc. Since half the values for N reduce to smaller fractions, this is where I leave off memorizing.

There you have them, the first 12 fractions for easy memorization. Amaze your friends! Astound your kids! Become even more of a know-it-all than you already are! I joke, but I guess you’d be surprised how often I use these, I know I am.

The Count


P.S. Please excuse my use of * to denote repeating decimals, I’d be happy to hear of a better symbol, since my font doesn’t allow lines across the top of text

5 comments to Fractured Memories

  • Joe

    Instead of an overbar to signify repetition, how about something like

    :||

    in imitation of the repeat-sign in sheet music?

    1/3 = 0.3:||

    1/6 = 0.1||:6:||

    1/7 = 0.||:142857:||

    A bit clunky, and the vertical line looks too much like the numeral 1.

    So how about just putting the repeating part in brackets the way chemists symbolize repeated parts of polymer molecules (e.g. http://en.wikipedia.org/wiki/Polyvinylchloride#Preparation)?

    1/3 = 0.[3]

    1/6 = 0.1[6]

    1/7 = 0.[142857]

  • I like it, put a star after the brackets and you’ve got the regular expression (betraying my programming background). We were thinking along the same lines, but your brackets are more exact since 0.16* isn’t as precise as 0.1[6]

  • Sipstaff

    Fractions of 7 is also very interesting if you look at the pattern they form on a calculator keypad (or your normal keyboard numpad)
    All you have to do is to memorize the pattern and know where to start for each of the 6 fractions, which is easy since every next fraction starts with the next bigger number.

  • Neil

    Some pretty nice shortcuts; I’ll have to remeber these. Just one little nitpick, a typo. In the ‘1/12’ description, you forgot a zero in the second sentence.

    I also like the idea of using brackets for repeating digits.

    Cheers :)

  • @Neil,

    Right you are! Fixing now.

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