Discrete Ideas » fractions http://www.discreteideas.com Thoughts count Thu, 21 Apr 2011 05:24:16 +0000 en hourly 1 http://wordpress.org/?v=3.1.3 Fractured Memories http://www.discreteideas.com/2010/07/fraction-memorization/ http://www.discreteideas.com/2010/07/fraction-memorization/#comments Sun, 04 Jul 2010 04:12:12 +0000 The Count http://www.discreteideas.com/?p=306 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 [...]]]>

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

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