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Math peeps, How do you solve...


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I always have trouble remembering rules. In this case, you really don't need the rule -- just a bit of common sense, and a determination to make the problem look simpler. Whatever part bothers you, focus on that part, and find some way to simplify it.

 

This is how I think it through:

  • There is some fraction that equals 0.63636363...
  • I don't know what that fraction is, so I'll just give it a name. Things are always easier to talk about if they have names. Call it F:

F = 0.6363636363...

  • But I don't know how to work with all that decimal junk. This would be a whole lot simpler to work with if I had a nice number like I'm used to, something I could round off and hold in my mind.
  • Let's multiply by enough to get a nice number on the "good" side of the decimal point.
  • For one repeating digit, I could multiply by 10, but to move two digits out of darkness into the light, I need 100.

100 × F = 63.63636363...

  • That's better, but I still have a bunch of decimal junk. It would be simpler to throw that all away, but wouldn't that be cheating?
  • At least it's still the SAME decimal junk. In fact, it exactly matches the decimal junk that I already gave a name to. That's cool!
  • So I can call this junk by the same name:

100 × F = 63 + F

  • Hey, now, that looks ever so much better!
  • If a hundred of something (our mystery fraction F) equals a number plus a single something, then the number (63) must be worth ninety-nine of the somethings.

99 × F = 63

  • And if 99 of something is 63, then if I cut 63 into 99 pieces, I will find out what my original something was...

F = 63 ÷ 99 = 63/99 = 21/33 = 7/11

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IIRC, you put the repeating digit(s) over a number with as many 9s as there are in the repeat.

 

0.6666... = 6/9 (reduces to 2/3)

0.636363... = 63/99 (reduces to 7/11)

0.678678678 = 678/999 (226/333?)

 

etc.

 

Be careful with this rule! It works in the examples you give, but what if only part of your number repeats? Or what if the repeat doesn't start at the decimal point? Are you going to make your student memorize new, different rules for every situation? What a load of arbitrary stuff to clutter up the memory!

 

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