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Dutchess_III's avatar

Will you explain this decimal-to-fraction question to me?

Asked by Dutchess_III (27689 points ) March 8th, 2012

I hope I don’t feel too dumb when we’re finished but…

The fractional equivalent of .6 is 3/10. However, my teaching program says that the fractional equivalent of .6… repeating is ⅔. I don’t understand how they came up with this.

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9 Answers

tom_g's avatar

.6 = 6/10 or 3/5

It does not = 3/10. It also does not = ⅔. Your program is wrong. ⅔ = 0.6666…

GladysMensch's avatar

The fractional equivalent of .6 is 6/10. That’s six tenths. The fractional equivalent of .666 repeating is .66666/10 or ⅔.

Dutchess_III's avatar

Guys…that was a typo. I know .6 = 3/5ths. I’m tired! Had a long day.

@tom_q read it again…my question was on the second one, the .666… repeating—I didn’t know how to indicate repeating but I do now and it equals ⅔

@GladysMensch at what point do you stop throwing 6’s into the division? Why wouldn’t your example of .6666… be .6666/10,000? My brain isn’t working right now.

GladysMensch's avatar

@Dutchess_III Sorry, typo. The fractional equivalent is ⅔
You’re right in that the larger the number, the closer we come to the actual answer of ⅔ or .6666…

Equivalent isn’t 6/10, because that’s .6
Equivalent isn’t 66/100, because that’s .66
Equivalent isn’t 666/1000, because that’s .666
Equivalent isn’t 6666/10000, because that’s .6666
We want to get to .66666…, so we will need an infinite number of 6’s on top with an infinite number of zero’s (after the 1) on the bottom.

To get to ⅔ we start by getting ⅓ (or dividing 1 by 3) which gives us .33333…
Since ⅓ is .33333…. ⅔ would be .6666…. and 3/3 would be .9999….

LuckyGuy's avatar

Is there a missing ”.” above the 6.? (A dot above the digit 6?) If there is a dot up there it means a repeating digit. That means .6666666666…. which is ⅔

Dutchess_III's avatar

@LuckyGuy Want to show me how to put a dot . or a line – ABOVE the six using a keyboard? I started to put .6>>> but that didn’t seem to convey it either, so I used ellipses… and used the word “repeating” after them.

I’m still pondering @GladysMensch‘s answer. She says, “To get to ⅔ we start by getting ⅓ (or dividing 1 by 3) which gives us .33333… ” but that seems to assume you know the answer and work backwards. I’m sure it will just click tomorrow when I’m not so tired, but I’ll keep working on it.

gasman's avatar

There’s a simple algorithm I learned in 7th grade for converting, into a fraction, any decimal that infinitely repeats a finite group of digits.

Applied here: Let x = 0.6666666…
Then 10x = 6.6666666…
Now subtract x from 10x and the infinite strings go away:
9x = 6
x = 6/9 = ⅔
Bulletproof!

@Dutchess_III As you & @LuckyGuy pointed out the standard notation is either dots or a horizontal bar over the final digit(s). You could end the number with an ellipsis ( ... ) but then it may be ambiguous as to which digits are repeating. I’ve not seen any typographical conventions for rendering this at a keyboard—the situation doesn’t arise very often.

It’s customary in approximating the fraction ⅔ by a decimal string of 6’s to terminate the string with a 7, thus rounding off to the appropriate precision, since ⅔ closer to .7 than to .6.

On a related note, the value of 0.999… where an infinite string of 9’s follows the decimal point, exactly equals one; it’s just another representation of what we usually symbolize by “1”. Ref.

LuckyGuy's avatar

@gasman Your description was perfect! I had forgotten that “proof” – most likely it was sucked out of my brain during sinus surgery decade or so ago.

Dutchess_III's avatar

I know how to indicate a repeating number on paper. I didn’t know how to indicate it with my keyboard.

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