Converting a Repeating Decimal to a Fraction

[kandrew93]

A friend of mine recently brought their 8th grade kid to me for homework help. They're playing with fractions, decimals, percentages, etc. Anyway, one of the questions was:

Convert 0.583̅ to a fraction.

I just split the decimal up, converted, then added the fractions:

0.583̅ = 0.58 + 0.003̅
= (58/100) + (1/300)
= (7/12)

But I feel like this confused her when I explained it. Is there a trick to these repeating decimals that I'm not thinking about? 'Cause it's bothering me that I can't give an 8th grade explanation to an 8th grade math problem...

 

[pka]

A friend of mine recently brought their 8th grade kid to me for homework help. They're playing with fractions, decimals, percentages, etc. Anyway, one of the questions was:
Convert 0.583̅ to a fraction.
I just split the decimal up, converted, then added the fractions:
0.583̅ = 0.58 + 0.003̅
= (58/100) + (1/300)
= (7/12)
But I feel like this confused her when I explained it. Is there a trick to these repeating decimals that I'm not thinking about? 'Cause it's bothering me that I can't give an 8th grade explanation to an 8th grade math problem...

\(\displaystyle \\N=0.58\overline{3}\\1000N=583.\overline{3}\\100N=58.\overline{3}\text{ Subtract}\\\overline{\text{____________}}\\900N=525\\ N= \dfrac{525}{900}\)

 

[kandrew93]

\(\displaystyle \\N=0.58\overline{3}\\1000N=583.\overline{3}\\100N=58.\overline{3}\text{ Subtract}\\\overline{\text{____________}}\\900N=525\\ N= \dfrac{525}{900}\)

That could've been easier to follow. Might be what they're taught. *shrugs* Thanks for the reply.

 

[Deleted member 4993]

Another way:

(583-58)/900

 

[HallsofIvy]

That could've been easier to follow. Might be what they're taught. *shrugs* Thanks for the reply.

You want more "words"?

Start with x= .58333333...

Multiplying by powers of 10 just moves the decimal point and multiplying by 100 moves it so that only the "repeating" part is to the right of the decimal point: 100x= 58.3333333... Multiplying by another 10 (so multiplying x by 1000) moves the decimal point one more place, 1000x= 583.3333333... but to the right of the decimal point we still have just ".3333333...". Notice that the "3"s never stop so the right side remains the same. Subtracting, the right sides cancel leaving only integers:
1000x- 100x= (583.33333...)- (58.333333...)
900x= 525

\(\displaystyle x= \frac{525}{900}\)

Of course, that can be reduced now
\(\displaystyle x= \frac{525}{900}= \frac{105}{180}= \frac{21}{36}= \frac{7}{12}\).