Decimal to fraction ?

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2 Answers
Jan 20, 2018

=33/101.

Explanation:

In general, when you have a number of the form 0.[repeating part] you can convert it to the fraction

n/(10^c-1)

where n is the repeating part (the integer created by the decimal digits of one repetition, in our case it is 3267) and c is the number of digits in the repeating part. It's equally easy with the form

a.[repeating part], you just get

a+n/(10^c-1)

The reason this works is because if you multiply that repeating decimal by 10^c, you'll get n plus the repeating decimal. In our case, c=4 and n=3267:

0.3267...*10^4=3267.3267...=3267+0.3267...

Therefore,

0.3267...*10^4-0.3267...=3267

0.3267...*(10^4-1)=3267

0.3267...=3267/(10^4-1)=3267/9999=33/101

It's also simple to find 10^c-1, just write c nines.

Jan 20, 2018

33/101

Explanation:

0. color(red)(3267)32673267....=0. bar(3267)

"the bar represents the digits being repeated"

"these form the terms of a "color(blue)"geometric sequence"

3267/10000,3267/100000000,...

"with "a=3267/10000

"and "r=a_2/a_1

rArrr=cancel(3267)/(10000cancel(0000))xx(1cancel(0000))/cancel(3267)=1/10000

•color(white)(x)S_oo=a/(1-r)

rArrS_oo=(3267/10000)/(1-1/10000)

=(3267/10000)/(9999/10000)

=3267/9999=33/101