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#