Question #65c13

2 Answers
Wataru
Apr 14, 2017

#133/99#

Explanation:

Recall (Sum of Geometric Series Formula):

#a + ar + ar^2+ar^3+cdots=a/(1-r)# if #|r| < 1#

We can view a repeated decimal as the sum of a geometric series.

#1.343434...=1+[0.34+0.0034+0.000034+cdots]#

#=1+[34/100+34/100(1/100)+34/100(1/100)^2+cdots]#

By applying the formula above with #a=34/100# and #r=1/100# starting with the second term,

#=1+(34/100)/(1-1/100) =1+(34/100)/(99/100) =99/99+34/99=133/99#

Jim G.
Apr 14, 2017

#133/99#

Explanation:

Obtain 2 equations with the same repeating part then subtract them.

#"We can represent the repeated part by " 1.bar34#

#x=1.bar34to(1)larrcolor(red)" multiply by 100"#

#100x=134.bar34to(2)#

#"Subtract " (2)-(1)#

#(100x-x)=(134.bar34-1.bar34)#

#rArr99x=133larrcolor(red)" repeating part eliminated"#

#rArrx=133/99larrcolor(red)"in simplest form"#

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