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Answer 1 · 2017-10-03T08:41:23

#0.21bar(21) = 21/99#

#5.142bar(142) =5137/999#

#color(blue)("Comment")#
The way to depict repeating parts of a decimal is to but a bar over the part that repeats.

so #5.142142142.... -> 5.bar142# or some variation on it.

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#color(blue)("Solution to part A")#

Set #x=0.212121 .......Equation(1)#

Then

#100x=21.212121 ....Equation(2)#

#Equation(2)-Equation(1)# gives

#100x-x=21#

#x(100-1)=21#

#99x=21#

#x=21/99#
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#color(blue)("Solution to part B")#

Set #x=5.142142142 ......Equation(1)#

Then

#1000x=5142.142142142 .....Equation(2)#

#Equation(2)-Equation(1)#

#1000x-x=5137#

#x(1000-1)=5137#

#x=5137/999#