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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

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

$Equation(2)-Equation(1)$

$1000x-x=5137$

$x(1000-1)=5137$

$x=5137/999$