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What is #3.363636...# as a fraction?

Answer 1

#3.bar(36) = 37/11#

Note that the length of the repeating pattern is two and we can shift the number two places to the left by multiplying it by #100#.

If we then subtract the original number, it will cancel out the repeating tail and leave us with an integer.

In case you have not encountered it, writing a bar (viniculum) above a sequence of digits indicates that they repeat. So we write:

#3.363636... = 3.bar(36)#

So:

#(100-1) 3.bar(36) = 336.bar(36) - 3.bar(36) = 333#

Then dividing both ends by #(100-1)# we find:

#3.bar(36) = 333/(100-1) = 333/99 = (37*color(red)(cancel(color(black)(9))))/(11*color(red)(cancel(color(black)(9)))) = 37/11#

Alternatively, note that #0.bar(99) = 1#, so:

#3.bar(36) = 3+36/99*0.bar(99) = 3+36/99 = 3+(4*color(red)(cancel(color(black)(9))))/(11*color(red)(cancel(color(black)(9)))) = 3+4/11 = 33/11+4/11 = 37/11#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.