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What is 0.166 (repeating) as a fraction?

Answer 1

It can be written as #166/999#. See expanation for details.

The task is not complete because you did not indicate which part of the number is repeating. I solve it as if #166# was the period.

Note: to indicate the period of such decimals you can either put it in brackets: #0.(166)# or write a horizontal bar over the fraction's period: #0.bar(166)# without hashtag it would be 0.bar(166)

Solution

#0.bar(166)=0.166166166166...#, so it can be written as an infinite sum:

#0.bar(166)=0.166+0.000166+0.000000166+...#

From the last sum you can see that it is a sum of an infinite geometrical sequence, where: #a_1=0.166, q=0.001#

Since #q in (-1;1)# the sequence is convergent, so you can use the formula to calculate the sum:

#S=a_1/(1-q)#

#S=0.166/(1-0.001)#

#S=0.166/0.999#

In order to convert the fraction's denominator and numerator to integer numbers, we must now expand it by 1000:

#S=166/999#

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Answer 2

Eva Adamson

0.166 (repeating) as a fraction is ( \frac{1}{6} ).

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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.