Answer:
8/3
Step-by-step explanation:
Let S represent the value of 2.666...(repeated). Then 10S will have the value 26.666...(repeated). Subtracting S from 10S, we have ...
10S -S = 26.666... - 2.666... = 24
9S = 24
S = 24/9 = 8/3
The improper fraction equivalent of 2.666...(repeated) is 8/3.
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In general, if the number of repeated digits is n, you do this procedure multiplying the number by 10^n before you do the subtraction. The result is a fraction with (10^n)-1 as a denominator. That is, for a 2-digit repeat, the denominator will be 99; for a 6-digit repeat, 999999.
Knowing this, you can immediately recognize that 0.6...(repeated) = 6/9 = 2/3. Then your number 2.666...(repeated) is 2 2/3 = 8/3.
