9514 1404 393
Answer:
0.666... = 6/9 = 2/3
Step-by-step explanation:
The repeating decimal 0.666... can be turned to a fraction as follows.
Let x represent the value of the number. Then 10x is ...
10x = 10(0.666...) = 6.666...
Subtracting x gives ...
10x -x = 6.666... - 0.666... = 6.000...
9x = 6
Dividing by 9 and simplifying, we have ...
x = 6/9 = 2/3
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It can be worthwhile to remember that 1/9 = 0.111... repeating.
Then 3/9 = 1/3 = 3(0.111...) = 0.333... repeating
and 6/9 = 2/3 = 6(0.111...) = 0.666... repeating
Most calculators will round the last digit of 2/3 so that it is a 7. So, you may see ...
2 ÷ 3 = 0.6666667 or 0.66666666667
or something similar, depending on your calculator. It is up to you to recognize this as probably a representation of 2/3. Some calculators will display the number as a fraction, so you don't have to guess.
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Additional comment
A similar method is used to convert repeating decimals of other lengths to fractions. In general you multiply by 10^n, where n is the number of repeating digits. Then proceed as above: subtract x, divide by the coefficient of x. You will find you are always dividing by a number that has as many 9's as there are repeating digits.
