Question

Express $0.\overline{5}$ as a common fraction in lowest terms.

Answer 1

Answer:  5/9

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Explanation:

$0.\overline{5}$ means that the 5's go on forever because of that horizontal bar over top. So we can write it as $0.\overline{5} = 0.55555\ldots$

The three dots indicate it goes on forever following that pattern.

Let

x = 0.55555....

Multiply both sides by 10 to move the decimal point 1 spot to the right

10x = 5.55555....

Notice how both x and 10x involve a decimal number such that we have a string of 5's going on forever. If we subtract the two equations, then 10x-x becomes 9x, while the (5.55555....) - (0.55555....) simplifies to 5. The decimal portions cancel out when we subtract since they line up perfectly. We're effectively subtracting 5-0 when we cross off the decimal portions.

After those subtractions, we're left with 9x = 5 which solves to x = 5/9 when you divide both sides by 9.

Use of a calculator should show that 5/9 = 0.555555.... to help confirm the answer. Your calculator may show the last digit to be a 6 instead of a 5, but this is due to rounding. Ideally you should have a string of infinitely many 5's, but the calculator can only how so many digits.