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$Convert the following repeating decimal to a fraction in simplest form. .9 bar(2) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.9 \overline{2}$ $\newline$ Answer:

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Partial sums of geometric series

Full solution

Q. Convert the following repeating decimal to a fraction in simplest form.

\newline

.9 \overline{2}

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Answer:

Define repeating decimal: Let $x$ be the repeating decimal $0.9$ with a bar over the $2$ . We can express this repeating decimal as $x = 0.9222\ldots$
Multiply by $10$ : To isolate the repeating part, we multiply $x$ by $10$ , since the repeating part starts after the first decimal place. This gives us $10x = 9.2222\ldots$
Multiply by $1000$ : Now, we multiply $x$ by $1000$ , because the repeating part is a single digit and we want to shift the decimal point three places to the right to align the repeating parts. This gives us $1000x = 922.2222\ldots$
Subtract to eliminate decimals: We subtract the equation from Step $2$ $10x$ from the equation in Step $3$ $1000x$ to get rid of the repeating decimals. This results in $990x = 922.2222... - 9.2222...$ which simplifies to $990x = 913$ .
Solve for x: We solve for x by dividing both sides of the equation by $990$ . This gives us $x = \frac{913}{990}$ .
Simplify fraction: We simplify the fraction by finding the greatest common divisor (GCD) of $913$ and $990$ . The GCD of $913$ and $990$ is $1$ , so the fraction is already in its simplest form.
Write final answer: We write down the final answer as a fraction in simplest form. The fraction is $\frac{913}{990}$ .

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