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

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

. \overline{2}

\newline

Answer:

Set $x$ as repeating decimal: Let $x$ equal the repeating decimal $0.2\overline{2}$ . We express this algebraically as: $\newline$ $x = 0.2\overline{2}$
Convert to fraction: To convert the repeating decimal to a fraction, we can use the fact that the digit $2$ repeats indefinitely. We multiply $x$ by $10$ to shift the decimal point to the right, which gives us: $\newline$ $10x = 2.\overline{2}$
Subtract equations: Now we have two equations: $\newline$ $1$ ) $x = 0.\overline{2}$ $\newline$ $2$ ) $10x = 2.\overline{2}$ $\newline$ We subtract equation $1$ from equation $2$ to eliminate the repeating decimals: $\newline$ $10x - x = 2.\overline{2} - 0.\overline{2}$
Solve for x: Performing the subtraction, we get: $9x = 2$
Check for simplification: To solve for $x$ , we divide both sides of the equation by $9$ : $x = \frac{2}{9}$
Check for simplification: To solve for $x$ , we divide both sides of the equation by $9$ : $x = \frac{2}{9}$ We check to see if the fraction $\frac{2}{9}$ can be simplified further. Since $2$ and $9$ have no common factors other than $1$ , the fraction is already in its simplest form.

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