Write the repeating decimal as a fraction. .199199199 ______

Denote Repeating Decimal: Let's denote the repeating decimal 0.199199199... by x. x = 0.199199199...Multiply by Power of 10: To convert this repeating decimal into a fraction, we multiply x by a power of 10 that matches the repeating pattern. In this case, the repeating pattern is 199, which is three digits long, so we multiply x by 10^3 (which is 1000). 1000x = 199.199199199...Subtract Original Decimal: Now we subtract the original x from 1000x to get rid of the repeating decimal part. 1000x - x = 199.199199199... - 0.199199199...Perform Subtraction: Perform the subtraction on the left side of the equation. 1000x - x = 999xSimple Equation Without Decimals: Perform the subtraction on the right side of the equation. 199.199199199... - 0.199199199... = 199Solve for x: Now we have a simple equation without decimals. 999x = 199Simplify the Fraction: To solve for x, we divide both sides of the equation by 999. x = 199 / 999We can simplify the fraction by finding the greatest common divisor (GCD) of 199 and 999. The GCD of 199 and 999 is 1, so the fraction is already in its simplest form.

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Write the repeating decimal as a fraction. $\newline$ $.199199199$

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Algebra 2

Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.199199199

Denote Repeating Decimal: Let's denote the repeating decimal $0.199199199\ldots$ by $x$ . $x = 0.199199199\ldots$
Multiply by Power of $10$ : To convert this repeating decimal into a fraction, we multiply $x$ by a power of $10$ that matches the repeating pattern. In this case, the repeating pattern is $199$ , which is three digits long, so we multiply $x$ by $10^3$ (which is $1000$ ). $\newline$ $1000x = 199.199199199\ldots$
Subtract Original Decimal: Now we subtract the original $x$ from $1000x$ to get rid of the repeating decimal part. $1000x - x = 199.199199199\ldots - 0.199199199\ldots$
Perform Subtraction: Perform the subtraction on the left side of the equation. $1000x - x = 999x$
Simple Equation Without Decimals: Perform the subtraction on the right side of the equation. $\newline$ $199.199199199\ldots - 0.199199199\ldots = 199$
Solve for x: Now we have a simple equation without decimals. $\newline$ $999x = 199$
Simplify the Fraction: To solve for $x$ , we divide both sides of the equation by $999$ . $x = \frac{199}{999}$
Simplify the Fraction: To solve for $x$ , we divide both sides of the equation by $999$ . $x = \frac{199}{999}$ We can simplify the fraction by finding the greatest common divisor (GCD) of $199$ and $999$ . The GCD of $199$ and $999$ is $1$ , so the fraction is already in its simplest form.