Write the repeating decimal as a fraction. .219219219 ______

Rephrase Problem: Let's first rephrase the ＂Convert the repeating decimal 0.219219219... to a fraction.＂Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits 219 repeat indefinitely, so we can express the decimal as 0.219219219...Define Variable: Let x be the value of the repeating decimal: x = 0.219219219...Shift Decimal Point: Multiply x by 1000 to shift the decimal point three places to the right, since the repeating block is three digits long: 1000x = 219.219219219...Subtract Decimals: Subtract the original x from the 1000x to eliminate the repeating decimals: 1000x - x = 219.219219219... - 0.219219219...Solve for x: Perform the subtraction: 999x = 219Simplify Fraction: Divide both sides of the equation by 999 to solve for x: x = 219/999Find GCD: Simplify the fraction by finding the greatest common divisor (GCD) of 219 and 999. The GCD of 219 and 999 is 3.Divide by GCD: Divide both the numerator and the denominator by the GCD to simplify the fraction: (219 ÷ 3) / (999 ÷ 3) = 73/333

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Write the repeating decimal as a fraction. $\newline$ $.219219219$

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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

.219219219

Rephrase Problem: Let's first rephrase the ＂Convert the repeating decimal $0.219219219\ldots$ to a fraction.＂
Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits $219$ repeat indefinitely, so we can express the decimal as $0.219219219\ldots$
Define Variable: Let $x$ be the value of the repeating decimal: $x = 0.219219219\ldots$
Shift Decimal Point: Multiply $x$ by $1000$ to shift the decimal point three places to the right, since the repeating block is three digits long: $1000x = 219.219219219\ldots$
Subtract Decimals: Subtract the original $x$ from the $1000x$ to eliminate the repeating decimals: $1000x - x = 219.219219219... - 0.219219219...$
Solve for x: Perform the subtraction: $999x = 219$
Simplify Fraction: Divide both sides of the equation by $999$ to solve for $x$ : $x = \frac{219}{999}$
Find GCD: Simplify the fraction by finding the greatest common divisor (GCD) of $219$ and $999$ . The GCD of $219$ and $999$ is $3$ .
Divide by GCD: Divide both the numerator and the denominator by the GCD to simplify the fraction: $\frac{219 \div 3}{999 \div 3} = \frac{73}{333}$