Write the repeating decimal as a fraction. .035035035 ______

Define x as decimal: Let x be the repeating decimal 0.035035035... We can write this as: x = 0.035035035...Create equation with repeating part: To convert the repeating decimal to a fraction, we need to create an equation that isolates the repeating part. Since the digits 035 repeat every three decimal places, we multiply x by 10^3 (which is 1000) to shift the decimal point three places to the right. So, we get: 1000x = 35.035035035...Subtract original x from 1000x: Now we subtract the original x from the 1000x to get rid of the repeating decimals: 1000x - x = 35.035035035... - 0.035035035... This simplifies to: 999x = 35Divide by 999 to find x: To find the value of x, we divide both sides of the equation by 999: x = 35 / 999Simplify fraction for x: We can simplify the fraction by looking for the greatest common divisor (GCD) of 35 and 999. The GCD of 35 and 999 is 1, so the fraction is already in its simplest form. x = 35 / 999

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Write the repeating decimal as a fraction. $\newline$ $.035035035$

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Precalculus

Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.035035035

Define $x$ as decimal: Let $x$ be the repeating decimal $0.035035035\ldots$ $\newline$ We can write this as: $\newline$ $x = 0.035035035\ldots$
Create equation with repeating part: To convert the repeating decimal to a fraction, we need to create an equation that isolates the repeating part. Since the digits $035$ repeat every three decimal places, we multiply $x$ by $10^3$ (which is $1000$ ) to shift the decimal point three places to the right. $\newline$ So, we get: $\newline$ $1000x = 35.035035035\ldots$
Subtract original x from $1000$ x: Now we subtract the original x from the $1000x$ to get rid of the repeating decimals: $\newline$ $1000x - x = 35.035035035... - 0.035035035...$ $\newline$ This simplifies to: $\newline$ $999x = 35$
Divide by $999$ to find x: To find the value of x, we divide both sides of the equation by $999$ : $\newline$ $x = \frac{35}{999}$
Simplify fraction for $x$ : We can simplify the fraction by looking for the greatest common divisor (GCD) of $35$ and $999$ . The GCD of $35$ and $999$ is $1$ , so the fraction is already in its simplest form. $\newline$ $x = \frac{35}{999}$