Write the repeating decimal as a fraction. .271271271 ______

Denote Repeating Decimal: Let's denote the repeating decimal 0.271271271... by x. x = 0.271271271... To convert this repeating decimal into a fraction, we can use the following strategy: multiply x by a power of 10 that matches the length of the repeating sequence to shift the decimal point to the right, then subtract the original number from this result to eliminate the repeating part.Convert to Fraction: First, we identify the repeating sequence, which is 271. The length of this sequence is 3 digits. Therefore, we multiply x by 10^3 (which is 1000) to shift the repeating sequence to the left of the decimal point. 1000x = 271.271271...Multiply by Power of 10: Next, we subtract the original number x from 1000x to get rid of the repeating decimal part. 1000x - x = 271.271271... - 0.271271271... This simplifies to: 999x = 271Subtract Original Number: Now, we solve for x by dividing both sides of the equation by 999. x = 271 / 999Solve for x: We should check if the fraction can be simplified. The numbers 271 and 999 do not have any common factors other than 1, so the fraction is already in its simplest form.

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Write the repeating decimal as a fraction. $\newline$ $.271271271$

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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

.271271271

Denote Repeating Decimal: Let's denote the repeating decimal $0.271271271\ldots$ by $x$ . $x = 0.271271271\ldots$ To convert this repeating decimal into a fraction, we can use the following strategy: multiply $x$ by a power of $10$ that matches the length of the repeating sequence to shift the decimal point to the right, then subtract the original number from this result to eliminate the repeating part.
Convert to Fraction: First, we identify the repeating sequence, which is $271$ . The length of this sequence is $3$ digits. Therefore, we multiply $x$ by $10^3$ (which is $1000$ ) to shift the repeating sequence to the left of the decimal point. $\newline$ $1000x = 271.271271\ldots$
Multiply by Power of $10$ : Next, we subtract the original number $x$ from $1000x$ to get rid of the repeating decimal part. $\newline$ $1000x - x = 271.271271... - 0.271271271...$ $\newline$ This simplifies to: $\newline$ $999x = 271$
Subtract Original Number: Now, we solve for $x$ by dividing both sides of the equation by $999$ . $x = \frac{271}{999}$
Solve for $x$ : We should check if the fraction can be simplified. The numbers $271$ and $999$ do not have any common factors other than $1$ , so the fraction is already in its simplest form.