Write the repeating decimal as a fraction. .82828282 ______

Denote Repeating Decimal: Let's denote the repeating decimal 0.82828282... by x. x = 0.82828282... To convert this repeating decimal into a fraction, we can use algebra to create an equation that we can solve for x.Multiply by 100: First, we multiply x by 100 to shift the decimal two places to the right, since the repeating pattern is two digits long. 100x = 82.828282... This step ensures that when we subtract x from 100x, the decimal parts will align and the repeating parts will cancel out.Subtract to Eliminate Repeating Decimal: Now, we subtract the original x from 100x to get rid of the repeating decimal part. 100x - x = 82.828282... - 0.82828282... 99x = 82 We are left with a simple equation without the repeating decimal.Solve for x: Next, we solve for x by dividing both sides of the equation by 99. x = 82 / 99 This gives us the fraction form of the repeating decimal.Check for Simplification: Finally, we check if the fraction can be simplified. However, since 82 and 99 have no common factors other than 1, the fraction is already in its simplest form.

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Write the repeating decimal as a fraction. $\newline$ $.82828282$

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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

.82828282

Denote Repeating Decimal: Let's denote the repeating decimal $0.82828282\ldots$ by $x$ . $x = 0.82828282\ldots$ To convert this repeating decimal into a fraction, we can use algebra to create an equation that we can solve for $x$ .
Multiply by $100$ : First, we multiply $x$ by $100$ to shift the decimal two places to the right, since the repeating pattern is two digits long. $\newline$ $100x = 82.828282...$ $\newline$ This step ensures that when we subtract $x$ from $100x$ , the decimal parts will align and the repeating parts will cancel out.
Subtract to Eliminate Repeating Decimal: Now, we subtract the original $x$ from $100x$ to get rid of the repeating decimal part. $\newline$ $100x - x = 82.828282... - 0.82828282...$ $\newline$ $99x = 82$ $\newline$ We are left with a simple equation without the repeating decimal.
Solve for x: Next, we solve for $x$ by dividing both sides of the equation by $99$ . $x = \frac{82}{99}$ This gives us the fraction form of the repeating decimal.
Check for Simplification: Finally, we check if the fraction can be simplified. However, since $82$ and $99$ have no common factors other than $1$ , the fraction is already in its simplest form.