Write the repeating decimal as a fraction. .642642642 ______

Denote Decimal as x: Let's denote the repeating decimal 0.642642642... as x. x = 0.642642642... To isolate the repeating part, we multiply x by 1000 because there are three digits in the repeating sequence. 1000x = 642.642642642... Now, we subtract the original x from 1000x to get rid of the decimal part. 1000x - x = 642.642642642... - 0.642642642... This simplifies to: 999x = 642 Now, we can solve for x by dividing both sides by 999. x = 642 / 999Isolate Repeating Part: We can simplify the fraction 642/999 by looking for the greatest common divisor (GCD) of 642 and 999. The GCD of 642 and 999 is 3. Now we divide both the numerator and the denominator by 3 to simplify the fraction. 642 ÷ 3 = 214 999 ÷ 3 = 333 So, x = 214 / 333Simplify Fraction: We check if the fraction 214/333 can be simplified further. The GCD of 214 and 333 is 1, which means that the fraction is already in its simplest form. Therefore, the repeating decimal 0.642642642... as a fraction is 214/333.

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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.

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.642642642

Denote Decimal as $x$ : Let's denote the repeating decimal $0.642642642\ldots$ as $x$ .
$x = 0.642642642\ldots$
To isolate the repeating part, we multiply $x$ by $1000$ because there are three digits in the repeating sequence.
$1000x = 642.642642642\ldots$
Now, we subtract the original $x$ from $1000x$ to get rid of the decimal part.
$1000x - x = 642.642642642\ldots - 0.642642642\ldots$
This simplifies to:
$0.642642642\ldots$ $0$
Now, we can solve for $x$ by dividing both sides by $0.642642642\ldots$ $2$ .
$0.642642642\ldots$ $3$
Isolate Repeating Part: We can simplify the fraction $\frac{642}{999}$ by looking for the greatest common divisor (GCD) of $642$ and $999$ . The GCD of $642$ and $999$ is $3$ . Now we divide both the numerator and the denominator by $3$ to simplify the fraction. $642 \div 3 = 214$ $999 \div 3 = 333$ So, $x = \frac{214}{333}$
Simplify Fraction: We check if the fraction $\frac{214}{333}$ can be simplified further. $\newline$ The GCD of $214$ and $333$ is $1$ , which means that the fraction is already in its simplest form. $\newline$ Therefore, the repeating decimal $0.642642642\ldots$ as a fraction is $\frac{214}{333}.$