To convert the repeating decimal [tex]\(0.\overline{721}\)[/tex] into a fraction, let's follow these steps:
1. Assign a variable to the repeating decimal:
Let [tex]\( x = 0.\overline{721} \)[/tex].
2. Multiply the equation to shift the decimal point:
Since the repeating part has three digits, multiply both sides of the equation by 1000 to shift the decimal point three places to the right.
[tex]\[
1000x = 721.\overline{721}
\][/tex]
3. Set up an equation to eliminate the repeating part:
We now have two equations:
[tex]\[
x = 0.\overline{721}
\][/tex]
[tex]\[
1000x = 721.\overline{721}
\][/tex]
Subtract the first equation from the second equation to eliminate the repeating decimal part:
[tex]\[
1000x - x = 721.\overline{721} - 0.\overline{721}
\][/tex]
This simplifies as:
[tex]\[
999x = 721
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{721}{999}
\][/tex]
5. Simplify the fraction if possible:
The fraction [tex]\(\frac{721}{999}\)[/tex] is already in its simplest form, as the greatest common divisor of 721 and 999 is 1.
Therefore, the repeating decimal [tex]\(0.\overline{721}\)[/tex] can be written as the fraction [tex]\(\frac{721}{999}\)[/tex].
