Certainly! Let's transform the repeating decimal [tex]\( 2.\overline{81} \)[/tex] into a fraction.
1. Let [tex]\( x \)[/tex] represent the repeating decimal:
[tex]\[
x = 2.818181\ldots
\][/tex]
2. To eliminate the repeating part, multiply both sides of the equation by 100 (since 81 repeats every two decimal places):
[tex]\[
100x = 281.818181\ldots
\][/tex]
3. Now subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[
100x - x = 281.818181\ldots - 2.818181\ldots
\][/tex]
4. This simplifies to:
[tex]\[
99x = 279
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{279}{99}
\][/tex]
6. Simplify the fraction by finding the greatest common divisor (GCD) of 279 and 99. The GCD of 279 and 99 is 9.
7. Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{279 \div 9}{99 \div 9} = \frac{31}{11}
\][/tex]
Therefore, the repeating decimal [tex]\( 2.\overline{81} \)[/tex] expressed as a quotient of two integers is [tex]\( \frac{31}{11} \)[/tex].
