To convert [tex]\( 9.\overline{49} \)[/tex] to a mixed number, we start by understanding what [tex]\( 9.\overline{49} \)[/tex] represents. The notation [tex]\( 9.\overline{49} \)[/tex] means that the digits "49" repeat indefinitely.
First, consider the repeating decimal [tex]\( 0.\overline{49} \)[/tex]:
1. Let [tex]\( x = 0.\overline{49} \)[/tex].
2. To eliminate the repeating part, multiply [tex]\( x \)[/tex] by 100 (because the repeating part is two digits long):
[tex]\[
100x = 49.\overline{49}
\][/tex]
3. Subtract [tex]\( x \)[/tex] from [tex]\( 100x \)[/tex] to remove the repeating part:
[tex]\[
100x - x = 49.\overline{49} - 0.\overline{49} \implies 99x = 49
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{49}{99}
\][/tex]
Now, we know that [tex]\( 9.\overline{49} \)[/tex] can be written as:
[tex]\[
9 + 0.\overline{49} = 9 + \frac{49}{99}
\][/tex]
Next, we simplify the fraction [tex]\( \frac{49}{99} \)[/tex]:
1. Find the greatest common divisor (GCD) of 49 and 99. The GCD is 1.
2. Simplify [tex]\( \frac{49}{99} \)[/tex] by dividing both the numerator and the denominator by their GCD:
[tex]\[
\frac{49}{99} = \frac{49}{99}
\][/tex]
(Since the GCD is 1, the fraction is already in its simplest form.)
Hence, the mixed number representation of [tex]\( 9.\overline{49} \)[/tex] is:
[tex]\[
9 \frac{49}{99}
\][/tex]
