To convert the repeating decimal [tex]\( 0.\overline{49} \)[/tex] into a fraction, follow these steps:
1. Let [tex]\( x = 0.\overline{49} \)[/tex]:
This means that [tex]\( x \)[/tex] is equal to the repeating decimal [tex]\( 0.494949494949... \)[/tex].
2. Multiply both sides of the equation by 100:
Multiplying by 100 shifts the decimal point two places to the right:
[tex]\[
100x = 49.494949494949...
\][/tex]
3. Subtract the original equation from this new equation:
[tex]\[
100x = 49.494949494949...
\][/tex]
[tex]\[
- \quad x = 0.494949494949...
\][/tex]
[tex]\[
99x = 49
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{49}{99}
\][/tex]
So [tex]\( 0.\overline{49} \)[/tex] as a fraction is [tex]\( \frac{49}{99} \)[/tex].
Next, you might simplify the fraction. In this case, [tex]\( \frac{49}{99} \)[/tex] is already in its simplest form because the greatest common divisor (GCD) of 49 and 99 is 1.
Thus, [tex]\( 0.\overline{49} \)[/tex] as a fraction is:
[tex]\[
\boxed{\frac{49}{99}}
\][/tex]
