To convert the repeating decimal [tex]\( 0.\overline{14} \)[/tex] into a fraction, follow these steps:
1. Set up the repeating decimal as a variable:
Let [tex]\( x = 0.141414... \)[/tex].
2. Shift the decimal point to the right:
To move the decimal two places to the right, multiply [tex]\( x \)[/tex] by 100:
[tex]\[
100x = 14.141414...
\][/tex]
3. Set up the two equations:
From the above steps, we have:
[tex]\[
x = 0.141414...
\][/tex]
[tex]\[
100x = 14.141414...
\][/tex]
4. Subtract the first equation from the second:
To eliminate the repeating part, subtract the equation [tex]\( x = 0.141414... \)[/tex] from [tex]\( 100x = 14.141414... \)[/tex]:
[tex]\[
100x - x = 14.141414... - 0.141414...
\][/tex]
[tex]\[
99x = 14
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by 99:
[tex]\[
x = \frac{14}{99}
\][/tex]
6. Simplify the fraction (if possible):
Identify the greatest common divisor (GCD) for the numerator and the denominator. Here, both 14 and 99 do not share any common divisor other than 1, so the fraction is already in its simplest form:
[tex]\[
\frac{14}{99}
\][/tex]
Therefore, the repeating decimal [tex]\( 0.\overline{14} \)[/tex] as a fraction is [tex]\( \frac{14}{99} \)[/tex].
