To convert the repeating decimal [tex]\(7.\overline{4}\)[/tex] to a fraction, follow these steps:
1. Let [tex]\( x \)[/tex] represent the repeating decimal: [tex]\( x = 7.\overline{4} \)[/tex].
2. To isolate the repeating part, multiply [tex]\( x \)[/tex] by 10. This gives us:
[tex]\[ 10x = 74.\overline{4} \][/tex]
3. Now we have two equations:
[tex]\[
\begin{cases}
x = 7.\overline{4} \\
10x = 74.\overline{4}
\end{cases}
\][/tex]
4. Subtract the first equation from the second equation to get rid of the repeating part:
[tex]\[
10x - x = 74.\overline{4} - 7.\overline{4}
\][/tex]
5. Simplifying the left side and the right side results in:
[tex]\[
9x = 67
\][/tex]
6. Solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{67}{9}
\][/tex]
Thus, the repeating decimal [tex]\(7.\overline{4}\)[/tex] converts to the fraction [tex]\(\frac{67}{9}\)[/tex].
To summarize, the fraction in its simplest form representing the repeating decimal [tex]\(7.\overline{4}\)[/tex] is [tex]\( \frac{67}{9} \)[/tex], and the repeating decimal part [tex]\(7.444444\)[/tex] simplifies directly to this fraction.
