To write [tex]\(3.\overline{7}\)[/tex] as a mixed number, follow these steps:
1. Let [tex]\( x = 3.\overline{7} \)[/tex].
2. Multiply by 10 to shift the repeating decimal:
[tex]\[
10x = 37.\overline{7}
\][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this value:
[tex]\[
10x - x = 37.\overline{7} - 3.\overline{7}
\][/tex]
[tex]\[
9x = 34
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{34}{9}
\][/tex]
5. Convert the improper fraction to a mixed number. Divide 34 by 9:
[tex]\[
34 \div 9 = 3 \text{ remainder } 7
\][/tex]
This results in the mixed number:
[tex]\[
3 \frac{7}{9}
\][/tex]
So, [tex]\(3.\overline{7}\)[/tex] is equal to:
[tex]\[
3 \frac{7}{9}
\][/tex]
