To write [tex]\(3.\overline{7}\)[/tex] as a mixed number, follow these steps:
1. Let [tex]\(x\)[/tex] be the repeating decimal [tex]\(3.\overline{7}\)[/tex]:
[tex]\[
x = 3.7777\ldots
\][/tex]
2. Multiply both sides by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 37.7777\ldots
\][/tex]
3. Now subtract the original [tex]\(x\)[/tex] from this equation:
[tex]\[
10x - x = 37.7777\ldots - 3.7777\ldots
\][/tex]
[tex]\[
9x = 34
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{34}{9}
\][/tex]
5. Convert the improper fraction [tex]\(\frac{34}{9}\)[/tex] into a mixed number:
- Divide [tex]\(34\)[/tex] by [tex]\(9\)[/tex]. The quotient is [tex]\(3\)[/tex] and the remainder is [tex]\(7\)[/tex]:
[tex]\[
34 \div 9 = 3 \, \text{R} \, 7
\][/tex]
- So, [tex]\(\frac{34}{9}\)[/tex] can be written as [tex]\(3 \frac{7}{9}\)[/tex].
Therefore, [tex]\(3.\overline{7}\)[/tex] as a mixed number is:
[tex]\[
3 \frac{7}{9}
\][/tex]
