To solve the problem [tex]\( 2.8 + 7.\overline{2} \)[/tex], follow these detailed steps:
1. Understand the Components:
- The first number is 2.8.
- The second number is 7.[tex]\(\overline{2}\)[/tex], which means 7.2222... where the 2 repeats infinitely.
2. Convert the Repeating Decimal to a Fraction:
- Recognize that 7.[tex]\(\overline{2}\)[/tex] can be expressed as 7 plus the repeating decimal part 0.[tex]\(\overline{2}\)[/tex].
- The repeating decimal 0.[tex]\(\overline{2}\)[/tex] can be converted to a fraction. By recognizing the pattern, 0.[tex]\(\overline{2}\)[/tex] is equivalent to [tex]\(\frac{2}{9}\)[/tex].
3. Add the Integer Part and the Fraction Together:
- Combine the integer part 7 with the fractional part [tex]\(\frac{2}{9}\)[/tex]: [tex]$ 7 + 0.\overline{2} = 7 + \frac{2}{9} $[/tex]
4. Combine Both Numbers:
- Now, add the first number [tex]\( 2.8 \)[/tex] to the sum of the second number’s parts: [tex]$ 2.8 + (7 + 0.\overline{2}) = 2.8 + 7 + 0.\overline{2} $[/tex]
- This simplifies to: [tex]$ 2.8 + 7.2222... $[/tex]
5. Perform Actual Addition:
- Adding these together:
- [tex]\( 2.8 \)[/tex]
- [tex]\( + 7.2222...\)[/tex]
- Results in [tex]\( 10.0222...\)[/tex] which continues indefinitely.
So, the sum [tex]\( 2.8 + 7.\overline{2} \)[/tex] is
[tex]\[
\boxed{10.022222222222222}
\][/tex]
