To find the result of the expression [tex]\(2.8 + 7.\overline{2}\)[/tex], we can break it down into more manageable parts and then combine those parts for the final sum.
1. Understanding the Numbers:
- [tex]\(2.8\)[/tex] is a straightforward decimal number.
- [tex]\(7.\overline{2}\)[/tex] is a repeating decimal where "2" repeats indefinitely. This can also be expressed as [tex]\(7 + 0.2222...\)[/tex].
2. Converting the Repeating Decimal:
- We know that [tex]\(0.\overline{2}\)[/tex] (that is, 0.2222...) equals [tex]\(\frac{2}{9}\)[/tex]. Therefore, [tex]\(7.\overline{2}\)[/tex] can be expressed as [tex]\(7 + \frac{2}{9}\)[/tex].
3. Breaking Down the Parts:
- [tex]\(2.8\)[/tex] remains as is.
- [tex]\(7.\overline{2}\)[/tex] now separates into [tex]\(7\)[/tex] (the whole part) and [tex]\(\frac{2}{9}\)[/tex] (the repeating decimal part).
4. Summing the Parts:
- Start by adding the integer and decimal parts separately.
- Combine [tex]\(2.8\)[/tex] with [tex]\(7\)[/tex]:
[tex]\[
2.8 + 7 = 9.8
\][/tex]
- Next, add the fractional part [tex]\(\frac{2}{9}\)[/tex] to [tex]\(9.8\)[/tex]:
[tex]\[
9.8 + \frac{2}{9}
\][/tex]
- Convert [tex]\(9.8\)[/tex] to a fraction to add easily:
[tex]\[
9.8 = \frac{98}{10}
\][/tex]
- Now we need to find a common denominator to add:
[tex]\[
9.8 = \frac{98}{10} = \frac{882}{90}
\][/tex]
[tex]\[
\frac{2}{9} = \frac{20}{90}
\][/tex]
- Sum the fractions:
[tex]\[
\frac{882}{90} + \frac{20}{90} = \frac{902}{90}
\][/tex]
- Simplify the fraction if possible:
[tex]\[
\frac{902}{90} = 10.0222222222...
\][/tex]
Thus, the final decimal value for [tex]\(2.8 + 7.\overline{2}\)[/tex] is:
[tex]\[
\boxed{10.0222222222}
\][/tex]
