To find the sum of [tex]\(2.8 + 7.\overline{2}\)[/tex], let's break down and combine the individual components step by step.
### Step 1: Understanding [tex]\(7.\overline{2}\)[/tex] as a Fraction
First, we need to express the repeating decimal [tex]\(7.\overline{2}\)[/tex] (which is [tex]\(7.2222...\)[/tex]) as a fraction.
1. Let [tex]\(x = 7.\overline{2}\)[/tex].
2. Then [tex]\(10x = 72.2222...\)[/tex].
3. Subtract the first equation from the second:
[tex]\[
10x - x = 72.2222... - 7.2222...
\][/tex]
[tex]\[
9x = 65
\][/tex]
4. Solving for [tex]\(x\)[/tex], we get:
[tex]\[
x = \frac{65}{9} = 7.2222...
\][/tex]
### Step 2: Converting 2.8 to a Fraction
Next, we convert the decimal [tex]\(2.8\)[/tex] to a fraction:
[tex]\[
2.8 = \frac{28}{10} = \frac{14}{5}
\][/tex]
### Step 3: Finding a Common Denominator
To add these fractions, we need a common denominator. The denominators are 9 and 5. The least common multiple of 9 and 5 is 45.
Converting the fractions to have this common denominator:
1. For [tex]\(\frac{14}{5}\)[/tex]:
[tex]\[
\frac{14}{5} \times \frac{9}{9} = \frac{126}{45}
\][/tex]
2. For [tex]\(\frac{65}{9}\)[/tex]:
[tex]\[
\frac{65}{9} \times \frac{5}{5} = \frac{325}{45}
\][/tex]
### Step 4: Adding the Fractions
Now, we can add the fractions:
[tex]\[
\frac{126}{45} + \frac{325}{45} = \frac{126 + 325}{45} = \frac{451}{45}
\][/tex]
### Step 5: Converting the Sum to Decimal
Finally, we convert the fraction [tex]\(\frac{451}{45}\)[/tex] to a decimal:
[tex]\[
\frac{451}{45} = 10.0222...
\][/tex]
### Conclusion
The result of [tex]\(2.8 + 7.\overline{2}\)[/tex] as a decimal is:
[tex]\[
\boxed{10.0222...}
\][/tex]
