Sure! Let’s break this down step-by-step for the given question which involves adding the fractions [tex]\(\frac{8}{9}\)[/tex], [tex]\(\frac{5}{9}\)[/tex], and [tex]\(\frac{2}{3}\)[/tex].
### Step 1: Identify the Common Denominator
Since two of the fractions already share a common denominator (9), we should aim to convert [tex]\(\frac{2}{3}\)[/tex] to a fraction with the same common denominator of 9.
### Step 2: Convert [tex]\(\frac{2}{3}\)[/tex] to a Fraction with Denominator 9
To convert [tex]\(\frac{2}{3}\)[/tex] so that its denominator is 9, multiply both the numerator and the denominator by 3:
[tex]\[
\frac{2 \times 3}{3 \times 3} = \frac{6}{9}
\][/tex]
Now we have:
[tex]\[
\frac{2}{3} = \frac{6}{9}
\][/tex]
### Step 3: Add the Fractions Together
Now that all fractions have a common denominator, add their numerators:
[tex]\[
\frac{8}{9} + \frac{5}{9} + \frac{6}{9}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{8 + 5 + 6}{9}
\][/tex]
### Step 4: Sum of the Numerators
Add the numerators together:
[tex]\[
8 + 5 + 6 = 19
\][/tex]
So, we have:
[tex]\[
\frac{19}{9}
\][/tex]
### Step 5: Simplify the Fraction (if applicable)
Since [tex]\(\frac{19}{9}\)[/tex] is an improper fraction (the numerator is greater than the denominator), we can also express it as a mixed number:
[tex]\[
19 \div 9 = 2 \text{ remainder } 1
\][/tex]
So, [tex]\(\frac{19}{9}\)[/tex] can be written as:
[tex]\[
2 \frac{1}{9}
\][/tex]
### Conclusion
Thus, the answer to the question [tex]\(\frac{8}{9} + \frac{5}{9} + \frac{2}{3}\)[/tex] is [tex]\(\frac{19}{9}\)[/tex] or equivalently [tex]\(2 \frac{1}{9}\)[/tex].
The final result is:
[tex]\[
2.111111111111111
\][/tex]
