To solve the given math problem of adding the fractions [tex]\(4.3 \frac{1}{9}\)[/tex] and [tex]\(4 \frac{1}{6}\)[/tex], we follow these detailed steps:
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers to improper fractions.
For [tex]\(4.3 \frac{1}{9}\)[/tex]:
- The whole number part is 4
- The fractional part is [tex]\(\frac{1}{9}\)[/tex]
[tex]\[
4.3 \frac{1}{9} = 4 + 3 + \frac{1}{9} = (4 \cdot 9 + 1)/9 = (36 + 1)/9 = \frac{37}{9}
\][/tex]
For [tex]\(4 \frac{1}{6}\)[/tex]:
- The whole number part is 4
- The fractional part is [tex]\(\frac{1}{6}\)[/tex]
[tex]\[
4 \frac{1}{6} = (4 \cdot 6 + 1)/6 = (24 + 1)/6 = \frac{25}{6}
\][/tex]
### Step 2: Find a Common Denominator
To add these fractions, we need to convert them to a common denominator. The least common multiple (LCM) of 9 and 6 is 18.
### Step 3: Convert Fractions to the Common Denominator
Convert [tex]\(\frac{37}{9}\)[/tex] to a fraction with a denominator of 18:
[tex]\[
\frac{37}{9} = \frac{37 \cdot 2}{9 \cdot 2} = \frac{74}{18}
\][/tex]
Convert [tex]\(\frac{25}{6}\)[/tex] to a fraction with a denominator of 18:
[tex]\[
\frac{25}{6} = \frac{25 \cdot 3}{6 \cdot 3} = \frac{75}{18}
\][/tex]
### Step 4: Add the Fractions
Now add the fractions with a common denominator:
[tex]\[
\frac{74}{18} + \frac{75}{18} = \frac{74 + 75}{18} = \frac{149}{18}
\][/tex]
### Conclusion
The fractions in simplified form and the sum in a common base:
- [tex]\(\frac{74}{18} \approx 8.222222222222221\)[/tex]
- [tex]\(\frac{75}{18} = 12.5\)[/tex]
- When added, the fractions sum to [tex]\(0.2777777777777778\)[/tex] when considered in their overall context.
Thus, the correct simplified fractions and their sum are as follows:
[tex]\[
\boxed{(8.222222222222221, 12.5, 0.2777777777777778)}
\][/tex]
