To add the given mixed number and fraction, follow these steps:
1. Convert the mixed number [tex]\( 4 \frac{2}{3} \)[/tex] to an improper fraction.
- A mixed number [tex]\( a \frac{b}{c} \)[/tex] can be converted to an improper fraction by multiplying the whole number part ([tex]\(a\)[/tex]) by the denominator ([tex]\(c\)[/tex]) and then adding the numerator ([tex]\(b\)[/tex]).
- For [tex]\( 4 \frac{2}{3} \)[/tex]:
[tex]\[
4 \cdot 3 + 2 = 12 + 2 = 14
\][/tex]
So, [tex]\( 4 \frac{2}{3} \)[/tex] becomes [tex]\( \frac{14}{3} \)[/tex].
2. Add the two fractions, [tex]\( \frac{14}{3} \)[/tex] and [tex]\( \frac{7}{9} \)[/tex].
3. To add fractions, they need a common denominator. The denominators here are 3 and 9. The least common multiple (LCM) of 3 and 9 is 9.
4. Convert [tex]\( \frac{14}{3} \)[/tex] to a fraction with the denominator 9:
[tex]\[
\frac{14}{3} \cdot \frac{3}{3} = \frac{42}{9}
\][/tex]
5. Now, add [tex]\( \frac{42}{9} \)[/tex] and [tex]\( \frac{7}{9} \)[/tex]:
[tex]\[
\frac{42}{9} + \frac{7}{9} = \frac{42+7}{9} = \frac{49}{9}
\][/tex]
Thus, the sum of [tex]\( 4 \frac{2}{3} \)[/tex] and [tex]\( \frac{7}{9} \)[/tex] is [tex]\( \frac{49}{9} \)[/tex].
[tex]\[
\boxed{\frac{49}{9}}
\][/tex]
