To solve the problem of adding [tex]\(4 \frac{2}{3} + \frac{7}{9}\)[/tex], we'll follow these steps:
1. Convert the mixed number to an improper fraction:
For [tex]\(4 \frac{2}{3}\)[/tex]:
- Multiply the whole number [tex]\(4\)[/tex] by the denominator [tex]\(3\)[/tex]: [tex]\(4 \times 3 = 12\)[/tex]
- Add the numerator [tex]\(2\)[/tex] to the result: [tex]\(12 + 2 = 14\)[/tex]
- The improper fraction is [tex]\(\frac{14}{3}\)[/tex].
2. Align the fractions to a common denominator:
Fractions [tex]\(\frac{14}{3}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex]:
- The denominator of [tex]\(\frac{14}{3}\)[/tex] is [tex]\(3\)[/tex] and the denominator of [tex]\(\frac{7}{9}\)[/tex] is [tex]\(9\)[/tex].
- The least common multiple of [tex]\(3\)[/tex] and [tex]\(9\)[/tex] is [tex]\(9\)[/tex].
So, we will convert [tex]\(\frac{14}{3}\)[/tex] to have a denominator of [tex]\(9\)[/tex]:
- Multiply both numerator and denominator of [tex]\(\frac{14}{3}\)[/tex] by [tex]\(3\)[/tex] to make the denominator [tex]\(9\)[/tex]: [tex]\(\frac{14 \times 3}{3 \times 3} = \frac{42}{9}\)[/tex].
3. Add the fractions:
[tex]\(\frac{42}{9}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex]:
- Since the denominators are the same, simply add the numerators: [tex]\(42 + 7 = 49\)[/tex].
- The resulting fraction is [tex]\(\frac{49}{9}\)[/tex].
4. Simplify if necessary:
The fraction [tex]\(\frac{49}{9}\)[/tex] is already in its simplest form.
So, the answer is:
[tex]\[
\boxed{\frac{49}{9}}
\][/tex]
