To solve the problem of adding the mixed numbers [tex]\( 7 \frac{7}{9} \)[/tex] and [tex]\( 2 \frac{5}{9} \)[/tex], we will follow these steps:
1. Separate the whole numbers and the fractions:
- For [tex]\( 7 \frac{7}{9} \)[/tex]: The whole number is [tex]\( 7 \)[/tex] and the fraction is [tex]\( \frac{7}{9} \)[/tex].
- For [tex]\( 2 \frac{5}{9} \)[/tex]: The whole number is [tex]\( 2 \)[/tex] and the fraction is [tex]\( \frac{5}{9} \)[/tex].
2. Add the fractions:
- The denominators for both fractions are the same ([tex]\( 9 \)[/tex]), so we can add the numerators directly:
[tex]\[
\frac{7}{9} + \frac{5}{9} = \frac{7 + 5}{9} = \frac{12}{9}
\][/tex]
- Simplify the fraction if possible. Here, we can simplify [tex]\( \frac{12}{9} \)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is [tex]\( 3 \)[/tex]:
[tex]\[
\frac{12}{9} \div \frac{3}{3} = \frac{4}{3}
\][/tex]
- Convert this improper fraction ([tex]\( \frac{4}{3} \)[/tex]) into a mixed number:
[tex]\[
\frac{4}{3} = 1 \frac{1}{3}
\][/tex]
This gives us a whole number [tex]\( 1 \)[/tex] and a fraction [tex]\( \frac{1}{3} \)[/tex].
3. Add the whole numbers:
- The whole numbers from our original mixed numbers are [tex]\( 7 \)[/tex] and [tex]\( 2 \)[/tex], and we also need to add the whole number [tex]\( 1 \)[/tex] from the fraction conversion:
[tex]\[
7 + 2 + 1 = 10
\][/tex]
4. Combine the results:
- The combined whole number is [tex]\( 10 \)[/tex].
- The remaining fraction is [tex]\( \frac{1}{3} \)[/tex].
Thus, the sum of [tex]\( 7 \frac{7}{9} + 2 \frac{5}{9} \)[/tex] is:
[tex]\[
10 \frac{1}{3}
\][/tex]
