Let's find the sum of these mixed numbers:
[tex]\(
-7 \frac{8}{9} + 3 \frac{5}{9}
\)[/tex].
### Step-by-Step Solution:
1. Convert the Mixed Numbers to Improper Fractions:
- For [tex]\(-7 \frac{8}{9}\)[/tex]:
[tex]\[
-7 \frac{8}{9} = -7 - \frac{8}{9} = -\left(7 + \frac{8}{9}\right)
\][/tex]
Let's convert the integer part to a fraction:
[tex]\[
7 = \frac{63}{9}
\][/tex]
Therefore:
[tex]\[
-7 \frac{8}{9} = -\left(\frac{63}{9} + \frac{8}{9}\right) = -\frac{71}{9}
\][/tex]
- For [tex]\(3 \frac{5}{9}\)[/tex]:
[tex]\[
3 \frac{5}{9} = 3 + \frac{5}{9}
\][/tex]
Let's convert the integer part to a fraction:
[tex]\[
3 = \frac{27}{9}
\][/tex]
Therefore:
[tex]\[
3 \frac{5}{9} = \frac{27}{9} + \frac{5}{9} = \frac{32}{9}
\][/tex]
2. Add the Improper Fractions:
[tex]\[
-\frac{71}{9} + \frac{32}{9}
\][/tex]
Since the fractions have the same denominator, we can directly add the numerators:
[tex]\[
\frac{-71 + 32}{9} = \frac{-39}{9}
\][/tex]
3. Simplify the Resulting Fraction:
- The fraction [tex]\(\frac{-39}{9}\)[/tex] can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 3.
[tex]\[
\frac{-39 \div 3}{9 \div 3} = \frac{-13}{3}
\][/tex]
Therefore, the sum of [tex]\( -7 \frac{8}{9} + 3 \frac{5}{9} \)[/tex] in its simplest form is [tex]\(\frac{-13}{3}\)[/tex] or, as a mixed number, [tex]\(-4 \frac{1}{3}\)[/tex].
