To solve the problem [tex]\( 7 \frac{1}{6} - 3 \frac{4}{9} \)[/tex], we will go through the steps to subtract these two mixed numbers and express the result in its simplest form.
1. Convert the mixed fractions to improper fractions:
[tex]\[ 7 \frac{1}{6} \rightarrow 7 + \frac{1}{6} = 7 + 0.166\overline{6} = 7.166\overline{6} \][/tex]
[tex]\[ 3 \frac{4}{9} \rightarrow 3 + \frac{4}{9} = 3 + 0.4\overline{4} = 3.4\overline{4} \][/tex]
2. Perform the subtraction:
[tex]\[ 7.166\overline{6} - 3.4\overline{4} = 3.7222222\overline{2} \][/tex]
3. Convert the result back to a mixed fraction:
The result we obtained is [tex]\( 3.7222222\overline{2} \)[/tex]. Breaking it down into a mixed number, we get:
[tex]\[ \text{Whole number part} = 3 \][/tex]
[tex]\[ \text{Fractional part} = 0.7222222\overline{2} \][/tex]
4. Convert the decimal part to a fraction:
[tex]\[ 0.7222222\overline{2} \][/tex]
First, recognize it as:
[tex]\[ 0.7222222\overline{2} = \frac{13}{18} \][/tex]
5. Combine the whole number with the fraction:
Thus, [tex]\( 3.7222222\overline{2} \)[/tex] can be written as:
[tex]\[ 3 \frac{13}{18} \][/tex]
Selecting the simplified and correct answer from the options:
[tex]\[ \boxed{3 \frac{13}{18}} \][/tex]
