Let's work through the given problem step-by-step.
### Step 1: Simplify the Numerator
The expression for the numerator is:
[tex]\[
\frac{2}{5} \div \frac{1}{2} \text { of } \frac{4}{9} - 1 \frac{1}{10}
\][/tex]
First, we convert the mixed number [tex]\(1 \frac{1}{10}\)[/tex] to an improper fraction:
[tex]\[
1 \frac{1}{10} = 1 + \frac{1}{10} = \frac{10}{10} + \frac{1}{10} = \frac{11}{10}
\][/tex]
Next, we handle the division and multiplication:
[tex]\[
\frac{2}{5} \div \frac{1}{2} = \frac{2}{5} \times 2 = \frac{2 \times 2}{5} = \frac{4}{5}
\][/tex]
Then, we multiply by [tex]\(\frac{4}{9}\)[/tex]:
[tex]\[
\frac{4}{5} \times \frac{4}{9} = \frac{4 \times 4}{5 \times 9} = \frac{16}{45}
\][/tex]
We now subtract the improper fraction:
[tex]\[
\frac{16}{45} - \frac{11}{10}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple of 45 and 10 is 90.
Convert each fraction to have a denominator of 90:
[tex]\[
\frac{16}{45} = \frac{16 \times 2}{45 \times 2} = \frac{32}{90}
\][/tex]
[tex]\[
\frac{11}{10} = \frac{11 \times 9}{10 \times 9} = \frac{99}{90}
\][/tex]
Now subtract:
[tex]\[
\frac{32}{90} - \frac{99}{90} = \frac{32 - 99}{90} = \frac{-67}{90}
\][/tex]
### Step 2: Simplify the Denominator
The expression for the denominator is:
[tex]\[
\frac{1}{8} - \frac{1}{6} \times \frac{3}{8}
\][/tex]
First, handle the multiplication:
[tex]\[
\frac{1}{6} \times \frac{3}{8} = \frac{1 \times 3}{6 \times 8} = \frac{3}{48} = \frac{1}{16}
\][/tex]
Then, we subtract:
[tex]\[
\frac{1}{8} - \frac{1}{16}
\][/tex]
To subtract these, we need a common denominator. The least common multiple of 8 and 16 is 16.
Convert each fraction to have a denominator of 16:
[tex]\[
\frac{1}{8} = \frac{1 \times 2}{8 \times 2} = \frac{2}{16}
\][/tex]
Now subtract:
[tex]\[
\frac{2}{16} - \frac{1}{16} = \frac{2 - 1}{16} = \frac{1}{16}
\][/tex]
### Step 3: Final Division
Now divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{\frac{-67}{90}}{\frac{1}{16}} = \frac{-67}{90} \times 16 = -\frac{67 \times 16}{90} = -\frac{1072}{90}
\][/tex]
Simplify the fraction:
[tex]\[
- \frac{1072}{90} = - \frac{536}{45}
\][/tex]
Convert it to a mixed number:
[tex]\[
- \frac{536}{45} = -11 \frac{41}{45}
\][/tex]
Thus, the final result, as a mixed number in its simplest form, is:
[tex]\[
-11 \frac{41}{45}
\][/tex]
