Let's solve the expression [tex]\(\frac{\frac{2}{9} - \frac{4}{3}}{\frac{7}{4} + \frac{2}{6}}\)[/tex] step-by-step.
### Step 1: Simplify the Numerator
The numerator is [tex]\(\frac{2}{9} - \frac{4}{3}\)[/tex].
First, find a common denominator for the fractions. The least common multiple of 9 and 3 is 9.
Rewriting [tex]\(\frac{4}{3}\)[/tex] with a denominator of 9:
[tex]\[
\frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9}
\][/tex]
Now, subtract the fractions:
[tex]\[
\frac{2}{9} - \frac{12}{9} = \frac{2 - 12}{9} = \frac{-10}{9}
\][/tex]
So, the numerator is [tex]\(\frac{-10}{9}\)[/tex].
### Step 2: Simplify the Denominator
The denominator is [tex]\(\frac{7}{4} + \frac{2}{6}\)[/tex].
First, find a common denominator for the fractions. The least common multiple of 4 and 6 is 12.
Rewriting both fractions with a denominator of 12:
[tex]\[
\frac{7}{4} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12}
\][/tex]
[tex]\[
\frac{2}{6} = \frac{2 \times 2}{6 \times 2} = \frac{4}{12}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{21}{12} + \frac{4}{12} = \frac{21 + 4}{12} = \frac{25}{12}
\][/tex]
So, the denominator is [tex]\(\frac{25}{12}\)[/tex].
### Step 3: Divide the Numerator by the Denominator
We now need to divide [tex]\(\frac{-10}{9}\)[/tex] by [tex]\(\frac{25}{12}\)[/tex]. We can do this by multiplying by the reciprocal of the denominator:
[tex]\[
\frac{\frac{-10}{9}}{\frac{25}{12}} = \frac{-10}{9} \times \frac{12}{25}
\][/tex]
Multiply the numerators and the denominators:
[tex]\[
\frac{-10 \times 12}{9 \times 25} = \frac{-120}{225}
\][/tex]
Simplify the fraction by finding the greatest common divisor (GCD) of 120 and 225, which is 15:
[tex]\[
\frac{-120 \div 15}{225 \div 15} = \frac{-8}{15}
\][/tex]
So, the result of [tex]\(\frac{\frac{2}{9} - \frac{4}{3}}{\frac{7}{4} + \frac{2}{6}}\)[/tex] is [tex]\(\frac{-8}{15}\)[/tex].
However, we should convert the answer to a decimal as to match the previously given context from the numerical result, which is approximately:
[tex]\[
\boxed{-0.5333333333333333}
\][/tex]
### Summary of Important Values
- Simplified Numerator: [tex]\(\boxed{-1.1111111111111112}\)[/tex]
- Simplified Denominator: [tex]\(\boxed{2.0833333333333335}\)[/tex]
- Final Result: [tex]\(\boxed{-0.5333333333333333}\)[/tex]
