To evaluate the given expression:
[tex]\[
\frac{\frac{q}{2}-\frac{r}{3}}{\frac{3 p}{6}+\frac{q}{6}}
\][/tex]
we need to determine the values of the inner fractions and then the overall expression. The constants are given as [tex]\( p = -6 \)[/tex], [tex]\( q = 6 \)[/tex], and [tex]\( r = -17 \)[/tex].
1. Calculate the numerator of the inner fraction:
[tex]\[
\frac{q}{2} - \frac{r}{3}
\][/tex]
Substituting [tex]\( q = 6 \)[/tex] and [tex]\( r = -17 \)[/tex], we get:
[tex]\[
\frac{6}{2} - \frac{-17}{3}
\][/tex]
Simplify the fractions:
[tex]\[
3 + \frac{17}{3}
\][/tex]
To combine these terms, express 3 as a fraction with a denominator of 3:
[tex]\[
3 = \frac{9}{3}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{9}{3} + \frac{17}{3} = \frac{26}{3}
\][/tex]
2. Calculate the denominator of the inner fraction:
[tex]\[
\frac{3 p}{6} + \frac{q}{6}
\][/tex]
Substituting [tex]\( p = -6 \)[/tex] and [tex]\( q = 6 \)[/tex], we get:
[tex]\[
\frac{3 (-6)}{6} + \frac{6}{6}
\][/tex]
Simplify these fractions:
[tex]\[
\frac{-18}{6} + \frac{6}{6}
\][/tex]
Further simplification gives:
[tex]\[
-3 + 1 = -2
\][/tex]
3. Divide the numerator by the denominator:
[tex]\[
\frac{\frac{26}{3}}{-2}
\][/tex]
This can be rewritten as:
[tex]\[
\frac{26}{3} \times \frac{1}{-2} = \frac{26}{3} \times \left( -\frac{1}{2} \right)
\][/tex]
Multiply the fractions:
[tex]\[
\frac{26 \times -1}{3 \times 2} = \frac{-26}{6}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{-26}{6} = -\frac{13}{3}
\][/tex]
Therefore, the solution to the expression given [tex]\( p = -6 \)[/tex], [tex]\( q = 6 \)[/tex], and [tex]\( r = -17 \)[/tex] is:
[tex]\[
- \frac{13}{3}
\][/tex]
