To determine which expression is equivalent to the given one, we need to simplify the following fraction:
[tex]\[ \frac{\frac{r-5}{9 r}}{\frac{5-r}{3 r^2}} \][/tex]
First, let's rewrite the overall fraction:
[tex]\[ \frac{\frac{r-5}{9 r}}{\frac{5-r}{3 r^2}} = \left( \frac{r-5}{9 r} \right) \div \left( \frac{5-r}{3 r^2} \right) \][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. Hence, we get:
[tex]\[ \left( \frac{r-5}{9 r} \right) \times \left( \frac{3 r^2}{5-r} \right) \][/tex]
Now, let's perform the multiplication:
[tex]\[ \frac{(r-5) \cdot 3 r^2}{9 r \cdot (5-r)} \][/tex]
Next, we simplify the numerator and the denominator before multiplying:
[tex]\[ \frac{3 r^2 (r-5)}{9 r (5-r)} \][/tex]
Notice that [tex]\(5 - r = - (r - 5)\)[/tex]. So:
[tex]\[ \frac{3 r^2 (r-5)}{9 r (- (r-5))} = \frac{3 r^2 (r-5)}{-9 r (r-5)} \][/tex]
Both the numerator and the denominator have the factor [tex]\((r-5)\)[/tex], so we can cancel them out:
[tex]\[ \frac{3 r^2}{-9 r} \][/tex]
Further simplify by dividing both the numerator and the denominator by [tex]\(r\)[/tex]:
[tex]\[ \frac{3 r^2}{-9 r} = \frac{3 r}{-9} \][/tex]
Perform the final arithmetic simplification:
[tex]\[ \frac{3 r}{-9} = -\frac{r}{3} \][/tex]
Thus, the given expression simplifies to:
[tex]\[ -\frac{r}{3} \][/tex]
Therefore, the equivalent expression is:
[tex]\[ C. -\frac{r}{3} \][/tex]
