Question

question #6
hunter and haley are determining the quotient of the following rational expressions. both students have the same answer, however one of them has an error in their work. determine who has an error in their work and explain why it is incorrect.
\\(\frac{8}{n + 4}\div\frac{40n - 72}{45n^{2}-81n}\\)

hunters work
step 1: \\(\frac{8}{n + 4}\cdot\frac{45n^{2}-81n}{40n - 72}\\)
step 2: \\(\frac{8}{n + 4}\cdot\frac{9n(5n - 9)}{8(5n - 9)}\\)
step 3: \\(\frac{1}{n + 4}\cdot\frac{9n(5n - 9)}{5n - 9}\\)
step 4: \\(\frac{1}{n + 4}\cdot\frac{9n}{1}\\)
step 5: \\(\frac{9n}{n + 4}\\)

haleys work
step 1: \\(\frac{8}{n + 4}\cdot\frac{45n^{2}+81}{40n + 72}\\)
step 2: \\(\frac{8}{n + 4}\cdot\frac{9n(5n + 9)}{8(5n + 9)}\\)
step 3: \\(\frac{1}{n + 4}\cdot\frac{9n(5n + 9)}{1(5n + 9)}\\)
step 4: \\(\frac{1}{n + 4}\cdot\frac{9n}{1}\\)
step 5: \\(\frac{9n}{n + 4}\\)

hunter made an error in step 2; she factored the second rational expression incorrectly.
haley made an error in step 1; she rewrote the second rational expression incorrectly when multiplying by the reciprocal.
haley made an error in step 3; she simplified the incorrectly.
hunter made an error in step 1; she should not have multiplied by the reciprocal of the second rational expression.

Explanation:

Step1: Recall division of rational - expressions rule

To divide two rational expressions $\frac{a}{b}\div\frac{c}{d}$, we multiply the first by the reciprocal of the second, i.e., $\frac{a}{b}\times\frac{d}{c}$. For $\frac{8}{n + 4}\div\frac{40n-72}{45n^{2}-81n}$, it should be $\frac{8}{n + 4}\times\frac{45n^{2}-81n}{40n - 72}$. Hunter's step - 1 is correct.

Step2: Factor the expressions

Factor $45n^{2}-81n=9n(5n - 9)$ and $40n-72 = 8(5n - 9)$. Hunter's step - 2 is correct: $\frac{8}{n + 4}\times\frac{9n(5n - 9)}{8(5n - 9)}$.

Step3: Simplify the common factors

Cancel out the common factors. In Hunter's work, canceling out the 8 and $(5n - 9)$ in step - 3 is correct, getting $\frac{1}{n + 4}\times\frac{9n}{1}$.

Step4: Analyze Haley's work

In Haley's step 1, when rewriting the division as multiplication by the reciprocal, she wrote $\frac{8}{n + 4}\times\frac{45n^{2}+81}{40n + 72}$ instead of $\frac{8}{n + 4}\times\frac{45n^{2}-81n}{40n - 72}$. She made a mistake in rewriting the second rational expression when multiplying by the reciprocal.

Answer:

Haley made an error in step 1; she rewrote the second rational expression incorrectly when multiplying by the reciprocal.