Question

question #9
donna is adding the following rational expressions: $\frac{4x}{2xy}+\frac{x - 1}{6y^{2}}$. donnas work is shown.
did donna make a mistake? if so, in which step was the mistake made? what was the mistake?
step 1: $lcd = 6xy$
step 2: $\frac{3(4x)}{3(2xy)}+\frac{x(x - 1)}{x(6y^{2})}$
step 3: $\frac{12x}{6xy}+\frac{x^{2}-x}{6xy}$
step 4: $\frac{x^{2}+11x}{6xy}$
step 5: $\frac{x + 11}{6y}$
no errors were made. the work is correct.
step 5; donna simplified the incorrectly when writing the sum in simplest form.
step 1; donna determined the wrong factors for the least common denominator.
step 3; donna simplified the numerators from step 2 incorrectly.

Explanation:

Step1: Find the correct LCD

The denominators are $2xy$ and $6y^{2}$. Prime - factorize: $2xy = 2\times x\times y$ and $6y^{2}=2\times3\times y\times y$. The least - common denominator (LCD) should be $6xy^{2}$, not $6xy$ as Donna calculated.

Step2: Analyze Donna's error

Donna made an error in Step 1 when determining the least - common denominator.

Answer:

C. Step 1; Donna determined the wrong factors for the least common denominator.