Question

1. a veterinarian is providing care to cattle. one bottle of medicine is 8 fluid ounces. how many \\(\frac{2}{3}\\)-fluid-ounce doses can the veterinarian get out of one bottle of medicine? how many fluid ounces of medicine will be left in the bottle after she draws the maximum number of doses out?

Explanation:

Step1: Identify the operation

We need to find the number of \(\frac{2}{3}\)-fluid-ounce doses in an 8 - fluid - ounce bottle. This is a division problem, where we divide the total volume (8 fluid ounces) by the volume per dose (\(\frac{2}{3}\) fluid ounces). The formula for the number of doses \(n\) is \(n=\frac{\text{Total Volume}}{\text{Volume per Dose}}\), so \(n = 8\div\frac{2}{3}\).

Step2: Perform the division

Recall that dividing by a fraction is the same as multiplying by its reciprocal. So \(8\div\frac{2}{3}=8\times\frac{3}{2}\).
Simplify the multiplication: \(8\times\frac{3}{2}=\frac{8\times3}{2}=\frac{24}{2} = 12\).

To find the remaining medicine, we first find the total volume used in 12 doses. The volume per dose is \(\frac{2}{3}\) fluid ounces, so the total volume used \(V_{used}=12\times\frac{2}{3}\).
\(12\times\frac{2}{3}=\frac{12\times2}{3}=\frac{24}{3}=8\) fluid ounces. Wait, that can't be right. Wait, no, the total volume of the bottle is 8 fluid ounces. Wait, maybe I made a mistake. Wait, the problem says "How many \(\frac{2}{3}\)-fluid-ounce doses can the veterinarian get out of one bottle of medicine? How many fluid ounces of medicine will be left in the bottle after she draws the maximum number of doses out?"

Wait, let's re - examine. The total volume is 8 fluid ounces, dose size is \(\frac{2}{3}\) fluid ounces.

Number of doses: \(n = 8\div\frac{2}{3}=8\times\frac{3}{2}=12\). But if we use 12 doses, the volume used is \(12\times\frac{2}{3}=8\) fluid ounces. So the remaining volume is \(8 - 8=0\) fluid ounces. But that seems odd. Wait, maybe the total volume is 8 fluid ounces, and we are to find how many \(\frac{2}{3}\) - ounce doses. Let's check the division again. \(8\div\frac{2}{3}=\frac{8}{1}\times\frac{3}{2}=\frac{24}{2} = 12\). So 12 doses, and since \(12\times\frac{2}{3}=8\), there is 0 fluid ounces left.

Answer:

The number of doses is 12, and the number of fluid ounces left is 0.