The point here is that the units of the problem do not matter -- if the units are the same entity, they disappear when you divide.
This brings us back to the new algorithm for division with fractions: To divide two fractions, find a common denominator and then divide the numerators.
Let's try a visual version of the problem we did before: 1/4 2/3. First, find a common denominator:
1/4 2/3 = 3/12 8/12
Next, divide the numerators:
38 = 3/8
Here is the model for this problem, called the common denominator model:
In this problem, in effect, the original question was "How many 2/3s are there in 1/4?" By finding a common denominator, we changed the question to "How many 8/12s are there in 3/12?" -- which is the same as asking "How many 8s are there in 3?" The answer to both questions is the same: "There is 3/8 of an 8 in 3."