Self-study
Conclusion In North America there has been an explosion of public interest in, and enjoyment of, opera over the last three decades. ███ ████████ ██ ████ █████████ ██ ████ ██ ███ ██ ██ ██ ████████████ █████ █████████ █████████ ██████ ██ █████ ████████ ██ ████ ███████ ████ ███ ██████ ██ ███ ████ ██ ██████
Structure: Phenomenon-Hypothesis
Notice that the stimulus provides a causal hypothesis explaining a certain phenomenon. The phenomenon is that most — 45 out of 70 — of the professional opera companies in North America have been founded in the last 30 years. The stimulus hypothesizes that this phenomenon is caused by an "explosion" in public interest and appreciation of opera over those 30 years.
Analysis: Necessary Assumption
The stimulus makes a number of assumptions. The most obvious one might be that the establishment of these new opera companies was primarily driven by public interest in, and enjoyment of, opera, and not some other factor — increased government subsidies for opera companies, for example. More seriously, the argument seems to assume that these 45 opera companies represent some kind of overall increase in the number of active opera companies, when the evidence doesn't actually say that. What if one hundred opera companies were founded in the last 30 years, in addition to however many older opera companies already existed, and most of them failed? In that case, the fact that only 70 or so opera companies survive, including 45 from the last 30 years, might not represent an increase in appreciation for opera, but rather a decrease in demand and appreciation for opera over the last three decades.
So either of these assumptions could work as a necessary assumption for this argument: first, that these new opera companies were not all opened because of some other factor than public demand for opera, and second, that the total number of opera companies has not actually decreased in the last 30 years, rather than increasing.
20.
The reasoning above assumes which ███ ██ ███ ██████████
a
All of the ██ ████████████ █████ █████████ ███ ████████████ ██████ ███████████
This isn't a necessary assumption. Notice that if we negated it, (A) would only tell us that some of the 70 opera companies were not commercially viable. But some of these companies not being viable wouldn't destroy the argument that public interest has driven the founding of the 45 new opera companies, so we can't say that (A) is a necessary assumption.
b
There were fewer ████ ██ ████████████ █████ █████████ ████ ███ ████ ██████ ██ █████ ███ ███ ████ ██████ ██████████ ██████ ███ ████ ██ ██████
This is a necessary assumption, because it tells us that the fact that 45 of the currently active opera companies were founded in the last 30 years actually represents a net increase. Let's think about what it would mean to negate (B): there were at least (i.e., more than or equal to) 45 professional opera companies that were active 30 years ago and ceased operating during the last 30 years. It's really important to see that we don't negate or change anything in the phrase "active 30 years ago and that ceased operations during the last 30 years" because that is a descriptor of the kind of company we're talking about: companies that existed 30 years ago and then shut down in the last three decades, so were removed from the equation.
If there were exactly 45 such companies, then the 45 new companies founded in the last 30 years that have lasted until today actually don't represent a net increase in the number of active opera companies: they perfectly balance out the number of opera companies that closed in the last three decades. If there were more than 45 such companies, then even though 45 new companies were founded and have lasted until today, the total number of opera companies has gone down over the last 30 years. This would destroy the argument, which points to the assumed increase in the number of opera companies as evidence for an increase in people's enjoyment of opera.
c
There has not ████ █ █████████████ ████████ ██ ███ ██████ ██ ████████████ █████████ ███████ ██ █████ ██████████ █████
Irrelevant. We're not interested in other performing arts. The stimulus only talks about opera.
d
The size of ███ ███████ ████████ ██ ████████████ ██ ████████████ █████ █████████ ███ █████████ ████ ███ ████ █████ ████████
This isn't a necessary assumption. Even if this weren't true — if the average audience size had stayed the same, or even decreased — this could be offset by the apparent increase in the number of opera companies, which might suggest that the total number of people viewing opera could still have increased. Audiences might not have increased in size, but there could still be more audiences overall. So negating (D) wouldn't destroy the argument.
e
The 45 most ████████ ███████ █████ █████████ ████ ███ ███████████ ██ █ ██████ ██ ██████████ ██ ███ ████ ██ █ █████████ █████████
This is a tricky answer choice, because it does point to one of the assumptions we identified: that the primary cause behind the founding of these new companies was audience enthusiasm. But it's crucial to see that this assumption does not require every single one of the 45 companies to have been founded because of audience enthusiasm. If 43 companies were founded because of audience enthusiasm, and 2 were founded just because an older company split, then it would still be fair to say that the increase in opera companies suggests an increase in audience enthusiasm.
So notice that negating this answer choice wouldn't actually destroy the argument: "the 45 most recently founded opera companies were not all established as a result of enthusiasm on the part of a potential audience." The key here is knowing how to negate "all." This statement would destroy the argument if it said that "none" of these 45 companies were founded because of audience enthusiasm. But "all" simply negates to "not all" or "some... not": some of these opera companies were not founded because of audience enthusiasm. As we saw above, this wouldn't destroy the argument, so we can't say that (E) is a necessary assumption.