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In professional soccer leagues, teams are ranked based on a system of points awarded for wins (3 points) and draws (1 point). In the event that two or more teams have earned the same number of points, the team with the greatest cumulative goal difference is ranked above the others. This statistic represents the difference between goals scored and goals allowed for a given team. For instance, a team that loses a match 1-3 will earn a -2 goal difference.
On the basis of the above information, which of the following must be true?
(A) The cumulative goal difference for all teams in the league together will always be zero.
(B) The team with the greatest cumulative goal difference will be the highest ranked.
(C) In a league of twenty teams, a team with a cumulative goal difference of 0 will be ranked 10th or 11th.
(D) No team with a negative cumulative goal difference can have a winning record.
(E) A team with 6 points from four matches will have a positive goal difference for those
If you replaced the text of the answer choices with numbers and altered the question stem, you would have a workable PS question. There are two types of numerical systems that we need to keep track of to tell what
must be true. The first is the points system for wins and draws, and the second is a sort of plus/minus relationship based on
cumulative goal difference. With an example in hand, there should be no confusion as to how goal difference works. How about we jump right in and get our hands dirty?
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(A) The cumulative goal difference for all teams in the league together will always be zero.
This might seem like an impossible task, to keep track of all the teams in a league at the same time across an entire season. We are also trained to be skeptical of absolute or overreaching language such as
always. But remember, a must-be-true framework deals in absolutes, so we just need to consider the logical implications of the statement. From the example provided at the end of the passage, we should be able to deduce that the winning team (which would walk away with 3 points) would consider the 1-3 loss for the other team a 3-1 personal victory, and that would mean the team would earn a +2 goal difference. It does not take a
Bunuel to see that the net sum of +2 and -2 is 0, and that, regardless of a given outcome for any game, the sum would remain 0 for that game: +4/-4, 0/0, and so on. Since we cannot find an issue with this answer choice, we know we have the one we need.
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(B) The team with the greatest cumulative goal difference will be the highest ranked.
This could or could not be true. Consider a team that won a match 5-0, thus earning 3 points and a +5 goal difference, and then lost a second match 0-1, keeping its points tally at 3 but dropping its cumulative goal difference to +4. A second team might win back-to-back matches 1-0 and thus earn 6 points and a +2 cumulative goal difference. The second team would outrank the first on points, despite having a lower cumulative goal difference.
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(C) In a league of twenty teams, a team with a cumulative goal difference of 0 will be ranked 10th or 11th.
You might be overwhelmed at the prospect of tracking twenty teams and simply reason that the top half of the table would probably have a positive cumulative goal difference while the bottom half would have a negative cumulative goal difference, leaving the teams in the middle to straddle that line. Again, this could or could not be true, depending on just how many goals a given team scored. There are so many possibilities here that it is not worth seeking to conjure up examples. (If you need some inspiration, though, feel free to borrow from the explanation above.)
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(D) No team with a negative cumulative goal difference can have a winning record.
An easy statement to disprove using no more than three matches. Say that a team wins the first and second matches 1-0 but loses a third 0-3. The goal difference for each match would be +1, +1, and -3, respectively, leading to a cumulative goal difference of -1. But that would not change the fact that the team would be sitting on a winning record.
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(E) A team with 6 points from four matches will have a positive goal difference for those matches.
In one scenario, this must be true: if the team wins one match and draws the other three (3 + 1 + 1 + 1 = 6 points). The cumulative goal difference for those matches would have to be positive, carrying over from the win. However, it could just as easily be the case that the team wins two matches and loses the other two by a larger cumulative margin, say 1-0, 1-0, 0-2, 0-3, or even 1-0, 1-0, 0-1, 0-1, the latter case leading to a cumulative goal difference of 0. (Quant reminder: 0 is neither positive nor negative, a one-of-a-kind number.)
Keep your approach to these numerical CR questions simple, and you will often find an answer with relative ease (
Moneyland casino questions aside).
I suspect this question will prove easier than others in the competition, since it is purely quantitative, but I cannot say the topic is unfitting.
- Andrew
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Please use
official questions from the Official Guide or Verbal Review to practice for the Verbal section.