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17% (03:01) correct
83%
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The village of Devis is hosting a summit attended by 100 diplomats. Of these diplomats, 80 speak Arabic, 85 speak French, 95 speak Spanish, and 85 speak English. Do at least 60 of the diplomats speak all four languages?
(1) Every diplomat who speaks Arabic also speaks English, and every diplomat who speaks English also speaks Spanish.
(2) There are 10 diplomats who speak only Spanish.
(1) Every diplomat who speaks Arabic also speaks English, and every diplomat who speaks English also speaks Spanish.
(2) There are 10 diplomats who speak only Spanish.
24 Mar 2026, 09:39
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Good Data Sufficiency question. The key concept here is subset constraints in overlapping sets.
The question: among 100 diplomats (A=80, F=85, S=95, E=85), do at least 60 speak all four languages?
Without any constraints, the lower bound from inclusion-exclusion is 80+85+95+85 - 3x100 = 45. So baseline, at least 45 speak all four. Not enough to answer yes or no to the 60 threshold.
Statement (1): A is a subset of E, and E is a subset of S. This tells us all 80 Arabic speakers also speak English, and all 85 English speakers also speak Spanish. So E and S are effectively merged for the English-speaking group - all 85 English speakers also speak Spanish.
Now for French: 85 French speakers. The minimum overlap between the English-speaking group (which is 85 people who all also speak Spanish) and the French group is: 85 + 85 - 100 = 70.
So at least 70 diplomats speak English, Spanish, AND French. Of those 70, what's the minimum who also speak Arabic? 80 Arabic speakers are all inside that E-group of 85. Min overlap of Arabic with that group of at least 70 F-speakers: 80 + 70 - 85 = 65.
Wait, let me be cleaner. We know all 80 Arabic speakers speak E and S. So 80 people speak A+E+S. Min of those 80 who also speak French: 80 + 85 - 100 = 65.
At minimum 65 speak all four. 65 ≥ 60. Answer is YES. Statement (1) alone is sufficient.
Statement (2): 10 diplomats speak only Spanish. So 85 speak Spanish plus at least one other language. This doesn't constrain the four-way overlap enough. We could construct scenarios where the all-four overlap is 45 (below 60) or 80 (above 60). Not sufficient.
Answer: A.
The trap is assuming Statement (1) is complicated to work with because of the chain of subsets. It's actually the opposite - chains make the math cleaner.
The question: among 100 diplomats (A=80, F=85, S=95, E=85), do at least 60 speak all four languages?
Without any constraints, the lower bound from inclusion-exclusion is 80+85+95+85 - 3x100 = 45. So baseline, at least 45 speak all four. Not enough to answer yes or no to the 60 threshold.
Statement (1): A is a subset of E, and E is a subset of S. This tells us all 80 Arabic speakers also speak English, and all 85 English speakers also speak Spanish. So E and S are effectively merged for the English-speaking group - all 85 English speakers also speak Spanish.
Now for French: 85 French speakers. The minimum overlap between the English-speaking group (which is 85 people who all also speak Spanish) and the French group is: 85 + 85 - 100 = 70.
So at least 70 diplomats speak English, Spanish, AND French. Of those 70, what's the minimum who also speak Arabic? 80 Arabic speakers are all inside that E-group of 85. Min overlap of Arabic with that group of at least 70 F-speakers: 80 + 70 - 85 = 65.
Wait, let me be cleaner. We know all 80 Arabic speakers speak E and S. So 80 people speak A+E+S. Min of those 80 who also speak French: 80 + 85 - 100 = 65.
At minimum 65 speak all four. 65 ≥ 60. Answer is YES. Statement (1) alone is sufficient.
Statement (2): 10 diplomats speak only Spanish. So 85 speak Spanish plus at least one other language. This doesn't constrain the four-way overlap enough. We could construct scenarios where the all-four overlap is 45 (below 60) or 80 (above 60). Not sufficient.
Answer: A.
The trap is assuming Statement (1) is complicated to work with because of the chain of subsets. It's actually the opposite - chains make the math cleaner.
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