At Consolidated Foundries, for a resolution to become policy, a quorum of at least half the 20 directors must pass the resolution by at least a two-thirds majority. At a meeting of the board of directors, did resolution X pass or fail? 

(1) Ten directors voted for the resolution. 
(2) Seven directors voted against the resolution

Hi chetan2u

I got B, as statement 2 sufficient to say that resolution did not get 2/3 majority.

However, i am confused at the wording - "at least half the 20 directors", can you please explain? is n't 2/3 already mentioning that it needs more than half?

Thanks

Given: A quorum of at least half the 20 directors must pass the resolution by at least a two-thirds majority

Target question: At a meeting of the board of directors, did resolution X pass or fail? 

Statement 1: Ten directors voted for the resolution. 
KEY: We don't know how many directors attended the meeting.
Let's test some possible scenarios that satisfy statement 1:
Case a: 12 directors attended, and 10 of them voted FOR the resolution. Since 10/12 > 2/3, the answer to the target question is YES, the resolution passed
Case b: 20 directors attended, and 10 of them voted FOR the resolution. Since 10/20 < 2/3, the answer to the target question is NO, the resolution did NOT pass
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Seven directors voted against the resolution
Let's examine all possible cases:
Case a: 7 directors attended the meeting. Since we need at least 10 directors to form a quorum, the resolution did NOT pass
Case b: 8 directors attended the meeting. Since we need at least 10 directors to form a quorum, the resolution did NOT pass
Case c: 9 directors attended the meeting. Since we need at least 10 directors to form a quorum, the resolution did NOT pass
Case d: 10 directors attended, and 7 voted AGAINST. So 3 voted FOR the resolution. Since 3/10 < 2/3, the resolution did NOT pass
Case e: 11 directors attended, and 7 voted AGAINST. So 4 voted FOR the resolution. Since 4/11 < 2/3, the resolution did NOT pass
Case f: 12 directors attended, and 7 voted AGAINST. So 5 voted FOR the resolution. Since 5/12 < 2/3, the resolution did NOT pass
.
.
.
.
Case n: 20 directors attended, and 7 voted AGAINST. So 13 voted FOR the resolution. Since 13/20 < 2/3, the resolution did NOT pass

Since the answer to the target question is the SAME every time, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
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