## There were 5 candidates (Alexa, Bill, Charlie, Dan, and Erni

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### There were 5 candidates (Alexa, Bill, Charlie, Dan, and Erni

by alanforde800Maximus » Thu Oct 13, 2016 11:20 am
There were 5 candidates (Alexa, Bill, Charlie, Dan, and Ernie) vying for student council,
and 100 total votes were cast. Everyone received at least one vote, and no two
Bill came in 2nd, Charlie in 3rd, Dan in 4th, and Ernie in last.

1. What is the greatest number of votes that Bill could have received? What is the
least number?

2. What is the greatest number of votes that Charlie could have received? What is
the least number?

3. What is the greatest number of votes that Dan could have received? What is
the least number?

4. What is the greatest number of votes that Ernie could have received? What is
the least number?

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by [email protected] » Thu Oct 13, 2016 11:40 am
Please post only one question per thread. Otherwise things can become pretty complicated when there are discussions on multiple questions.

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by [email protected] » Thu Oct 13, 2016 11:55 am
This also feels a little more LSAT-y. In general, try to post actual GMAT questions, or questions that test concepts in a GMAT-like way.
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Save $100 off any live Veritas Prep GMAT Course GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:20 am Let's break these into separate responses. What is the greatest number of votes that Bill could have received? What is the least number? To maximize Bill, we'll minimize everybody else. There are 60 votes for B, C, D, and E, and we need B < 40. If we make B = 39, we can make C = 8, D = 7, and E = 6. Success! We met the conditions, and Bill got as close to Alexa as possible. To minimize Bill, we'll maximize everybody else. In this case, B + C + D + E = 60, as before, but we want B = C + 1, C = D + 1, and D = E + 1. Unfortunately this gives us a non-integer value of B (B = 16.5), so we'll round up: B = 17. GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:23 am What is the greatest number of votes that Charlie could have received? What is the least number? We've got B + C + D + E = 60, with B > C > D > E. To maximize Charlie, we'll minimize everybody else. That means Bill needs to be as close to Charlie as possible: B = C + 1 and D and E need to be as small as possible (let's start with D = 2, E = 1). Then we have (C + 1) + C + 2 + 1 = 60, or C = 28. This checks out: B = 29, C = 28, D = 2, E = 1. To minimize Charlie, we need to maximize Bill and get Charlie as close to 0 as possible. We'll make B = 39. That leaves us with C + D + E = 21. To get C as far down as we can, we'll put it next to D and E, or C = D + 1 = E + 2. That gives us C = 8, D = 7, and E = 6, which checks out. GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:27 am What is the greatest number of votes that Dan could have received? What is the least number? To maximize Dan, we minimize everybody else. That means we want B, C, and D as close as possible B = C + 1 = D + 2 and E = 1. That gives us (D + 2) + (D + 1) + D + 1 = 60. That makes D not an integer, so we round down, and find D = 18. (To see that D can't be 19, realize that if D is 19, C must be â‰¥Â 20 and B â‰¥Â 21, which forces E â‰¤Â 0, which isn't allowed.) To minimize Dan, we maximize B and C. Since B + C + D + E = 60, and E must be â‰¥Â 1, we'll say E = 1, D = 2, and B + C = 57. This works with many combinations (e.g. B = 29, C = 28), so D = 2 is fine. Obviously D can't = 1, as this would force E = 0, so D = 2 is the minimum. GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:29 am What is the greatest number of votes that Ernie could have received? What is the least number? Same idea as above. To maximize, minimize everyone else: B = C + 1 = D + 2 = E + 3 so (E + 3) + (E + 2) + (E + 1) + E = 60, or E = 13.5. Obviously this must be an integer, so we round down, and find E = 13. (If E = 14, we'd have D â‰¥Â 15, C â‰¥Â 16, and B â‰¥Â 17, which is too big.) To minimize, use any of the E = 1 cases we've used in the previous responses. GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:30 am If Bill received 25 votes, did Charlie get at least 13 votes? Suppose that C < 13. We know B + C + D + E = 60. B = 25, so C + D + E = 35. If C < 13, then our max is C = 12, D = 11, E = 10. But 12 + 11 + 10 < 35, so this is impossible. Hence C â‰¥Â 13. GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:32 am If Charlie received 12 votes, did Dan get at least 5 votes? Algebraically, this is B + C + D + E = 60 and C = 12. From this we have B + D + E = 48. Suppose D < 5. Our max for D + E then is 4 + 3, which forces B = 41. But B must be less than 40, since Alexa's 40 votes were more than whatever Bill's tally is, so B = 41 is impossible. Hence D â‰¥Â 5. GMAT Instructor Posts: 2630 Joined: 12 Sep 2012 Location: East Bay all the way Thanked: 625 times Followed by:118 members GMAT Score:780 by [email protected] » Fri Oct 14, 2016 12:34 am ... and I think these questions are fine! They're posted as exercises to get you thinking about logic and min/max's, so they don't LOOK like GMAT problems, but the concepts they test could absolutely appear on the test. They don't seem all that LSATy to me - the LSAT doesn't do arithmetic, does it? - unless that test has changed a bit in the last two years. (I haven't kept tabs on it!) Master | Next Rank: 500 Posts Posts: 187 Joined: 13 Sep 2016 by alanforde800Maximus » Fri Oct 14, 2016 12:52 am Thanks Matt for detail explanation on below questions. Legendary Member Posts: 2663 Joined: 14 Jan 2015 Location: Boston, MA Thanked: 1153 times Followed by:128 members GMAT Score:770 by [email protected] » Fri Oct 14, 2016 8:00 am Here are a couple of official questions that tests the min/max concept: https://www.beatthegmat.com/six-countries-t33874.html and https://www.beatthegmat.com/gmat-prep-av ... 75647.html Veritas Prep | GMAT Instructor Veritas Prep Reviews Save$100 off any live Veritas Prep GMAT Course

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### Re: There were 5 candidates (Alexa, Bill, Charlie, Dan, and Erni

by GMATGuruNY » Wed Jan 20, 2021 1:38 pm
There were 5 candidates (Alexa, Bill, Charlie, Dan, and Ernie) vying for student council, and 100 total votes were cast. Everyone received at least one vote, and no two candidates received the same number of votes. Alexa won the election with 40 votes, Bill came in 2nd, Charlie in 3rd, Dan in 4th, and Ernie in last. What is the least number of votes that Bill could have received?

A. 12
B. 15
C. 16
D. 17
E. 18
Since Alexa wins 40 of the 100 votes, the remaining 60 votes must be won by B, C, D and E.

We can PLUG IN THE ANSWERS, which represent the least number of votes that B could have received.
To MINIMIZE the votes for B, we must MAXIMIZE the votes for C, D and E, bearing in mind that no two candidates may receive the same number of votes.
When the correct answer is plugged in, the four vote tallies for B, C, D and E will sum to 60 or more.

A: 12+11+10+9 = 42
B: 15+14+13+12 = 54
C: 16+15+14+13 = 58

Since only two more votes are needed to reach the threshold of 60, the next largest answer choice is viable.