Problem Solving

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.

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?

5. If Bill received 25 votes, did Charlie get at least 13 votes?

6. If Charlie received 12 votes, did Dan get at least 5 votes?

Please assist with above problems.

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.

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.

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.

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 ≥ 20 and B ≥ 21, which forces E ≤ 0, which isn't allowed.)

To minimize Dan, we maximize B and C. Since B + C + D + E = 60, and E must be ≥ 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.

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 ≥ 15, C ≥ 16, and B ≥ 17, which is too big.)

To minimize, use any of the E = 1 cases we've used in the previous responses.

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 ≥ 13.

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 ≥ 5.

... 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!)

Thanks Matt for detail explanation on below questions.

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

When you post a problem, please include the answer choices.

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.
Since the correct answer must be the least viable option, start with the smallest answer choice.
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.

The correct answer is D.