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Tricky questions - Needs logic rather than Algebra

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Tricky questions - Needs logic rather than Algebra

by nikhilgmat31 » Mon Aug 10, 2015 4:25 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
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?
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by [email protected] » Mon Aug 10, 2015 8:56 am
HI nikhilgmat31,

These questions can all be solved by TESTing VALUES and thinking about the 'limits' involved (how big or small one number can be AND how that would impact the other numbers).

From the prompt, we know a number of facts:

1) There are 100 total votes.
2) Everyone got AT LEAST 1 vote.
3) A+B+C+D+E = 100
4) A>B>C>D>E
5) A = 40

The first question asks for the maximum and minimum number of votes that Bill could have received....We know that Bill got FEWER than 40, so let's start with 39 and see if we can make all the facts 'fit' together...

A = 40
B = 39

That leaves 21 votes for the other 3 people...there are a number of ways to distribute those votes. Here's one...

C = 15
D = 5
E = 1

So, Bill COULD have gotten 39 votes.

The minimum number of votes will be a little trickier to determine (we have to maximize all of the other numbers while still making Bill the biggest of those 4 numbers)...

B+C+D+E = 60

This means the average would be 60/4 = 15. Using that information, we can 'bunch' the numbers close...this means that the minimum number of votes for Bill would be a little more than 15....

IF....
B = 16
C = 15
D = 14
E = 13
Total = 58, which is NOT enough votes. Thus, Bill had to have received MORE than 16 votes...

IF....
B = 17
C = 16
D = 15
E = 12
Total = 60, so the minimum number of votes for Bill would have to be 17.

You can use a similar approach for each of the other questions.

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by nikhilgmat31 » Tue Aug 11, 2015 3:39 am
Thanks Brent,
To continue on finding max & min for C

1) There are 100 total votes.
2) Everyone got AT LEAST 1 vote.
3) A+B+C+D+E = 100
4) A>B>C>D>E
5) A = 40


A = 40
E =1
D =2

A + E + D = 43 ; Remaining votes = 57
So B + C = 57

B = 29

C = 28 is maximum value for C. Please suggest If I am correct or not.


To find maximum for C & minimum of votes C

A = 40
B = 39

C + D + E = 100- 79 = 21

average is 7
so the value can be 8,7,6

C = 8, D = 7 , E = 6

C = 8 is minimum value for C. Please suggest If I am correct or not.
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by nikhilgmat31 » Tue Aug 11, 2015 3:48 am
Thanks Brent,
To continue on finding max & min for D

1) There are 100 total votes.
2) Everyone got AT LEAST 1 vote.
3) A+B+C+D+E = 100
4) A>B>C>D>E
5) A = 40


B + C + D + E = 60

E =1

B + C + D = 59

average = 59/3

possible numbers for B,C,D can 21,20,18

Maximum value for D is 18 , Please suggest if I am correct or not


To Find minimum value of D

E =1
D = 2

Minimum value for D is 2 , Please suggest if I am correct or not
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by nikhilgmat31 » Tue Aug 11, 2015 3:52 am
Thanks Brent,
To continue on finding max & min for E

1) There are 100 total votes.
2) Everyone got AT LEAST 1 vote.
3) A+B+C+D+E = 100
4) A>B>C>D>E
5) A = 40


B + C + D + E = 60

Average is 15

17,16,14,13

Maximum value for E is 13 , Please suggest if I am correct or not


To Find minimum value of E

E =1


Minimum value for E is 1 , Please suggest if I am correct or not
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by nikhilgmat31 » Tue Aug 11, 2015 3:54 am
If Bill received 25 votes, did Charlie get at least 13 votes?

A = 40
B = 25

C + D + E = 35

we need to maximize C , 13 12 10 C is always greater than 13.

20 10 5

Answer is YES
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by nikhilgmat31 » Tue Aug 11, 2015 3:57 am
If Charlie received 12 votes, did Dan get at least 5 votes?

A = 40
maximum for B = 29
C = 12

A + B + C = 40 + 39 + 12 = 91

remaining votes = 9

D + E = 9
5,4 or 6,3 or 7,2 or 8,1

YES D get atleast 5 votes.
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