There are three candidates for one position as a class monitor and 130 students are voting. Ali has 24 votes so far, while Tef has 29 and Ninja has 37,how many more votes does ninja need to be elected

Answer 1

The answer depends on the nature of the election. There are actually many different strategies for evaluating the winner of a vote, which are discussed here: https://plato.stanford.edu/entries/voting-methods/

However, assuming for the sake of simplicity that the winner of your election is determined by simple plurality, you need to ask, "What is the minimum number of votes that would make it impossible for a candidate to be 'caught'?" Note that in a vote determined by simple plurality, the candidate with the most votes will win. This is sometimes mistakenly called winning by "majority", but, technically, a majority win would require greater than 50% of the votes cast.

Well, if a candidate has half of the votes, in other words 65 votes, that candidate can still be tied unless some other votes have already been taken by a third candidate. So 66 would be the bare minimum number of votes required to guarantee victory (by either plurality or majority) before the election is in progress.

However, in this case, many of the votes are taken by a third candidate, so we have to figure out how that might affect things. Ali and Tef have 53 votes altogether, which might seem to make their votes irrelevant... 66 votes would still guarantee Ninja victory. But things are not so simple, because 65 would also guarantee Ninja victory, as would 64, 63, 62, 61... Why? Because neither Ali nor Tef can get infinite votes if Ninja also gets more votes. There are only 130 votes to be had.

So how low could Ninja really go? Well, 130 - 24 - 29 - 37 = 40, so there are 40 votes left. If Ninja gets even half of these, Ninja is still in the lead, because Ninja is already in the lead. To figure out exactly how few votes Ninja would need in order to win, we must simply compare Ninja to Tef, the runner-up. With 29 votes, Tef is closest to catching Ninja. Therefore, Ninja is guaranteed to win as soon as there are no longer enough votes remaining for Tef to catch Ninja.

This will happen when the sum of Tef's current votes and the number of remaining voters is less than the sum of Ninja's current votes and some critical number of voters between 0 and the remaining 40, which we could call "X".

Algebraically, we could say the previous paragraph like this:

29 + (40 - X) < 37 + X

...where X is the number of additional votes needed by Ninja to win. (Notice how much more concisely we can represent the previous paragraph as a simple inequality... This is why we math.)

Solving this inequality, we get:

69 - X < 37 + X

32 - X < X

32 < 2X

16 < X

So what does this mean in normal-people speak?

Well, it means that Ninja is guaranteed victory after receiving more than 16 votes, or at least 17 votes.

Does this answer work? Let's check. If Ninja receives 17 more votes, Ninja will have 54 votes. At that point, there will be 23 votes remaining, but Tef will require 25 votes (54 - 29) just to tie Ninja, rendering Ninja the guaranteed victor.

But maybe Ninja could get away with 16? Let's see. If Ninja only receives 16 more votes, Ninja has 53 votes, and there are 24 votes remaining, and Tef needs 24 votes (53 - 29) to catch Ninja. This would be possible, resulting in a tie, which would not give Ninja a victory in a simple plurality election. Therefore we have proven that we were correct before: Ninja needs at least 17 more votes to win by plurality.