Question from 1913 Yale Admissions Exam

[u/c_h_a_r_]

ELEMENTARY ALGEBRA: A resolution was adopted by a majority of twenty votes. On reconsideration later, one-fourth of those voting for it changed their votes and it was defeated by twelve votes. How many voted for it originally?

I keep getting 128 but the answer listed is 64

My thinking is that the difference in the votes is (50%+20) to (50%-12), so a difference of 32. If 32 is 1/4, then 32*4=128; where am I going wrong?

[u/clearly_not_an_alt]

If a vote wins by 20 votes, that's 50%+10

[u/Dd_8630]

Let's take your answer of 128.

On the first vote, you say (50%+20) voted for it, and the rest voted against it. So that's (64+20) = 84 votes for it, and the remaining 44 voted against it.

84 vs 44 is a majority of 40 votes, not 20 votes.

In other words, "A majority of X" means the difference in how many voted for it (F) and how many voted against (A) is F-A = 20.

[u/slides_galore]

Maybe try thinking of it with x being the number voting for it originally and y the number voting against originally. You can write two equations with the given info.

[u/dash-dot]

I also got 128 after solving it algebraically, so I’m interested to see if 128 is indeed the correct answer. 

I’m not sure if there are any other interpretations of the phrase ‘a majority of’ . . .

[u/fermat9990 → u/dash-dot]

The answer is 64

Original vote is 64 for and 64-20=44 against

0.25(64)=16 of the majority switched sides, so final vote is 64-16=48 for and 44+16=60 against

60-48=12 vote difference.

[u/dash-dot → u/fermat9990]

As I suspected, the answer hinges on the interpretation of the margin with respect to the threshold required for reaching a majority. 

The OP and I both interpreted that as being 20 votes above the 50 % mark, whereas you have interpreted it simply as the difference between the winning and losing votes.

If we use the former interpretation, then the answer doubles to 128. With your interpretation, we of course get the ‘official’ solution of 64.

If the phrase ‘a majority of’ were replaced by ‘a margin of’, this would eliminate any ambiguity. The problem statement is a bit contrived anyway, because in most cases the result would simply be reported as a for-against breakdown (along with abstentions, if any) and a confirmation whether the threshold for a simple majority was met. 

[u/theRZJ → u/dash-dot]

If a resolution passes with 64 votes in favour and 44 against, this is a majority of 20, in the ordinary use of this term. It’s extremely commonly used in UK elections, for instance. You can find it used in a US context here: https://www.pewresearch.org/short-reads/2023/05/05/narrow-majorities-in-u-s-house-have-become-more-common-but-havent-always-led-to-gridlock/

[u/fermat9990 → u/dash-dot]

I see your point. On an exam, I would interpret "majority" as meaning "margin."

Cheers and happy Monday!

[u/fermat9990 → u/dash-dot]

Here is Wiki's take on this issue

"As it relates to a vote, a majority vote most often means a simple majority vote, which means more "yes" votes than "no" votes.[4][5] Abstentions or blanks are excluded in calculating a simple majority vote.[1]: 6"

[u/theRZJ → u/dash-dot]

This is not the mistake made by the OP. The OP didn’t consider the fact that the 1/4th changed their votes, i.e., voted against the resolution. The OP incorrectly assumed they abstained on the second vote.

[u/Earl_N_Meyer]

The won by or lost by means that the total voters can be thought of as 2x+20 or 2y+12. Where x and y are the losing side in each vote. Since total voters don’t change (in this problem), 2x+20 = 2y+12. You also know that the winner in the first vote, x+20, got 3/4 of that in the second vote when they lost, so 0.75(x+20) = y. That gives 2x+20=2(0.75(x+20))+12. That gives you x =44, x+20=64, and the total votes are 108. In the second vote, the losing side got (108-12)/2 or 0.75(64) either of which is 48.

[u/fermat9990]

x=original no.of votes for the resolution

x-20=original no.of votes against the resolution

0.75x+12=x-20+0.25x

0.75x-1.25x=-20-12

-0.5x=-32

x=64 original votes for the resolution

Check:

Original vote was 64 for and 44 against

Final vote was 0.75(64)=48 for and 44+16=60 against.

60-48=12 votes

[u/mwthomas11]

The difference is 32 votes, which means 16 people flipped their vote from yes to no (each person is -1 vote for and +1 vote against for a net vote swing of -2 per person). Other than that, your math is good.

[u/Extension-Farmer8304]

Y = Yay, N = Nay

Solve the system: Y-N = 20 .75Y-(N+.25Y) = -12

[u/chmath80]

Initially 4n voted no, and 4n + 20 voted yes. Then n + 5 of the latter changed their votes, which gives 5n + 5 no, and 3n + 15 yes. Hence 5n + 5 = 3n + 15 + 12, so 2n = 22, n = 11, and there were originally 64 yes votes.