Calculating the total from known percentages

[AlabamaTiger]

I'm trying to find a formula to calculate the total number of people who took part in a survey based on knowing the percentage of replies before and after a known response.

For example: Before my vote, the score was Yes 38.46% No 61.54%. I voted No and the scores changed to Yes 33.33% No 66.67% what is the total number of votes cast?

So:

Y
_____ = 0.3846
Y+N

and:

Y
_______ = 0.3333
Y+(N+1)


I had thought about finding the difference in the results and somehow relating that to the increase of 1. For example 0.3846 - 0.3333 = 0.0513 so my "1" is 5.13% of the total but that doesn't help when I try to plug that in to a known solution...

e.g. If we know 3 votes have been cast and there are 2 for yes and 1 for no it would be:

Yes = 2/3 = 0.3333 (33.33%)

Add a No vote and it would be

Yes = 2/4 = 0.5 (50%)

So my 1 "no" vote is worth a change of 16.66% - but then I get stuck...

Help!

Thanks

 

[wjm11]

I'm trying to find a formula to calculate the total number of people who took part in a survey based on knowing the percentage of replies before and after a known response.

For example: Before my vote, the score was Yes 38.46% No 61.54%. I voted No and the scores changed to Yes 33.33% No 66.67% what is the total number of votes cast?

So:

Y
_____ = 0.3846
Y+N

and:

Y
_______ = 0.3333
Y+(N+1)

You are off to a good start. You have two equations and two unknowns, Y and N. You simply need to solve. I suggest you start by multiplying each equation by the denominator in each equation. For example, your first equation would become Y = .3864(Y + N). Once you have gotten rid of the denominators by this method, solve the system of equations. I would recommend the substitution method in this case.

Hope that helps.

 

[soroban]

Hello, AlabamaTiger!

Is there a typo?
Your set-up is correct . . . but I'm getting an unacceptable answer.

Before my vote, the score was: Yes 38.46%, No 61.54%.
I voted No and the scores changed to: Yes 33.33%, No 66.67%.
What is the total number of votes cast?

I changed the percents to fractions . . . much easter to work with.


Your work is correct!

. . \(\displaystyle \frac{Y}{Y+N} \:=\:\frac{5}{13} \quad\Rightarrow\quad N \:=\:\tfrac{8}{5}Y \;\;[1]\)

. . \(\displaystyle \frac{Y}{Y+N+1} \:=\:\frac{1}{3} \quad\Rightarrow\quad 2Y \:=\:N + 1 \;\;[2]\)


\(\displaystyle \text{Substitute }[1]\text{ into }[2]\!:\;\;2Y \:=\:\tfrac{8}{5}Y + 1 \qyad\Rightarrow\quad \tfrac{2}{5}Y \:=\:1 \quad\Rightarrow\quad Y \:=\:2\tfrac{1}{2}\;(??)\)


Before your vote, two-and-a-half Yes-votes were cast.
. . How is that possible?


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Our answer do check out . . . LOL!

There were 2.5 Yes-votes and 4 No-votes . . . a total of 6.5 votes.
. . \(\displaystyle \text{And: }\:\frac{2.5}{6.5} \:=\:0.384615385 \:\approx\:38.46\%\)

\(\displaystyle \text{After your No-vote, we have: }\:\frac{2.5}{7.5} \:=\:0.33333333 \;\approx\;33.33\%\)

 

[Mrspi]

I worked the problem out too, and also got 2.5 yes votes to 4 no votes in the initial situation.

I figured I MUST have made an error (even though those numbers CHECKED....add 1 "no" vote, so that you have 7.5 total votes, 2.5 yes votes, and 5 no votes. Percentages work out just fine)

I figured I must have made an error, but maybe not!

I look forward to "enlightenment" as others comment on this problem.

No one says that a problem will always yield solutions that make sense!

(And, having voted in some online surveys myself, I'm well aware that in the few milleseconds it takes me to vote, OTHERS may well have voted also, and those votes may also be counted in those final percentages.)

In other words...if I see an online vote in which the Yes percentage is 38.46% and the NO percentage is 61.54% AND

I cast my "no" vote, and find that the "yes" percentage has changed to 33.33% and the "no" percentage has changed to 66.67%. Does that mean MY one "no" vote is the sole contributor to that change in the percentages? I surely don't THINK so. In an "online voting" situation, MANY votes could be cast and recorded in just a second or two.

So...........unless I have some more very specific information about this situation, I'd hate to place a monetary bet on ANYTHING anyone opines.

 

[AlabamaTiger]

Thanks for some great input. Yes, I guess someone could have voted in the time it took for me to vote again from another machine so lets try the equation with some known numbers and try to figure out the answer. Ultimately what I would like to have is an Excel spreadsheet where I can enter the known data and click "refresh" to generate the total number of replies.

This one I worked out easily by trial and error because I knew the survey was new and very few people had voted;

I voted No and the results after my vote showed 33.33% yes, 66.66% no.

I then voted yes and the results changed to 50% yes, 50% no.

Is there any other conclusion other than:

before my first (no) vote the score was 50/50 with just 2 votes: 1 yes, 1 no

My no vote tipped the balance to 66% No (2n/3y * 100)

My yes vote even things back at 50/50 (2/4 * 100)


I thought this would be so easy to plug in to a spreadsheet but my brain just can't get around it...!