SOLUTION: After leaving the polls, many people are asked how they voted. Concerning propositions A,B, and C the following info was obtained. 294 voted yes on A, 90 voted yes only on A, 346 v

After leaving the polls, many people are asked how they voted.
Concerning propositions A,B, and C the following info was obtained.
294 voted yes on A, 90 voted yes only on A, 346 voted yes on B, 166 
voted yes only on B, 517 voted yes on A or B, 339 voted yes on C, 
no one voted yes on all three and 72 voted no on all three. What percent
of the voters voted yes on more than one proposition. 

First draw a big square or rectangle to hold all the voters:


 
Next draw a circle to hold all the voters who voted yes on A
and label it A:
 

 
Next draw a circle overlapping the first to hold all 
the voters who voted yes on B and label
it B. (The overlapping part will contain the voters
who voted yes on both A and B).
 

 
 
 
Next draw a circle overlapping the first two circles
to hold all the voters who voted for C and label it C. 
 

 
This gives us 8 regions to classify voters with
according to what they voted yes on:

We will assume that every voter voted either yes or
no on all three.

The statements which immediately give us complete 
information about any of the 8 regions are these:

90 voted yes only on A
166 voted yes only on B
no one voted yes for all three
72 voted no on all three

So we will fill these numbers in the proper regions:


Then we'll assign unknowns x,y,z, and w for
the numbers of voters in the remaining regions:



"...294 voted yes on A..."

So we get our first equation by
adding up the parts of circle A
and setting it equal to 294

which simplifies to


"...346 voted yes on B..."

So we get our second equation by
adding up the parts of circle B
and setting it equal to 346

which simplifies to


"...517 voted yes on A or B..."

So we get our third equation by
adding up all the parts of both 
circles A and B and setting it 
equal to 517

which simplifies to


"...339 voted yes on C..."

So we get our fourth equation by
adding up the parts of circle C 
and setting it equal to 339:



So we have his system of 4 equations 
in 4 unknowns:



Solve that system and get



Now we replace the unknown letters by these numbers:



Now we add up all the numbers to
get the total number of voters:



So there were 790 voters.

The question asks:

>>...What percent of the voters voted yes on 
more than one proposition..."

123 voted yes on A and B
 81 voted yes on A and C
 57 voted yes on B and C

So 

The fraction who voted on more than
one proposition is therefore  
Changing this to a percent,

33.03797468% 

Edwin