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 Mars November 3rd, 2009 03:18 PM

possibly a venn diagram solution??

Hello, I could use some help with this problem. I've trying a Venn diagram, not sure if this is the way to go???

In the forth Dodgeville precinct the Democrats, and the Republicans both had one candidate each for the House of Representatives, for the senate, and for the president. There was however one additional candidate for congress, who was registered as a Democrat and running a write-in campaign.
In the voting, one-seventh of the voters did not vote. Seven wrote in the congressional candidate and voted the rest straight Democratic ( voted
for the on write-in candidate for Congress, the Democratic candidate for senator and the Democratic candidate for president).
Forty-one people voted for at least one Republican candidate, though none of
those were the ones who voted for the write in candidate. Thirty-seven voted for at least
one Democrat, including the eighteen who split their ticket between Democrats and
Republicans. How many people in all were registered in the fourth precinct?

 aswoods November 3rd, 2009 04:26 PM

Re: possibly a venn diagram solution??

You have three choices for House of Representatives, two choices for Senate, and two choices for President.
That makes twelve groups to keep track of. In the diagram below, the left-hand table contains people who voted for the Democratic candidate for President, and the right-hand table shows people who voted for the Republican candidate for President.

$\begin{array}{|c|c|c|}\hline\\
\mathrm{(D-pres)} & \mathrm{D-sen} & \mathrm{R-sen} \\
\hline
\mathrm{D-rep} & w_0 & x_0 \\
\hline
\mathrm{R-rep} & y_0 & z_0 \\
\hline
\mathrm{d-rep} & 7 & 0 \\
\hline
\end{array}$
... $\begin{array}{|c|c|c|}\hline\\
\mathrm{(R-pres)} & \mathrm{D-sen} & \mathrm{R-sen} \\
\hline
\mathrm{D-rep} & w_1 & x_1 \\
\hline
\mathrm{R-rep} & y_1 & z_1 \\
\hline
\mathrm{d-rep} & 0 & 0 \\
\hline
\end{array}$

See if you can come up with equations. For example, "37 people voted for at least one Democrat" means:

$7+w_0+x_0+y_0+z_0+w_1+x_1+y_1= 37$

 Mars November 3rd, 2009 05:17 PM

Re: possibly a venn diagram solution??

I do not understand the format you used, it seems confusing. I rreally thought that a venn diagram would work better, but I still struggle with it. Any ideas to help make a venn work.

 aswoods November 3rd, 2009 05:54 PM

Re: possibly a venn diagram solution??

A Venn diagram isn't ideal because it won't include the write-in candidate.

If you want to do that, draw three overlapping circles. The first contains people who voted for the Dem congressman, the second contains people who voted for the Dem senator, and the third contains people who voted for the Dem candidate for President. Anyone outside any given circle is assumed to have voted for the Republican candidate for that office.

Have another look at the table; it is simpler. The "D-rep" row contains people who voted for the Democratic congressman, the "R-rep" row contains people who voted for the Republican congressman, and the "d-rep" row contains people who voted for the write-in candidate. The "D-sen" columns contain people who voted for the Dem senator, and so on. The left-hand table contains people who voted for the Democratic candidate for President.

Who voted only for Republican candidates? Answer: $z_1$, because it's in the R-rep row, the R-sen column, and the R-pres table.

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