Logic Puzzle

A survey of a group of 116 tourists was taken in St. Louis. The survey showed the following:

64 of the tourists plan to visit Gateway Arch;
49 plan to visit the zoo;
11 plan to visit the Art Museum and the zoo, but not the Gateway Arch;
14 plan to visit the Art Museum and the Gateway Arch, but not the zoo;
16 plan to visit the Gateway Arch and the zoo, but not the Art Museum;
9 plan to visit the Art Museum, the zoo, and the Gateway Arch;
16 plan to visit none of the three places.

How many plan to visit the Art Museum only?

OK... did some Venn diagrams. I think the answer is that 12 people are just going to the museum?

EDIT: My solution... 16 people are total slackers, so we'll throw them out, and say that the universe is 100 people. Let's start with the Arch. We have 9 people doing all three things, 14 people also going to the Museum, and 16 also going to the Zoo; these are mutually exclusive. So the arch total is 64, subtracting 14, 16, and 9, we have 25 people just visiting the Arch. Now for the Zoo. Again, 9 are doing all three, 11 are also going to the Museum, and 16 are also going to the Arch; all of which are mutually exclusive. So the Zoo total is 49, subtracting 16, 9, and 11, we have that 13 are just going to the Zoo. Finally, we have a universe of 100 people. 25 are just going to the Arch, and 13 are just going to the Zoo, 14 are doing the Museum and the Arch, 16 the Arch and Zoo, 11 the Museum and Zoo, and 9 people doing all three. Which, by my count, leaves 12 people just going to the Museum.

I think...?

Here's my approach:

Spoiler:

Throw out the 16 who don't visit any place. There are seven different combinations (all three (1x), two (3x), only one (3x)). 14 + 9 + 16 tourists have plans to visit the Gateway Arch plus some other place. In sum, 64 plan to visit the Gateway Arch, so 64 -14 -9 -16 = 25 visit the Gateway Arch and only the Gateway Arch. Same approach for the zoo yields 13 tourists who visit the zoo and only the zoo. For the seven different cominations, we now know the results for six of them. It has to add up to 100. That leaves those who visit the museum and only the museum: 100 - 14 - 11 -9 - 16 - 25 - 13 = 12. Or so I think.

Alzo Sparch Zarathustra. You are correct.

And AlephZero, so are you.

After I posted it, I realized it probably wasn't that challenging a'tall.

I like the way you threw out the 16 slackers, AlephZero. Get rid of the bums.