Image found at this…awesome site.

This will be the last Weekend Challenge question that has access to the Math Guide Beta Program as its prize. I’m going to be closing the Beta to new users soon, as the book nears completion. If, by the way, you’re interested in the day-to-day progress I’m making on the book, you can check in on me at Google+.

When I write questions (and I have to imagine this is the same way every question writer does it) I just put in placeholder numbers while I write and then I go back and solve. If the numbers don’t work out nicely (say, a fractional child, or something) I’ll change them around. But when I wrote this one, the numbers worked out perfectly the first time. That feels awesome. Anyhoo, Beta access to the first non-anonymous commenter to PWN the following question:

At Masuk High School, 200 people are in the chorus, 130 people are in the band, and 45 people take AP calculus. If, in total, 92 people take two of the three classes, and 80 take just one of the three, how many students take all three classes?

I’ll post the solution Monday. Good luck, and have a great weekend.

UPDATE: Nice work, Jeffery. You’re officially the last person into the Beta. I hope you enjoy it.

Solution below (although Jeffery and JD both posted rather nice ones in the comments).

If you were thinking Venn diagram here, then you’re eVenn more clever than I thought!

chorus: 200 = c + p + n + x
band: 130 = b + p + m + x
AP calc: 45 = a + m + n + x
Add those all up, and you get:
375 = a + b + c + 2m + 2n + 2p + 3x
375 = a + b + c + 2(m + n + p) + 3x

From the question you know that 92 people take 2 of the three classes, so:

m + n + p = 92

You also know 80 people take only one of the three, so:

a + b + c = 80
Substitute:
375 = 80 + 2(92) + 3x
111 = 3x
37 = x

Awesome, right?

Jeffery Jung says:

variables:
a = people who take chorus and band
b = people who take AP Calculus and band
c = people who take AP Calculus and chorus
d = people who take all three

People in only chorus = 200 – (c+a+d)People in only band = 130 – (a+b+d)
People in only AP Calculus = 45 – (c+d+b)
a+c+b = 92
(200 – (c+a+d)) + (130 – (a+b+d)) + (45 – (c+d+b)) = 80
200 – c – a – d + 130 – a – b – d + 45 – c – d – b = 80
375 – 2c – 2a – 2b – 3d = 80
375 – 2(c+a+b) – 3d = 80; substitute a+c+b = 92
375 – 2(92) – 3d = 80
375 – 194 – 3d = 80
181 – 3d = 80
111 = 3d
d = 37

JD says:

200+130+45 = 375 total classes taken

92*2 + 80 = 264 classes taken by doubles and singles

therefore:

375 – 264 = 111 classes taken by triples

111 / 3 = 37 students

no algebra necessary!

JD

PS good luck with book sales PWN, seems to be an awesome product. unfortunately I have been unable to get through it yet but I will!