Question

Two questions:

(1) In a class every boy is friends with exactly 3 girls. Every girl is friends with exactly 2 boys. There are only 19 desks (each holding at most two students), and 31 of the students in the class study French. How many students are there?

(2) A smaller circle rolls along the inside curve of a larger circle. The diameter of the larger circle is 3 times that of the smaller circle. How many revolutions does the small circle have to make to complete the inside curve of the larger circle?

I think the answer is 3. If I cut the circles and lay them flat on the ground, the larger circle is 3 times longer than the smaller circle. So the smaller circle would have to revolve 3 times in order to equal the distance inside the larger circle. Is that correct?

Answer 1

yes on #2

Answer 2

What is the answer and solution to #1?

Answer 3

I can't quite get my head around this the way it is stated.

I see four restrictions:
(b -- boys, g -- girls)

b+g ≥ 31
b+g ≤ 38
g ≤ 3b
b ≥ 2g

If I graph this on a b-g axes graph, there is a region with about 80 ordered pairs that obey all 4 restrictions

e.g. b = 12 g = 22
b+g = 34 , which is less than 38 and greater than 31, check first two restrictions

third: is 22 < 3(12) ? yes
fourth: is 12 < 2(22) yes.

What don't I see ?

Answer 4

36?