300 students participated in a symposium with 3 conferences in sequence.
Half of the students that attended the first conference, attended neither the other two.
One-third of the students that attended the second conference, attended neither the other two.
And one-fourth of the students that attended the third conference, attended neither the other two.
Knowing that the three conferences were attended by the same number of students, and that each of the 300 students attended at least one conference :
a) how many students attended each conference ?
b) how many students attended only one conference ?
c) how many students attended only two conferences ?
d) how many students attended all 3 conferences ?
One unique solution !
I solved this problem with a system of equations, however, I had 8 unknowns and 7 equations, which leads to a parametric equation. This parametric equation was resticted by the fact that the variables all had to be positive integers, but it did require some trial and error to find the correct solution, so hopefully this will satisfy you that some of us youngsters are still going ahead with logical solution methods.
My variables are a,b,c,d,e,f,g,n for x(1),x(2),x(3),x(12),x(13),x(23),x(123),number of attendees each.
Using the information, we see that:
A=D+E+G
2B=D+F+G
3C=E+F+G
A+D+E+G=N
B+D+F+G=N
C+E+F+G=N
A+B+C+D+E+F+G=300
Which reduces to a parametric solution with one variable left (I chose n)
a=1/2*n
b=1/3#n
c=1/4*n
d=300*22/12*n
e=300-21/12*n
f=300-19/12*n
g=49/12*n-600
Now, my first selection of n (144) yielded a negative g, and my second value (trial'd) was 156, yielding all positive integers results. A check of the next n (168) yields some negatives and so I am convinced this is this single answer
a,b,c,d,e,f,g,n=78,52,39,14,27,53,27,156
And no computer was used. :)
Ok then
