Initially we have two pairs of simultaneous equations and 3 variables. If A is the number of adults, C the number of children and P the number of pensioners then we find:
3.5A + 0.85C + P = 100
A + C + P = 100
We realised that the number of pensioners would be equal to the amount the pensioners paid so really there are only two variables:
3.5A +0.85C = 100-p = A + C
2.5A = 0.15C
We decided to make the coefficients whole numbers, since A, C and P are all discrete variables.
250A = 15C
50A = 3C
We can see then that one solution is A=3 and C=50. This will force P = 47. All other whole number solutions will be in the ratio 3:50 adults to children and this means other solutions have C greater than 100 which cannot be true as there are only 100 seats in the cinema.