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Let the number of adults be A.
Let the number of pensioners be P.
Let the number of children be C.
We have:
A + P + C = 100
3.50A + P + 0.85C = 100
P < 100
Also A, P and C are integers.
So A + P + C = 3.5A + P + 0.85C
A + C = 3.5A + 0.85C
C = 2.5A + 0.85C
0.15C = 2.5A
3C = 50A
C = 50A/3
Since C is an integer, 50A/3 is an integer
The smallest value of A satisfying this is 0. But then C is 0, so P = 100, which contradicts P < 100.
We turn to the next smallest value for A, which is 3. So C = 50. So P= 47. This is a solution.
The next smallest value for A is 6. But then C = 100, and A + C = 106, exceeding the number of seats.
So A = 3
C = 5
P = 47
This does indeed satisfy the equation 3.50A + P + 0.85C = 100