I am facing the following problem:
A number of a adults, b children older than 12, and c children younger than twelve attend an event. The sum of all people a+b+c=100. The prices are \$6 per adult, \$3 per old child, and \$1 per young child. The cumulative money spent 6a+3b+c=310. So far, so good. But from these two equations
a + b + c = 100 6a + 3b + c = 310
How do I determine the variables (a, b, c) for a, b and c, respectively? My (failed) approach so far has been to solve by the parameter, and I got
3b + 5c = 290
but that wasn't much help. Could you help me get on the right track for solving this problem? Thanks.
EDIT: I now managed to solve for maximum a using following operation: By subtracting the upper equation from the equation system from the lower one, I got:
5a + 2b = 210
Solving this by a returned
a = 42 - 0.4b
The maximum value for a is for b = 0, thus a = 42. With b = 0, the only remaining solution for c was 100-a-b = 100-42-0 = 58.
EDIT 2: I now also managed to solve for maximum b and c, too. For b: Using the same previous equation, 5a + 2b = 210, I now solve it for b, and get
b = 105 - 2.5a
Since the static part of the equation is bigger than 100, I add:
b + a ≤ 100, therefore: b + a = 105 - 1.5a ≤ 100 -1.5a ≤ -5 a ≥ 3.333333, and since a needs to be a natural number, a = 4. Thus: b = 105 - 6 - 4 = 95 a = 4 c = 100 - a - b = 100 - 95 - 4 = 1.