Question

Owen made 100 sandwiches which
she sold for exactly $100. She sold
caviar sandwiches for $5.00 each, the
bologna sandwiches for $2.00, and
the liverwurst sandwiches for 10 cents.
How many of each type of sandwich
did she make?

Answer 1

Ok, so the key insight to this problem: the fact that the solution must be non-negative integer values for each type of sandwich is an additional constraint on the problem. There doesn't seem to be a purely analytical solution to this problem; one must do some sort of searching to find it.

So here's the setup.

5x + 2y + 0.1z = 100
x + y + z = 100

Let's convert this to:
50x + 20y + z = 1000
x + y + z = 100

let eq 1 = eq 1 - eq 2 -> 49x + 19y = 900
So, x = (900 - 19y)/49

y is a free parameter.

x = (900 - 19y)/49
y = y
z = 100 - x - z = 100 - (900 - 19y)/49 - y

So, now, iterate through values of y until all x, y, and z (as a function of y as defined above) is a non-negative integer.

This happens when y = 19, therefore x = 11 and z = 70.

Answer 2

Btw, I just made a simple Python script to iterate through a bunch of values for y and plugging those values into the equations to see that all of them are non-negative integers. There was only one valid solution, as already given: x = 11, y = 19, z = 70

Answer 3

Btw, this question is just a slight dip into Operations Research, which was basically invented during WW2 to help solve logistic problems and other sorts of decision problems.