Solution for #21

Let x equal the number of marbles in a small bag, and y represent the number of marbles in a large bag. We know that the following equation holds:

        7x + 18y = 233 

Ordinarily, it is not possible to solve for two unknowns with a single equation; however, we also know that x and y are positive integers and that x is less than y. We can narrow the search by solving for the highest and lowest possible values for y:

     Assume y = x:

        7y + 18y = 233
             25y = 233
               y = 9.32 

Since y must be an integer, y ≥ 10.

     Assume y = 1:

        7 + 18y = 233
            18y = 226
              y = 12.555 

Since y must be an integer, y ≤ 12.

So the only possible values for y are 10, 11, and 12. Substituting each of these values into the original equation, we can find the corresponding values of x. Respectively, they are 7.57, 5, and 2.42. Since x must be an integer, the correct solution is x = 5 and y = 11. So the small bags of marbles contained five each, and the large bags contained eleven each.