An alloy contains zinc and copper in the ratio 3 : 8 and another alloy contains zinc and copper in the ratio 7 : 15.
- When the two alloys are melted together in the ratio p:q, then the ratio of zinc and copper in the resulting alloy is p:q. Determine, with proof, the ratio p:q.
- What is the ratio p:q, if keeping all the other conditions in (i) unaltered, the ratio of zinc and copper in the resulting alloy is q:p?
Note: Assume that p and q does not have any common factor > 1.
Normalize the ratios so they add to 1.
Alloy 1 has a ratio of [Zn, Cu] = [3/11, 8/11]
Alloy 2 has a ratio of [Zn, Cu] = [7/22, 15/22]
Then the alloy ratio formed by mixing Alloy 1 : alloy 2 in the ration p : p is given by the matrix equation:
[3/11 7/22] * [p] = [r]
[8/11 15/22] [q] [s]
Then for part 1 [r, s] = k*[p, q]. Then the matrix equation describes the eigenvector [p, q] for eigenvalue k.
The eigenvalue/eigenvector pairs for the matrix are
k=1; [p,q]=[7/16, 1]
k=-1/22; [p,q]=[-1, 1]
The second is to be discarded leaving [p,q]=[7/16,1]. Renormalizing to have p and q integers yields the answer of p:q = 7:16.
Now for part 2 we have [r, s] = k*[q, p].
Left-multiply both sides by the permutation matrix to swap the rows, then the matrix equation becomes
[8/11 15/22] * [p] = k*[p]
[3/11 7/22] [q] [q]
This time the eigenvalue/eigenvector pairs for the matrix are
k=1; [p,q]=[5/2, 1]
k=1/22; [p,q]=[-1, 1]
The second is to be discarded leaving [p,q]=[5/2,1]. Renormalizing to have p and q integers yields the answer of p:q = 5:2.
Another solution
