Let us consider three alloys of which the first alloy contains zinc, tin, and copper in the ratio 2:3:4; the second alloy contains zinc, tin, and copper in the ratio 3:4:5 and, the third alloy contains zinc, tin, and copper in the ratio 4:5:6
- When the three alloys are melted together in the ratio p:q:r, the ratio of zinc, tin, and copper in the resulting alloy is also p:q:r. Determine the ratio p:q:r
- What is the ratio p:q:r, if keeping all the other conditions unaltered, the ratio of zinc, tin, and copper in the resulting alloy is r:q:p?
Note: Assume that each of p, q, and r is a positive integer with gcd(p,q,r)=1
Alloy 1 has a ratio (Zn, Sn, Cu) = (2/9, 1/3, 4/9).
Alloy 2 has a ratio (Zn, Sn, Cu) = (1/4, 1/3, 5/12).
Alloy 3 has a ratio (Zn, Sn, Cu) = (4/15, 1/3, 2/5).
We are to mix them in the ratio (p, q, r).
There are two results sought for the two parts of the problem:
Alloy R1 with a ratio (Zn, Sn, Cu) = (kp, kq, kr) for some k.
Alloy R2 with a ratio (Zn, Sn, Cu) = (mr, mq, mp) for some m.
Treat all these as column vectors. And for now focus on part 1. Then we can combine the three base Alloy vectors into a matrix and form a matrix equation:
[ 2/9 1/4 4/15 ] [p] [p]
[ 1/3 1/3 1/3 ] * [q] = k*[q]
[ 4/9 5/12 2/5 ] [r] [r]
Then the potential mixing ratio vectors (p,q,r) are the eigenvectors of the matrix. Just about any good math software has capabilities to do this, including Wolfram Alpha.
The eigenvalue/eigenvectors are:
k1=1 -> (p,q,r)=(3/5, 4/5, 1)
k2=-2/45 -> (p,q,r)=(-1, 0, 1)
k3=0 -> (p,q,r)=(3/5, -8/5, 1)
The second and third eigenvectors must be rejected as we are seeking positive solutions. So the one solution is (p,q,r)=(3/5, 4/5, 1). We can scale this so that p, q, and r are integers yielding p:q:r = 3:4:5.
Now part 2. Its matrix equation is:
[ 2/9 1/4 4/15 ] [p] [r]
[ 1/3 1/3 1/3 ] * [q] = m*[q]
[ 4/9 5/12 2/5 ] [r] [p]
This can be modified by left-multiplying both sides by a permutation matrix
[ 0 0 1 ]
[ 0 1 0 ]
[ 1 0 0 ]
Then the new equation becomes
[ 4/9 5/12 2/5 ] [p] [p]
[ 1/3 1/3 1/3 ] * [q] = m*[q]
[ 2/9 1/4 4/15 ] [r] [r]
Now same as before, find the eigenvalues/eigenvectors:
m1=1 -> (p,q,r)=(219/125, 172/125, 1)
m2=2/45 -> (p,q,r)=(-1, 0, 1)
m3=0 -> (p,q,r)=(3/5, -8/5, 1)
Again, the second and third eigenvectors must be rejected as we are seeking positive solutions. So the one solution is (p,q,r)=(219/125, 172/125, 1). We can scale this so that p, q, and r are integers yielding p:q:r = 219:172:125.
Solution
