Alloy 1 contains copper and zinc in the ratio N-1:N and Alloy 2 contains copper and zinc in the ratio N:N+1, where N is a positive integer > 1.
(i) Alloy 1 and Alloy 2 are melted together in the ratio P:Q, so that the ratio of copper and zinc in the resulting alloy is P:Q.
(ii) Alloy 1 and Alloy 2 are melted together in the ratio P:Q, so that the ratio of copper and zinc in the resulting alloy is Q:P.
In each of the cases (i) and (ii) - determine p:q in terms of N.
*** Each of p and q is a positive integer.
(In reply to
Solution by Brian Smith)
After solving Ratio Resolution VII, I remembered this problem and realized I made a major mistake solving it. Biggest error is that I never normalized the alloy proportions like I did in the other problem.
Alloy 1 has (Cu, Zn) = ( (N-1)/(2N-1), N/(2N-1) ) and Alloy 2 has (Cu, Zn) = ( N/(2N+1), (N+1)/(2N+1) ).
Then part 1 has (Cu, Zn) = (p*(N-1)/(2N-1) + q*N/(2N+1), p*N/(2N-1) + q*(N+1)/(2N+1))
So the ratio we actually want is [p*(N-1)/(2N-1) + q*N/(2N+1)]/[p*N/(2N-1) + q*(N+1)/(2N+1)] = p/q.
This simplifies to (2N^2+N)(p/q)^2 + (2N)(p/q) - (2N^2-N) = 0
Then taking the positive root for p/q, we get p/q = (2N-1)/(2N+1) to which we can take p=2N-1 and q=2N+1.
Then part 2 has [p*(N-1)/(2N-1) + q*N/(2N+1)]/[p*N/(2N-1) + q*(N+1)/(2N+1)] = q/p.
This simplifies to (2N^2-N-1)(p/q)^2 - (2N)(p/q) - (2N^2+N-1) = 0
Then taking the positive root for p/q, we get p/q = (2N^2+N-1)/(2N^2-N-1) to which we can take p=2N^2+N-1 and q=2N^2-N-1.
re: Solution - correction
