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Ratio Resolution IV (Posted on 2016-10-20) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 3 of 3 |
There is p*(n-1)+q*n copper and p*n+q*(n+1) zinc in the final alloy in both cases.  In the first case the ratio equals p/q and in the second case the ratio equals q/p.

In case 1 the equation [p*(n-1)+q*n]/[p*n+q*(n+1)] = p/q.  This simplifies to n*(p/q)^2 + 2(p/q) - n = 0. Then by the quadratic formula p/q = -1 + sqrt(n^2+1).

In case 2 the equation [p*(n-1)+q*n]/[p*n+q*(n+1)] = q/p.  This simplifies to (p/q)^2 = (n+1)/(n-1).  Then p/q = sqrt[(n+1)/(n-1)].

In both cases the ratio p/q can never be rational when n is a positive integer, therefore there are no solutions satisfying the final statement "Each of p and q is a positive integer."

  Posted by Brian Smith on 2017-07-24 20:27:09
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