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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? | Comment 1 of 3
Unless I erred, there don't seem to be solutions to either part.

The ratio of the mixture of the alloys in the ratio p:q can be expressed as [p(n-1)+qn] : [pn + q(n-1)]

(i) Setting this equal to p:q and cross-multiplying gives
pq(n-1)+nq^2 = np^2+pq(n-1)
0=n(p^2+2pq-q^2)
since n cannot be zero
p^2+2pq^2=0 
and solving for p gives
p=(-2+/-sqrt(2))q
which is impossible since p and q have opposite signs but neither can be negative

(ii) Setting equal to q:p and cross-multiplying gives
(n-1)p^2+pqn=pqn+(n+1)q^2
p^2/q^2=(n+1)/(n-1)
there are solutions here but not positive integer ones.  Either p or q is irrational.

** My dumb calculator thinks there are some solutions but we've seen this problem before.  For values of n from 37 to 50 it gives a fraction answer.

For example n=50 the actual p/q=sqrt(51/49)=sqrt(51)/7 which is clearly irrational but the calculator converts it to 10099/9899.
The decimals tell the tale: 
sqrt(51.000000000000000000000000)/7
1.0202040612204071425713428
10099.000000000000000000000/9899
1.0202040610162642691180927
using http://apfloat.appspot.com/



  Posted by Jer on 2016-10-20 11:31:39
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