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.
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)
since n cannot be zero
and solving for p gives
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
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:
Posted by Jer
on 2016-10-20 11:31:39