An alloy contains zinc and copper in the ratio 3 : 8 and another alloy contains zinc and copper in the ratio 7 : 15.

1. When the two alloys are melted together in the ratio p:q, then the ratio of zinc and copper in the resulting alloy is p:q. Determine, with proof, the ratio p:q.
2. What is the ratio p:q, if keeping all the other conditions in (i) unaltered, the ratio of zinc and copper in the resulting alloy is q:p?

Note: Assume that p and q does not have any common factor > 1.

Charlie's solution is much simpler but I thought I'd share my method anyway.

Start with the fractions

(3/11)p +  (7/22)q = p
(8/11)p + (15/22)q = q

Cross multiplying and gathering terms gives the quadratic

0 = 16p^2 + 9pq - 7 q^2

If you solve this for p using the quadratic formula

p = (9 ± 23)q/32

only the addition give a positive value p = 7q/16 so
p/q = 7/16

The second part eventually gives

p = (9 ± 21)q/12 so p/q = 5/2

Posted by Jer on 2012-12-14 16:10:35