Let us consider three alloys of which the first alloy contains zinc, tin, and copper in the ratio 2:3:4; the second alloy contains zinc, tin, and copper in the ratio 3:4:5 and, the third alloy contains zinc, tin, and copper in the ratio 4:5:6
- When the three alloys are melted together in the ratio p:q:r, the ratio of zinc, tin, and copper in the resulting alloy is also p:q:r. Determine the ratio p:q:r
- What is the ratio p:q:r, if keeping all the other conditions unaltered, the ratio of zinc, tin, and copper in the resulting alloy is r:q:p?
Note: Assume that each of p, q, and r is a positive integer with gcd(p,q,r)=1
Part 1:
For simplicity let r=1. If the solution for (p,q,1) is rational, it can be scaled up to make them integers.
The extended ratio becomes
2p+3q+4:3p+4q+5:4p+5q+6=p:q:1
This implies 3 equations, but only two will be needed for the two unknowns. (The third has the same intersections in the graph https://www.desmos.com/calculator/c8w9xyri4s)
To solve the system
p(4p+5q+6)=2p+3q+4
q(4p+5q+6)=3p+4q+5
Rather than solve this system of two hyperbolas, I let wolframalpha do the work. The only positive solution is
p=(-5+2sqrt(42))/13
q=(4+sqrt(42))/13
This is the approximate solution (0.61,0.81,1) but there is not exact solution in positive integers.
(There's an integer solution (1,-2,1) but it makes no sense in this context.)
Part 2 is very similar. https://www.desmos.com/calculator/ljteyvymzs
p=(5+sqrt(33))/8
q=(1+sqrt(33))/4
So again, no exact solution in integers, just an approximation (1.34, 1.67,1)
(This one also has the integer solution (1,-2,1))
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Posted by Jer
on 2022-08-16 12:38:48 |
Solution?
