1974 AHSME Problems/Problem 6
Problem
For positive real numbers 
and 
define 
' then
Solution
First, let's check for commutivity. We have ![\[y*x=\frac{y\cdot x}{y+x}=\frac{x\cdot y}{x+y}=x*y\]](http://latex.artofproblemsolving.com/1/4/4/144ffa2f41a12855644c53fbb76cd0cbb710824b.png)
, so 
is commutative.
Now we check for associativity. We have ![\[(x*y)*z=\left(\frac{x\cdot y}{x+y}\right)*z=\frac{\frac{x\cdot y}{x+y}\cdot z}{\frac{x\cdot y}{x+y}+z}=\frac{x\cdot y\cdot z}{x\cdot y+z(x+y)}=\frac{x\cdot y\cdot z}{x\cdot y+y\cdot z+x\cdot z}.\]](http://latex.artofproblemsolving.com/8/0/5/8058038396e45c0c82addc82b1b6292de87a03b9.png)
Also, ![\[x*(y*z)=x*\left(\frac{y\cdot z}{y+z}\right)=\frac{x\cdot\frac{y\cdot z}{y+z}}{x+\frac{y\cdot z}{y+z}}=\frac{x\cdot y\cdot z}{x(y+z)+y\cdot z}=\frac{x\cdot y\cdot z}{x\cdot y+y\cdot z+x\cdot z}=(x*y)*z ,\]](http://latex.artofproblemsolving.com/f/8/b/f8be0e79766a4e840b681f9a676912da4c868d1e.png)
and so is also associative. 
See Also
| 1974 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
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