Here is the question: A a number x is said to be the harmonic mean of y and z if 1/x is the average of 1/y and 1/z.
a. Write an equation for which the solution is the harmonic mean of 30 and 45.
b. Find the harmonic mean of 30 and 45.
My book doesnt give an explanation of what a harmonic mean is, how to make an equation for one or how to solve one therefore please make your explanation as detailed as possible.
Let $a=\frac{1}{y}$ and $b=\frac{1}{z}.$ The average of $a,b$ is $$ \frac{(a+b)}{2}=\frac{\frac{1}{y}+\frac{1}{z}}{2}=\frac{1}{2}\left(\frac{1}{y}+\frac{1}{z}\right)=\frac{(y+z)}{2yz}. $$ The Harmonic mean $x$ is the reciprocal of this average so $$ x=\frac{2yz}{(y+z)}.$$ So, the answer to (a) is (letting $y=30$, $z=45$), $$ x=\frac{2yz}{(y+z)}=\frac{2\times 30\times 45}{30 + 45}. $$ This should be enough to enable you to answer (b).