Question

Find the mean proportion between $4$ and $25$.

Answer 1

We have to find b such that 4 : b = b : 25. Thus we must have $\frac{4}{b}=\frac{b}{25}$.
Therefore $b^2=4\times 25=100$, which gives $b=\sqrt{100}=10$. (We can also take $b=-10$, since $4:-10=-10:25$)
If three numbers $a,b,c$ are such that $a:b=b:c$, we say $c$ is the third proportion.