20. The ratio of two positive numbers equals the ratio of their sum to their difference. What is this ratio?
A $(1+\sqrt{3}): 2$
B $\sqrt{2}: 1$
C $(1+\sqrt{5}): 2$
D $(2+\sqrt{2}): 1$
$\mathrm{E}(1+\sqrt{2}): 1$
Solution
E Let the two numbers be $x$ and $y$ and suppose that $x: y=k: 1$, so that $x=k y$. Since the ratio $x+y: x-y$ is also $k: 1$, it follows that $x+y=k(x-y)$ and so $k y+y=k(k y-y)$. Since $y$ is positive, we may divide through by $y$ to obtain $k+1=k(k-1)$.
It follows that $k^{2}-2 k-1=0$, and so, using the standard formula for the roots of a quadratic equation, we get
$$
k=\frac{2 \pm \sqrt{4+4}}{2}=1 \pm \sqrt{2} .
$$

Since $x$ and $y$ are both positive $k>0$, and it follows that $k=1+\sqrt{2}$.