In a triangle ABC, a:b:c = 4:5:6. The ratio of the radius of the circumcircle to that of the incircle is

Let $\frac{a}{4}=\frac{b}{5}=\frac{c}{6}=k$

$\begin{array}{r}R=\frac{abc}{4\mathrm{\Delta }}\text{ and }r=\frac{\mathrm{\Delta }}{s}\\ \Rightarrow \frac{R}{r}=\frac{abc}{4\mathrm{\Delta }}\times \frac{s}{\mathrm{\Delta }}=\frac{sabc}{4{\mathrm{\Delta }}^2}\end{array}$

$\begin{array}{l}=\frac{s\mathit{ }abc}{4s(s-a)(s-b)(s-c)}\\ =\frac{4k\times 5k\times 6k}{4\left(\frac{15}{2}k-4k\right)\left(\frac{15}{2}k-5k\right)\left(\frac{15}{2}k-6k\right)}\\ =\frac{5\times 6\times 8}{7\times 5\times 3}=\frac{16}{7}.\end{array}$