Given $a: b: c=4: 5: 6$
To find R : r


Where, R is circumradius and r is inradius.
Let a + 4k, b = 5k and c = 6k
$\begin{aligned} & s=\frac{a+b+c}{2}=\frac{15 k}{2} \\ & \therefore s-a=\frac{15 k}{2}-4 k=\frac{7 k}{2}, s-b=\frac{5 k}{2} \text { and } s-c=\frac{3 k}{2} \\ & \therefore \quad \Delta=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{\frac{15 k}{2} \cdot \frac{7 k}{2} \cdot \frac{5 k}{2} \cdot \frac{3 k}{2}} \\ & \Delta=\frac{15 \sqrt{7}}{4} k^2 \\ & \therefore \quad R=\frac{a b c}{4 \Delta}=\frac{4 k \cdot 5 k \cdot 6 k}{4 \cdot \frac{15 \sqrt{7}}{4} k^2}=\frac{8}{\sqrt{7}} k \\ & \quad r=\frac{\Delta}{s}=\frac{\frac{15 \sqrt{7}}{\frac{15}{2}} k}{k^2}=\frac{\sqrt{7}}{2} k \\ & \therefore R: r=\frac{8}{\sqrt{7}} k: \frac{\sqrt{7}}{2} k=16: 7\end{aligned}$