In a  $\Delta ABC$, the sides are in the ratio is  $4:5:6$.  The ratio of  $R,r$

$a:b:c=4:5:6$

$a=4k,b=5k,c=6k$

$s=\frac{15k}{2}$

$\Delta =\sqrt{\frac{15k}{2}(\frac{15k}{2}-4k)(\frac{15k}{2}-5k)(\frac{15k}{2}-6k)}=\frac{15\sqrt{7}}{4}{k}^2$

$R=\frac{abc}{4\Delta }=\frac{4k\cdot 5k\cdot 6k}{\frac{15\sqrt{7}}{4}{k}^2\cdot 4}=\frac{8}{\sqrt{7}}k$

$r=\frac{\Delta }{s}=\frac{\frac{15\sqrt{7}}{4}{k}^2}{\frac{15k}{2}}=\frac{\sqrt{7}}{2}k$

$\therefore R:r=\frac{8}{\sqrt{7}}k:\frac{\sqrt{7}}{2}k$

$=16:7$