Given, sides of triangle are in the ratio 4:5:7.
So, \(a=4k,b=5k,c=7k\)
Semi-perimeter is\[ s=\frac{\left(a+b+c\right)}{2}=8k \]
Area of triangle is \[\Delta=\sqrt{(\mathrm{s})(\mathrm{s}-\mathrm{a})(\mathrm{s}-\mathrm{b})(\mathrm{s}-\mathrm{c})}\]\[ =\sqrt{8k\left(8k-4k\right)\left(8k-5k\right)\left(8k-7k\right)} \]\[=\sqrt{8k\left(4k\right)\left(3k\right)\left(k\right)}=4\sqrt6k^2\]
Now, \[\mathrm{R}=\frac{\mathrm{abc}}{4 \Delta}=\frac{4.5.7.k^3}{4.4\sqrt6k^2}=\frac{5\left(7k\right)}{4\sqrt6}\]
and \[\mathrm{r}=\frac{\Delta}{\mathrm{s}}=\frac{4\sqrt6k^2}{8k}=\frac{\sqrt6k}{2}\]
Thus,
\[\frac{\mathrm{r}}{\mathrm{R}}=\frac{\sqrt6k}{2}\times\frac{4\sqrt6}{5\left(7k\right)}=\frac{12}{35}\]
Therefore, \[ r:R=12:35 \]