Answer
460
Work Step by Step
The times rotate by the wheel is
$a_{1}+a_{2}+a_{3}+...=400+\frac{2\times3(10)}{4}+\frac{2\times3^{2}(10)}{4}+\frac{2\times3^{3}(10)}{4}+..$
$=400+2\times10[\frac{3}{4}+(\frac{3}{4})^{2}+(\frac{3}{4})^{3}+...]$
$=400+20(\frac{\frac{3}{4}}{1-\frac{3}{4}})=460$
