2014/113) find the number of solutions of $\sin\,x = \dfrac{x}{100}$

first let us look at positive x and same number of solutions shall be for -ve x

as $\sin,x$ and $x$ both are odd functions.

as $\sin\,x$ is less than 1 so x shall be less than 100 and so if we draw a sin  curve there shall be $\dfrac{100}{2\pi}$ or 15.91 ( around 16) units so there shall be 16 parts( 1/2 oscilaltions) in positive side and 16 in -ve side.

for each on the curve y = x shall intersect the sin curve 2 times so 32 in positive side including zero) so there is 1 value at 0, 31 positive values and 31 -ve values or 63 values