Answer
The ratio of the red pinhole's diameter to that of the violet pinhole should be $~~1.63$
Work Step by Step
We can write an expression for the width of the central maximum:
$w = \frac{2.44~\lambda~L}{d}$
We can equate this expression for the two wavelengths to find the required ratio $\frac{d_r}{d_v}$:
$w = \frac{2.44~\lambda_v~L}{d_v} = \frac{2.44~\lambda_r~L}{d_r}$
$\frac{d_r}{d_v} = \frac{\lambda_r}{\lambda_v}$
$\frac{d_r}{d_v} = \frac{670~nm}{410~nm}$
$\frac{d_r}{d_v} = 1.63$
The ratio of the red pinhole's diameter to that of the violet pinhole should be $~~1.63$
