What is the ratio of the number of men in the production department to the number of men in the marketing department ?

Total employees given= 4600

Total employees in production = 15% of 4600= 690.

Ratio of men and women in Production= 3:2

Number of men= $$\ \ \frac{\ 3}{5}\times\ 690=3\times\ 138=414$$

Total employees in Marketing= 22% of 4600= 22$$\times\ $$46= 1012. 

Ratio of men and women in Production= 1:1

Therefore number of men= $$\ \frac{\ 1}{2}\times\ 1012=\ 506$$

So, Number of men in Production: Number of men in Marketing= $$\ \frac{\ 414}{506}=\ \frac{\ 9}{11}$$