The ratio of the third proportional to 16 & 40 and the mean proportional between 10 & 40 is:

5 : 1

Third proportion to a, b = $\frac{b\; \times\; b}{a}$

Third proportion to 16, 40 = \(\frac{40\; \times\; 40}{16}\) 

                                         = 100

Mean proportion to a, b = $\sqrt{a\; \times\; b}$

Mean proportion to 10, 40 = \(\sqrt{10\; \times\;40}\) 

                                       = 20    

Ratio = 100 : 20 = 5 : 1