Four positive numbers, a, b, c, and d, are related in a specific way.  The first number, a, is to the second number, b, as 3 is to 5.  Interestingly, the ratio between b and c is the same as the ratio between c and d.  Furthermore, the ratio of a to d is given as 3 to 8.  Given that d has a value of 16, determine the numerical value of b.