If the number of hardbacks is 1/3 the number of paperbacks, the ratio of hardbacks to paperbacks is 1:3. That's a part-to-part ratio; more useful here would be a part-to-whole ratio. The ratio of hardbacks to paperbacks to total is 1:3:4, meaning that hardbacks make up 1/4 of the total and paperbacks make up 3/4 of the total. That's (1/4)b and (3/4)b.
3/4 of the paperbacks are biographies. The paperbacks are (3/4)b, so multiply that number by 3/4, and we have the number of biographies:
(3/4)(3/4)b = (9/16)b, choice (E).