BDD-based SMT solvers have recently shown to be competitive for solving satisfiability of quantified bit-vector formulas. However, these solvers reach their limits when the input formula contains complicated arithmetic. Hitherto, this problem has been alleviated by approximations reducing efficient bit-widths of bit-vector variables. In this paper, we propose an orthogonal abstraction technique working on the level of the individual instances of bit-vector operations. In particular, we compute only several bits of the operation result, which may be sufficient to decide the satisfiability of the formula. Experimental results show that our BDD-based SMT solver Q3B extended with these abstractions can solve more quantified bit-vector formulas from the smt-lib repository than state-of-the-art SMT solvers Boolector, CVC4, and Z3.