The multiple-choice continuous knapsack problem is defined as follows: maximize z =Σ^n_<i=1>Σ^<mi>_<j=1>c_<ij>x_<ij> subject to (1) Σ^n_<i=1>Σ^<mi>_<j=1> a_<ij>x_<ij> 〓 b, (2) 0 〓 x_<ij> 〓 1, i = 1 , 2, …, n, j = 1. 2, …, m_i, (3) at most one of x_<ij>(j = 1, 2, …, m_i) is positive for each i = 1 , 2, …, n, where n, m_i are positive integers and a<ij> are nonnegative integers. In this paper, it is proved that this problem is NP-complete (which strongly suggests the computational intractability to obtain exact optimal solutions). Then, starting with an approximate solution obtained from the LP optimal solution by rounding, two approximate algorithms are proposed and analyzed. The first one (called the breadth-1 search method) obtains an approximate solution with the (worst case) relative error less than 25%. The required computation time is o (mN logN), where N = Σ^n_<i=1>m_i and m = max m_i. The second one (called the breadth-K search method) obtains an approximate solution within an arbitrarily specified (worst case) error bound lt;isinsgt; x 100% in 0 (m┌1/4(&i;sins;-&i;sins;^2)┐N┌l/4(&i;sins;-&i;sins;^2)┐)time.