Write $x_i = a_i/b_i$, $y_i= t_i b_i$. Your inequality then reduces to the question of if there is some $C$ such that sometimes <center> $$ C \sum_i x_i y_i \le \sum_i x_i \cdot \sum_i y_i, $$ </center> and if so, under what conditions. One answer to this question goes by the name of Chebyshev's inequality in math olympiad circles; it's sometimes called Chebyshev's _order_ inequality or Chebyshev's _sum_ inequality to distinguish it from the (apparently unrelated?) result in probability theory. It can be found in Hardy–Littlewood–Pólya, Steele's book <i>The Cauchy–Schwarz Masterclass</i>, and various places on the internet.