In this paper we establish the weak law of large numbers for randomly weighted partial sums of random variables and study conditions imposed on the triangular array of random weights {$W_{nj}{;}:{;}1{leq}j{leq}n,{;}n{geq}1$} and on the triangular array of random variables {$X_{nj}{;}:{;}1{leq}j{leq}n,{;}{geq}1$} which ensure that $sum_{j=1}^{n}{;}W_{nj}{mid}X_{nj}{;}-{;}B_{nj}{mid}$ converges In probability to 0, where {$B_{nj}{;}:{;}1{;}{leq}{;}j{;}{leq}{;}n,{;}n{;}{geq}{;}1$} is a centering array of constants or random variables.