In　this　paper,　we　consider　the　k-prize-collecting　Steiner　tree　problem　(k-PCST),　extending　both　the　prize-collecting　Steiner　tree　problem　(PCST)　and　the　k-minimum　spanning　tree　problem　(k-MST).　In　this　problem,　we　are　given　a　connected　graph　G=(V,E),　a　root　vertex　r　and　an　integer　k.　Every　edge　in　E　has　a　nonnegative　cost.　Every　vertex　in　V　has　a　nonnegative　penalty　cost.　We　want　to　find　an　r-rooted　tree　F　that　spans　at　least　k　vertices　such　that　the　total　cost,　including　the　edge　cost　of　the　tree　F　and　the　penalty　cost　of　the　vertices　not　spanned　by　F,　is　minimized.　Our　main　contribution　is　to　present　a　5-approximation　algorithm　for　the　k-PCST　via　the　methods　of　primal-dual　and　Lagrangean　relaxation.