First, we show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the complete tripartite multi-graph λ Kn1,n2,n3 is (i) n1=n2=n3 0 (mod 6) for λ 1,5 (mod 6), (ii) n1=n2=n3 0 (mod 3) for λ 2,4 (mod 6), (iii) n1=n2=n3 0 (mod 2) for λ 3 (mod 6), and (iv) n1=n2=n3 2 for λ 0 (mod 6). Next, we show that the necessary and sufficient condition for the existence of a balanced trefoil decomposition of the complete tripartite multi-graph λ Kn1,n2,n3 is (i) n1=n2=n3 0 (mod 9) for λ 1,2,4,5,7,8 (mod 9), (ii) n1=n2=n3 0 (mod 3) for λ 3,6 (mod 9), and (iii) n1=n2=n3 3 for λ 0 (mod 9).