What is important about the two halves of the song Jing-Mei discovers in Two Kinds by Amy Tan?

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It is highly significant that Jing Mei only discovers the second part of the song at the end of the short story and after her mother has died and she has become an adult. This is because Jing Mei's discovery of the partner-song to "Pleading Child" indicates her own development as an individual and her arrival at a stage where she is happy with who she is and is no longer trying to be someone she is not or live her life for someone else (namely her mother). The title, "Perfectly Contented" clearly suggests that having gone through a stage where Jing Mei was a "Pleading Child", desperate for her mother's approval, she is now happy with herself.

Jing Mei's realisation that they were "two halves of the same song" perhaps indicates that this is a universal struggle that all must go through: we all go through a stage when we are a "Pleading Child", wanting our parents' approval and aprobation, yet eventually have to learn to live our own life and make our own choices, and become "Perfectly Contented."