The ending of the story suggests that both of these pieces of music are symbolically related to our lives and the process of growing up, with the titles being very significant in terms of how we develop and teh various stages of relationships that we have with our parents. Let us look at the last paragraph:
And for the first time, or so it seemed, I noticed the piece on the right-hand side. It was called "Perfectly Contented." I tried to play this one as well. It had a lighter melody but the same flowing rhythm and turned out to be quite easy. "Pleading Child" was shorter but slower; "Perfectly Contented" was longer but faster. And after I played them both a few times, I realised they were two halves of the same song.
Both of these pieces of music then relate to specific stages. Note how "Pleading Child" is actually quite short, but because it is slower, it appears to be longer. However, "Perfectly Contented" is longer. Jing-Mei has passed through her "Pleading Child" stage, and now, having gone through the conflict with her mother, is able to reach the stage of being "Perfectly Contented," when she is at peace with herself and with her mother. The story ends with the realisation that both of these stages are necessary and related to each other.