Location: The Hideaway (ship)
Petitioner: Shirleigh, Tutor
Unlocked: Complete Things Fall Apart
Reward: 1 Black Blood, 1 Gil Bugs, 25 Renown, 800 gil
In order to unlock this side quest you need to more or less complete Things Fall Apart, a main story quest with three sub-quests attached. Once you've completed the three sub-quests and reported back to Otto at the Hideaway you can begin On Balance.
Enter the Shelves in the Hideaway and speak to Shirleigh, a tutor. She's standing on the right side of the room, and she has a mystery: The scales Mid lent her for lessons are gone. Shirleigh suspects that her students have something to do with its sudden absence, and wants Clive to question the kids.
Head to the Atrium, on the west side of the Hideaway. There are three kids standing at a table, and it is obvious at a glance that they broke the scale. Speak to Josselin, the kid in brown, to learn what happened, then inspect the scales. Have a closer look at the three parts of the scales to wrap up the lesson, then return to Shirleigh to end the quest.