Pet Aging and Maturity
Maturity is measured in hours and comes in four categories: Eggling, Hatchling, Adolescent, and Adult.

It takes roughly 72 hours (3 Days) for a newborn Eggling to become a Hatchling.
It then takes roughly an additional 144 hours (6 Days) for a Hatchling to become an Adolescent.
It then takes an additional 18 Days for an Adolescent to become an Adult.
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You can either wait a few days for the egg to hatch, or nurture it, but it takes a few successful nurtures per percent point to get it to hatch. You nurtured it once. The only way to tell if it's successful is if it matches with the description it had when you were nurturing it (ie if the egg is cool, warm it/if it's hot, cool it).

It'll take awhile, so it's mostly waiting.