Perhaps he was influenced by some new friends he met who had a different lifestyle or view of the world. They might have convinced him that running away from home was an exciting adventure and a way to break free from the restrictions at home.
He might first find a place to hide, like an abandoned building. There, he starts to realize how difficult it is to be on his own. He has to find food and water by himself.
One possible reason could be that he feels misunderstood. In the camp, everyone has high expectations of him as a son of Poseidon. If he makes a small mistake, it might be blown out of proportion. For example, if he accidentally causes a small flood during a training session, instead of being understood as still learning control over his powers, he could be harshly criticized. Another reason might be that he's tired of the constant danger at the camp. Every day there seems to be a new threat looming, whether it's a monster attack or some sort of internal camp conflict. He might just want to get away and have some peace for a while.
It could be due to a conflict with a particular authority figure at the camp. If he has a major disagreement with a camp leader over how to handle a situation, like dealing with a new enemy. Percy, being headstrong, might decide to run away and handle it his own way instead of following the camp's rules.
One possible reason could be the constant feeling of being an outcast in the village. Despite his efforts, many villagers still look at him with suspicion because of the Nine - Tails inside him. He might feel that running away would give him a chance to find a place where he is truly accepted.
Another possible reason is a sense of not belonging. Maybe she is different from the rest of her family in terms of beliefs or interests, and she runs away to find a place where she can be herself without judgment.
One possible reason could be that she feels misunderstood at home. In the wizarding world, Hermione is a very intelligent and studious witch. If her family doesn't understand or support her magical pursuits, she might feel the urge to run away. For example, if they constantly tell her to focus more on 'normal' things and less on her spells and potions, it could drive her away.
Her friends would be extremely worried. Harry and Ron, especially, would be very concerned about her safety. They might start searching for her immediately, thinking that she could be in some kind of trouble or danger. They know Hermione well and understand that she wouldn't run away without a good reason.
She might run away because of a conflict with her friends or team members. Maybe there was a misunderstanding or a disagreement that she couldn't resolve at that moment, and running away seemed like the best way to avoid further confrontation.
One possibility in 'Percy Runs Away Fanfiction' is that Percy runs away because he is being hunted. There could be a powerful enemy after him, and he decides that running away is the best way to keep his loved ones out of danger. So he leaves his home and goes into hiding, maybe in a far - off land or a hidden magical place, and while he's away, he has to learn to survive on his own and also figure out how to defeat the enemy that's chasing him.
One possible reason could be family pressure. In the Weasley family, there are high expectations and a lot of chaos. Percy might feel suffocated and decide to run away to find his own path. Maybe he's tired of always being in the shadow of his more popular siblings and wants to prove himself on his own terms.