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The reward for the deeds of a child who has not reached puberty – meaning his good deeds – go to the child himself, not to his parents or anyone else, but his parents will be rewarded for teaching and guiding and helping him to do good, because of the report in Saheeh Muslim from Ibn ‘Abbaas (may Allaah be pleased with him) who said, “A woman held up a boy and said, ‘O Messenger of Allaah, will his Hajj be counted?’ He said, ‘Yes, and you will have a reward.’” (Reported by Muslim, 2378)
The Prophet (peace and blessings of Allaah be upon him) stated that the Hajj would be counted for the boy, and that his mother would be rewarded for taking him on Hajj.
Similarly, people other than the parents may also be rewarded for good deeds, such as teaching those under their care such as orphans, relatives, servants and other people, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever guides others to do good will have a reward similar to that of the one who does it …” (reported by Muslim in his Saheeh, 3509), and because this is a form of co-operating in righteousness and piety, which Allaah urges us to do.
Al-Tirmidhi Hadith Number 131 - Narrated Anas ibn Malik
The Prophet (peace be upon him) said: If a young man honours an older person on account of his age, Allah will appoint someone who will honour him, when he attains old age.'
Transmitted by Tirmidhi.
Fiqh-us-Sunnah Volume 3 Number 114b
Though the young are not required to fast, it is proper for their guardians to encourage them to fast so they will become accustomed to it at an early age. They may fast as long as they are able to and then may break it. ArRabi'a bint Mu'awiyyah reported: "The Messenger of Allah sent a man, on the morning of the day of 'Ashurah, to the residences of the Ansar, saying: 'Whoever has spent the morning fasting is to complete his fast. Whoever has not spent this morning fasting should fast for the remainder of the day.' We fasted after that announcement, as did our young children. We would go to the mosque and make toys stuffed with cotton for them to play with. If one of them started crying due to hunger, we would give them a toy to play with until it was time to eat." This is related by al-Bukhari and Muslim.