Abu Dawud

Benefits and Issues:
From the aforementioned hadith, it is understood that whoever takes excellent care of the family of a mujahid (one striving in the path of Allah), he too receives the full reward like that of the mujahid.
And in the other hadith, mention is made of half the reward.
The reconciliation between these is as follows:
Hafiz Ibn Hajar (rahimahullah) states:
If the total reward of both individuals is divided in half, then it becomes equal for both.
And in this way, there remains no contradiction.
However, the opinion of the present translator (raqim mutarjim) is:
If the one who stayed behind preferred this act of staying (to serve the family), then he will receive half the reward.
But if both are equally eager to participate in the fighting, and the leader (amir) selects one for fighting and assigns the other to serve the family of the fighter,
then in this way, both will be equal in reward.
And Allah knows best.

Hadith Commentary: Benefits and Issues: The person who actually participates in jihad receives two rewards: the original reward and the reward of bounty (increase and addition). As for the one who deputizes for him in good, he receives a reward equal to half of the original reward, and the increase or multiplication is only for the one who actually participates.