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As the title says, is the damage from a draconic sorcerer's elemental affinity halved when a spell's damage is halved? For example, if a target of the fireball spell makes their save.

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Yes, all the spell's damage is halved. What Elemental Affinity does is alters the damage done by the spell, it doesn't deal its own damage separately. So, anything that later halves the damage done by the spell (such as a saving throw) halves all the spell's damage, including the extra from Elemental Affinity. Just add up all the fireball dice and any damage modifiers to get the final damage amount, and then halve that for targets that made their save.

If Elemental Affinity caused damage separately (such as if it said “when a spell … deals damage, Elemental Affinity deals damage to the target equal to your Charisma modifier…”) then that would be separate and not affected by anything reducing the damage done by the spell. That's not how it's worded though — the wording makes the spell do all the damage.

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