When you upgrade your subscription from regular Premium to Premium for Family, any remaining time you've already paid for is converted into a Premium for Family subscription. This can bring your next payment date forward. (See your subscription page.)
And when they accept the invite, they're added to Premium for Family straight away. They won't be charged for their account any further. Their future payments are handled by the Premium for Family administrator (you). Any Premium time that they have remaining will resume if they leave Premium for Family.
Note: If they purchased your Premium subscription via a gift/e-card, they'll lose any remaining Premium time when accepting an invite to Premium for Family.
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