Step 1: Think about the direction of the share movement:
In sacrifice, share moves away from old partners towards someone joining the firm; in gain, share moves towards continuing partners from someone leaving the firm — opposite directions of flow.
Step 2: Tie each ratio to its formula and use:
Sacrificing Ratio = Old Ratio − New Ratio (used only on admission, to split the premium for goodwill among old partners); Gaining Ratio = New Ratio − Old Ratio (used only on retirement/death, to decide how much each remaining partner pays for goodwill).
Final Answer:
They are mirror concepts used at opposite events — sacrifice ratio (admission, old partners lose share) versus gaining ratio (retirement/death, remaining partners gain share).