Husband and wife together with mother buy a house $452,855 purchase price in 2005. Mother puts down $152,000 downpayment H&W hold mortgage of $300,855.Mother has 33.6% ownership and H&W have 66.4% ownership. In 2009 a refi gave mother $20,000 and H&W $20,000. Mother $152,000 minus $20,000 now gives her 29.1% ownership and H&W 70.9% ownership. 2015 house appraises at $400,000. Does Mother get $400k x 29.1% ownership for a buyout or is her percentage reduced to 26.35% because the value of the house went down?
When these three people bought the house, they should have signed a document saying what fraction of the house each owns, and how it's value would be divided between them when sold. If they didn't then there may be a default division that is applied legally.
If it isn't defined in law or the agreement then it is up to them to decide how the value is split on sale. They can decide to do it however they want and whatever they think is fair. There is no "correct" way.