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He simply set up an ICE table with the provided information and the assumption that the initial concentrations of AD and Pi were 0 initially. With the initial and equilibrium concentrations of ADP provided in the problem, he calculated the change in concentration for ADP. Because the molar ratios of all reactants and products were 1:1, we can assume that the concentrations of both AD and Pi changed by the same amount. This gives you the equilibrium molarity of all reactants and products.
For this example, the mole ratios of ADP to AD and ADP to Pi are 1, so the change in moles of products is the same as change in moles of reactants. If the mole ratios were different, you would have to multiply by stoichiometric coefficients before finding equilibrium concentrations.
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