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You cannot use the ratio alone to solve for the half-lives of a second order reaction, because the half life of a second order reaction is not dependent only on the rate constant, but also on initial concentration ([A]0). 15.27 deals with the half life of a first order reaction, so it is only dependent on the rate constant, and that is why multiplying it by the ratio works. For 15.35, you must take into account the initial concentrations of the reactants as well as the rate constant in order to get the right answer.
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