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They're using the ratios to see how the rate changes with respect to activation energy. I believe you could have plugged in the actual values given in the problem but simply keeping the Ea as a variable makes the calculation easier.
Think about it like this: You have two reaction rates, and you want to compare how the factor is affected by the addition of a catalyst. You know that adding a catalyst changes the activation energy; you know that changing the activation energy changes the reaction rate; you know the formula for the reaction rate. Therefore, in order to calculate the factor by which adding a catalyst changes the reaction rate, you must calculate the ratio of the catalyzed/uncatalyzed reaction rates (Aexp(-Ea[cat]/RT)/Aexp(-Ea[uncat]/RT)). A cancels out.
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