How to predict one equilibrium constant given two others?
Remember that when you reverse the reaction the equilibrium constant changes. For the general gas-phase reaction $\ce{A(g) + B(g) &lt;=&gt; C(g) + D(g)}$ the equilibrium constant expression is $$K_\text{f} = \frac{p_\ce{C} p_\ce{D} }{p_\ce{A} p_\ce{B}}$$ (Strictly, it's activities, not partial pressures, but the principle is the same.)
The reversed reaction $\ce{C(g) + D(g) &lt;=&gt; A(g) + B(g)}$ has the equilibrium constant expression $$K_\text{r} = \frac{p_\ce{A} p_\ce{B} }{p_\ce{C} p_\ce{D}} = \frac{1}{K_\text{f}}$$

You are correct in multiplying the equilibrium constants to get the equilibrium constant for the combined reaction.