Video solution:

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Written solution:

For the first $15$ miles, Qiang drove for $\dfrac{15}{30} = \dfrac{1}{2}$ an hour.

Let $x$ be the distance that Qiang must drive to satisfy the condition. It will take him $\dfrac{x}{55}$ hours to drive this distance.

The total trip is now $15 + x$ miles. We know the average speed is $50$ miles per hour, so this will take him $\dfrac{x + 15}{50}$ hours.

Setting the two times equal to each other, we get $$\dfrac{1}{2} + \dfrac{x}{55} = \dfrac{x + 15}{50}.$$

Cross-multiplying and simplifying yields $$15 + x = 25 + \dfrac{10x}{11} \Rightarrow x = 110.$$

Thus, the correct answer is D.