Mary, James, and Carlos sold $\dfrac{1}{4}$-page advertisements for the school yearbook. Mary sold twice as many as Carlos did, and James sold 3 times as many as Mary did. What fraction of these advertisements did Carlos sell?

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Mary sold twice the amount as Carlos:

$$M = 2C$$

James sold three times as much as Mary:

$$J = 3M$$ $$J = 3(2C)$$ $$J = 6C$$

The total:

$$C + M + J$$ $$C + 2C + 6C$$ $$= 9C$$

Therefore, Carlos sold:

$$\frac{C}{9C} = \boxed{\frac{1}{9}}$$