Find 5 rational numbers between $\frac{1}{6}$ and $\frac{2}{5}$.

LCM(6, 5) = 30.

$$\frac{1}{6} = \frac{5}{30} \text{ and } \frac{2}{5} = \frac{12}{30}.$$

Only 6 integers (6 to 11) lie strictly between 5 and 12 in the numerator, so we enlarge further.

Multiply both by 2: $\frac{1}{6} = \frac{10}{60}$ and $\frac{2}{5} = \frac{24}{60}$.

Five rationals: $\frac{11}{60}, \frac{13}{60}, \frac{15}{60}, \frac{17}{60}, \frac{19}{60}$.