Find three rational numbers between:

$\dfrac{-1}{2} \text{ and } \dfrac{-1}{4}$

Write both the rational numbers with a common denominator, then multiply the numerator and the denominator of each by one more than the number of rational numbers required.

$[\text{ LCM of } 2 \text{ and } 4 = 4] \\[1em] \Rightarrow \dfrac{-1}{2} = \dfrac{-1 \times 2}{2 \times 2} = \dfrac{-2}{4} \\[1em] \Rightarrow \dfrac{-1}{4} = \dfrac{-1}{4}$

To insert 3 rational numbers, multiply both the numerator and the denominator of each rational number by 3 + 1 i.e. 4.

$\Rightarrow \dfrac{-2}{4} = \dfrac{-2 \times 4}{4 \times 4} = \dfrac{-8}{16} \\[1em] \Rightarrow \dfrac{-1}{4} = \dfrac{-1 \times 4}{4 \times 4} = \dfrac{-4}{16}$

As -8 < -7 < -6 < -5 < -4,

$\therefore \dfrac{-8}{16} \lt \dfrac{-7}{16} \lt \dfrac{-6}{16} \lt \dfrac{-5}{16} \lt \dfrac{-4}{16}$

Hence, three rational numbers between $\mathbf{\dfrac{-1}{2} \text{ and } \dfrac{-1}{4}}$ are $\mathbf{\dfrac{-7}{16}, \dfrac{-6}{16} \text{ and } \dfrac{-5}{16}}$.