Find five rational numbers between:

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

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

First convert the mixed fraction into an improper fraction and write both the numbers with a common denominator.

$\Rightarrow -1\dfrac{1}{2} = \dfrac{-3}{2} \\[1em] \Rightarrow -2 = \dfrac{-4}{2}$

To insert 5 rational numbers, multiply both the numerator and the denominator of each rational number by 5 + 1 i.e. 6.

$\Rightarrow \dfrac{-4}{2} = \dfrac{-4 \times 6}{2 \times 6} = \dfrac{-24}{12} \\[1em] \Rightarrow \dfrac{-3}{2} = \dfrac{-3 \times 6}{2 \times 6} = \dfrac{-18}{12}$

As -24 < -23 < -22 < -21 < -20 < -19 < -18,

$\therefore \dfrac{-24}{12} \lt \dfrac{-23}{12} \lt \dfrac{-22}{12} \lt \dfrac{-21}{12} \lt \dfrac{-20}{12} \lt \dfrac{-19}{12} \lt \dfrac{-18}{12}$

Hence, five rational numbers between -2 and $\mathbf{-1\dfrac{1}{2}}$ are

$\mathbf{\dfrac{-23}{12}, \dfrac{-22}{12}, \dfrac{-21}{12}, \dfrac{-20}{12} \text{ and } \dfrac{-19}{12}}$.