Find five rational numbers between:

-3 and -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.

To insert 5 rational numbers between -3 and -2, multiply both the numerator and the denominator of each number by 5 + 1 i.e. 6.

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

As -18 < -17 < -16 < -15 < -14 < -13 < -12

$\therefore \dfrac{-18}{6} \lt \dfrac{-17}{6} \lt \dfrac{-16}{6} \lt \dfrac{-15}{6} \lt \dfrac{-14}{6} \lt \dfrac{-13}{6} \lt \dfrac{-12}{6}$

Hence, five rational numbers between -3 and -2 are

$\mathbf{\dfrac{-17}{6}, \dfrac{-16}{6}, \dfrac{-15}{6}, \dfrac{-14}{6} \text{ and } \dfrac{-13}{6}}$.