Find 12 rational numbers between -1 and 2.

To insert 12 rational numbers, write both the numbers with the denominator 12 + 1 i.e. 13.

$\Rightarrow -1 = \dfrac{-1 \times 13}{1 \times 13} = \dfrac{-13}{13} \\[1em] \Rightarrow 2 = \dfrac{2 \times 13}{1 \times 13} = \dfrac{26}{13}$

We note that -12, -11, -10, …, 24, 25 are the integers between -13 and 26

i.e. -13 < -12 < -11 < -10 < … < -2 < -1 < 0 < … < 25 < 26

Choosing any twelve of these integers as numerators over the common denominator 13, we get the required rational numbers.

Hence, twelve rational numbers between -1 and 2 are

$\mathbf{\dfrac{-12}{13}, \dfrac{-11}{13}, \dfrac{-10}{13}, \dfrac{-9}{13}, \dfrac{-8}{13}, \dfrac{-7}{13}, \dfrac{-6}{13}, \dfrac{-5}{13}, \dfrac{-4}{13}, \dfrac{-3}{13}, \dfrac{-2}{13} \text{ and } \dfrac{-1}{13}}$.