Find a rational number between the following pair of rational numbers:

$\dfrac{-3}{5} \text{ and } \dfrac{-4}{7}$

If $x$ and $y$ are two rational numbers, then $\dfrac{1}{2}(x + y)$ is a rational number lying between them.

Finding the required rational number,

LCM of 5 and 7 = 35

$\dfrac{1}{2} \times \left(\dfrac{-3}{5} + \dfrac{-4}{7}\right) = \dfrac{1}{2} \times \left(\dfrac{(-3) \times 7 + (-4) \times 5}{35}\right) \\[1em] = \dfrac{1}{2} \times \dfrac{-21 - 20}{35} \\[1em] = \dfrac{1}{2} \times \dfrac{-41}{35} \\[1em] = \dfrac{-41}{70}$

Hence, $\mathbf{\dfrac{-41}{70}}$ is a rational number lying between $\mathbf{\dfrac{-3}{5} \text{ and } \dfrac{-4}{7}}$.