Insert three rational numbers between $\dfrac{1}{3}$ and $\dfrac{4}{5}$, and arrange in descending order.

The L.C.M of 3 and 5 is 15.

$\dfrac{1}{3} = \dfrac{1 \times 5}{3 \times 5} = \dfrac{5}{15} \\[0.5em] \dfrac{4}{5} = \dfrac{4 \times 3}{5 \times 3} = \dfrac{12}{15} \\[0.5em] \text{Since } 5 \lt 12, \dfrac{1}{3} \lt \dfrac{4}{5}$

A rational number between $\dfrac{1}{3}$ and $\dfrac{4}{5}$

$= \dfrac{\dfrac{1}{3} + \dfrac{4}{5}}{2} \\[0.5em] = \dfrac{\dfrac{5 + 12}{15}}{2} \\[0.5em] = \bold{\dfrac{17}{30}} \\[0.5em]$

A rational number between $\dfrac{1}{3}$ and $\dfrac{17}{30}$

$= \dfrac{\dfrac{1}{3} + \dfrac{17}{30}}{2} \\[0.5em] = \dfrac{\dfrac{10 + 17}{30}}{2} \\[0.5em] = \bold{\dfrac{27}{60}} \\[0.5em]$

A rational number between $\dfrac{17}{30}$ and $\dfrac{4}{5}$

$= \dfrac{\dfrac{17}{30} + \dfrac{4}{5}}{2} \\[0.5em] = \dfrac{\dfrac{17 + 24}{30}}{2} \\[0.5em] = \bold{\dfrac{41}{60}} \\[0.5em]$

Numbers in descending order are:

$\bold{\dfrac{4}{5}}, \bold{\dfrac{41}{60}}, \bold{\dfrac{17}{30}}, \bold{\dfrac{27}{60}}, \bold{\dfrac{1}{3}}$