We can start at the end
The third prince ate 1/2 and left behind 3 mangoes. Therefore, there were 6 mangoes before he ate 1/2.
Next, the second prince ate 1/3, so there were 9 before he ate the mangoes.
The first prince ate 1/4, so there were 12 mangoes before he ate his share.
Similarly, there were 15 before the Queen ate.
Finally, there were 18 mangoes before the King and everyone ate their share of mangoes.