From a basket of mangoes, the king took 1/6, the queen took 1/5 of the remainder, the three chief princes to ¼, 1/3, and ½ of the successive remainders, respectively. the youngest child took the remaining 3 mangoes. how many mangoes were in the basket originally

let
x---------> quantity of mangoes that were in the basket originally

we know that
the king took 1/6
(1/6)*x-------> remainder is (5/6)*x

the queen took 1/5 of the remainder
(1/5)*(5/6)*x---> (1/6)*x----> remainder is (5/6-1/6)*x----> (4/6)*x

one chief princes to ¼ of the remainder
(1/4)*(4/6)*x-----> (1/6)*x------> remainder is (4/6-1/6)*x----> (3/6)*x

second chief princes to 1/3 of the remainder
(1/3)*(3/6)*x-----> (1/6)*x-------> remainder is (3/6-1/6)*x----> (2/6)*x

third chief princes to 1/2 of the remainder
(1/2)*(2/6)*x-----> (1/6)*x--------> remainder is (2/6-1/6)*x----> (1/6)*x

we know 
the remaining is 3 mangoes
so
(1/6)*x=3-------> x=6*3------> x=18 mangoes

the answer is
18 mangoes