A and B each have a certain number of marbles. A says to B, “If you give 30 to me, I will have twice as many as left with you.” B replies, “If you give me 10, I will have thrice as many as left with you.” How many marbles does each have?

Let the number of marbles A has be x, and the number of marbles B has be y.

After B gives 30 marbles to A,

⇒ x + 30 = 2(y - 30)

⇒ x + 30 = 2y - 60

⇒ x = 2y - 60 - 30

⇒ x = 2y - 90     …….(1)

After A gives 10 marbles to B,

⇒ y + 10 = 3(x - 10)

⇒ y + 10 = 3x - 30

⇒ y = 3x - 30 - 10

⇒ y = 3x - 40     ……..(2)

Substituting value of x from equation (1) in (2), we get :

⇒ y = 3(2y - 90) - 40

⇒ y = 6y - 270 - 40

⇒ y = 6y - 310

⇒ y - 6y = -310

⇒ -5y = -310

⇒ y = $\dfrac{-310}{-5} = 62$.

Substituting value of y in equation (1), we get :

⇒ x = 2y - 90

⇒ x = 2(62) - 90

⇒ x = 124 - 90

⇒ x = 34.

Hence, A has 34 marbles and B has 62 marbles.