If 2 is added to each of two given numbers, then their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers.

Let two numbers be x and y.

Given,

If 2 is added to each, the ratio becomes 1 : 2.

⇒ $\dfrac{x + 2}{y + 2} = \dfrac{1}{2}$

⇒ 2(x + 2) = y + 2

⇒ 2x + 4 = y + 2

⇒ y = 2x + 4 - 2

⇒ y = 2x + 2     ….(1)

Given,

If 4 is subtracted from each number, the ratio becomes 5 : 11.

⇒ $\dfrac{x - 4}{y - 4} = \dfrac{5}{11}$

⇒ 11(x - 4) = 5(y - 4)

⇒ 11x - 44 = 5y - 20

⇒ 11x - 5y = -20 + 44

⇒ 11x - 5y = 24     ….(2)

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

⇒ 11x - 5(2x + 2) = 24

⇒ 11x - 10x - 10 = 24

⇒ x - 10 = 24

⇒ x = 24 + 10

⇒ x = 34.

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

⇒ y = 2x + 2

⇒ y = 2 × 34 + 2

⇒ y = 68 + 2

⇒ y = 70.

Hence, the numbers are 34 and 70.