If 1 is added to each of the two certain numbers; their ratio is 1 : 2 and if 5 is subtracted from each of the two numbers, their ratio becomes 5 : 11. Find the numbers.

Let x and y be two numbers.

Given:

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

Cross multiplying both sides,

⇒ (x + 1) $\times$ 2 = (y + 1) $\times$ 1

⇒ 2x + 2 = y + 1

⇒ y = 2x + 2 - 1

⇒ y = 2x + 1 ……………….(1)

Given:

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

Cross multiplying both sides,

⇒ (x - 5) $\times$ 11 = (y - 5) $\times$ 5

⇒ 11x - 55 = 5y - 25

⇒ 5y = 11x - 55 + 25

⇒ 5y = 11x - 30

On solving equations (1) and (2), we get :

⇒ 5 $\times$ (2x + 1) = 11x - 30

⇒ 10x + 5 = 11x - 30

⇒ 11x - 10x = 5 + 30

⇒ x = 35

From equation (1),

⇒ y = 2 $\times$ 35 + 1

= 70 + 1

= 71

Hence, the two numbers are 35 and 71.