Four times a certain two digit number is seven times the number obtained on interchanging its digits. If the difference between the digits is 4; find the number.

Let x be the digit at ten's place and y be the digit at unit's place.

Number = 10(x) + y = 10x + y

On reversing digits, the number becomes = 10(y) + x = 10y + x

Given,

Difference between digits = 4

⇒ x - y = 4

⇒ x = y + 4 …..(1)

Given,

Four times a certain two digit number is seven times the number obtained on interchanging its digits.

⇒ 4(10x + y) = 7(10y + x)

⇒ 40x + 4y = 70y + 7x

⇒ 40x - 7x = 70y - 4y

⇒ 33x = 66y

⇒ x = $\dfrac{66\text{y}}{33}$

⇒ x = 2y ………(2)

From (1) and (2), we get :

⇒ 2y = y + 4

⇒ 2y - y = 4

⇒ y = 4

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

⇒ x = y + 4 = 4 + 4 = 8.

Number = 10x + y = 10(8) + 4 = 80 + 4 = 84.

Hence, number = 84.