Question

A two-digit number becomes `5/6` of the reversed number obtained when the digits are interchanged. The difference between the digits is 1. Find the number.

Answer 1

Let the ten’s digit be a and the unit’s digit be b,

Original number = 10a + b; reversed number = 10b + a,

Given:

`10a + b = 5/6(10b + a)`     ...(1)

Here, multiplying equation (1) by 6:

`6(10a + b) = 6(5/6(10b + a))`

60a + 6b = 5(10b + a)

60a + 6b = 50b + 5a

60a − 5a = 50b − 6b

55a = 44b     ...(Dividing expression by 11)

5a = 4b

∴ b = `5/4a`     ...(2)

Since b = `5/4a` > a, we must have b − a = 1 (because the digits differ by 1). Using equation (2):

`b - a = 5/4a - a`

`1 = 5/4a - 4/4a`

`1 = (5/4 - 4/4)a`

`1 = 1/4a`   ...(Multiplying both sides by 4)

∴ a = 4

Substituting a = 4 in equation (2),

`b = 5/4a`

`b = 5/4(4)`

∴ b = 5

Therefore, the number = 10a + b

= 10(4) + 5

= 45