Question

A number consists of two digits whose sum is 9. If 27 is subtracted from the original number, its digits are interchanged. Find the original number

Answer 1

Let the units/digit of a number be ‘u’ and tens digit of the number be ‘t’

Given that sum of it’s digits is 9

∴ t + u = 9  ...(1)

If 27 is subtracted from original number, the digits are interchanged

The number is written as 10t + u

[Understand: Suppose a 2 digit number is 21]

it can be written as 2 × 10 + 1

∴ 32 = 3 × 10 + 2

45 = 4 × 10 + 5

tu = t × 10 + u = 10t + u]

Given that when 27 is subtracted, digits interchange
10t + u – 27 = 10u + t ...(number with interchanged digits)

∴ By transposition and bringing like variables together

10t – t + u – 10u = 27

∴ 9t – 9u = 27

Dividing by 9 throughout, we get

`(9"t")/9 - (9"u")/9 = 27/9`

⇒ t – u = 3 ........(2)

Solving (1) and (2)

t + u = 9
t − u = 3 
2t      = 12

t = `12/2` = 6

∴ u = 3

t = 6 substitute in (1)

t + u = 9

⇒ 6 + u = 9

⇒ u = 9 – 6 = 3

Hence the number is 63.