Question

Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?​

Answer 1

       $\underline{\bold{Answer:}}$

       $36$

       $\underline{\bold{Step-by-step-explaination:}}$

       $\text{Let the one's digit of the number be }x \text{ and the ten's digit be }y$

$\implies 10y + x \text{ is the two-digit number}$        

       $\text{According to the question:}$

$\implies \boxed{x + y = 9} \text{ ------(i)}$

       $\text{The question also states that:}$

       $\text{When we interchange the digits, the resulting new number is}\\\text{greater than the original by 27}$

$\implies \boxed{10x + y = 10y + x + 27}$

$\implies \boxed{10x + y - 10y - x - 27 = 0}$

$\implies \boxed{9x - 9y - 27 = 0}$

$\implies \boxed{9(x-y-3) = 0}$

$\implies \boxed{x-y = 3} \text{ ------(ii)}$

       $\text{Adding (i) and (ii)}$

$\implies \boxed{x+y + x - y = 9 + 3}$

$\implies \boxed{2x = 12}$

$\implies \boxed{x = 6}$

       $\text{Substituting the value of }x \text{ in (ii)}$

$\implies \boxed{6 - y = 3}$

$\implies \boxed{-y = 3 - 6 }$

$\implies \boxed{-y = -3}$

$\implies \boxed{y = 3}$

       $\text{Now putting the values of }x \text{ and }y \text{ in }(10y + x)$

$\implies 10(3) + 6 = 30 + 6 = 36$

   $\therefore \text{ The number is }36$

Answer 2

Answer:

$\huge\green{\boxed{\sf Answer}}$

Given

The sum of the two digits = 9

On interchanging the digits, the resulting new number is greater than the original number by 27.

Let us assume the digit of units place = x

Then the digit of tens place will be = 9–x

Thus the two-digit number is 109–x + x

Let us reverse the digit

the number becomes 10x + 9–x

As per the given condition

10x + 9–x = 109–x + x + 27

⇒ 9x + 9 = 90 – 10x + x + 27

⇒ 9x + 9 = 117 – 9x

On rearranging the terms we get,

⇒ 18x = 108

⇒ x = 6

So the digit in units place = 6

Digit in tens place is

⇒ 9 – x

⇒ 9 – 6

⇒ 3

Hence the number is 36