Question

The sum of two digits of a two digit number is $12$.The number obtained by interchanging the digits exceeds the given number by $18$.Find the number.

Answer 1

Hint: Assume two digit number with variable as $xy$ or use any another variable
and apply the given conditions to find the number.

Complete step-by-step answer:
Let the tens digit of required number be $x$ and units digit be $y$
Given sum of two digit of two digit number = $12$
$ \Rightarrow x + y = 12 \to (1)$
Here required number is in the form = $(10x + y) \to (2)$
Number obtained on reversing the digit = $(10y + x)$
Here the number obtained by interchanging the digits exceeds the given number by $18$
Therefore the number = $(10x + y)$$ + 18$$ \to (3)$
On equating equation $(2)\& (3)$ we get
$
   \Rightarrow 10x + y + 18 = 10y + x \\
   \Rightarrow 9y - 9x = 18 \\
   \Rightarrow y - x = 2 \to (4) \\
 $
On adding $(1)\& (4)$ we get
$
  x + y = 12 \\
  \dfrac{{ - x + y = 2}}{{2y = 14}} \\
 $
Here $2y = 14$
$ \Rightarrow y = 7$
On substituting $'y'$ value in equation $(1)$ we get
$
   \Rightarrow x + y = 12 \\
   \Rightarrow x + 7 = 12 \\
   \Rightarrow x = 12 - 7 \\
   \Rightarrow x = 5 \\
  \therefore x = 5 \\
$
Here we have consider the required number as $10x + y$
Now let us substitute the $x\& y$ values in the required number to get the answer
$ \Rightarrow 10(5) + 7 = 57$
Therefore the required number is $57$

Note: In this problem make a note that after getting the assumed required number we have to interchange digits of required number and should be considered as a number. Add here the required number is exceeded with the number $18$ not the interchanged number.