Question

The sum of the digits of a two digit number is 11 . The number obtained by interchanging the digits exceeds the original number by 27. Find the number.

Answer 1

Hint: At first let's assume the two digit number to be $10x + y$ and by the the condition the sum of the digits is 11 , we get $x + y = 11$ and by using the condition that when the digits are interchanged it exceeds the original number by 27 we get $x - y = - 3$.Now we need to solve this pair of linear equations using substitution method and find the value of x and y .

Complete step-by-step answer:
We need to find a two digit number
Let the two digit number be $10x + y$
In the first condition , we are given that the sum of the digits is 11
$ \Rightarrow x + y = 11$ …………..(1)
The next condition states that when the digits are interchanged the new number exceeds the original number by 27
That is ,
$
   \Rightarrow 10y + x = 10x + y + 27 \\
   \Rightarrow 10x + y + 27 - 10y - x = 0 \\
   \Rightarrow 9x - 9y = - 27 \\
$
Divide the equation by 9 on both sides
$ \Rightarrow x - y = - 3$ ………….(2)
Now we have a pair of linear equation and hence we can solve it using substitution method
From (1) we have $x = 11 - y$
Substituting this (2) we get
$
   \Rightarrow 11 - y - y = - 3 \\
   \Rightarrow 11 - 2y = - 3 \\
   \Rightarrow - 2y = - 3 - 11 \\
   \Rightarrow - 2y = - 14 \\
   \Rightarrow y = \dfrac{{ - 14}}{{ - 2}} = 7 \\
$
Substituting the value of y in (1) we get
$
   \Rightarrow x + 7 = 11 \\
   \Rightarrow x = 11 - 7 \\
   \Rightarrow x = 4 \\
$
Using the values of x and y we get our two digit number to be
$
   \Rightarrow 10(4) + 7 \\
   \Rightarrow 40 + 7 \\
   \Rightarrow 47 \\
$
Hence the two digit number is 47.

Note: Many students assume the two digit number to be xy but that is wrong . A two digit number is always represented as $10x + y$. If it is a three digit number it is represented as $100x + 10y + z$
Here the pair of linear equations can also be solved using elimination method
That is let's add the equations (1) and (2)
$
  x + y = 11 \\
  \underline {x - y = - 3} \\
  2x + 0 = 8 \\
$
From this we get
$
   \Rightarrow 2x = 8 \\
   \Rightarrow x = \dfrac{8}{2} = 4 \\
$
Substituting this in any one of the equations we get
$
   \Rightarrow 4 + y = 11 \\
   \Rightarrow y = 11 - 4 \\
   \Rightarrow y = 7 \\
$
Hence we get the two digit number to be
$
   \Rightarrow 10(4) + 7 \\
   \Rightarrow 40 + 7 \\
   \Rightarrow 47 \\
$