Question

The sum of the digits of a two digit number is 9 .Also , nine times this number is twice the number obtained by reversing the order of the digits. 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 9, we get $x + y = 9$ and by using the condition that when the digits are interchanged twice the new number is nine times the original number we get $8x - y = 0$.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 solution:
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 9
$ \Rightarrow x + y = 9$ …………..(1)
The next condition states that when the digits are interchanged two times the new number is nine times the original number
That is ,
$
   \Rightarrow 2\left( {10y + x} \right) = 9\left( {10x + y} \right) \\
   \Rightarrow 20y + 2x = 90x + 9y \\
   \Rightarrow 90x + 9y - 20y - 2x = 0 \\
   \Rightarrow 88x - 11y = 0 \\
 $
Dividing by 11 we get
$ \Rightarrow 8x - y = 0$ ………..(2)
Now we have a pair of linear equation and hence we can solve it using substitution method
From (1) we have $x = 9 - y$
Substituting this (2) we get
$
   \Rightarrow 8\left( {9 - y} \right) - y = 0 \\
   \Rightarrow 72 - 8y - y = 0 \\
   \Rightarrow - 9y = - 72 \\
   \Rightarrow 9y = 72 \\
   \Rightarrow y = \dfrac{{72}}{9} = 8 \\
 $
Substituting the value of y in (1) we get
$
   \Rightarrow x + 8 = 9 \\
   \Rightarrow x = 9 - 8 \\
   \Rightarrow x = 1 \\
 $
Using the values of x and y we get our two digit number to be
$
   \Rightarrow 10(1) + 8 \\
   \Rightarrow 10 + 8 \\
   \Rightarrow 18 \\
 $

Hence the two digit number is 18.

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 = 9 \\
  \underline {8x - y = 0} \\
  9x + 0 = 9 \\
 $
From this we get
$
   \Rightarrow 9x = 9 \\
   \Rightarrow x = \dfrac{9}{9} = 1 \\
 $
Substituting this in any one of the equations we get
$
   \Rightarrow 1 + y = 9 \\
   \Rightarrow y = 9 - 1 \\
   \Rightarrow y = 8 \\
 $
Hence we get the two digit number to be
$
   \Rightarrow 10(1) + 8 \\
   \Rightarrow 10 + 8 \\
   \Rightarrow 18 \\
 $