Question

The sum of the digits of a two – digit number is 9. If 9 is added to the number formed by reversing the digits, then the result is thrice the original number. Find the original number.

Answer 1

Hint: Assume that the original two digit number is xy with x as its tens’ place digit and y as its units’ place digit. Take the sum of the face value of these digits and equate it with 9 to form the first equation. Now, reverse the digits of xy and write them along with their place values as yx = 10y + x. Add 9 to this expression as equate it with three times the expression xy, i.e. 3 (10x + y). Form the second equation and solve for the values of x and y to get the original number.

Complete step by step answer:
Let us assume that the original two digit number is xy, where x is the tens’ place digit and y is the units’ place digit of the number. Let us form two linear equations to find the values of x and y.
(1) Now, in the first case we have been told that the sum of the digits of this number is 9, so we have to consider only the face values of the digits, therefore mathematically we have,
$\Rightarrow x+y=9............\left( i \right)$
(2) In the second case we have been told that we are adding 9 to the number that will be formed by reversing the digits of the original number and this will be equal to three times the original number.
So, on reversing the digits of the original number we will get yx. Adding 9 to it and equating it with three times of xy we get,
$\Rightarrow yx+9=3\left( xy \right)$
Here we have to consider the place values of the digits because we are adding and multiplying the overall numbers and not just the digits, so xy can be written as (10x + y) and similarly yx can be written as (10y + x). Therefore we get,
$\begin{align}
  & \Rightarrow 10y+x+9=3\left( 10x+y \right) \\
 & \Rightarrow 10y+x+9=30x+3y \\
 & \Rightarrow 29x-7y=9................\left( ii \right) \\
\end{align}$
Multiplying equation (i) with 7 and adding with equation (ii) we get,
$\begin{align}
  & \Rightarrow 29x+7x=9+63 \\
 & \Rightarrow 36x=72 \\
 & \therefore x=2 \\
\end{align}$
Substituting the above obtained value of x in equation (i) we get,
$\begin{align}
  & \Rightarrow 2+y=9 \\
 & \therefore y=7 \\
\end{align}$
Hence, the original two digit number is 27.

Note: Remember the difference between the place value and face value of a number and the process to write a number in terms of its place value. For this you must remember the Indian and International system of numeration. It was necessary to form two equations in x and y because we assumed two variables, one for the tens’ place and one for the ones’ place.