Question

The sum of digits of a two-digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. What is the number?

Answer 1

Hint: We will start solving this question by assuming the digits of the number to be $x\And y$. The digit of one's place is x and the digit of tens place is y. Then we will form equations by using the given information i. e. the sum of digits of a two-digit number is 13 and if the number is subtracted from the one obtained by interchanging the digits, the result is 45. Then by solving the obtained equations we will get the desired answer.

Complete step by step solution:
We have been given that the sum of digits of a two-digit number is 13 and if the number is subtracted from the one obtained by interchanging the digits, the result is 45.
We have to find the number.
Let us assume that the ones place digit of the number be x and digit of tens place be y.
Then the number will be $10y+x$.
So we have been given the sum of digits is 13. Then we will get
$\Rightarrow x+y=13..........(i)$
Now, if we interchange the digits the number will be
$\Rightarrow 10x+y$
Now, we are given in the question that if the original number is subtracted from the one obtained by interchanging the digits, the result is 45. Then we will get
$\Rightarrow 10x+y-\left( 10y+x \right)=45$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 10x+y-10y-x=45 \\
 & \Rightarrow 9x-9y=45 \\
\end{align}$
Now, dividing the above obtained equation by 9 we will get
$\Rightarrow x-y=5.........(ii)$
Now, adding the equation (i) and (ii) we will get
$\Rightarrow x+y+x-y=13+5$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 2x=18 \\
 & \Rightarrow x=\dfrac{18}{2} \\
 & \Rightarrow x=9 \\
\end{align}$
Now, substituting the value of x into equation (i) we will get
$\Rightarrow 9+y=13$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow y=13-9 \\
 & \Rightarrow y=4 \\
\end{align}$
Therefore the number will be
$\begin{align}
  & \Rightarrow 10\times 4+9 \\
 & \Rightarrow 49 \\
\end{align}$

Hence we get the number as 49.

Note: The possibility of mistake is while assuming the number. Students may assume the number as $xy$ by simply writing the digits. The digits have to be written in the place value form starting from right to left. Also the point to be remembered is that we have given the sum of digits so no need to include 10 in the sum of digits.