Question

The sum of two digit numbers and the number obtained by interchanging the digits is 99. If the digits differ by 3, find the number.

Answer 1

Hint: Let the digits of the numbers be x and y. Also, as one digit will occupy the tens position while the other will occupy the ones position. So, the numbers are 10x+y and 10y+x. Now we just have to apply the conditions given in the question to form the two equations and solve them to get the values of x and y.

Complete step-by-step answer:
Let the digits of the numbers be x and y. Also, as one digit will occupy the tens position while the other will occupy the ones position. So, the numbers are 10x+y and 10y+x.
Now it is given that the difference between the digits is 3, this can be written in the form of an equation as:
$\begin{align}
  & x-y=3 \\
 & x=y+3.......(i) \\
\end{align}$
The other condition that is given in the question states that the sum of two digit numbers and the number obtained by interchanging the digits is 99. If we write this in form of an equation, we get
$10x+y+10y+x=99$
$\Rightarrow 11x+11y=99$
$\Rightarrow 11\left( x+y \right)=99$
$\Rightarrow x+y=\dfrac{99}{11}$
$\Rightarrow x+y=9$
Now, if we substitute the value of x using equation (i), we get
$y+3+y=9$
$\Rightarrow 2y=6$
$\Rightarrow y=3$
Now, we will substitute y in equation (i), we get
$x=3+3=6$
So, the digits are 6 and 3, hence, the numbers are 63 and 36.

Note: The key thing in such a question is using the concept of place values and representing the number in terms of its digits, as in the above question we represented the number as 10x+y and 10y+x. Once you are done with this part, you just have to form the right equations from the conditions given in the question. After getting the number, you can always check the conditions, i.e if their sum is 99 and if the digits differ by 3 to confirm.