Question

The sum of the digits of a two-digit number is 14. The number obtained on interchanging its digits exceeds the given number by 18. Find the number.

Answer 1

Hint: In this question, we will first let the tens digit and the units digit as variables and then we will find the answer of this question using the elimination method of solving equations to solve the equations that will be formed by using the information that is provided in the question.
For solving the above question, we would be requiring the knowledge of solving the system of linear equations in two variables. In this question we would be using an elimination method.

Complete step by step answer:
In elimination method, we first try to make the coefficient of any one variable of the two as equal and then subtract or add the new equations accordingly.
Then, we will get the equation which will be having only one variable.
Then we can solve the equation to get the value of that variable which is left and after getting the value of any one variable, we can plug in that value in any of the equations and then get the value of the other variable as well.
As mentioned in the question, we have to find the number that is to be found.
Now, let the tens digit be x and the units digit be y.
Now, as mentioned in the question, we can write as follows
\[x+y=14~~\ \ \ \ \ ...(a)\]
Now, in the question, there is some more information regarding the number and its reverse which we can use to form the following equation
The original number \[=10x+y\ \ \ \ \ \ ...(b)\] .
Number formed by reversing the digits \[~=10y+x\ \ \ \ ...(c)\] .
Now, it is given in the question that the reversed number is less than the original number by 18, so, we can write as follows
\[\begin{align}
  & \left( 10y+x \right)-\left( 10x+y \right)=18 \\
 & 9y-9x= 18\\
 & y-x=2\ \ \ \ \ ...(d) \\
\end{align}\]
Now, on adding equations (d) and (a), we get
\[\begin{align}
  & 2y=16 \\
 & y=8 \\
\end{align}\]
Now, on using the value of x, we can get the value of y as
\[\begin{align}
  & 8+x=14 \\
 & x=6 \\
\end{align}\]

Hence, the original number is 68 and the reversed number is 86.

Note: For questions in which there are more than 2 variables, in order to know whether the equations are solvable or whether we will be able to get the values of the variables by just counting the number of variables and number of the equations. If the number of equations and the number of variables involved in the question is equal then we can surely say that every variable will be having a unique value. If these numbers are not equal, then we do not comment on that.