Question

The tens digits of a two-digit number exceeds the units digit by 5. If the digits are reversed, the new number is less by 45. If the sum of their digits is 9, find the numbers.

Answer 1

Hint: In this question, let consider tens digit be x and unit digit be y of a two digit number then the original number becomes $10x+y$. The tens digit exceeds the units digit by 5 means, the difference of tens digit and units digit is 5 i.e x- y=5. When digits are reversed means we have to replace tens and units place i.e reversed number is $10y+x$ and the given difference of original number and reversed number is 45 and sum of tens digit and unit digit is 9 i.e x+y =9 we get an equation. Using the elimination method, solve the equations that will be formed by using the information to get the required answer.

Complete step-by-step answer:
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.
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 have 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 that the tens digits of a two-digit number exceeds the units digit by 5 i.e the difference between tens digit and unit digit is 5 ,we can write as follows
\[x-y=5~~\ \ \ \ \ ...(a)\]
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 45, so, we can write as follows
\[\begin{align}
  & \left( 10x+y \right)-\left( 10y+x \right)=45 \\
 & 9x-9y=45 \\
 & x-y=5\ \ \ \ \ ...(d) \\
\end{align}\]
Another equation that can be made is that
The sum of the digits of the number is 9, so, we can write as follows
\[x+y=9\ \ \ \ \ ...(e)\]
Now, on adding equations (d) and (e), we get
\[\begin{align}
  & 2x=14 \\
 & x=7 \\
\end{align}\]
Now, on using the value of x, we can get the value of y as
\[\begin{align}
  & 7+y=9 \\
 & y=2 \\
\end{align}\]
Hence, the original number is 72 and the reversed number is 27.

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.
There are two other methods of solving a 2 variable system of equations:-
i) substitution method
ii) cross multiplication method