Question

The ten’s digit of a two digit number is thrice the unit digit. The new number obtained by interchanging the digits is 54 less than the original number. Find the original number.

Answer 1

Hint:We are going to take the original number as xy and then we will apply the given condition in the question to find the relation between x and y. And then we will interchange the digits and equate the difference of two numbers to 54.

Complete step-by-step answer:
Let xy is our original number where x is tens digit and y is unit digit so it can be represented as $10x+y$
Given that ten’s digit is thrice the unit digit.
Therefore, x = 3y using this in both the numbers we get
$\begin{align}
  & xy=10x+y \\
 & =30y+y \\
 & =31y \\
\end{align}$
Now after interchanging the digits we get new number as yx,
$\begin{align}
  & yx=10y+x \\
 & =10y+3y ({\text{ substituting value of x=3y}}) \\
 & =13y \\
\end{align}$
Therefore, as per given in the question that the new number obtained by interchanging the digits is 54 less than the original number
$xy = yx +54$
$\begin{align}
  & 31y=13y+54 \\
 & 18y=54 \\
 & y=3 \\
\end{align}$
Now substituting the value of y in x = 3y we get,
x = 9.
Now we have calculated the value of both x and y,
Hence, the original number will be xy which will be 93.

Note: We have done as given in the question and taken the original number variable, but one can also solve this question by taking the interchanged number as xy, and again interchange it to get the original number yx. And we can proceed as we have as we have done in the solution and we will arrive at the same answer.Verify the result by interchanging the digits place of number and taking the difference of original and new number whether we will get 39 or not.