Question

The difference between a two-digit number obtained by interchanging the digits is 27. What is the difference between the two digits of the number?
(A) 9
(B) 6
(C) 12
(D) 3

Answer 1

Hint: Assume that the unit place and tenth place of the required number is \[x\] and \[y\] respectively. The number must be equal to \[10y+x\] . After interchanging the digits, the unit place and tenth place of the required number is \[y\] and \[x\] respectively. The interchanged number is \[10x+y\] . It is given that the difference between the original number and the interchanged number is 27. Now, solve it further and calculate the difference between the digits of the original number.

Complete step-by-step answer:
According to the question, we are given a two-digit number and its difference with the number obtained by the interchanging the digits is 27.
First of all, let us assume that the unit place and tenth place of the required number is \[x\] and \[y\] respectively.
Our number will look like,

Since the place value of \[x\] and \[y\] is unit and tenth place so, the above number must be equal to,
\[10y+x\] .
The original number = \[10y+x\] …………………………………………….(1)
After interchanging the digits, our number will look like,

Here, we can observe that the place value of \[x\] and \[y\] is tenth and unit place so, the above number must be equal to, \[10x+y\] .
The number after the interchange of digits = \[10x+y\] ………………………………………………(2)
We are also given one more information that is the difference between the original number and the number obtained by interchanging the digits is 27 …………………………………….(3)
Using equation (3) and subtracting equation (2) from equation (1), we get

\[\begin{align}
  & \Rightarrow \left( 10y+x \right)-\left( 10x+y \right)=27 \\
 & \Rightarrow 10y+x-10x-y=27 \\
 & \Rightarrow 9y-9x=27 \\
 & \Rightarrow 9\left( y-x \right)=27 \\
 & \Rightarrow \left( y-x \right)=\dfrac{27}{9} \\
\end{align}\]
\[\Rightarrow \left( y-x \right)=3\] …………………………………….(4)
Earlier, we have assumed that \[x\] and \[y\] are the unit and tenth place digit of the original number.
In equation (4), \[\left( y-x \right)\] means the difference between the digits of the number.
Therefore, the difference between the digits is 3.

So, the correct answer is “Option D”.

Note: In this type of question, one basic thing we have to take into consideration, that is if a and b are the unit and tenth place of a number, then the number is \[10b+a\] . This basic thing makes the question easy to approach.