Question & Answer
QUESTION

A number consists of two digits whose sum is 9. If 27 is subtracted from the number its digits are reversed. Find the number.

ANSWER Verified Verified
Hint: Here, we will first consider the two digits of the number as x and y. Then, we will try to form equations using the given conditions to obtain the values of x and y and hence, we can find the number.

Complete Step-by-Step solution:
We know that if the digits of a two digit number are known then we can easily find the number.
If the first digit of the two digit number is x and the second digit is y, then we can write the given number as $10x+y$ .
Since, it is given that the sum of the digits of the number is 9. So, we can write the following equation:
$x+y=9...........\left( 1 \right)$
It is also given that if we subtract 27 from the number, its digit gets reversed. The reversed number will have now y as its first digit and x as its second digit.
So, the number formed after reversing the digits can be written as $10y+x$ .
So, on subtracting 27 from the given number and then equating it to its reverse, we get:
\[\begin{align}
  & 10x+y-27=10y+x \\
 & \Rightarrow 10x+y-10y-x=27 \\
 & \Rightarrow 9x-9y=27 \\
 & \Rightarrow 9\left( x-y \right)=27 \\
 & \Rightarrow x-y=\dfrac{27}{9}=3 \\
\end{align}\]
Therefore, we have another equation as:
$x-y=3..........\left( 2 \right)$
On adding equation (1) and equation (2), we get:
$\begin{align}
  & x+y+x-y=9+3 \\
 & \Rightarrow 2x=12 \\
 & \Rightarrow x=\dfrac{12}{2}=6 \\
\end{align}$
So, the value of x comes out to be = 6.
On substituting x = 6 in equation (1), we get:
$\begin{align}
  & 6+y=9 \\
 & \Rightarrow y=9-6=3 \\
\end{align}$
So, the value of y is =3.
Since, the number is of the form of $10x+y$, we can write that the number is :
$\begin{align}
  & =10\times 6+3 \\
 & =60+3 \\
 & =63 \\
\end{align}$
Hence, the required number is 63.

Note: Students should keep in mind that a two digit number is always represented in the form of $10x+y$, where x and y are the first and second digits of the number respectively. It is not necessary to find the value of y using equation (1), it can also be found by using equation (2) also.