Question

A number consists of two digits whose sum is $9$. If $9$ is subtracted from the number, the digits interchanged their places. Find the number?

Answer 1

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 answer:
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:
$x + y = 9 - - - - (1)$
It is also given that if we subtract $9$ from the number, its digit interchanged its places. The reversed number will now have 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 $9$ from the given number and then equating it to its reverse, we get:
\[10x + y - 9 = 10y + x\]
\[ \Rightarrow 10x + y - 10y - x = 9\]
\[ \Rightarrow 9x - 9y = 9\]
\[ \Rightarrow 9(x - y) = 9\]
\[ \Rightarrow x - y = \dfrac{9}{9} = 1\]
Therefore, we have another equation as:
\[x - y = 1..........(2)\]
On adding equation (1) and equation (2), we get:
\[x + y + x - y = 1 + 9\]
\[ \Rightarrow 2x = 10\]
\[ \Rightarrow x = \dfrac{10}{2} = 2\]
So, the value of x comes out to be \[ = {\text{ 5}}\].
On substituting $x = 5$ in equation \[\left( 1 \right),\]we get:
\[5 + y = 9\]
\[ \Rightarrow y = 9 - 5 = 4\]
So, the value of y is \[ = 4\].
Since, the number is of the form of \[10x + y\], we can write that the number is :
\[ = 10 \times 5 + 4\]
\[ = 50 + 4\]
\[ = 54\]
Hence, the required number is $54$.

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.