Question

Sum of the digits of a two digit number is 9. When the digits are reversed, it is found that the resulting number is greater than the original number by 27. Find the number.

Answer 1

Hint:
Here, we need to find the original number. We will assume the digit at ten’s place to be \[x\] and the digit at unit’s place to be \[y\]. A two-digit number can be written as 10 \[ \times \] the digit at ten’s place \[ + \] the digit at unit’s place. We will write the original and the reversed number in terms of \[x\] and \[y\]. Then, using the given information, we can form two linear equations in two variables. We will solve these equations to find the values of \[x\] and \[y\], and use these values to find the original number.

Complete step by step solution:
We will use two variables \[x\] and \[y\] to form a linear equation in two variables using the given information.
A two-digit number can be written as 10 \[ \times \] the digit at ten’s place \[ + \] the digit at unit’s place.
For example, 28 can be written as \[2 \times 10 + 8\].
Let us assume the digit at ten’s place be \[x\] and the digit at unit’s place to be \[y\].
Therefore, we get the original number as
\[10 \times x + y = 10x + y\]
When the digits are reversed, the digit at ten’s place becomes \[y\] and the digit at unit’s place becomes \[x\].
We can write the number when the digits are reversed as
\[10 \times y + x = 10y + x\]
Now, it is given that the sum of the digits is 9.
Thus, we get
\[ \Rightarrow x + y = 9 \ldots \ldots \ldots \left( 1 \right)\]
The number obtained by reversing the digits is greater than the original number by 27.
Thus, we get
\[ \Rightarrow 10y + x = 10x + y + 27\]
Subtracting \[10y\] and \[x\] from both sides of the equation, we get
\[\begin{array}{l} \Rightarrow 10y + x - 10y - x = 10x + y + 27 - 10y - x\\ \Rightarrow 0 = 9x - 9y + 27\end{array}\]
Dividing both sides by 9, we get
\[\begin{array}{l} \Rightarrow \dfrac{{9x - 9y + 27}}{9} = \dfrac{0}{9}\\ \Rightarrow x - y + 3 = 0 \ldots \ldots \ldots \left( 2 \right)\end{array}\]
We can observe that the equations \[\left( 1 \right)\] and \[\left( 2 \right)\] are a pair of linear equations in two variables.
We will solve the equations to find the values of \[x\] and \[y\].
Rewriting equation \[\left( 1 \right)\], we get
\[ \Rightarrow x = 9 - y\]
Substituting \[x = 9 - y\] in equation \[\left( 2 \right)\], we get
\[ \Rightarrow 9 - y - y + 3 = 0\]
Adding and subtracting the like terms, we get
\[ \Rightarrow 12 - 2y = 0\]
Rewriting the equation, we get
\[ \Rightarrow 2y = 12\]
Dividing both sides by 2, we get
\[\begin{array}{l} \Rightarrow \dfrac{{2y}}{2} = \dfrac{{12}}{2}\\ \Rightarrow y = 6\end{array}\]
Substituting \[y = 6\] in the equation \[x = 9 - y\], we get
\[ \Rightarrow x = 9 - 6 = 3\]
Now, we will use the values of \[x\] and \[y\] to find the original number.
Substituting \[x = 3\] and \[y = 6\] in the expression \[10x + y\], we get
\[ \Rightarrow \]Original Number \[ = 10\left( 3 \right) + 6\]
Simplifying the expression, we get
\[ \Rightarrow \]Original Number \[ = 30 + 6 = 36\]

Therefore, the original number is 36.

Note:
We have formed two linear equations in two variables and simplified them to find the number. A linear equation in two variables is an equation of the form \[ax + by + c = 0\], where \[a\] and \[b\]are not equal to 0. For example, \[2x - 7y = 4\] is a linear equation in two variables.
We can verify our answer by using the given information.
The sum of the digits of the number 36 is 9.
The number obtained by reversing the digits of 36 is 63.
We can observe that \[63 = 36 + 27\].
Thus, the number obtained by reversing the digits is greater than the original number by 27.
Hence, we have verified our answer.