Complete step-by-step answer:
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 the digit at ten’s place be \[x\] and the digit at the unit's place be \[y\].
Assume that \[x > y\].
Therefore, we get the first number as
\[10 \times x + y = 10x + y\]
When the digits are interchanged, 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 interchanged as
\[10 \times y + x = 10y + x\]
Now, it is given that the difference between the two digit number and the number obtained by interchanging the digits is 36.
Thus, we get
\[ \Rightarrow \left[ {10x + y} \right] - \left[ {10y + x} \right] = 36\]
Simplifying the expression, we get
\[ \Rightarrow 10x + y - 10y - x = 36\]
Adding and subtracting the like terms, we get
\[ \Rightarrow 9x - 9y = 36\]
Factoring the number 9, we get
\[ \Rightarrow 9\left[ {x - y} \right] = 36\]
Dividing both sides of the equation by 9, we get
\[ \Rightarrow x - y = 4 \ldots \ldots \ldots \left[ 1 \right]\]
It is given that the ratio of the digits of the two digit number is 1: 2.
Since \[x > y\], we get
\[ \Rightarrow y:x = 1:2\]
Rewriting the equation, we get
\[ \Rightarrow \dfrac{y}{x} = \dfrac{1}{2}\]
Multiplying both sides of the equation by 2, we get
\[ \Rightarrow 2y = x\]
Rewriting the equation, we get
\[ \Rightarrow x = 2y \ldots \ldots \ldots \left[ 2 \right]\]
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 \[y\].
Substituting \[x = 2y\] in equation \[\left[ 1 \right]\], we get
\[ \Rightarrow 2y - y = 4\]
Subtracting the like terms, we get
\[\therefore y = 4\]
Substituting \[y = 4\] in the equation \[x = 2y\], we get
\[ \Rightarrow x = 2\left[ 4 \right]\]
Multiplying the terms in the expression, we get
\[\therefore x = 8\]
Therefore, we get the original two digit number as
\[10x + y = 10\left[ 8 \right] + 4 = 80 + 4 = 84\]
Now, we will find the sum and difference of the digits.
Adding the digits of the number 84, we get
Sum of digits \[ = 8 + 4 = 12\]
Subtracting the digits of the number 84, we get
Difference of digits \[ = 8 - 4 = 4\]
Finally, subtracting the difference of the digits from the sum of the digits, we get
Difference between the sum and the difference of the digits of the number \[ = 12 - 4 = 8\]
Therefore, we get the difference between the sum and the difference of the digits of the number as 8.
Thus, the correct option is option [b].
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 number obtained by reversing the digits of 84 is 48.
We can observe that \[84 - 48 = 36\].
Thus, the difference of the number and the number formed by interchanging the digits is 36.
The ratio of the digits 4 and 8 is 1: 2.
Hence, we have verified our answer.