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The difference between a two digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2?
(a) 4
(b) 8
(c) 16
(d) None of these

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: Here, we need to find the difference between the sum and the difference of the digits of the number. Let the digit at ten’s place be \[x\] and the digit at unit’s place 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 thus, the original two digit number. Finally, we will use the digits to find the difference between the sum and the difference of the digits of the number.

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.


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