Question

The difference between a 2-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

Answer 1

The correct option is B

8

Determine the difference between the sum and the difference of digits of the number.

Explanation of the correct option:

B. Assume that the number is $10x+y$.

The number after interchanging its digits is $10y+x$.

The difference between the digits is $36$.

Simplify the expression of the difference of numbers:

$\begin{array}{rcl}10x+y-10y-x& =& 36\\ & \Rightarrow & 9x-9y=36\\ & \Rightarrow & 9\left(x-y\right)=36\\ & \Rightarrow & x-y=4\end{array}$

The ratio of the digits is $1:2$.

Since $x>y$, $y:x=1:2$.

This implies that $x=2y$.

Substitute $x=2y$ into $\begin{array}{rcl}x-y& =& 4\end{array}$ and solve:

$\begin{array}{rcl}2y-y& =& 4\\ & \Rightarrow & y=4\end{array}$

So, $x=8$.

The sum of the digits $x\mathrm{and}y$ is $8+4=12$.

The difference of the digits is $8-4=4$.

So, the difference between the sum and the difference of the digits of the number is $12-4=8$.

Thus, option (B) is correct.

Explanation of the incorrect options.

Since the difference between the sum and the difference of the digits of the number is $8$, so options (A), (C), and (D) are incorrect.

Hence,option (B) is correct.