Question

Seven times a two-digit number is equal to four times the number obtained by reversing the order of the digits. Find the number, if the difference between its digits is 3.
$
  {\text{A}}{\text{. 14}} \\
  {\text{B}}{\text{. 25}} \\
  {\text{C}}{\text{. 36}} \\
  {\text{D}}{\text{. 47}} \\
 $

Answer 1

Hint: Here, we will proceed by assuming the digits at the ones place and tens places of the original number as y and x respectively. Then, we will obtain two equations in these two variables (i.e., x and y) in order to find the values of x and y.

Complete step-by-step answer:
Let the original number consist of digit y at the ones place and the digit x at the tens place.
i.e., Original Number = 10(Digit in the tens place) + 1(Digit in the ones place)
$ \Rightarrow $Original Number = 10(x)+1(y) = 10x + y
When the original number is reversed then, the digit in the ones place of the original number will become the digit in the tens place of the reversed number and the digit in the tens place of the original number will become the digit in the ones place of the reversed number.
In reversed numbers, the digit x is at the ones place and the digit y is at the tens place.
So, Reversed Number = 10(Digit in the tens place) + 1(Digit in the ones place)
$ \Rightarrow $ Reversed Number = 10(y)+1(x) = 10y + x
Given, 7(original number) = 4(reversed number)
$
   \Rightarrow 7\left( {10x + y} \right) = 4\left( {10y + x} \right) \\
   \Rightarrow 70x + 7y = 40y + 4x \\
   \Rightarrow 70x - 4x = 40y - 7y \\
   \Rightarrow 66x = 33y \\
   \Rightarrow x = \dfrac{{33y}}{{66}} \\
   \Rightarrow x = \dfrac{y}{2}{\text{ }} \to (1) \\
 $
Also, given that the difference between the digits of the original number is 3. Let us assume that the digit y is greater than the digit x.
So, $y - x = 3{\text{ }} \to {\text{(2)}}$
By substituting equation (1) in equation (2), we get
$
   \Rightarrow y - \dfrac{y}{2} = 3 \\
   \Rightarrow \dfrac{y}{2} = 3 \\
   \Rightarrow y = 6 \\
 $
Put y=6 in equation (1), we get
\[
   \Rightarrow x = \dfrac{6}{2} \\
   \Rightarrow x = 3 \\
 \]
Clearly, y=6 is greater than x=3 which satisfies the assumption taken.
Original Number $ = 10x + y = 10\left( 3 \right) + 6 = 30 + 6 = 36$
Therefore, the original number is 36.
Hence, option C is correct.

Note: In this particular problem, we have assumed that the digit y is greater than the digit x because if we would have assumed the digit x to be greater than the digit y then both the values of x and y obtained will come negative and since, we take any digit to be positive only that’s why that particular assumption is taken.