Question

A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

Answer 1

Hint- Here, we will proceed by assuming the digits at the tens place and the ones place of the two-digit number as x and y respectively. Then, we will form two linear equations in these two variables and solve them with the help of the substitution method.

Complete step-by-step solution-
Let us suppose the digits at the tens place and the ones place are x and y respectively.
As we know that any two-digit number is given as
Two-digit number = 10(Digit at the tens place) + Digit at the ones place $ \to \left( 1 \right)$
Using the above formula, we can write
Two-digit number = 10x + y $ \to \left( 2 \right)$
Given,
Two-digit number = 4(x + y)
By using equation (2), we can write
\[
   \Rightarrow 10x + y = 4\left( {x + y} \right) \\
   \Rightarrow 10x + y = 4x + 4y \\
   \Rightarrow 10x - 4x = 4y - y \\
   \Rightarrow 6x = 3y \\
   \Rightarrow x = \dfrac{{3y}}{6} \\
   \Rightarrow x = \dfrac{y}{2}{\text{ }} \to {\text{(3)}} \\
 \]
The reversed two-digit number of the given two-digit number will be having y and x as the digits at the tens place and the ones place respectively.
Using the formula given by equation (1), we get
Reversed two-digit number = 10y + x $ \to \left( 4 \right)$
Also given,
Two-digit number + 18 = Reversed two-digit number
Using equations (2) and (4) in the above equation, we get
$
   \Rightarrow \left( {10x + y} \right) + 18 = 10y + x \\
   \Rightarrow 10x + y + 18 - 10y - x = 0 \\
   \Rightarrow 9x - 9y = - 18 \\
   \Rightarrow 9\left( {x - y} \right) = - 18 \\
   \Rightarrow x - y = - 2 \\
 $
By substituting the value of x from equation (3) in the above equation, we get
\[
   \Rightarrow \dfrac{y}{2} - y = - 2 \\
   \Rightarrow \dfrac{{y - 2y}}{2} = - 2 \\
   \Rightarrow y - 2y = 2\left( { - 2} \right) \\
   \Rightarrow - y = - 4 \\
   \Rightarrow y = 4 \\
 \]
By putting y = 4 in equation (3), we get
\[ \Rightarrow x = \dfrac{4}{2} = 2\]
So, the digit at the tens place and the ones place of the given two-digit number are 2 and 4 respectively i.e., x = 2 and y = 4
Using equation (2), we get
Two-digit number = 10(2) + 4 = 20 + 4 = 24
Therefore, the given two-digit number is 24.

Note- In this particular problem, in order to solve the two linear equations in two variables we can also proceed by elimination method. In elimination method, the coefficients of any one variable (either x or y) in both the linear equations are made the same and then we will subtract these obtained equations.