Question

In a two-digit number, the digit at the ten’s place is twice the digit at units’ place. If the number obtained by interchanging the digits is added to the original number, the sum is 66. Find the number.

Answer 1

Hint: First assume the digits at the unit’s and ten’s place and thus form the two-digit number. Then we use the given conditions in the problem to form equations between the two assumed. Solve them to get the desired answer.

Complete step-by-step answer:
Given the problem a two-digit number such that the digit at the ten’s place value is twice the digit at one's place value.
Also, if the number obtained by interchanging the place value of the digits is added to the original number, the sum is equal to 66.
Our aim is to find the two-digit number.
Let us assume the digit at one’s place be x and the digit at ten’s place be y.
Then the two-digit number can be written as $10 \times y + x = 10y + x$
The first condition given in the problem states that the digit at the ten’s place value is twice the digit at one’s place value.
$ \Rightarrow y = 2x{\text{ (1)}}$
Hence, we obtain our first equation.
Similarly using the second condition given in the problem.
The number obtained by interchanging the place value of the digits is added to the original number, the sum is equal to 66.
Number after interchanging the one’s and ten’s digits $ = 10x + y$
$
   \Rightarrow 10y + x + 10x + y = 66 \\
   \Rightarrow 11y + 11x = 66 \\
   \Rightarrow y + x = 6{\text{ (2)}} \\
$
Hence, we obtain our second equation.
Now we need to solve the two equations.
\[
  y = 2x{\text{ (1)}} \\
  y + x = 6{\text{ (2)}} \\
\]
Using equation (1) in equation (2), we get
$
   \Rightarrow 2x + x = 6 \\
   \Rightarrow 3x = 6 \\
   \Rightarrow x = 2 \\
$
Using the above obtained value of $x$in equation (1), we get
$ \Rightarrow y = 2x = 4$
Hence the original two-digit number is $10y + x = 10 \times 4 + 2 = 42$.

Note: The amount of equations needed to find the value of unknown quantities is always equal to the number of unknown quantities. The system of linear equations obtained in the solution above are solved by substitution method. The other two methods, elimination and cross multiplication could also be used to do the same.