Question

The sum of the digits of a 2-digit number is 11. The number obtained by adding 4 to this number is 41 less than the reversed number. Find the original number.

Answer 1

Hint:
Here, we will assume the number in the tens place and the number in units place to be some variable and form an expression for the original number. Then we will use the given information and form a linear equation. Then we will reverse the number and again form a linear equation using the given information. We will then add these two equations and solve it further to find the values of the variable. We will then use the variables to find the original number.

Complete step by step solution:
Let the number in tens place be $x$and the number in units place be $y$.
Now, as we know, the place value of tens digit is $10 \times x = 10x$
Therefore, this number can be written as:
Original Number $ = 10x + y$
Now, it is given that the sum of the digits of a 2-digit number is 11
Therefore, mathematically, we get,
$x + y = 11$………………………..$\left( 1 \right)$
Now, since the given number is $10x + y$
Hence, if the digits of this number are reversed i.e. the number present at tens place is written at the units place and vice-versa, we get the reversed number as:
Reversed number $ = 10y + x$
It is given that the number obtained by adding 4 to this number is 41 less than the reversed number.
Hence, this can be written mathematically as:
$\left( {10x + y} \right) + 4 = \left( {10y + x} \right) - 41$
$ \Rightarrow 10x + y + 4 = 10y + x - 41$
Adding and subtracting the like terms, we get
$ \Rightarrow 9x - 9y = - 45$
Dividing both sides by 9, we get
$ \Rightarrow x - y = - 5$…………………………$\left( 2 \right)$
Now, we will add the equations $\left( 1 \right)$ and $\left( 2 \right)$. Therefore, we get
$x + y + x - y = 11 - 5$
Adding and subtracting the terms, we get
$ \Rightarrow 2x = 6$
Dividing both sides by 2, we get
$ \Rightarrow x = 3$
Substituting this in equation $\left( 1 \right)$, we get
$3 + y = 11$
$ \Rightarrow y = 8$

So, the original number \[ = 10\left( 3 \right) + 8 = 30 + 8 = 38\]
Therefore, the original number is 38.

Note:
Here, we can make mistakes if we understand the meaning incorrectly and write the equation as $\left( {10x + y} \right) - 41 = \left( {10y + x} \right) + 4$, hence, making our equation completely wrong. We can also make another mistake by writing the reversed digit as $y + 10x$, which is wrong because if the digits are reversed then the unit digit will become tens digit and multiplied by 10 and the unit digit will not be multiplied by 10.