Question

The sum of digits of two-digit numbers is $\;11$. If the new number formed by reversing the digits is more than the original number by $9$. Then find the original number.
A. $\;55$
B. $\;60$
C. $\;56$
D. $\;70$

Answer 1

Hint: To solve such questions start by assuming two variables. Next, write the two-digit number using those variables. After that write the two-digit number formed when the digits are reversed. Then compute those equations to find the original number.

Complete step-by-step solution:
Given that the sum of digits of two-digit numbers is $\;11$ .
Also given that the new number formed by reversing the digits is more than the original number by $9$.
Assume that the variables of the two digits number are $x$ and $y$.
So the two-digit number can be written as
$\Rightarrow$$10x + y$
As the sum of digits of a two-digit number is $\;11$ , so,
$\Rightarrow$$x + y = 11……….....(1)$
Also, the number formed when the digits are reversed will be $10y + x$ .
As the new number formed by reversing the digits, is more than the original number by $9$, it can be written as
$\Rightarrow$$10y + x = 10x + y + 9$
$\Rightarrow$$10y + x - 10x - y = 9$
Simplifying further we get,
$\Rightarrow$$9y - 9x = 9$
Dividing both RHS and LHS by $9$ , that is,
$\Rightarrow$$y - x = 1$
Substitute equation $\left( 1 \right)$ in the above equation, that is,
$\Rightarrow$$\left( {11 - x} \right) - x = 1$
$\Rightarrow$$11 - 2x = 1$
Simplifying further we get,
$\Rightarrow$$- 2x = 1 - 11$
$\Rightarrow$$- 2x = - 10$
Cancel out the minus sign from both sides. That is,
$\Rightarrow$$2x = 10$
Divide throughout by $2$ , that is,
$\Rightarrow$$\dfrac{{2x}}{2} = \dfrac{{10}}{2}$
$\Rightarrow$$x = 5$
Next substitute the value of $x$ in equation $\left( 1 \right)$ , that is,
$\Rightarrow$$5 + y = 11$
$\Rightarrow$$y = 11 - 5$
Further simplifying the RHS we get,
$\Rightarrow$$y = 6$
Therefore, the original number is
$= \left( {10 \times 5} \right) + 6$
That is by substituting the values of $x$ and $y$
$= 50 + 6$
$= 56$

Option C is the correct answer.

Note: It is possible to check whether the obtained number is correct or not by putting it in the given condition that is, the sum of the digits is equal to $\;11$ . To solve these types of problems also remember that a two-digit number with a first digit $x$ and second digit $y$ can be written as $10x + y$.