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# A number consists of two digits whose sum is $11$. The number formed by reversing the digits is $9$ less than the original number. Find the number.

Last updated date: 20th Jun 2024
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Hint: This word problem is reduced to a problem of a pair of linear equations in two variables. It is convenient to solve this equation by elimination method.

Elimination method- In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are of opposite sign you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

Let the one’s place digit of the number be $x$ and the ten’s place digit of the number be $y$.
Therefore the number is $= 10y + x$
The number formed by reversing the digit is$= 10x + y$
Given condition is,
Sum of the two digits is$11$.
Mathematically,
$x + y = 11$ …equation (1)
Original number$- 9 =$ Number formed by reversing the digits
Mathematically,
$(10y + x) - 9 = (10x + y)$
On simplifying the equation,
$\Rightarrow 10y - y - 9 = 10x - x$
$\Rightarrow 9y - 9 = 9x$
On dividing the equation by $9$,
$\Rightarrow \dfrac{{9(y - 1)}}{9} = \dfrac{{9x}}{9}$
$\Rightarrow y - 1 = x$
$\Rightarrow - x + y = 1$ …equation (2)
These two are the pair of linear equations in two variables. There are many methods to solve these equations; here we will use the Elimination method.
On adding the equation (1) and equation (2),
$(x + y) + ( - x + y) = 11 + 1$
$\Rightarrow 2y = 12$
$\Rightarrow y = \dfrac{{12}}{2}$
$\Rightarrow y = 6$
It means the ten’s place digit is 6.
On substituting the value of $y$in equation (1),
$\therefore x + 6 = 11$
$\Rightarrow x = 11 - 6$
$\Rightarrow x = 5$
It means the one’s place digit is 5.
The number is$= 10y + x$
On substituting the value of $x$ and $y$,
$\therefore 10y + x = 10 \times 6 + 5$
$\Rightarrow 10y + x = 65$
Hence the original number is $65$.

Note:
Generally in this type of question, someone can get confused with the one’s place digit and the ten’s place digit. It means mistakenly they write the one’s place digit on the place of ten’s place digit and vice-versa. So this thing can be kept in mind while solving the question.