Question

(linear equations in 1 variable) sum of the digits of a two digit number is 9. when we interchange the digits, it is found that the resulting new number is greater than the original number by 27 . what is the two digit number?

Answer 1

Answer:

Let the digits at tens place and ones place: x and 9−x respectively.

∴ original number =10x+(9−x)

=9x+9

Now Interchange the digits: Digit at ones place and tens place: x and 9−x respectively.

∴ New number: 10(9−x)+x

=90−10x+x

=90−9x

AS per the question

New number = Original number +27

90−9x=9x+9+27

90−9x=9x+36

18x=54

x=1854

x=3

Digit at tens place ⇒3 and one's place : 6

∴ Two digit number: 36

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

Sum of the digits of a two digit number is 9

So, Let we assume that

$\begin{gathered}\begin{gathered}\bf\: Let-\begin{cases} &\sf{digit \: at \: tens \: place \: be \: x} \\ \\ &\sf{digits \: at \: ones \: place \: be \: 9 - x} \end{cases}\end{gathered}\end{gathered}$

Thus,

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 10 (9 - x) + x = 90 - 9x} \\ \\ &\sf{reverse \: number = 10x + 9 - x= 9x + 9} \end{cases}\end{gathered}\end{gathered}$

According to statement

When we interchange the digits, it is found that the resulting new number is greater than the original number by 27.

$\rm\implies \:9x + 9 - (90 - 9x) = 27$

$\rm\implies \:9x + 9 - 90 + 9x = 27$

$\rm\implies \:18x - 81 = 27$

$\rm\implies \:18x = 27 + 81$

$\rm\implies \:18x = 108$

$\rm\implies \:\boxed{\tt{ x = 6}}$

So,

$\begin{gathered}\begin{gathered}\bf\:\rm :\longmapsto\:\begin{cases} &\sf{digit \: at \: tens \: place \: be \: 6} \\ \\ &\sf{digits \: at \: ones \: place \: be \: 3} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 90 - 9x = 90 - 54 = 36} \\ \\ &\sf{reverse \: number = 9x + 9 = 54 + 9 = 63} \end{cases}\end{gathered}\end{gathered}$

Thus, 2 digit number is 36

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Basic Concept Used :-

Writing System of Linear Equation from Word Problem.

1. Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find.

2. Translate the problem to an equation.

• Assign a variable (or variables) to represent the unknown.

• Clearly state what the variable represents.

3. Carry out the plan and solve the problem.