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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?​

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SalmonWorldTopper

Step-by-step explanation:

[tex]\large\underline{\sf{Solution-}}[/tex]

Given that,

Sum of the digits of a two digit number is 9

So, Let we assume that

[tex]\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}[/tex]

Thus,

[tex]\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}[/tex]

According to statement

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

[tex]\rm\implies \:9x + 9 - (90 - 9x) = 27[/tex]

[tex]\rm\implies \:9x + 9 - 90 + 9x = 27[/tex]

[tex]\rm\implies \:18x - 81 = 27[/tex]

[tex]\rm\implies \:18x = 27 + 81[/tex]

[tex]\rm\implies \:18x = 108[/tex]

[tex]\rm\implies \:\boxed{\tt{ x = 6}}[/tex]

So,

[tex]\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}[/tex]

[tex]\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}[/tex]

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.

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