Question

The digits of a 2-digit number differ by 5. If the digits are interchanged and the resulting number is added to the original number, we get99. Find the original number.

Answer 1

$\bf {Solution} :-$

Let the digit in ten's place be x and the digit in one's place be y

x – y = 5 ____ eq(1)

Two digit number = 10x + y

Two digit after reversing the digits = 10y + x

According to the question,

10x + y + 10y + x = 99

=> 11x + 11y = 99

=> 11(x + y) = 99

=> x + y = 9 ____ eq(2)

On Adding equation (1) and (2) we get,

2x = 14

=> x = $\frac {\cancel{14}\: \: 7}{\cancel2}$

=> x = 7

Putting the value of x in equation 1,

x – y = 5

=> 7 – y = 5

=> y = 2

Now,

Two digit number = 10x + y = 10 × 7 + 2 = 72

$\bf The \: original \: number \: is \: 72 \: [Ans]$

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Answer 2

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

$\begin{gathered}\begin{gathered}\bf \: So - \begin{cases} &\sf{number \: formed \: be \: 10y + x} \\ &\sf{reverse \: number \: be \: 10x + y} \end{cases}\end{gathered}\end{gathered}$

$\large \underline{\tt \: \red{ According \: to \: statement }}$

• If the digits are interchanged and the resulting number is added to the original number, we get99.

$\rm :\implies\:10y + x + 10x + y = 99$

$\rm :\implies\:11x + 11y = 99$

$\rm :\implies\:\:\boxed{ \green{ \bf \:x + y = 9 }} - - - (1)$

Again,

$\large \underline{\tt \: \purple{ According \: to \: statement }}$

• The digits of two digit number differ by 5.

So, we have two cases,

$\begin{gathered}\begin{gathered}\bf \:either - \begin{cases} &\bf{x - y = 5 - - - (2)} \\ &\bf{y - x = 5 - - - (3)} \end{cases}\end{gathered}\end{gathered}$

Now,

• Solving equations (1) and (2), on adding we get

$\tt \longmapsto\:x + y + x - y = 9 + 5$

$\tt \longmapsto\:2x = 14$

$\rm :\implies\: \red{ \tt \: x \: = \: 7}$

• On substituting x = 7, in equation (1), we get

$\tt \longmapsto\:7 + y = 9$

$\rm :\implies\: \red{ \tt \: y \: = \: 2}$

Hence,

$\begin{gathered}\begin{gathered}\bf \: Number \: is\begin{cases} &\tt{10y + x = 10(2) + 7 = 27}  \end{cases}\end{gathered}\end{gathered}$

Now,

• Solving equations (1) and (3), on adding we get

$\tt \longmapsto\:x + y + y - x = 9 + 5$

$\tt \longmapsto\:2y = 14$

$\rm :\implies\: \red{ \tt \: y \: = \: 7}$

• On substituting y = 7 in equation (1), we get

$\tt \longmapsto\:x + 7 = 9$

$\rm :\implies\: \red{ \tt \: x \: = \: 2}$

Hence,

$\begin{gathered}\begin{gathered}\bf \: Number \: is\begin{cases} &\tt{10y + x = 10(7) + 2 = 72}  \end{cases}\end{gathered}\end{gathered}$

$\rm :\implies\:\:\boxed{ \purple{ \bf \: Hence, \: the \: number \: is \: 72 \: or \: 27}}$