# Ex.2.4 Q3 Linear Equations in One Variable Solution - NCERT Maths Class 8

## Question

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?

## Text Solution

**What is known?**

i) Sum of the digits of a two-digit number is \(9\)

ii) Interchanging the digits result in a new number greater than the original number by \(27\)

**What is unknown?**

Number

**Reasoning:**

Assume one of the digits of two-digit as variable then use other conditions and form a linear equation.

**Steps:**

Let the digits at tens place and ones place be *\(x\)* and \(9{\text{ }} - {\text{ }}x\) respectively.

Therefore, original number \( = {\text{ }}10x{\text{ }} + {\text{ }}\left( {9{\text{ }} - {\text{ }}x} \right){\text{ }} = {\text{ }}9x{\text{ }} + {\text{ }}9\)

On interchanging the digits, the digits at ones place and tens place will be *\(x\)* and \(9 - x\) respectively.

Therefore, new number after interchanging the digits:

\[\begin{align}&= 10\left( {9 - {\text{ }}x} \right) + x \\&= 90 - 10x + x \\&= 90 - 9x\end{align}\]

According to the given question,

New number \(=\) Original number \(+\; 27\)

\[\begin{align}90 - 9x &= 9x + 9 + 27 \\90 - 9x &= 9x + 36\\\end{align} \]

Transposing \(9x\) to RHS and \(36\) to LHS, we obtain

\[\begin{align}90 - 36 = 18x \\54 = 18x\end{align}\]

Dividing both sides by \(18\), we obtain

\(3=x\) and \(9 - x=6\)

Hence, the digits at tens place and ones place of the number are \(3 \) and \(6\) respectively.

Therefore, the two - digit number is \(9x + 9\)

\[\begin{align}&= {\text{ }}9{\text{ }} \times {\text{ }}3{\text{ }} + {\text{ }}9{\text{ }} \\& = {\text{ }}36 \\\end{align}\]