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

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## 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}

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