Question

A two digit number is such that the product of the digit is 35, when 18 is added to the number, the digit interchange their places. Find the number.

Answer 1

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

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

So,

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

Now,

According to statement,

The product of the digits = 35

$\bf\implies \:\boxed{ \bf{ xy = 35}}$

Also,

According to second condition,

When 18 is added to the number, the digit interchange their places.

It means, Original number + 18 = Reverse number

$\rm :\longmapsto\:10y + x + 18 = 10x + y$

$\rm :\longmapsto\:10x + y - 10y - x = 18$

$\rm :\longmapsto\:9x - 9y = 18$

$\rm :\longmapsto\:9(x - y) = 18$

$\rm :\longmapsto\:x - y= 2$

$\rm :\implies\:x = y + 2 - - - (2)$

On substituting (2) in (1), we get

$\rm :\longmapsto\:(y + 2)y = 35$

$\rm :\longmapsto\: {y}^{2} + 2y = 35$

$\rm :\longmapsto\: {y}^{2} + 2y - 35 = 0$

$\rm :\longmapsto\: {y}^{2} + 7y - 5y - 35 = 0$

$\rm :\longmapsto\:y(y + 7) - 5(y + 7) = 0$

$\rm :\longmapsto\:(y + 7)(y - 5) = 0$

$\bf\implies \:y = 5 \: \: \: as \: y \cancel < \: 0$

On substituting y = 5 in equation (2), we get

$\bf\implies \:x = 5 + 2 = 7$

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{digit \: at \: tens \: place \: be \: 5} \\ &\sf{digits \: at \: ones \: place \: be \: 7} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 10 y + x = 50 + 7 = 57} \\ &\sf{reverse \: number = 10x + y = 70 + 5 = 75} \end{cases}\end{gathered}\end{gathered}$

Hence,

• Number formed = 57

Basic Concept Used :-

Writing Systems of  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.