# A Two-digit Number is Such that the Product of Its Digits is 20. If 9 is Added to the Number, the Digits Interchange Their Places. Find the Number. - Mathematics

Definition

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.

#### Solution

Let the digits at units and tens place of the given number be x and y respectively. Thus, the number is 10y + x.

The product of the two digits of the number is 20. Thus, we have  xy =20

After interchanging the digits, the number becomes 10x + y.

If 9 is added to the number, the digits interchange their places. Thus, we have

(10y+x)+9=10x+y

⇒ 10y +x +9 =10x +y

 ⇒ 1=x + y -10y -x =9

 ⇒ 9x -9y =9

⇒ 9( x - y)=9

 ⇒ x -y 9/9

 ⇒ x - y =1

So, we have the systems of equations

xy=20

x-y=1

Here x and y are unknowns. We have to solve the above systems of equations for xand y.

Substituting  x =1+y from the second equation to the first equation, we get

(1+y )y=20

⇒ y + y^2 = 20

⇒ y^2 + y = 20 =0

 ⇒ y^2 + 5y -4y -20 =0

⇒ y (y + 5)-4(4+5)=0

⇒ ( y + 5)(y - 4)=0

⇒ y = -5 Or y = 4

Substituting the value of in the second equation, we have

Hence, the number is 10 xx4 + 5= 45.

Note that in the first pair of solution the values of x and y are both negative. But, the digits of the number can’t be negative. So, we must remove this pair.

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 3 Pair of Linear Equations in Two Variables
Exercise 3.7 | Q 12 | Page 86