Question

Seven times a two-digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3. Find the number.

Answer 1

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

The difference between the two digits of the number is 3. Thus, we have `x-y=+-3`

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

Seven times the number is equal to four times the number obtained by reversing the order of the digits. Thus, we have

` 7 (10y+x)=4(10x +y)`

` ⇒ 70 y + 7x = 40x +4y`

`⇒ 40x +4y -70y -7x =0`

` ⇒ 33x - 66y=0`

` ⇒ 33(x-2y)=0`

` ⇒ x - 2y =0`

So, we have two systems of simultaneous equations

` x-y =3`

` x- 2y =0`

`x- y = -3`

` x- 2y =0`

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

(i) First, we solve the system

`x - y = 3`

` x- 2y =0`

Multiplying the first equation by 2 and then subtracting from the second equation, we have

`(x-2y)-2(x-y)=0-2xx3`

`⇒ x - 2y -2x +2y =-6`

` ⇒ -x =-6`

` ⇒ x=6`

Substituting the value of x in the first equation, we have

` 6 -y =3`

`⇒ y = 6-3 `

`⇒ y = 3`

Hence, the number is `10xx3+6 =36`

(ii) Now, we solve the system

`x -y =-3`

` x - 2y =0`

Multiplying the first equation by 2 and then subtracting from the second equation, we have

`( x - 2y)- 2( x -y)=0 - (-3xx2)`

`⇒ x - 2y -2x +2y =6 `

` ⇒ -x =6`

` ⇒ x =-6`

Substituting the value of x in the first equation, we have

` -6 -y =-3`

`⇒ y = -6 +3`

` ⇒ y = -3`

But, the digits of the number can’t be negative. Hence, the second case must be removed.