Question

A number is 3 more than 4 times the sum of its digit . If 18 is added to the number, its digit are reversed. Find the no.

Answer 1

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

The number is 3 more than 4 times the sum of the two digits. Thus, we have:

$\bold{10y + -4 ( x + y ) + 3}$

⇒ $\bold{10y + x = 4x + 4y + 3}$

⇒ $\bold{4x + 4y - 10y - x = -3}$

⇒ $\bold{3x - 6y = -3}$

⇒ $\bold{3 ( x - 2y ) = -3}$

⇒ $\bold{x - 2y = -3/3}$

⇒ $\bold{x - 2y = -1}$

After interchanging the digits, the number becomes $\bold{10x + y.}$

If 8 is added to the number are reversed. Thus, we have:

$\bold{( 10y + x ) + 18 = 10x + y}$

⇒ $\bold{10x + y -10y -x = 18}$

⇒ $\bold{9x - 9y = 18}$

⇒ $\bold{9(x -y ) = 18}$

⇒ $\bold{x - y = 18/9}$

⇒ $\bold{x - y = 2}$

So, we have the systems of equations

$\bold{x - 2y = -1,}$

$\bold{x - y = 2}$

Here x and y are unknowns. So, we have to solve the above systems of equations for x and y.
Subtracting the first equation from the second, we have:

$\bold{( x - y ) - ( x - 2y ) = 2- (-1)}$

⇒ $\bold{x - y - x + 2y = 3}$

⇒ $\bold{y = 3}$

Substituting the value of y in the first equation, we have:

$\bold{x - 2 × 3 = -1}$

⇒ $\bold{x - 6 = -1}$

⇒ $\bold{x = -1 + 6}$

⇒ $\bold{x = 5}$

Hence, the number is $\bold{\fbox{\red{10 × 3 + 5 = 35}}}$

[ Based on R.D Sharma's solution ]