Question

A two-digit number is $4$ times the sum of its digits. If $18$ is added the number, the digits are reversed. Find the number.

Answer 1

Let the tens and unit digit of number are =$x,y$ respectively

So the number $=10x+y$

If the digits are reversed then the number is $10y+x$.

First condition :

$10x+y=4(x+y)$

$\Rightarrow 10x+y=4x+4y$

$\Rightarrow 10x+y-4x-4y=0$

$\Rightarrow 6x-3y=0$

$\Rightarrow 3(2x-y)=0$

$\Rightarrow 2x-y=0$

$\Rightarrow 2x=y$------(1)

Second condition :

$\Rightarrow 10x+y+18=10y+x$

$\Rightarrow 10x+y+18-10y-x=0$

$\Rightarrow 9x-9y=-18$

$\Rightarrow x-y=-2$

$\Rightarrow x-2x=-2$ [Since from (1)]

$\Rightarrow x=2$

and $y=2x=2\left(2\right)=4$

$\Rightarrow y=4$

$\therefore$ Two digit number $=24$