We can write the unknown number as [tex]10x+y[/tex], where both [tex]x,y[/tex] are in the set [tex]\{1,2,\ldots,8,9\}[/tex]. (Neither can be 0)
Interchanging the digits makes the number [tex]10y+x[/tex]. So
[tex]10x+y+10y+x=11(x+y)=132\implies x+y=12[/tex]
Adding 12 to the number makes it 5 times the sum of the number's digits, which means
[tex]10x+y+12=5(x+y)\iff 5x-4y=-12[/tex]
Now we can solve the system,
[tex]\begin{cases}x+y=12,5x-4y=-12\end{cases}\implies x=4,y=8[/tex]
so the original number is 48.
