SOLUTION: A two digit number is such that its value equals four times the sum of its digits.If 27 is added to the number the results is equal to the value of the number obtained when the dig

N = the two-digit number = 10a + b    (1)



We are told  

 4a+4b = N             (2)

  and  

 10a+b + 27 = 10b + a  (3)  (27 added to the number results in reversal of digits)





(1) and (2) can be used to eliminate N:

   10a+b = 4a+4b



This simplfies to  6a-3b = 0

(3) simplifies to  9a-9b = -27



These last two equations can be solved easily for  a=3, b=6  ==> 



Check:

3(4)+6(4) = 12+24 = 36  (ok)

36 + 27 = 63    (digits reversed, ok)



 

Let the tens and units digits be T, and U, respectively
Then we get: 10T + U = 4(T + U)
6T = 3U
2T = U ------ eq (i)

Also, 10T + U + 27 = 10U + T
10T - T + U - 10U = - 27
9T - 9U = - 27
9(T - U) = 9(- 3)
T - U = - 3 ------ eq (ii)

T - 2T = - 3 ------ Substituting 2T for U in eq (ii)
- T = - 3
Tens digit, or 
2(3) = U ------ Substituting 3 for T in eq (i)
Units digit or 
Number: