The sum of a man's age and his son is 66. What are their ages if the digits are reveresed ? The teacher said there are 3 answers.

Let 10x+y be the fathers
age.

The sun's age is in reverse digits by data. So the
sun's age = 10y+x.

Naturally father's age  is more than
sun's age.

=> 10x+y > 10y
+x.

=> 10x+y-10y-x >
0

=> 9x-9y >
0.

=> 9(x-y) >
0.

=> x-y >
0.

 x > y.....(1)

Also
the sum of their ages = (10x+y)+(10y+x) = 11(x+y) which is 66 by
data.

So 11(x+y) = 66.

We
divide both sides of 11(x+y) = 66 by 11:

x+y =
6.....(2)

So x> y and x+y = 6. So we have the
choice: x= 3, x= 4, x = 5, x = 6.

Then  the corresponding y
values are  y= 3,  y = 2, y = 1 and y = 0.

So  father's age
= 33 ,  42,  51, 60.

Sun's age : 33 , 24, 15,
06.

We exclude 33 as it is not
practical.

So father and sun's age are one of
the pairs:  (42 , 24) , (51, 15) , (60, 06).