The original number is **34 **.

The sum of its digits is (3 + 4) = 7.

Reversing its digits makes it 43. 9 more than 34.

How I did it: (There must be a much more elegant mathematical method)

There are only 6 possibilities for the original number.

They are.

16

25

34

43

52

61.

Instead of trying to dream up a sophisticated equation,

I decided it was faster and easier to just go down the list

and look for one that worked.

The third one I tried worked.

Let the two numbers be x and y.

According to your question;

x + y = 7

10y + x = 10x + y + 9

By equation 1 ; x = 7-y

Substituting the value of x ;

10y + ( 7 -y) = 10(7-y) + y + 9

9y + 7 = 70 -10y + y + 9

9y + 7 = 70 - 9y + 9

=> 18y = 70 -7 + 9

=> 18y = 72

=> y = 4

Substituting for x ;

x = 7 - y

=> x = 7 -4

=> x = 3

Thus, x = 3 and y = 4;

=> The number is 34.