When you reverse the digits in a certain two-digit number you increase its value by 18. What is the number if the sum of its digits is 12?

Let's call the number ab with digits a and b

a+b=12 = > b=12-a

a*10+b + 18 = b*10+a = > 9a + 18 = 9b

9a + 18 = 9 (12-a) = > 18a = 108-18 = 90 = > a=5

b = 12-5 = 7

The number is 57. Reversed is 75 and 57+18 is indeed 75.