Let x be the larger number and y be the smaller number. Then I have the equations x + y = 12 and x - y = 20. To eliminate y, I can add the two equations and get 2x = 32. Dividing both sides by 2, I get x = 16. To find y, I can substitute x = 16 into either equation. For example, using x + y = 12, I get 16 + y = 12. Subtracting 16 from both sides, I get y = -4. This means the smaller number is -4. 
The sum of two numbers is $12$ and their difference is $20$. What is the smaller number?