The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

Step 1 (Define Variables): Let the two original numbers be $x$ and $y$. We are given that their sum is $S$:

Step 2 (Apply Operations to Each Number):

1. Add 3 to each number:

• First number becomes: $x+3$

• Second number becomes: $y+3$

2. Double each resulting number:

• First resulting number: $2(x+3)=2x+6$

• Second resulting number: $2(y+3)=2y+6$

Step 3 (Sum the Final Numbers): We find the sum of these two final numbers:

Step 4 (Substitute S): Since $x+y=S$:

Step 5 (Conclusion): The sum of the final two numbers is exactly 2S + 12 (or 2S+12).