Let's solve the problem step by step.

The problem states: "One number is 2 more than another number. The sum of three times the larger number and three times the smaller is 36. What are the numbers?"

1. Define the variables:

   • Let $x$ be the smaller number.
   • Let $y$ be the larger number.

2. Set up the equations based on the problem:

   • Equation 1: $y = x + 2$ (since one number is 2 more than another number)
   • Equation 2: $3x + 3y = 36$ (since the sum of three times the larger number and three times the smaller is 36)

3. Substitute Equation 1 into Equation 2: $3x + 3(x + 2) = 36$

4. Simplify the equation: $3x + 3x + 6 = 36$ $6x + 6 = 36$

5. Solve for $x$: $6x = 36 - 6$ $6x = 30$ $x = 5$

6. Find $y$ using Equation 1: $y = x + 2$ $y = 5 + 2$ $y = 7$

7. Conclusion:

   • The smaller number $x$ is 5.
   • The larger number $y$ is 7.

So, the numbers are 5 and 7.

Would you like further details or have any questions?

Here are some related questions:

1. How would you solve if the sum of twice the larger number and three times the smaller number is 46?
2. If the difference between the larger number and twice the smaller number is 1, what are the numbers?
3. What if the larger number is 3 more than the smaller number, and the sum of twice the larger and four times the smaller number is 40?
4. If one number is 3 less than another number and the sum of four times the larger number and twice the smaller number is 50, what are the numbers?
5. How do you solve for the numbers if the sum of four times the larger number and five times the smaller number is 63?
6. If the larger number is 5 less than twice the smaller number, and the sum of five times the larger and three times the smaller number is 61, what are the numbers?
7. What are the numbers if one number is 4 more than the other, and the sum of three times the smaller number and twice the larger number is 38?
8. How would the problem change if the larger number is 4 less than three times the smaller number, and the sum of the numbers is 22?

Tip: When dealing with word problems, always start by defining the variables and translating the words into mathematical equations.