Stuck on a Logical Puzzle Involving Two Numbers

Question

I'm working through a thinking problem and could really use a fresh perspective.

Here's the problem:

I am thinking of two numbers. The sum of the two numbers is 12. One number is twice the other. What are the numbers?

I understand how to set up the equations, but I’d like to learn how to solve it step by step using both algebra and logical reasoning (i.e., without algebra, just thinking it through). I'm especially interested in the logical approach - how would you explain this to a younger student who doesn't know algebra yet?

By solitaire bliss

Answer 1

x+y=12, x=2y

2y+y = 12

3y =12

y =12/3 =4

x =2(4) = 8

(x,y) = (8,4)

Answer 2

I am thinking of two numbers that add up to twelve and the larger number is twice the smaller number.

Smaller Larger

1 11 Sum is 12, but 11 is 11 times as large as 1

2 10 Sum is 12, but 10 is 5 times as large as 2 (10 divided by 2 equals 5)

3 9 Sum is 12, but 9 divided by 3 is 3 (3 times as large)

4 8 Sum is 12 AND 8 is twice 4 -- these are the numbers!

x 2x Let x = smaller number, then two times x is represented by 2x

x + 2x = 12 The two numbers add to 12

3x = 12

x = 4

2x = 8