Question 19967
Actually, they said no such thing! But if they had.....
Let's use two letters for the unknown weights of Tweedledum and Tweedledee. To keep them straight, use the LAST letter of their names:  m and e.
So,
e + 2m = 361
This is the first statement by Tweedledum: "your weight" [e] plust twice MY weight [2m] is 361.
The second statement, likewise, is rendered: 
m + 2e = 362

Now just solve. 
e + 2m = 361
2e + m = 362
2e + 4m = 722 [multiplying the first equation by 2]
3m = 360 [subtracting the second equation from the third one]
m = 120 [dividing both sides by 3]
e + 240 = 361 [substituting the value of m in the first equation]
e = 121 [solving for e]

Checking:
121 + 240 = 361 [first equation]
242 + 120 = 362 [second equation]

They don't seem very fat!