Question

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The sum of ages of Tom and Elizabeth is 49. Tim is twice as old as Elizabeth was when Tom was as old as Elizabeth is now. How old are they?

Answer 1

We have two moments of history here: "now" and "then" (at particular moment in the past). Let's mark as the following:
T - Tom's age now
E - Elizabeth's age now
To - Tom's age then
Eo - Elizabeth's age then

So we have:
T - E =  To - Eo (their difference is constant)   (1)
T + E = 49                                                         (2)
T = 2Eo                                                            (3)
To = E                                                               (4)

Using (4) into (1): 
T - E =  E - Eo  
Then using (3) here:
2Eo - E =  E - Eo  
2Eo + Eo =  E + E
3Eo =  2E
E = 3/2 Eo                                                         (5)

Using (5) and (3) into (2):
T + E = 49
2Eo + 3/2Eo = 49  
3 1/2Eo = 49
7/2Eo = 49
Eo = 49 : 7/2
Eo = 14

So, Elizabeth was 14. It means now Tom is 2*14=28. Elizabeth now is 49 - 28 = 21

Answer: Elizabeth is 21, Tom is 28