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(Disclaimer: I remember a puzzle like this I did a long time ago, but this isn't a direct copy of the puzzle, only it's formatting. If this is appears to be a copy of another riddle, I will simply say that this mimics only the format, while not the actual puzzle. Please enjoy.)

Jim and John are brothers. One day, they go to the local drugstore to pick up some medicine for their sick father (their mother being at work). When they arrive, they pick the proper medicine and place it, along with $5, on the counter. The clerk, who sees these two young boys unattended in a drugstore, immediately gets suspicious and asks for their age. Jim and John, after huddling for a moment, respond with the following:

John: Well, sir, I'm one-seventh my age older than my brother.

Jim: And twice my age is five-sixths my age more than my brother's.

John: But there is no more than 4 years between us.

Jim: And we aren't quite mature in-law's, yet.

The clerk was befuddled, and distracted enough for the boys to snatch the medicine up and run out with it. After a moment, the clerk recovers and chuckles to himself, having decided that boys these days are very clever.

How old are Jim and John?

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    $\begingroup$ This is very close to being a pure math problem - only the framing makes it any different from one. $\endgroup$
    – Deusovi
    May 5 at 19:13
  • $\begingroup$ Is that a bad thing? $\endgroup$ May 5 at 19:32
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    $\begingroup$ Yes. See e.g. puzzling.meta.stackexchange.com/questions/2783/… and puzzling.meta.stackexchange.com/questions/6030/… (which I think is the single clearest statement of current policy on such things). $\endgroup$
    – Gareth McCaughan
    May 5 at 20:30
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    $\begingroup$ (Of course it's not a bad thing in the abstract; there's nothing wrong with mathematics exercises. But being a mathematical exercise rather than a puzzle makes something inappropriate for posting here.) $\endgroup$
    – Gareth McCaughan
    May 5 at 20:31
  • $\begingroup$ After seeing Deusovi's answer, it is clear to me that this is not really a standard maths problem and really does have some puzzle elements to it. $\endgroup$ May 6 at 7:01
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We can just translate their statements into algebra:

Where $x$ is John's age and $y$ is Jim's:
$$x = y+x/7$$ $$2y = 5y/6 + x$$
But these equations are equivalent! So one of them doesn't give any new information.
Solving one equation gives $y=6x/7$.
Assuming their ages are integers, that means that the possibilities are (0,0), (7,6), (14,12), (21,18), and (28,24). Since their ages are in question, it's probably not (0,0) or (7,6). If "not quite mature" means "not age 18 or more", then their ages must be 14 and 12.

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