# Typical age problem that I can't figure

Bob and Alice have together a sum of $103$ years old. In $4$ years, Alice will have $2$ times the age of Bob.

What age will they have ?

I'm trying different equation and doing my substitution but I can't get a valid answer (a integer).

$$x+y = 103 ,$$

$$(x+4) + (y+4) = 2x$$

etc..

Thanks for help, it's appreciated !

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The second equation is not quite right. This question is straight-forward so I think you would benefit most by just thinking about this a bit more, and ask for help in a while, but only if you get stuck again. –  JavaMan Sep 19 '11 at 1:53
It might help you here a bit if you use the names "Alice" and "Bob" instead of the awfully uninformative variable names $x$ and $y$... so for instance, $\text{Alice}+\text{Bob}=103$. –  Ｊ. Ｍ. Sep 19 '11 at 2:02
In any solution, if you introduce variables, you should say what they stand for. That will help both you and the reader. It looks as if $x$ is Bob's current age, and $y$ is Alices's (not a great choice, potentially confusing to the reader and you). But let's go on. In terms of $x$, how old will Bob be in $4$ years? In terms of $y$, how old will Alice be in $4$ years? Alice will be twice Bob's age. What equation does that give us? –  André Nicolas Sep 19 '11 at 6:38

Consider an analogous problem: Art and Bill's answers currently have $31$ total upvotes. If both receive $4$ more upvotes then Art will have twice as many upvotes as Bill. How many is that?

Let $\rm\:A,B\:$ be the upvotes Art, Bill have after the $4$ upvotes. We are given $\rm\:A = 2\:B\:,\:$ and before the $4$ upvotes they had sum $31\:,\:$ i.e. $\rm\: A\!-\!4\:+\:B\!-\!4\: =\: 31\:.\:$ Hence $\rm\ 39 = A + B = 2\:B + B = 3\:B\:.\:$ Therefore $\rm\:B = 13\:$ and $\rm\:A = 2\:B = 26\:.\:$

Extra credit: Whose answer will be accepted?

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Nice phrasing! :D –  Ｊ. Ｍ. Sep 19 '11 at 2:40
+1 for the extra credit problem, which from empirical evidence seems to be independent of all known axioms. –  Rahul Sep 19 '11 at 6:16

Let $A$ and $B$ represent the original age of alice and bob.

You have to solve the following system of equations.

$A + B = 103$

$2(B + 4) = A + 4$

Thus $A = 103 - B$

So

$2B + 8 = 107 - B$

Therefore

$3B = 99$

$B = 33$ and $A = 70$.

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