Mary is 24 years old. When Mary was Ann’s current age, Ann was 12 years old (half Mary’s current age).

This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.

perl -e '$_=24;print+(y///c,$m=$_)&&$_*3/4 .$/'

Let M_n := Mary's age now = 24, A_n := Ann's age now, and let M_p, A_p be the ages of Mary and Ann at a certain point in the past. We have that M_n = 24 = 2(A_p) from "Mary is 24 years old. She is twice as old as Ann was", immediately implying that A_p = 12. At the point in time p when M_p, A_p were the ages of Mary and Ann, respectively, we have that M_p = A_n from "when Mary was as old as Ann is now". Also, let x := | M_n - M_p | = | A_n - A_p | be the amount of time that has passed between the point p in time and now. Since M_n > M_p and A_n > A_p by construction, we can drop absolute values, yielding M_n = M_p + x, A_n = A_p + x. Substituting, we have M_p = 24 - x, and A_n = 12 + x, yielding x = 24 - M_p = A_n - 12, yielding A_n = 36 - M_p. But, since M_p = A_n, we can write 2(A_n) = 36 yielding A_n = 18.

This problem, however, completely ignores the fact that Ann boarded an interstellar spaceship 6 years ago at a reasonable fraction of the speed of light c. To account for this missing detail, we need to use Lorentz factors. Since Ann was the one traveling in space, we have to adjust her current age by calculating A_n = A_p + x * sqrt(1 - (v/c)^2)), where v is Ann's velocity Given that the problem is missing the critical detail of how fast Ann is hurtling towards the outer boundary of the universe, we really can't calculate their age at all. Nonetheless, given the completely reasonable and plausible assumption that Ann has been traveling at 95% of the speed of light because spaceships totally can do that, Ann is obviously about 13.87 years old. So 18 years old is clearly the wrong answer.

Note that the fact that Ann is not aging as much due to traveling at an enormous velocity does NOT change Mary's age. Mary remains exactly twice as old as Ann was at the given point in the past, so she's 24 and 6 years have passed from her perspective. Only Ann's age changes. Obviously.

I love this response so much.

    x: Ann's current age
    y: The age difference between Ann and Mary

    24 = x + y      (ie. their current ages)
    x = (24/2) + y  (ie. their age in the past)

    Solve for y in one equation and plug it into the other. You'll get x = 18.

I can't see a way to do it without algebra.

> Mary is 24 years old. She is twice as old as Ann was

So Ann was 12

> when Mary was as old as Ann is now

So Mary is older than Ann.

The age of Ann is somewhere between 12 and 24. Without much thinking I'd say that probably 12 and 24 are not included and probably it is an even number.

We can test each of them.

If Ann is 14 now, when Mary was 14 (10 years ago) Ann was 4 year old and Mary would be 8 now, this is not the solution.

If Ann is 16 now, when Mary was 16 (8 years ago) Ann was 8, Mary would be 16 now, nope.

If Ann is 18 now, when Mary was 18 (6 years ago) Ann was 12, Mary would be 24 now and she is. This is the solution.

Ann is 18.

This is related to "When all you have is a hammer, everything looks like a nail."

It's slightly different, because you've supercharged your hammer. Everything you ever understood about how to solve word problems has been subsumed into what your brain labels as "algebra."

But look at it this way:

1) Did you use algebra to convert the problem to algebraic form? Probably not; algebra says nothing about word problems.

2) Once it was in algebraic form, did you need to repeatedly apply algebraic rules in order to reduce the problem, or could you glance at it and figure it out?

You may also be hampered by your choice of variable, because you chose "X" to be an intermediate variable.

If, instead, you choose "X" to be what you are searching for, Ann's current age, then the problem setup is:

  24 - x = x - 12