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Ann is 18.
I think what makes it confusing is the "variable overloading" referring to both the current ages of Mary and Ann (M & A), and their ages at some point in the past "when Mary was as old as Ann is now" (M' & A').
So, what we're given is:
M = 24
M' = A
M = 2A' => A' = M/2 = 12
Since the age gap between Mary and Ann is constant, we know:
M - A = M'- A'
So, substituting in the known values:
24 - A = A - 12
2A = 36
A = 18
We can double check the result. Since Mary (24) is 6 years older than Ann (18), then when Mary was 18 Ann would have been 12, so Mary is now twice that age as given.
I think it’s easier to see if you drop the time indices and just introduce a variable for the elapsed time. So:
Mary := 24 = 2(A - Δ)
A = Mary - Δ
And then it’s just substitution. But of course what appeals to one’s intuition is very personal
Thanks for sharing this, it actually makes it easy. I somehow convinced myself it was a trick having to do with ages being rounded off to whole numbers, so delta-age can be off by one... anyway I want my day back.
Rephrasing it makes it easier to grasp:
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.
Ah, that makes sense. I originally read it differently.
> Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?
Which I read as:
> Mary is 24 years old. When Mary was Ann's current age, Anne was half the age she currently is.
Which would mean Anne is 16 (because when Mary was 24-8=16, Anne's current age, then Anne was 16/2=8, half Anne's current age).
But re-reading it, then for that to be true the original would have needed to be phrased:
> Mary is 24 years old. She was twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?
> This makes it a bit easier to realize that Ann’s age has to be right in the middle of 12 and 24.
How did you actually make that deduction? I can only see it by writing down an equation. I can't see anything in the problem that directly implies it.
Personally, I would rephrase it to:
Ann is 6 years younger than Mary, who is 24 years old.
Or even better, Ann is 18 and Mary is 24.
But then again, I always disliked word-based logic problems that didn't correspond to genuine complexity. If something CAN be stated in simple terms, I believe it should be. If you want to create a word-based logic puzzle then present a genuine problem that can be engaged with rather than a poorly formulated explanation of the mundane.
That being said, I have no idea how I would present the same fundamental logical puzzle using a material problem that doesn't have a simpler distillation that solves the problem for you. Maybe I'm just not cut out for the realm of intellectual puzzles.
Without rephrasing it… I determined that Ann is 16.
I'm not sure why it needs rephrasing—it seems pretty straightforward? What am I missing?
16 works (as another pointed out). Ann is 16 and Mary would have been that age 8 years ago—when of course Ann would then have been 8 (Mary, at 16, was twice her age).
I figured it out by knowing that Ann had to be 12 in the past since Mary is 24 now which is twice Ann’s age. 6 years have passed since Ann was 12 and Mary was 18 making them 18 and 24, respectively.
I don't know why but I really struggled to internalize what the question was asking. Here's my take on a rewrite that's clearer:
Mary is 24, which is twice as old as Ann was some time ago. At that time, Mary was the age that Ann is now. What is that age?
This may help people:
> Mary is 24 years old. She is twice as old as Ann was (12) when Mary was as old (18) as Ann is now (18). How old is Ann?
So 6 years ago, Mary was 18, and Ann was 12.
Today, Mary is twice the age that Ann was (12 * 2) at the time that she (Mary) was 18, making her 24.
What makes it confusing is that the sentence is comparing Mary’s present age with Ann’s past age (which I did not catch until mulling it over a bit).
Yeah, I had to spell it out to get it to click for me
> mary is 24
> 8 years apart: ann is 16 now, when ann was 12 mary was 20
> 6 years apart: ann is 18 now, when ann was 12 mary was 18
> 4 years apart: ann is 20 now, when ann was 12 mary was 16
Mary IS 24. So Ann WAS 12. Say that was X years ago. Then Mary WAS 24-X and Ann IS 12+X. So 24-X=12+X, X=6, Ann is 18.
I can't see a way to do it without algebra.
there's 12 years between 12 and 24. it is divided equally between OldAnn-OldMary and OldMary-NowMary. 12/2=6, 12+6=18
It's a kind of cheating because it means not understanding the essence of the problem but with such small numbers enumeration is a viable strategy.
> 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.
> I can't see a way to do it without algebra.
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
"Any eighth-grade boy or girl who is 'all there' should solve it in about thirty seconds" is hyperbole (it took me a couple of minutes), but in general the public reaction to this seems like a demonstration of the Flynn effect.
> it took me a couple of minutes
Are you in the 8th grade? Those of us a couple of decades past the 8th grade are maybe slower at algebra than someone who is actively drilling it...
Couple of minutes to understand the english, and an editor to write the formulas :-)
Not the Flynn effect; just people having fun, and selection bias. I (the blogger) deliberately left out many boring responses, except for linking to the Evening World's letters page of 1903-11-11.[1] I filtered out scores of replies from schoolchildren walking through the algebra to get the correct answer, and dozens of (usually short) replies from people of various ages giving wrong answers with and without reasoning, some almost certainly joking, some almost certainly not. But that's what you should expect from a Letters to the Editor section on any given math meme! Our discourse hasn't changed that much over 120 years. See my other post,[2] linked at the bottom of the current one, on "What is 8÷2(2+2)?" — that's not as fair a question, but the discourse is pretty much the same in unfiltered Letters-to-the-Editor sections: "I think it's X. Well, I think it's Y."
The whole point of this kind of meme is for the populace to perform disagreement about the answer! (See also: the Monty Hall problem; sports; comments sections.)
[1] - https://www.loc.gov/resource/sn83030193/1903-11-11/ed-1/?sp=...
[2] - https://quuxplusone.github.io/blog/2019/08/01/what-is-8-divi...
It's amazing to me how much effort people invest in elaborating wrong answers.
The key insight here is to remember that time moves the same for both Mary and Ann. Let's call the number of years between past and present X. Then
24 - X = 12 + X => 24 - 12 = 2X => X = 6.
Ah yes. The famous problem that is ambiguous because it ignores space travel!
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.
Here's the simplest form I could distill it to. The other comments I find quite verbose or confusing, no offense intended.
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.That's not how I read it. I read it as:
Now : Mary is 24 = 2y, Ann is x
Past: Mary is x, Ann is 12 = y
> When Mary had a little lamb, she was twice as old as Ann was when Ann was as old as Mary now claims to be. How old is the sheep? —James Barton Adams.
I love this response so much.
The closing problem is more fun.
> A man is twice the age his wife was when he was the age she is now. When she reaches his present age, their combined years will be 100. Find the age of each.
This seemed initially like something amenable to guessing and checking a few ratios. <narrator voice>It was not.</narrator voice>
My answer: The man is 44 4/9 years old, the wife 33 1/3 years.
I am surprised no one spotted the obvious solution:
perl -e '$_=24;print+(y///c,$m=$_)&&$_*3/4 .$/'
Like all of these things that get posted on facebook regularly, they almost intentionally use vague or improper language to be confusing so realistically there is no valid answer. You have to rewrite the problem in clear, unambiguous language and then they are usually easily solved.
Wait so how old is she?
well into the post-geriatric era by now
18
Barely legal
67
My approach at parsing this:
There's two point in time with two ages each:
M, M0, and A, A0
Then the sentence can be expressed as:
* M = M0 + X and A = A0 + X and (X is the time difference)
* M = 24 and ("Mary is 24")
* M = 2 * A0 and ("Mary is twice as old as Ann was")
* M0 = A ("Mary was as old as Ann is now")
Yes, this was exactly what I had, pretty much. A system of equations, 4 equations, 4 variables.
MO (mary old), MN (mary now), AO (ann old), AN (ann now)
MN = 24
MO = AN
2AO = MN
MN - AN = MO - AO
How interesting!
This discussion itself offers insight on the actual need for technical professionals to be able to logically reason through non-obvious [information] structures.
I've anecdotally suspected that the ability to synthesize information into a navigable data structure and subsequently analyze it, both from internal and external POV, is not a hard requirement for the majority SWE positions.
This has flashbacks to GMAT problems.
83. This is an AI verification referring to my aunt. I didn't read the article and skimmed the comments.
what the true answer? oh, there is no?
My intuition says that Ann now is 24
> Dozens of people declare solemnly that Mary and Ann are twins, though the slightest attempt to "prove" this shows it does not fulfil the conditions of the problem. [...] "About Mary and Ann, I think they are eight years old," says a woman, disregarding the first words of the question, which state that Mary is twenty-four.
[1] - https://www.loc.gov/resource/sn88085488/1903-10-31/ed-1/?sp=...
18.
Ann is 7 years old.
Opus says Ann is 18
Sounds about right.
How I understand it:
Mary is 24, Ann is a few (n) years younger
M = 24
A = M-n
M-2n = M/2 = 12
M-2n = 12 --> n=6
Hallucinations like these make it pretty clear that these agents do not really understand or think, IMHO.