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## #1 2022-05-18 04:16:13

Median Joe
Member
Registered: 2019-12-04
Posts: 5

### Tough word problem

The combined ages of Mary and Ann are 44 years and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann was.

How old is Mary?

Translating into equations seems to be the way to go with this but the difficulty is where to start after the first one, which is clearly :  m + a = 44.

Any suggestions?

Last edited by Median Joe (2022-05-18 04:29:30)

“I like work: it fascinates me. I can sit and look at it for hours.”

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## #2 2022-05-19 01:12:28

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Hi Median Joe;

I'd suggest structuring from right to left.

That's the method I used for my equations, and I think it worked coz my solution makes sense at each stage (well, to me it does, anyway).

I cheated by using Excel's Goal Seek for the final solve computation, but I'm working on adapting my equations to obtain an algebraic solution to replace Goal Seek's 'what-if analysis' approach.

Struggling, though, and I may not get there!

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #3 2022-05-19 02:29:28

Median Joe
Member
Registered: 2019-12-04
Posts: 5

### Re: Tough word problem

Hi phrontister,

Thanks for your input. I should have perhaps mentioned that the text I got this from gives the answer, which is 27.5, but not a detailed solution. I posted it on another site and some bright spark came up with this:

M+A=44.
k years ago, Mary's age was M−k and Ann's age was A−k
You know that M−k=3(A−k)

So, now you have 2 linear equations, in the 3 unknowns, M,A,k
When Ann is 3(M−k), and Mary is (1/2) that age, Mary is (3/2)(M−k)
At that point, Ann was (3/2)(M−k)−(M−A).
Mary is now twice that age.
Therefore,

M=3(M−k)−2(M−A)⟹2A=3k.
The other two equations are

M+A=44
M−k=3(A−k)⟹[M−(2/3)A]=3A−2A⟹
M=(5/3)A.

From which, M = 27.5 and A = 16.5.

Like you, he started from right to left, but I must admit I don't quite understand this line :

At that point, Ann was (3/2)(M−k)−(M−A).

Not quite sure why you subtract M - A?

“I like work: it fascinates me. I can sit and look at it for hours.”

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## #4 2022-05-19 18:21:07

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Hi Median Joe;

Yes, I also got the answer 27.5.

I'd forgotten about the algebraic method I showed in these two threads I opened back in 2009 - Rita's age & Edna's age - and instead went straight to Excel's Goal Seek to solve your puzzle.

Below is my solution to your puzzle, adapting the algebraic method I posted here:
============================================================
Let c = Ann's age when Mary was 3 times as old as Ann.

Note: Their age difference at each stage is a constant, which in the first stage is established as being 2c (ie, 3c - c).

``````                                           Mary    Ann
when Mary was 3 times as old as Ann was      3c      c
when Ann is 3 times as old as Mary was              9c
when Mary was half as old as Ann will be   4.5c   2.5c
Mary is twice as old as Ann was              5c     3c``````

So Mary's present age is 5c and Ann's is 3c, from which 5c + 3c = 44 (their given combined present age).

Solving: c = 5.5

∴ Mary is 27.5 (5*c = 5*5.5) & Ann is 16.5 (3*c = 3*5.5).
============================================================
The following equation is based on the above info, and evaluates to c = 5.5:

I haven't looked at your solution yet, nor your "Not quite sure why you subtract M - A?" question, but I will...

Last edited by phrontister (2022-05-19 20:42:50)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #5 2022-05-20 00:39:05

Median Joe
Member
Registered: 2019-12-04
Posts: 5

### Re: Tough word problem

Hi phrontister,

Thanks for the links - very interesting. I solved the problem using the algebraic method as described in the Edna's age problem.

m = Mary's present age
a = Ann's present age
x = some years ago
y = some other years ago
z = some years hence

1. m + a = 44               Mary and Ann's combined age is 44 and
2. m = 2(a - x)             Mary is twice as old as Ann was (x years ago)
3. m - x = (a + z)/2        when Mary was half as old as Ann will be (z years hence)
4. a + z = 3(m - y)         when Ann is three times as old as Mary was (y years ago)
5. m - y = 3(a - y)         when Mary was three times as old as Ann was. (y years ago)

Maple solution:

solve({m+a=44,m=2*(a-x),m-x=(a+z)/2,a+z=3*(m-y),m-y=3*(a-y)},[m,a,x,y,z]);
[[m = 55/2, a = 33/2, x = 11/4, y = 11, z = 33]]

Although it's not really much harder than solving the simple age problems you see in elementary algebra (at least, in terms of setting up the equations), what confused me was demarcating the different times correctly.

I don't have Excel, but I believe there is a similar solver in Libreoffice Calc, which I do have. I'll try to replicate the method you posted above using it.

Last edited by Median Joe (2022-05-20 00:40:46)

“I like work: it fascinates me. I can sit and look at it for hours.”

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## #6 2022-05-20 05:23:49

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Hi Median Joe;

I didn't think of trying that Edna method, but your Maple code sparked my interest.

I don't know Maple (nor have it), but the following works in WolframAlpha here and in Mathematica, with the same result as yours: Solve[{m+a==44,m==2(a-x),m-x==(a+z)/2,a+z==3(m-y),m-y==3(a-y)},{m,a,x,y,z}]

I should be in bed!

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #7 2022-05-22 03:18:26

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Hi Median Joe,

...I don't quite understand this line :

At that point, Ann was (3/2)(M−k)−(M−A).

Not quite sure why you subtract M - A?

I'm confused. Did you deliberately omit the brackets?

Anyway, if (M−A) weren't subtracted, M & A's ages would be identical at that point (which they're not). Mary's age at that point is stated by the 'bright spark' as being (3/2)(M−k).

I used the same method: see post #4's "Note: Their age difference at each stage is a constant, which in the first stage is established as being 2c (ie, 3c - c)."

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #8 2022-05-23 16:56:35

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Hi Median Joe;

Here's an image of the worksheet for my solution using Excel's What-If Analysis (Goal Seek).

Goal Seek's settings:-
Set cell: \$A\$11
To value: 44
By changing cell: \$B\$3

As I mentioned in D3, the solution can be obtained by changing B3 until A11 becomes 44. Doing that manually doesn't take long at all.

Last edited by phrontister (2022-05-23 19:44:39)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #9 2022-05-23 17:35:40

Median Joe
Member
Registered: 2019-12-04
Posts: 5

### Re: Tough word problem

Hi phrontister,

I'm confused. dizzy Did you deliberately omit the brackets?

Not deliberately, but I guess I should have included them, although arguably  "subtract M - A" isn't ambiguous.

I used the same method: see post #4's "Note: Their age difference at each stage is a constant, which in the first stage is established as being 2c (ie, 3c - c)."

Yes, I see now. Looking for the invariant is often a good strategy.

Thanks for the goal seek screenshot. Haven't tried it yet with Libreoffice Calc, but I will.

“I like work: it fascinates me. I can sit and look at it for hours.”

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## #10 2022-05-23 19:48:38

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Sorry, Median Joe, but the Goal Seek image had some errors in column D. Fixed now.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #11 2022-05-24 12:56:38

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,715

### Re: Tough word problem

Hi Median Joe;

I opened my Excel file in LibreOffice Calc, and it ran perfectly...including the Goal Seek function.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #12 2022-05-25 19:23:40

Median Joe
Member
Registered: 2019-12-04
Posts: 5

### Re: Tough word problem

Hi phrontister,

It took me a little while to figure out how to use goal seek as I'd never used it before, but it worked and gave the correct results. This is cool, and it's nice to have another problem solving tool in my armoury.

Thanks again.

“I like work: it fascinates me. I can sit and look at it for hours.”

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