How old is Mary Enigma

Enigma: How old is Mary? (Note: it does not have to be an integer)

Ok...We know that Mary's age now ( F ) plus Ann's age now ( G ) is equal to 44...

F + G = 44

Now we can also make the assumption that Mary's age now ( F ) minus Ann's age now ( G ) will equal the age difference between them ( X )...

F - G = X

(We know that Mary is the older one because of the last line in the enigma..."...when Mary was three times older that Ann.")

Now that we have formed these two initial equations we can move on to dealing with the actual wording of the problem itself.

I found that it was easier to approach this problem backwards. By starting with the last line first it made things alot easier.

In the last line it says that Mary ( at age A )was three times as old as Ann ( at age B )was. When one person is three times as old as the other person the age difference between them ( X ), a constant, will be equal to twice as much as the younger persons age, obviously Ann's age ( 2B )...

X = 2B

We also know that Mary ( at age A ) was three times as old as Ann ( at age B ) was...

A = 3B

Next it says that Ann ( at age C ) will be three times as old as Mary ( at age A ) was...

C = 3A

Next it says that Mary ( at age D ) was half as old as Ann ( at age C ) was...

D = C/2

Next it says that Ann was ( age E ) when Mary was ( age D )... This brings us back to our age difference ( X )...

D - E = X

Finally, it says that Mary ( at age F ) is twice as old as Ann ( at age E ) was...

F = 2E

So just to summarize things to this point that leaves us with the following equations...

F + G = 44

F - G = X

X = 2B

A = 3B

C = 3A

D = C/2

D - E = X or E = D - X

F = 2E

Now if we start to substitute the equations backwards we move towards solving in terms of F ( Mary's current age )...

F = 2E

F = 2(D - X)

F = 2(C/2 - X)

F = 2((3A)/2 - X)

F = 2((3(3B))/2 - X)

F = 2((3(3(X/2)))/2 - X)

F = 2((3(3X/2))/2 - X)

F = 2((9X/2)/2 - X)

F = 2(9X/4 - X)

F = 18X/4 - 2X

F = 4.5X - 2X

F = 2.5X

So now we can solve for X in terms of F...

X = F/2.5

We can also solve for G in terms of F from the equation...

F + G = 44

To come to the following...

G = 44 - F

Now we can substitute them all into this equation...

F - G = X

To come up with this...

F - (44 - F) = F/2.5

F - 44 + F = F/2.5

2F - 44 = F/2.5

2.5(2F - 44) = F

5F - 110 = F

4F - 110 = 0

4F = 110

F = 27.5

So Mary's current age is 27.5

(incidentally...that would make Ann 16.5 because their age difference ( X ) is 11 years...)

(My Solution was generic. So, I used this better and more detailed version sent in by Paul G. Thanks!)

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