SOLUTION: Mary is twice as old as Ann was when Mary was half as old as Ann will be when she is three times as old as Mary was when she was three times as old as Ann.

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Question 774783: Mary is twice as old as Ann was when Mary was half as old as Ann will be when she is three times as old as Mary was when she was three times as old as Ann.
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
Mary is now M years old
Ann is now A years old
x = the first mentioned number of years in the past
y = the mentioned number of years in the future
z = the second mentioned number of years in the past

Mary is twice as old as Ann was (x years ago)

M = 2(A-x)

(x years ago) was when Mary was half as old as Ann will be (y years from now)

M-x = 1%2F2(A+y)

(y years from now is) when she (Ann) is three times as old as Mary was 
(z years ago) 

A+y = 3(M-z)

when she was three times as old as Ann (was z years ago)

M-z = 3(A-z)

M = 2(A-x)
M-x = 1%2F2(A+y)
A+y = 3(M-z)
M-z=3(A-z)


That's 4 equations in 5 unknowns, so there probably are
more than one answer

  M - 2A          +  x = 0
 2M -  A - y      - 2x = 0
-3M +  A + y + 3z      = 0
  M - 3A     + 2z      = 0

Solving that by matrices gives:

(M,A,y,z) = (6x,10x,12x,4x)

The youngest they could be is when x=1 year ago, y=12 years from now,
and z=4 years ago:  Mary is now 6, Ann is now 10.

The next youngest they could be is when x=2 year ago, y=24 years from now,
and z=8 years ago:  Mary is now 12, Ann is now 20.

The next youngest they could be is when x=3 year ago, y=36 years from now,
and z=12 years ago:  Mary is now 18, Ann is now 30.

etc. etc.

Edwin