Complicated Ages (Andrea, Bonnie)

Andrea and Bonnie are comparing their ages; they find that Bonnie is as old as Andrea was when Bonnie was as old as Andrea had been when Bonnie had been half as old as Andrea is now. If the sum of their present ages is 44 years, then what is Andrea’s age?


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Let a and b be the current ages of Andrea and Bonnie. Let “was” be x years ago and “had been” be y years ago.

Then “Bonnie is as old as Andrea was”:  b = ax  (1)
When “Bonnie was as old as Andrea had been”:  bx = ay  (2)
When “Bonnie had been half as old as Andrea is now”:  by = ½a  (3)
Also, we know that:  a + b = 44  (4)

So, here we go (there are probably other paths).

(4)  a = 44 – b
 
(1)  b = (44 – b) – x
2b = 44 – x

 
(2) 
44 – x  – x = 44 – by
2

44 – x – 2x = 88 – 2b – 2y

44 – 3x = 88 – 2(  44 – x  ) – 2y
2

44 – 3x = 88 – 44 + x – 2y
44 – 4x = 44 – 2y
4x = 2y
2x = y

 
(3) 
44 – x  – 2x (44 – b)
2 2

44 – x – 4x = 44 – b

44 – 5x = 44 –  44 – x
2
5x 44 – x
2

10x = 44 – x
11x = 44
x = 4
y = 8

 
(1)  b = a – 4
 
(4)  a + (a – 4) = 44
2a = 48
a = 24 (Andrea)
b = 24 – 4 = 20 (Bonnie)
 
check:  (1) 20 = 24 – 4
(2) 20 – 4 = 24 – 8
(3) 20 – 8 = ½(24)
(4) 24 + 20 = 44

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