Solution
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 = a - x |
(1) |
| When "Bonnie was as old as Andrea had been": |
b - x = a - y |
(2) |
| When "Bonnie had been half as old as Andrea is now": |
b - y = 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 - b - y |
| 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
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
|