Question 1188740: The combined ages of Mark and Andrew are 44, and Mark is twice as old as Andrew was when Mark was half as old as Andrew will be when Andrew is three times as old as Mark was when Mark was three times as old as Andrew. How old is Mark?
Answer by VFBundy(438)   (Show Source):
You can put this solution on YOUR website! Mark's age now = M
Andrew's age now = 44 - M
Work backwards. When was Mark three times as old as Andrew? Let x = the number of years ago that Mark was three times as old as Andrew:
M - x = 3(44 - M - x)
M - x = 132 - 3M - 3x
4M + 2x = 132
2(2M + x) = 132
2M + x = 66
x = 66 - 2M
"...will be when Andrew is three times as old as Mark was when (x) Mark was three times as old as Andrew", which was 66 - 2M years ago:
44 - M - (66 - 2M) = 3(M - (66 - 2M))
44 - M - 66 + 2M = 3(M - 66 + 2M)
-22 + M = 3M - 198 + 6M
M - 22 = 9M - 198
-8M = - 176
M = 22
So, in 22 years, Andrew will be three times as old as Mark was when Mark was three times as old as Andrew.
"...when Mark was half as old as Andrew will be", which we just learned will occur in 22 years:
M + 22 = 0.5(44 - M + 22)
M + 22 = 22 - 0.5M + 11
1.5M = 11
M = 22/3
So, 22/3 years ago, Mark was half as old as Andrew will be when Andrew is three times as old as Mark was when Mark was three times as old as Andrew.
"Mark is twice as old as Andrew was when Mark was..." We just learned this was 22/3 years ago:
M - 22/3 = 2(44 - M - 22/3)
M - 22/3 = 88 - 2M + 44/3
3M = 88 - 22
3M = 66
M = 22
Mark's age now = M = 22
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