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5. Ages

Alice and Ben's ages add up to 91. Alice is three times as old as Ben was when Alice was twice as old as Ben is now. What are their ages?
 

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Let Alice'a age be a and Ben's age be b. The first equation is obtained easily as

a + b = 91

Now take yourself back, say, n years so that Alice's age was a - n and Ben's was b - n

At this time 'Alice is [now] three times as old as Ben was...' and Alice was [then] 'twice as old as Ben is now'.

So a = 3(b - n)
and a - n = 3b

You can eliminate n from these two equations to get

4a - 9b = 0

Using this and the original equation, we get b = 28 and a = 63
 

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