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28. Bob and Alice

Alice's age is the same as that of Bob if the digits are reversed. A year ago Alice was twice as old as Bob. How old are they?
 

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If Alices age is represented by the digits ab and Bob's by ba then the value of Alice's age is given by

10a + b

and Bob's age is

10b + a

Using these values, then one year ago we have

10a + b - 1 = 2(10b + a - 1)

so  10a + b - 1 = 20b + 2a - 2
so  8a - 1 = 19b - 2
so  19b = 8a + 1

and the only integer values (between 1 and 9) that satisfy this are a = 7 and b = 3. So Alice is 73 and Bob is 37.
 

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