Cab driver Bob mentioned to his friend Jim that he recently drove three passengers in his cab, that the product of their (the passengers') ages was 2450, and that the sum of their ages was exactly twice Jim's age.

From this, Jim couldn't deduce what their three ages were.

But when Bob added that he was younger than at least one of the passengers, Jim, who knew Bob's age, was able to deduce all the passengers' ages.

What were Bob's and Jim's ages, and the ages of the passengers?

The prime factors of 2450 are 2, 5, 5, 7 and 7. From this information the passengers' ages would need be one of the following:
2 [2],  5 [ 5 ] and 245 [5x7x7] with sum 252
2 [2],  7 [ 7 ] and 175 [5x5x7] with sum 175
2 [2], 25 [5x5] and  49 [ 7x7 ] with sum  76
2 [2], 35 [5x7] and  35 [ 5x7 ] with sum  72
5 [5],  7 [ 7 ] and  70 [2x5x7] with sum  82
5 [5], 10 [2x5] and  49 [ 7x7 ] with sum  64
5 [5], 14 [2x7] and  35 [ 5x7 ] with sum  54
7 [7],  7 [ 7 ] and  50 [2x5x5] with sum  64
7 [7], 10 [2x5] and  35 [ 5x7 ] with sum  52
7 [7], 14 [2x7] and  25 [ 5x5 ] with sum  46

Given that Jim could not deduce the ages from the product and sum of the passengers' ages, Jim's age must be 32 as it is the only age that is not distinct as half the sum of the ages of the passengers. Yet, with Jim being able to discern the ages after given that Bob was younger than at least one of the passengers, the ages must then be Bob (49), Jim (32), and the passengers (50, 7 and 7) as any other age would not limit Jim to a distinct solution.

Posted by Dej Mar on 2009-03-02 17:52:14