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| Bob's Cab (Posted on 2009-03-02) |
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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?
Explanation to Puzzle Answer
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Comment 5 of 5 | 
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(In reply to Puzzle Answer by K Sengupta)
At the outset we need to determine 3 positive integer whose product is 2450. Case 1: Precisely one of the numbers is 1 Only 1, 1, 2450 is possible
Case 2: Precisely two of the numbers is 1 Eight triplets are possible, and these are: 1, 2, 1225 1, 14, 175 1, 5, 490 1, 25, 90 1, 7, 350 1, 35, 70 1, 10, 245 1, 49, 50
Case 3: None of the three numbers is a 1 Since 2450 = 2* 5^2 * 7^2, it follows that the prime factors are 2, 5 and 7 So, the valid triplets are posited hereunder as follows: 2, 4, 5 5, 10, 49 2, 7, 175 5, 14, 35 2, 25, 49 7, 1, 50 2, 35, 35 7, 10, 35 5, 5, 98 7, 14, 25 5, 7, 70
We now consider the sum of each of the 20 triplets as follows:
T r I p l e t SUM ---------------- -------- 1, 1, 2450 2452 1, 2, 1225 1228 1, 5, 490 496 1, 7, 350 358 1, 10, 245 256 1, 14, 175 190 1, 25, 98 124 1, 35, 70 106 1, 49, 50 100 2, 5, 245 252 2, 7, 175 184 2, 25, 49 76 2, 35, 35 72 5, 5, 98 108 5, 7, 70 82 5, 10, 49 64 5, 14, 35 54 7, 7, 50 64 7, 10, 35 52 7, 14, 25 46
Now, Bob mentioned to Jim that the sum of the ages of the passengers is precisely twice the age of Jim whereupon Jim expressed his inability to deduce their age. Since Jim knows his own age, and therefore the sum of the three ages, this means the sum is NOT sufficient to identify the three ages. In other words, the sum corresponds to two different triplets. The only sum that is repeated is 64, corresponding to: • 5+10+49=64 • 7+7+50= 64 So there is precisely one adult passenger, whose correct age corresponds to the third element of one of the abovementioned triplets. We now remember that Bob mentioned that he was younger than one of the passengers. So the incorrect age of the adult passenger (which is equal to Bob's age) is less than the correct age of the said passenger. This is only possible when: Passenger's age = 50 Bob's age = 49
Also, ages of other 2 cab passengers = 7, and 7 Also, 2* (Jim's age) = 64 => Jim's age = 32
Consequently, summarizing the foregoing we have: • Jim's age = 32 • Bob's age = 49 • Ages of the three passengers = 7, 7, and 50 Edited on June 15, 2022, 9:02 am
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