SOLUTION: Driving between City A and City B, the ratio of the time Sam takes to the time Richard takes is 5 : 4. If Sam leaves City A and Richard leaves City B at the same time toward to eac

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Question 1163672: Driving between City A and City B, the ratio of the time Sam takes to the time Richard takes is 5 : 4. If Sam leaves City A and Richard leaves City B at the same time toward to each other, they meet in 40 minutes, and they continue to drive. After Richard arrives at City A, how many minutes later will Sam arrives at City B?
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52782) About Me  (Show Source):
You can put this solution on YOUR website!
.

Let T%5BS%5D be the time Sam     takes to drive from A to B, and

Let T%5BR%5D be the time Richard takes to drive from B to A.


We are given

    T%5BS%5D%2FT%5BR%5D = 5%2F4,  or  T%5BS%5D = %285%2F4%29%2AT%5BR%5D    (1)



Hence, the ratio of their rates per minute is reciprocal ratio

    V%5BS%5D%2FV%5BR%5D = 4%2F5,  or  V%5BS%5D = %284%2F5%29%2AV%5BR%5D = 0.8%2AV%5BR%5D.    (2) 



The total distance equation is

    40%2AV%5BS%5D + 40%2AV%5BR%5D = d     (3)


Replace here  V%5BS%5D = 0.8%2AV%5BR%5D  based on (2).  You will get

    40%2A0.8%2AV%5BR%5D + 40%2AV%5BR%5D = d,   or

    32%2AV%5BR%5D + 40%2AV%5BR%5D = d

    72%2AV%5BR%5D = d


which implies that the time  d%2FV%5BR%5D = 72  minutes.


Thus, Richard completes his trip in 72 minutes:  T%5BR%5D = 72 minutes.


Then the time by Sam is  T%5BR%5D = see (1) = %285%2F4%29%2AT%5BR%5D = %285%2F4%29%2A72 = 90 minutes.


Thus Sam reaches B  90-72 = 18 minutes after Richard reaches A.    ANSWER

Solved.

---------------

Surely, this problem is two levels higher than a regular Math/Physics problem in any high school.

It is, actually, an Olympiad level problem in Physics.

Could you tell me please where is it from, from which source ?


Do not forget to post your "THANKS" to me for my teaching.




Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Driving between City A and City B, the ratio of the time Sam takes to the time Richard takes is 5 : 4. If Sam leaves City A and Richard leaves City B at the same time toward to each other, they meet in 40 minutes, and they continue to drive. After Richard arrives at City A, how many minutes later will Sam arrives at City B?
Let multiplicative factor be x, and distance from A to B, or B to A, D

Then times Sam and Richard take to complete journey are 5x and 4x, respectively

Therefore, Sam’s speed = D%2F%285x%29, and Richard’s is: D%2F%284x%29 

With them meeting in 40 minutes, we get:  

matrix%281%2C3%2C+2%2F%2815x%29+%2B+1%2F%286x%29%2C+%22=%22%2C+1%29 ------- Factoring out GCF, D, in numerators

4 + 5 = 30x ------ Mulltiplying by LCD, 30x

9 = 30x

Multiplicative factor, or matrix%281%2C6%2C+x%2C+%22=%22%2C+9%2F30%2C+%22=%22%2C+3%2F10%2C+hour%29

Therefore, the difference in times for them to complete their respectively journeys is highlight_green%28matrix%281%2C5%2C+3%2F10%2C+%22hour%2C%22%2C+or%2C+18%2C+minutes%29%29