Lesson Olympiad level problems on Travel and Distance

Olympiad level problems on Travel and Distance

Problem 1

Let  be the time Sam     takes to drive from A to B, and

Let  be the time Richard takes to drive from B to A.


We are given

     = ,  or   =     (1)



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

     = ,  or   =  = .    (2) 



The total distance equation is

     +  = d     (3)


Replace here   =   based on (2).  You will get

     +  = d,   or

     +  = d

     = d


which implies that the time   = 72  minutes.


Thus, Richard completes his trip in 72 minutes:   = 72 minutes.


Then the time by Sam is   = see (1) =  =  = 90 minutes.


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

Problem 2

The scheme of placing points is shown in the Figure below.


    +--------------------------------|-----------|--------------+

    A                                M           F              B 


Points A and B are cities;  point M is the 1st meeting point;  point F is the 2-nd meeting point.

So, AM = 120 kilometers.

Let x be the distance from the 1st meeting point to B.

Thus the total distance is 120+x kilometers.



The times driving from the starting points to the 1st meeting pojnt are the same

     = ,     (1)

where    and   are the rates of Sam and Richard, respective.



It is our first equation, which I want to rewrite in this form as the ratio of their rates

     = .     (2)



The distance Sam     drove from the 1st meeting point to the second meeting point was   km.

The time Sam     spent covering this distance was   =   hours.



The distance Richard drove from the 1st meeting point to the second meeting point was   km.

The time Richard spent covering this distance was   =   hours.


The times    and   are the same;  it leads to equation

   
     = .



In the right side, replace    by  ,  based on (2).  You will get then

     = .



Simplify it step by step

     = 

     = 

    x*(3x + 60) = 120*(540+2x)

    3x^2 + 60x = 120*540 + 240x 

    3x^2 - 180x - 120*540 = 0

      x^2 - 60x - 21600 = 0    

      (x+120)*(x-180) = 0


Of the two roots,  -120 and 180,  only positive 180  makes sense.


So, the distance x is 180 kilometers, and the total distance from A to B is  120+180 = 300 kilometers.     ANSWER

Problem 3

Let x be that distance from the start, the problems asks for.


Then the time of the whole journey for the first friend is

     +   hours.     (1)


Then the scooter returns back to meet the second frient.  Let's find time "t" for this particular trip.


We write the distance equation for the scooter returning back and the second frieng walking toward him

    20t + 5t = x.


From this equation, the time to return for the scooter is  t =  hours.


During this time  t =  hours, the seconf frient travels   =  kilometers.


So, now they together should cover   =  kilometers at the rate of 20 km/h.


It will take   =  hours.


Thus the total time traveling for the second friend is

     +  +  hours.    (2)


You should identify each addend in this expression, based on my descriptions above.


Lastly, our final equation says that the time (1)  is equal to the time  (2)

     +  =  +  + .


At this point, I just completed setup for you.


You have now a simple linear equation to find x.


The rest is just a technique.