John runs in a race against Matthew Matthew runs twice as fa

John runs in a race against Matthew. Matthew runs twice as fast as John for the first two kilometers. But during the remainder of the race, he runs half as fast as John. If the race ends in a tie , and if John runs at a constant rate, how far do they each run?   

Solution

John DATA:
rate = x kph ; distance = d km ; time = d/x hrs
---------------------------
Mathew DATA:
For 2 kilometer you have the following:
distance = 2 km ; rate = 2x ; time = d/r = 2/2x = 1/x hrs
----
For the remainder of the race:
distance = d-2 km ; rate = x/2 kph ; time = d/r = (d-2)/(x/2) = (2d-4)/x hrs
-----------------------------------------------
Equation based on \"the race ends in a tie\":
John time = Matthew time
d/x = (1/x) + (2d-4)/x
Multiply thru by \"x\" to get:
d = 1 + 2d-4
d = 3 km (this is the distance they each run)