Asked by XXX

In a 24 km race, during which each runner maintains a constant speed throughout, Alice crosses the finish line while Becky is still 8 km from finishing and Caitlin is 12 km from finishing. When Becky crosses the finish line, Caitlin still has d kilometers to go. Find all possible values of d.
Answered by Reiny
Suppose it took Alice 4 hrs, when she finished
Alice went at 6 km/h
Becky went 16 km at 4 km/h
Caitlin went 12 km at 3 km/h
So for Becky to finish her remaining 8 km would take an extra 2 hrs, but in 2 hrs Caitlin only went 6 more km of her remaining 12 km, so she still has 6 km to go , d = 6

suppose Alice took 5 hrs.
Alice's rate = 24/5 or 4.8 km/h
Becky's rate = 16/5 or 3.2 km/h
Cailin's rate = 12/5 = 2.4 km/h
So for Becky to finish her remaining 8 km would take an extra 2.5 hrs, but in 2.5 hrs Caitlin only went 6 more km of her remaining 12 km, so she still has 6 km to go , d = 6

Suppose Alice took t hrs
Alice went 24 km , her rate is 24/t km/h
Becky went 16 km , her rate is 16/t km/h
Caitlin went 12 km , her rate is 12/t km/h
So for Becky to finish her remaining 8 km would take an extra 8÷(16/t) or t/2 hrs, but in t/2 hrs Caitlin only went (t/2)(12/t) or 6 more km of her remaining 12 km, so she still has 6 km to go , d = 6

Ahhh, d = 6
Answered by Jon
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Answered by aops cop
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Answered by Wonka
Busted!
Answered by TK24NC
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Answered by AoPSSheriff
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Answered by AoPS person
we;; mp
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