Question

Help!

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. What are all the possible values of d?

Answer 1

This question seems a bit tricky, but once you write down the information you know, you'll see it's actually much simpler than it appears! Also, v=d/t is your best friend here :P

What we know:
- The race is 24 km long (total d=24)
- When Alice finishes the race, Becky has run only 24-8=16 km, and Caitlin has run 24-12=12 km

We don't know anything about speeds or times. However, now we can set up some relationships using the information we have and v=d/t. Let's call Alice's speed v(A), and the time it takes her to finish the race t(A). Her speed is v(A)=d/t=24/t(A).

When Alice finishes the race and t=t(A), Becky has run only 16 km. That means Becky's speed is v(B)=d/t=16/t(A). If you look back at Alice though, you can see that by rearranging her equation you get t(A)=24/v(A), so another way to express Becky's speed is as 16/(24/v(A)) or, more simply, ⅔ v(A). In other words, Becky only covers ⅔ the distance Alice covers because Becky is running at ⅔ the speed!

By the same method, you'll find that Caitlin's speed, v(C), is only ½ v(A).

Now, when Becky finishes the race, more time has passed. Let's call the time is took Becky to finish the race t(B), and remember, Becky's speed was ⅔ v(A). Using v=d/t, we have ⅔ v(A)=24/t(B). Rearranging for time, you get t(B)=36/v(A). This is the time it took Becky to finish the race (and it's a bigger number, as we'd expect - 36/v(A) is bigger than 24/v(A), no matter what v(A) is).

We want to know how much distance there is left for Caitlin to run at this stage. We can figure that out by rearranging v=d/t to d=vt. For Caitlin, d(C)=v(C)t(B) (remember, we're using the time it took Becky to finish the race). This equation doesn't look to useful right now, but we already have expressions for v(C) and t(B)! From before, v(C)=½ v(A), and t(B)=36/v(A). Throw this back into the equation for d(C) and you get d(C)=(½ v(A))(36/v(A))=18.

That's pretty convenient - the v(A)'s reduced out leaving us with a constant! So Caitlin has traveled 18 km at the time Becky has finished the race! And we didn't even need to figure out what anyone's speeds or times actually were! That means the distance remaining for Caitlin, which is what the question asks for, must be 24-18=6 km! And this is the only possible distance because everyone is running at a constant speed the entire time.

Does that make sense? if you have any questions, let me know!

Answer 2

Thank you so much, Matt! Very thourough!