Math CentralQuandaries & Queries


Question from: Angel

Four friends ran a race:
Ben finished seven seconds ahead of Mike.
Noel finished three seconds behind Sam.
Mike finished five seconds behind Noel.
What will be the formula to support their place?

Hi Angel,

I'm not sure what you mean by "the formula to support their place", but here is what I would do. I am going to use the number line and put each runner on the line at the time it takes them to finish the race. Time is in seconds. I am going to call the runners B, N, S, and M for Ben, Noel, Sam, and Mike. First Ben.


Ben's time is somewhere on the line but I'm not sure where. Now put Mike on the line. "Ben finished seven seconds ahead of Mike." so Ben took 7 seconds less time than Mike to complete the race. Hence we have


The second clue compares Noel and Sam and we don't have either on the diagram. The third clue says "Mike finished five seconds behind Noel." so Noel took 5 seconds less time that Mike and thus we have


Now we can use the fact that "Noel finished three seconds behind Sam." and hence


Thus Sam took the least amount of time so he won. But what about a formula?

Suppose Sam took $t$ seconds to complete the race. How many seconds did Ben take to complete the race? What about Noel? and Mike?

I hope this helps,

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