# Solution of the Minute

October 25th, 2009 by ben

“Lewis and Carol travel together on a road from A to B, then return on the same road, with the entire trip taking 3 hours. Sometimes that road goes uphill, sometimes downhill, and sometimes it is level. When the road goes uphill, their rate is 40 mph; downhill their rate is 60 mph; on level road, their rate is x mph.”

Even if you were given a numerical value for x, the distance from A to B would (in most cases) not be uniquely determined. But there is one value for x that would determine that distance uniquely. Compute this value of x [Note: uphill going is downhill returning!]

Credits – Southern California ARML, Oct. 2009, Team round

Original post here.

Solution follows.
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Let $l$ be the distance spent on level road, and let $s$ be the distance spent on sloped road (which way, uphill or downhill, doesn’t matter because it’ll be reversed when they turn back).

$s \cdot \dfrac{1}{40} + s \cdot \dfrac{1}{60} + 2 l \cdot \dfrac{1}{x} = 3$

This equation can be simplified to:

$\dfrac{s}{24} + \dfrac{2l}{x} = 3$

Or, multiplying both sides by 24:

$s + \dfrac{48l}{x} = 72$

The only value of $x$ for which $s+l$ can be uniquely defined is $x=\boxed{48}$.

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