bonanova Posted March 22, 2009 Report Share Posted March 22, 2009 In Hill County there are two kinds of roads: hilly and level. The cabs in Andy's Hilltop Taxi fleet travel 72 mph downhill, 63 mph on level roads, and 56 mph going uphill. It took Andy 4 hours to pick up his fare in Green Valley and 4 hours and 40 minutes to return him to Hilltop. How far is it between the two cities? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 22, 2009 Report Share Posted March 22, 2009 168miles Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 22, 2009 Report Share Posted March 22, 2009 Is it 273 miles? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 22, 2009 Report Share Posted March 22, 2009 271.75miles Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 22, 2009 Report Share Posted March 22, 2009 Couldn't this problem have many solutions? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 22, 2009 Report Share Posted March 22, 2009 It appears that although you could have varying amounts of hilly and flat roads, the total distance would always roughly be the same, which seems to be about 274 miles (if I am correct, it varies between about 274 and 274.3 miles. If, on the way over, there were 160.3138 miles of downhill road and 0 miles of uphill road, 113.647 miles of flat road. The total mileage would be 273.96 miles. The other extreme would be on the way over, there are 217.313 miles of downhill road and 57 miles of uphill road, and no flat road. Total mileage would be 274.313 miles. So I can't settle on 1 definite value, but I'm reasonably certain it's between 274 and 274.3 miles. Unfortunately, I just figured that out with 1 equation and used trial-and-error with my calculator. There's probably a more elegant way to find the solution. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 22, 2009 Report Share Posted March 22, 2009 (edited) 273 miles Let x equal the number of miles traveled downhill on the way there. Let y equal the number of miles traveled on level roads on the way there. Let z equal the number of miles traveled uphill on the way there. On the way there we get the equation: x/72 + y/63 + z/56 = 4 On the way back, the rate of travel over distances x & z is reversed, yielding this equation: x/56 + y/63 + z/72 = 14/3 By multiplying each of these equations by 504, they can be simplified to the following: 7x + 8y + 9z = 2016 9x + 8y + 7z = 2352 Now add the two equations together, yielding: 16x + 16y + 16z = 4368 Divide by 16: x + y + z = 273 Unless I'm mistaken, there are infinite possible solutions for the individual values x, y, & z, however the sum of all 3 is all we care about here. Edited March 22, 2009 by Ungoliant Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 23, 2009 Report Share Posted March 23, 2009 there is no definant answer it depends on what u want to say ratio of flat to downhill to uphill is Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 23, 2009 Author Report Share Posted March 23, 2009 there is no definant answer it depends on what u want to say ratio of flat to downhill to uphill is Hi cddude, and welcome to the Den. Can you find two distances that work? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 23, 2009 Report Share Posted March 23, 2009 (edited) Here's another way to look at it, which arrives at the same result I posted above. For the total trip, lasting 8 hours and 40 minutes, we know that the hilly roads will have to be traveled over twice, once uphill, once downhill. Therefore, it doesn't matter what the ratio of uphill vs downhill is in the one-way trip, we know that there is an equal amount of uphill & downhill travel over the entire round trip. So what is the average speed of travel over these hilly sections over the round trip? Total distance / total time. If total distance is x, half of which is traveled at 72mph and half at 56mph, average speed equals: x / (0.5x/72 + 0.5x/56) With some simple algebra, the x's cancel out and we find the average speed over the hilly regions is 63mph. What a coincidence! That is also the average speed over the level roads. Therefore we can say, regardless of the distribution of uphill, downhill, and level roads, the average speed of the entire round trip will always be 63mph. Now simply multiply 63mph by 8 hours 40 minutes to arrive at 546 miles for the round trip, or 273 miles one-way. Edited March 23, 2009 by Ungoliant Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 23, 2009 Author Report Share Posted March 23, 2009 Here's another way to look at it, which arrives at the same result I posted above. For the total trip, lasting 8 hours and 40 minutes, we know that the hilly roads will have to be traveled over twice, once uphill, once downhill. Therefore, it doesn't matter what the ratio of uphill vs downhill is in the one-way trip, we know that there is an equal amount of uphill & downhill travel over the entire round trip. So what is the average speed of travel over these hilly sections over the round trip? Total distance / total time. If total distance is x, half of which is traveled at 72mph and half at 56mph, average speed equals: x / (0.5x/72 + 0.5x/56) With some simple algebra, the x's cancel out and we find the average speed over the hilly regions is 63mph. What a coincidence! That is also the average speed over the level roads. Therefore we can say, regardless of the distribution of uphill, downhill, and level roads, the average speed of the entire round trip will always be 63mph. Now simply multiply 63mph by 8 hours 40 minutes to arrive at 546 miles for the round trip, or 273 miles one-way. Nice. And welcome to the Den. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 23, 2009 Report Share Posted March 23, 2009 273 miles 168 hilly and 105 plain Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 23, 2009 Author Report Share Posted March 23, 2009 273 miles 168 hilly and 105 plain Hi imran, ltns Is that distribution unique? Not that is has to be, mind you, just wondering. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 23, 2009 Author Report Share Posted March 23, 2009 Some puzzles fall into the category of "Excuse me, but there's not enough information." Two that come to mind are this one, of course, and "Hole in the Sphere". Here we're not told what fraction of the journey is on level ground. In Hole in the Sphere, we're not told the diameter of the hole. In both cases, the "missing" information itself is the puzzle. We conclude [1] the answer does not depend on the missing info or [2] the puzzle is flawed. Either way, we do no harm by imagining the simplest case and solving it. Here we imagine all the ground is level. In "Hole" we imagine the diameter is zero. The answers immediately become clear. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 24, 2009 Report Share Posted March 24, 2009 Hi imran, ltns Is that distribution unique? Not that is has to be, mind you, just wondering. Yes, IMO this distribution is unique. Quote Link to comment Share on other sites More sharing options...
Question
bonanova
In Hill County there are two kinds of roads: hilly and level.
The cabs in Andy's Hilltop Taxi fleet travel 72 mph downhill, 63 mph on level roads, and 56 mph going uphill.
It took Andy 4 hours to pick up his fare in Green Valley and 4 hours and 40 minutes to return him to Hilltop.
How far is it between the two cities?
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