Difference between revisions of "2017 AMC 10B Problems/Problem 7"
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Revision as of 14:33, 20 January 2020
Contents
Problem
Samia set off on her bicycle to visit her friend, traveling at an average speed of kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at kilometers per hour. In all it took her minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
Solution 1
Let's call the distance that Samia had to travel in total as , so that we can avoid fractions. We know that the length of the bike ride and how far she walked are equal, so they are both , or . She bikes at a rate of kph, so she travels the distance she bikes in hours. She walks at a rate of kph, so she travels the distance she walks in hours. The total time is . This is equal to of an hour. Solving for , we have: Since is the distance of how far Samia traveled by both walking and biking, and we want to know how far Samia walked to the nearest tenth, we have that Samia walked about .
Solution 2 (Quick, Make Approximations)
Notice that Samia walks times slower than she bikes, and that she walks and bikes the same distance. Thus, the fraction of the total time that Samia will be walking is Then, multiply this by the time 34 minutes is a little greater than of an hour so Samia traveled The answer choice a little greater than 2.5 is . (Note that we could've multiplied by and gotten the exact answer as well)
Solution 3 (Harmonic Mean)
Since the distance that both rates travel are equivalent, we can find the harmonic mean of the two rates to find the average speed. Therefore, let and
Now, we can find the overall distance by multiply the average speed by the time(hours), 44 minutes. Let
Because the distance Samia walks is half the total distance, the distance she walks is . With some knowledge of sixths, one finds that Samia walked, in kilometers,
~mn28407
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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