2014 AMC 10A Problems/Problem 15
- The following problem is from both the 2014 AMC 12A #11 and 2014 AMC 10A #15, so both problems redirect to this page.
Problem
David drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?
Solution 1
Note that he drives at miles per hour after the first hour and continues doing so until he arrives.
Let be the distance still needed to travel after the first hour. We have that , where the comes from hour late decreased to hours early.
Simplifying gives , or .
Now, we must add an extra miles traveled in the first hour, giving a total of miles.
Solution 2 (Answer Choices)
Instead of spending time thinking about how one can set up an equation to solve the problem, one can simply start checking the answer choices. Quickly checking, we know that neither choice or choice work, but does. We can verify as follows. After hour at , David has miles left. This then takes him hours at . But . Since is hours less than , our answer is .
Solution 3
Let us call the total number of miles after David changes speed . We realize that if David is driving 50 miles an hour, and he arrives 30 minutes early, . The only solution that fits the description is .
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 |
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.