2014 AMC 12B Problems/Problem 10
Contents
[hide]Problem
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, miles was displayed on the odometer, where is a 3-digit number with and . At the end of the trip, the odometer showed miles. What is .
Solution 1
We know that the number of miles she drove is divisible by , so and must either be equal or differ by . We can quickly conclude that the former is impossible; otherwise, she would have travelled 0 miles. Therefore, and must be apart. Because we know that and and , we find that the only possible values for and are and , respectively. Because , . Therefore, we have
Solution 2
Let the number of hours Danica drove be . Then we know that = . Simplifying, we have , or . Thus, k is divisible by . Because , must be , and therefore . Because and , , and , and our answer is , or .
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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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