2014 AMC 10B Problems/Problem 14
Danica drove her new car on a trip for a whole number of hours, averaging miles per hour. At the beginning of the trip, miles was displayed on the odometer, where is a -digit number with and . At the end of the trip, the odometer showed miles. What is ?
Let be the number of hours Danica drove. Note that can be expressed as . From the given information, we have . This can be simplified into by subtraction, which can further be simplified into by dividing both sides by . Thus we must have . However, if , then , which is impossible since must be a digit. The only value of that is divisible by and less than or equal to is .
From this information, . Combining this with the inequalities and , we have , which implies , so , , and . Thus
Danica drives miles, such that and is a multiple of 55. Therefore, must have an units digit of either or If the units digit of is then which would imply that Danica did not drive at all. Thus, Therefore, and because we have Finally, then must be due to and
We can set up an algebraic equation for this problem.
From what's given, we have that
This simplifies to be
Factoring, we get that
Hence, notice that we want so that
The only pair that works for this problem that satisfies the original requirements is
Checking, we have that
Hence, the answer is
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