2014 AMC 10B Problems/Problem 14
Problem
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
?
Solution
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 $a^2+b^2+c^2=1+0+36=\boxed{37 \textbf {(D) }}
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AMC 10 Problems and Solutions |
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