Difference between revisions of "2014 AMC 12B Problems/Problem 10"
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==Solution 2== | ==Solution 2== | ||
Let the number of hours Danica drove be <math>k</math>. Then we know that <math>100a + 10b + c + 55k</math> = <math>100c + 10b + a</math>. Simplifying, we have <math>99c - 99a = 55k</math>, or <math>9c - 9a = 5k</math>. Thus, k is divisible by <math>9</math>. Because <math>55 * 18 = 990</math>, <math>k</math> must be <math>9</math>, and therefore <math>c - a = 5</math>. Because <math>a + b + c \leq{7}</math> and <math>a \geq{1}</math>, <math>a = 1</math>, <math>c = 6</math> and <math>b = 0</math>, and our answer is <math>a^2 + b^2 + c^2 = 6^2 + 0^2 + 1^2 = 37</math>, or <math>\boxed{D}</math>. | Let the number of hours Danica drove be <math>k</math>. Then we know that <math>100a + 10b + c + 55k</math> = <math>100c + 10b + a</math>. Simplifying, we have <math>99c - 99a = 55k</math>, or <math>9c - 9a = 5k</math>. Thus, k is divisible by <math>9</math>. Because <math>55 * 18 = 990</math>, <math>k</math> must be <math>9</math>, and therefore <math>c - a = 5</math>. Because <math>a + b + c \leq{7}</math> and <math>a \geq{1}</math>, <math>a = 1</math>, <math>c = 6</math> and <math>b = 0</math>, and our answer is <math>a^2 + b^2 + c^2 = 6^2 + 0^2 + 1^2 = 37</math>, or <math>\boxed{D}</math>. | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2014|ab=B|num-b=9|num-a=11}} | {{AMC12 box|year=2014|ab=B|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:46, 17 January 2021
Contents
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 the equal or differ by . We can quickly conclude that the former is impossible, so 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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