Difference between revisions of "2009 AMC 8 Problems/Problem 14"
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==Problem== | ==Problem== | ||
− | Austin and Temple are <math> 50</math> miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging <math> 60</math> miles per hour. Leaving the car with her daughter, Bonnie | + | Austin and Temple are <math> 50</math> miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging <math> 60</math> miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged <math> 40</math> miles per hour on the return trip. What was the average speed for the round trip, in miles per hour? |
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<math> \textbf{(A)}\ 46 \qquad | <math> \textbf{(A)}\ 46 \qquad | ||
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\textbf{(D)}\ 52 \qquad | \textbf{(D)}\ 52 \qquad | ||
\textbf{(E)}\ 54</math> | \textbf{(E)}\ 54</math> | ||
+ | |||
+ | ==Solution== | ||
+ | The way to Temple took <math>\frac{50}{60}=\frac56</math> hours, and the way back took <math>\frac{50}{40}=\frac54</math> for a total of <math>\frac56 + \frac54 = \frac{25}{12}</math> hours. The trip is <math>50\cdot2=100</math> miles. The average speed is <math>\frac{100}{25/12} = \boxed{\textbf{(B)}\ 48}</math> miles per hour. | ||
+ | |||
+ | ==Solution 2== | ||
+ | This question simply asks for the [[harmonic mean]] of <math>60</math> and <math>40</math>, regardless of how far Austin and Temple are. | ||
+ | Plugging in, we have: <math>\frac{2ab}{a+b} = \frac{2 \cdot 60 \cdot 40}{60 + 40} = \frac{4800}{100} = \boxed{\textbf{(B)}\ 48}</math> miles per hour. | ||
==See Also== | ==See Also== | ||
+ | 2015 Problem 17 | ||
{{AMC8 box|year=2009|num-b=13|num-a=15}} | {{AMC8 box|year=2009|num-b=13|num-a=15}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:32, 14 August 2021
Contents
Problem
Austin and Temple are miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
Solution
The way to Temple took hours, and the way back took for a total of hours. The trip is miles. The average speed is miles per hour.
Solution 2
This question simply asks for the harmonic mean of and , regardless of how far Austin and Temple are. Plugging in, we have: miles per hour.
See Also
2015 Problem 17
2009 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 AJHSME/AMC 8 Problems and Solutions |
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