Difference between revisions of "2009 AMC 8 Problems/Problem 14"
(→Problem) |
Themathgood (talk | contribs) m (→Solution 3: Just adding parentheses) |
||
| (15 intermediate revisions by 9 users not shown) | |||
| Line 11: | Line 11: | ||
==Solution== | ==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. | 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== | ||
| + | We calculate the harmonic mean of Austin and Temple. | ||
| + | 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. | ||
| + | |||
| + | ==Solution 3== | ||
| + | The 50 miles part of this question is redundant. We find that the average speed is skewed towards the lower speed, since you spend more time in the lower speed. So, we take the LCM of 40 and 60, getting 120. So, we do (40 + 40 + 40 + 60 + 60)/5, calculation the ratio of the parts and taking the mean. Solving, we get 240/5 = <math>\boxed{\textbf{(B)}\ 48 } \qquad</math> | ||
| + | -themathgood | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/CnWuadfT3xA Soo, DRMS, NM | ||
| + | |||
| + | https://www.youtube.com/watch?v=-O5zQT9mp2I ~David | ||
==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}} | {{MAA Notice}} | ||
Latest revision as of 20:23, 20 March 2024
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
We calculate the harmonic mean of Austin and Temple.
Plugging in, we have: 
miles per hour.
Solution 3
The 50 miles part of this question is redundant. We find that the average speed is skewed towards the lower speed, since you spend more time in the lower speed. So, we take the LCM of 40 and 60, getting 120. So, we do (40 + 40 + 40 + 60 + 60)/5, calculation the ratio of the parts and taking the mean. Solving, we get 240/5 = 
-themathgood
Video Solution
https://youtu.be/CnWuadfT3xA Soo, DRMS, NM
https://www.youtube.com/watch?v=-O5zQT9mp2I ~David
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
