Difference between revisions of "2017 AMC 10A Problems/Problem 9"
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==Problem== | ==Problem== | ||
| − | Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at 30 kph, and uphill at 5 kph. Penny rides on a flat road at 30 kph, downhill at 40 kph, and uphill at 10 kph. Minnie goes from town A to town B, a distance of 10 km all uphill, then from town B to town C, a distance of | + | Minnie rides on a flat road at <math>20</math> kilometers per hour (kph), downhill at <math>30</math> kph, and uphill at <math>5</math> kph. Penny rides on a flat road at <math>30</math> kph, downhill at <math>40</math> kph, and uphill at <math>10</math> kph. Minnie goes from town <math>A</math> to town <math>B</math>, a distance of <math>10</math> km all uphill, then from town <math>B</math> to town <math>C</math>, a distance of <math>15</math> km all downhill, and then back to town <math>A</math>, a distance of <math>20</math> km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the <math>45</math>-km ride than it takes Penny? |
| + | |||
| + | <math>\textbf{(A)}\ 45\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 65\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 95</math> | ||
==Solution== | ==Solution== | ||
| − | The distance from town A to town B is | + | The distance from town <math>A</math> to town <math>B</math> is <math>10</math> km uphill, and since Minnie rides uphill at a speed of <math>5</math> kph, it will take her <math>2</math> hours. Next, she will ride from town <math>B</math> to town <math>C</math>, a distance of <math>15</math> km all downhill. Since Minnie rides downhill at a speed of <math>30</math> kph, it will take her half an hour. Finally, she rides from town <math>C</math> back to town <math>A</math>, a flat distance of <math>20</math> km. Minnie rides on a flat road at <math>20</math> kph, so this will take her <math>1</math> hour. Her entire trip takes her <math>3.5</math> hours. |
| − | Secondly, Penny will go from town A to town C, a flat distance of | + | Secondly, Penny will go from town <math>A</math> to town <math>C</math>, a flat distance of <math>20</math> km. Since Penny rides on a flat road at <math>30</math> kph, it will take her <math>\frac{2}{3}</math> of an hour. Next Penny will go from town <math>C</math> to town <math>B</math>, which is uphill for Penny. Since Penny rides at a speed of <math>10</math> kph uphill, and town <math>C</math> and <math>B</math> are <math>15</math> km apart, it will take her <math>1.5</math> hours. Finally, Penny goes from Town <math>B</math> back to town <math>A</math>, a distance of <math>10</math> km downhill. Since Penny rides downhill at <math>40</math> kph, it will only take her <math>\frac{1}{4}</math> of an hour. In total, it takes her <math>29/12</math> hours, which simplifies to <math>2</math> hours and <math>25</math> minutes. |
| − | Finally, Penny's 2 Hour 25 Minute trip was 65 minutes less than Minnie's 3 Hour 30 Minute Trip | + | Finally, Penny's <math>2</math> Hour <math>25</math> Minute trip was <math>\boxed{\textbf{(C)}\ 65}</math> minutes less than Minnie's <math>3</math> Hour <math>30</math> Minute Trip. |
| + | Review: <math>\boxed{\textbf{(C)}\ 65}</math> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/ZzXmY5mG-rQ | ||
| + | |||
| + | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=A|num-b=8|num-a=10}} | {{AMC10 box|year=2017|ab=A|num-b=8|num-a=10}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 16:44, 22 July 2020
Contents
Problem
Minnie rides on a flat road at 
kilometers per hour (kph), downhill at 
kph, and uphill at 
kph. Penny rides on a flat road at kph, downhill at 
kph, and uphill at 
kph. Minnie goes from town 
to town 
, a distance of km all uphill, then from town to town 
, a distance of 
km all downhill, and then back to town , a distance of km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the 
-km ride than it takes Penny?
Solution
The distance from town to town is km uphill, and since Minnie rides uphill at a speed of kph, it will take her 
hours. Next, she will ride from town to town , a distance of km all downhill. Since Minnie rides downhill at a speed of kph, it will take her half an hour. Finally, she rides from town back to town , a flat distance of km. Minnie rides on a flat road at kph, so this will take her 
hour. Her entire trip takes her 
hours.
Secondly, Penny will go from town to town , a flat distance of km. Since Penny rides on a flat road at kph, it will take her 
of an hour. Next Penny will go from town to town , which is uphill for Penny. Since Penny rides at a speed of kph uphill, and town and are km apart, it will take her 
hours. Finally, Penny goes from Town back to town , a distance of km downhill. Since Penny rides downhill at kph, it will only take her 
of an hour. In total, it takes her 
hours, which simplifies to hours and 
minutes.
Finally, Penny's Hour Minute trip was 
minutes less than Minnie's 
Hour Minute Trip.
Review:
Video Solution
~savannahsolver
See Also
| 2017 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
