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Difference between revisions of "2017 AMC 10A Problems/Problem 9"

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==Solution==
 
==Solution==
 
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.
 
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.
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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.
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Then, 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 <math>2</math>hr <math>25</math>min trip was <math>\boxed{\textbf{(C)}\ 65}</math> minutes less than Minnie's <math>3</math>hr <math>30</math>min trip.
 
Finally, Penny's <math>2</math>hr <math>25</math>min trip was <math>\boxed{\textbf{(C)}\ 65}</math> minutes less than Minnie's <math>3</math>hr <math>30</math>min trip.
  

Latest revision as of 01:55, 18 December 2024

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 $15$ km all downhill, and then back to town $A$, a distance of $20$ 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 $45$-km ride than it takes Penny?

$\textbf{(A)}\ 45\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 65\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 95$

Solution

The distance from town $A$ to town $B$ is $10$ km uphill, and since Minnie rides uphill at a speed of $5$ kph, it will take her $2$ hours. Next, she will ride from town $B$ to town $C$, a distance of $15$ km all downhill. Since Minnie rides downhill at a speed of $30$ kph, it will take her half an hour. Finally, she rides from town $C$ back to town $A$, a flat distance of $20$ km. Minnie rides on a flat road at $20$ kph, so this will take her $1$ hour. Her entire trip takes her $3.5$ hours.

Then, Penny will go from town $A$ to town $C$, a flat distance of $20$ km. Since Penny rides on a flat road at $30$ kph, it will take her $\frac{2}{3}$ of an hour. Next Penny will go from town $C$ to town $B$, which is uphill for Penny. Since Penny rides at a speed of $10$ kph uphill, and town $C$ and $B$ are $15$ km apart, it will take her $1.5$ hours. Finally, Penny goes from Town $B$ back to town $A$, a distance of $10$ km downhill. Since Penny rides downhill at $40$ kph, it will only take her $\frac{1}{4}$ of an hour. In total, it takes her $29/12$ hours, which simplifies to $2$ hours and $25$ minutes. Finally, Penny's $2$hr $25$min trip was $\boxed{\textbf{(C)}\ 65}$ minutes less than Minnie's $3$hr $30$min trip.

Video Solution

https://youtu.be/ZzXmY5mG-rQ

~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

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