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Difference between revisions of "2017 AMC 10B Problems/Problem 7"

(Problem Type)
(→‎Solution 1)
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<cmath></cmath>
 
<cmath></cmath>
 
Since <math>x</math> is the distance of how far Samia traveled by both walking and biking, and we want to know how far Samia walked to the nearest tenth, we have that Samia walked about <math>\boxed{\bold{(C)} 2.8}</math>.
 
Since <math>x</math> is the distance of how far Samia traveled by both walking and biking, and we want to know how far Samia walked to the nearest tenth, we have that Samia walked about <math>\boxed{\bold{(C)} 2.8}</math>.
 +
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==Solution 2 (Quicker, Make Approximations)==
 +
 +
Notice that Samia walks <math>\frac {17}{5}</math> times slower than she bikes, and that she walks and bikes the same distance. Thus, the fraction of the total time that Samia will be walking is
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<cmath>\frac{\frac{17}{5}}{\frac{17}{5}+\frac{5}{5}} = \frac{17}{22}</cmath>
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Then, multiply this by the time:
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<cmath>\frac{17}{22} \cdot 44 \text{minutes} = 34 \text{minutes} </cmath>
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34 minutes is a little greater than <math>\frac {1}{2}</math> of an hour so Samia traveled
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<cmath>\sim \frac {1}{2} \cdot 5 = 2.5 \text{kilometers}</cmath> The answer choice a little greater than 2.5 is <math>\boxed{\bold{(C)} 2.8}</math>. (Note that we could've multiplied <math>\frac {34}{60}=\frac {17}{30}</math> by <math>5</math> and gotten the exact answer as well)
  
 
==See Also==
 
==See Also==

Revision as of 09:15, 25 August 2018

Problem

Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?

$\textbf{(A)}\ 2.0\qquad\textbf{(B)}\ 2.2\qquad\textbf{(C)}\ 2.8\qquad\textbf{(D)}\ 3.4\qquad\textbf{(E)}\ 4.4$

Solution 1

Let's call the distance that Samia had to travel in total as $2x$, so that we can avoid fractions. We know that the length of the bike ride and how far she walked are equal, so they are both $\frac{2x}{2}$, or $x$. \[\] She bikes at a rate of $17$ kph, so she travels the distance she bikes in $\frac{x}{17}$ hours. She walks at a rate of $5$ kph, so she travels the distance she walks in $\frac{x}{5}$ hours. \[\] The total time is $\frac{x}{17}+\frac{x}{5} = \frac{22x}{85}$. This is equal to $\frac{44}{60} = \frac{11}{15}$ of an hour. Solving for $x$, we have: \[\] \[\frac{22x}{85} = \frac{11}{15}\] \[\frac{2x}{85} = \frac{1}{15}\] \[30x = 85\] \[6x = 17\] \[x = \frac{17}{6}\] \[\] Since $x$ is the distance of how far Samia traveled by both walking and biking, and we want to know how far Samia walked to the nearest tenth, we have that Samia walked about $\boxed{\bold{(C)} 2.8}$.

Solution 2 (Quicker, Make Approximations)

Notice that Samia walks $\frac {17}{5}$ times slower than she bikes, and that she walks and bikes the same distance. Thus, the fraction of the total time that Samia will be walking is \[\frac{\frac{17}{5}}{\frac{17}{5}+\frac{5}{5}} = \frac{17}{22}\] Then, multiply this by the time: \[\frac{17}{22} \cdot 44 \text{minutes} = 34 \text{minutes}\] 34 minutes is a little greater than $\frac {1}{2}$ of an hour so Samia traveled \[\sim \frac {1}{2} \cdot 5 = 2.5 \text{kilometers}\] The answer choice a little greater than 2.5 is $\boxed{\bold{(C)} 2.8}$. (Note that we could've multiplied $\frac {34}{60}=\frac {17}{30}$ by $5$ and gotten the exact answer as well)

See Also

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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
2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 5
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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