Difference between revisions of "2017 AMC 10B Problems/Problem 7"
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| − | <math>\boxed{\ | + | Let's call the distance that Samia had to travel in total as <math>2x</math>, 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 <math>\frac{2x}{2}</math>, or <math>x</math>. |
| − | + | <cmath></cmath> | |
| + | She bikes at a rate of <math>17</math> kph, so she travels the distance she bikes in <math>\frac{x}{17}</math> hours. She walks at a rate of <math>5</math> kph, so she travels the distance she walks in <math>\frac{x}{5}</math> hours. | ||
| + | <cmath></cmath> | ||
| + | The total time is <math>\frac{x}{17}+\frac{x}{5} = \frac{22x}{85}</math>. This is equal to <math>\frac{44}{60} = \frac{11}{15}</math> of an hour. Solving for <math>x</math>, we have: | ||
| + | <cmath></cmath> | ||
| + | <cmath>\frac{22x}{85} = \frac{11}{15}</cmath> | ||
| + | <cmath>\frac{2x}{85} = \frac{1}{15}</cmath> | ||
| + | <cmath>30x = 85</cmath> | ||
| + | <cmath>6x = 17</cmath> | ||
| + | <cmath>x = \frac{17}{6}</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>. | ||
{{AMC10 box|year=2017|ab=B|num-b=6|num-a=8}} | {{AMC10 box|year=2017|ab=B|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 15:18, 16 February 2017
Problem
Samia set off on her bicycle to visit her friend, traveling at an average speed of 
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 
kilometers per hour. In all it took her 
minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
Solution
Let's call the distance that Samia had to travel in total as 
, 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 
, or 
.
![\[\]](http://latex.artofproblemsolving.com/5/7/f/57f1406b65f9f2e6a365b7356da731780b42cceb.png)
She bikes at a rate of kph, so she travels the distance she bikes in 
hours. She walks at a rate of kph, so she travels the distance she walks in 
hours.
The total time is 
. This is equal to 
of an hour. Solving for , we have:
![\[\frac{22x}{85} = \frac{11}{15}\]](http://latex.artofproblemsolving.com/1/0/4/1047df3d263df2f4101416c6bbd2350f5b3fe49f.png)
![\[\frac{2x}{85} = \frac{1}{15}\]](http://latex.artofproblemsolving.com/a/b/e/abea6c21f569b9e8506517f21781841f11a2cf64.png)
![\[30x = 85\]](http://latex.artofproblemsolving.com/7/d/d/7ddd1cf0135652196e90aa16b630babae2eb6214.png)
![\[6x = 17\]](http://latex.artofproblemsolving.com/6/0/8/608bd4ededb3cc5b4bdfb775e9ac91ab4633320a.png)
![\[x = \frac{17}{6}\]](http://latex.artofproblemsolving.com/d/9/3/d9360642a7e793f542984bb113dba97831767793.png)
Since 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 
.
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
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| All AMC 10 Problems and Solutions | ||
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