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Difference between revisions of "2003 AMC 12A Problems/Problem 4"

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{{duplicate|[[2003 AMC 12A Problems|2003 AMC 12A #4]] and [[2003 AMC 10A Problems|2003 AMC 10A #4]]}}
 
== Problem ==
 
== Problem ==
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It takes Mary <math>30</math> minutes to walk uphill <math>1</math> km from her home to school, but it takes her only <math>10</math> minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?  
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It takes Anna <math>30</math> minutes to walk uphill <math>1</math> km from her home to school, but it takes her only <math>10</math> minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?  
  
 
<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5 </math>
 
<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5 </math>
  
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== Solution ==
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==Solution 1==
 
Since she walked <math>1</math> km to school and <math>1</math> km back home, her total distance is <math>1+1=2</math> km.  
 
Since she walked <math>1</math> km to school and <math>1</math> km back home, her total distance is <math>1+1=2</math> km.  
  
 
Since she spent <math>30</math> minutes walking to school and <math>10</math> minutes walking back home, her total time is <math>30+10=40</math> minutes = <math>\frac{40}{60}=\frac{2}{3}</math> hours.  
 
Since she spent <math>30</math> minutes walking to school and <math>10</math> minutes walking back home, her total time is <math>30+10=40</math> minutes = <math>\frac{40}{60}=\frac{2}{3}</math> hours.  
  
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Therefore her average speed in km/hr is <math>\frac{2}{\frac{2}{3}}=3 \Rightarrow A</math>
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Therefore her average speed in km/hr is <math>\frac{2}{\frac{2}{3}}=\boxed{\mathrm{(A)}\ 3}</math>.
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==Solution 2==
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The average speed of two speeds that travel the same distance is the [[harmonic mean]] of the speeds, or <math>\dfrac{2}{\dfrac{1}{x}+\dfrac{1}{y}}=\dfrac{2xy}{x+y}</math> (for speeds <math>x</math> and <math>y</math>). Mary's speed going to school is <math>2\,\text{km/hr}</math>, and her speed coming back is <math>6\,\text{km/hr}</math>. Plugging the numbers in, we get that the average speed is <math>\dfrac{2\times 6\times 2}{6+2}=\dfrac{24}{8}=\boxed{\mathrm{(A)}\ 3}</math>.
  
 
== See Also ==
 
== See Also ==
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*[[2003 AMC 12A Problems]]
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{{AMC10 box|year=2003|ab=A|num-b=3|num-a=5}}
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*[[2003 AMC 12A/Problem 3|Previous Problem]]
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{{AMC12 box|year=2003|ab=A|num-b=3|num-a=5}}
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*[[2003 AMC 12A/Problem 5|Next Problem]]
 
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 03:55, 7 November 2020

The following problem is from both the 2003 AMC 12A #4 and 2003 AMC 10A #4, so both problems redirect to this page.

Problem

It takes Anna $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?

$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5$

Solution 1

Since she walked $1$ km to school and $1$ km back home, her total distance is $1+1=2$ km.

Since she spent $30$ minutes walking to school and $10$ minutes walking back home, her total time is $30+10=40$ minutes = $\frac{40}{60}=\frac{2}{3}$ hours.

Therefore her average speed in km/hr is $\frac{2}{\frac{2}{3}}=\boxed{\mathrm{(A)}\ 3}$.

Solution 2

The average speed of two speeds that travel the same distance is the harmonic mean of the speeds, or $\dfrac{2}{\dfrac{1}{x}+\dfrac{1}{y}}=\dfrac{2xy}{x+y}$ (for speeds $x$ and $y$). Mary's speed going to school is $2\,\text{km/hr}$, and her speed coming back is $6\,\text{km/hr}$. Plugging the numbers in, we get that the average speed is $\dfrac{2\times 6\times 2}{6+2}=\dfrac{24}{8}=\boxed{\mathrm{(A)}\ 3}$.

See Also

2003 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		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 10 Problems and Solutions
2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		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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