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Difference between revisions of "2008 AMC 10A Problems/Problem 6"
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Thus, the average speed is <math>3d\div\left(\frac{29d}{60}\right)=\frac{180}{29}</math>. This is closest to <math>6</math>, so the answer is <math>\mathrm{(D)}</math>.
 
Thus, the average speed is <math>3d\div\left(\frac{29d}{60}\right)=\frac{180}{29}</math>. This is closest to <math>6</math>, so the answer is <math>\mathrm{(D)}</math>.
 +
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==Solution 2==
 +
Since the three segments are all the same length, the triathlete's average speed is the [[harmonic mean]] of the three given rates. Therefore, the average speed is <cmath>\frac{3}{\frac{1}{3}+\frac{1}{120}+\frac{1}{10}}=\frac{3}{\frac{29}{60}}=\frac{180}{29}\approx6\Rightarrow\boxed{\mathrm{(D)}\ 6}</cmath>.
 
   
 
   
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=A|num-b=5|num-a=7}}
 
{{AMC10 box|year=2008|ab=A|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:05, 29 December 2019

Problem

A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?

$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 5\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 7$

Solution

Let $d$ be the length of one segment of the race.

Average speed is total distance divided by total time. The total distance is $3d$, and the total time is $\frac{d}{3}+\frac{d}{20}+\frac{d}{10}=\frac{29d}{60}$.

Thus, the average speed is $3d\div\left(\frac{29d}{60}\right)=\frac{180}{29}$. This is closest to $6$, so the answer is $\mathrm{(D)}$.

Solution 2

Since the three segments are all the same length, the triathlete's average speed is the harmonic mean of the three given rates. Therefore, the average speed is \[\frac{3}{\frac{1}{3}+\frac{1}{120}+\frac{1}{10}}=\frac{3}{\frac{29}{60}}=\frac{180}{29}\approx6\Rightarrow\boxed{\mathrm{(D)}\ 6}\].

See also

2008 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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All AMC 10 Problems and Solutions

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