https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Jing.han&feedformat=atom AoPS Wiki - User contributions [en] 2022-01-27T11:14:41Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2022_AIME_I_Problems/Problem_1&diff=150682 2022 AIME I Problems/Problem 1 2021-03-31T20:22:50Z <p>Jing.han: Created page with &quot;Don't Cheat Kids&quot;</p> <hr /> <div>Don't Cheat Kids</div> Jing.han https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_8_Problems/Problem_14&diff=111228 2009 AMC 8 Problems/Problem 14 2019-11-11T23:37:35Z <p>Jing.han: /* See Also */</p> <hr /> <div>==Problem==<br /> <br /> Austin and Temple are &lt;math&gt; 50&lt;/math&gt; miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging &lt;math&gt; 60&lt;/math&gt; miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged &lt;math&gt; 40&lt;/math&gt; miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 46 \qquad<br /> \textbf{(B)}\ 48 \qquad<br /> \textbf{(C)}\ 50 \qquad<br /> \textbf{(D)}\ 52 \qquad<br /> \textbf{(E)}\ 54&lt;/math&gt;<br /> <br /> ==Solution==<br /> The way to Temple took &lt;math&gt;\frac{50}{60}=\frac56&lt;/math&gt; hours, and the way back took &lt;math&gt;\frac{50}{40}=\frac54&lt;/math&gt; for a total of &lt;math&gt;\frac56 + \frac54 = \frac{25}{12}&lt;/math&gt; hours. The trip is &lt;math&gt;50\cdot2=100&lt;/math&gt; miles. The average speed is &lt;math&gt;\frac{100}{25/12} = \boxed{\textbf{(B)}\ 48}&lt;/math&gt; miles per hour.<br /> <br /> ==Solution 2==<br /> This question simply asks for the [[harmonic mean]] of &lt;math&gt;60&lt;/math&gt; and &lt;math&gt;40&lt;/math&gt;, regardless of how far Austin and Temple are. <br /> Plugging in, we have: &lt;math&gt;\frac{2ab}{a+b} = \frac{2 \cdot 60 \cdot 40}{60 + 40} = \frac{4800}{100} = \boxed{\textbf{(B)}\ 48}&lt;/math&gt; miles per hour.<br /> <br /> ==See Also==<br /> 2015 Problem 17<br /> {{AMC8 box|year=2009|num-b=13|num-a=15}}<br /> {{MAA Notice}}</div> Jing.han