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Difference between revisions of "2021 WSMO Accuracy Round Problems/Problem 4"

(Created page with "==Problem 4== A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minut...")
 
 
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A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of <math>m,</math> the tip of the second hand travels a distance of <math>s,</math> and the tip of the hour hand travels a distance of <math>h.</math> The value of <math>\frac{m^2}{hs}</math> can be expressed as <math>\frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
 
A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of <math>m,</math> the tip of the second hand travels a distance of <math>s,</math> and the tip of the hour hand travels a distance of <math>h.</math> The value of <math>\frac{m^2}{hs}</math> can be expressed as <math>\frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
  
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==Solution==
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==Solution 1==
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WLOG, assume that the length of the minute hand is 2. This means that the length of the second hand is 2 and the length of the hour hand is 1. In a day, there are 24 hours, which means that the minute hand travels <math>24\cdot4\pi</math> each day. Also, since there are <math>24=2\cdot12</math> hours, the hour hand travels <math>2\cdot2\pi=4\pi</math> each day. Finally, there are <math>24\cdot60</math> minutes in a day, which means that the second hand travels <math>24\cdot60\cdot4\pi</math> each day. Thus, the final answer is <cmath>\frac{(24\cdot4\pi)\cdot(24\cdot4\pi)}{(4\pi)\cdot(24\cdot60\cdot4\pi)}=\frac{(\cancel{24}\cdot\cancel{4\pi})\cdot(24\cdot\cancel{4\pi})}{(\cancel{4\pi})\cdot(\cancel{24}\cdot60\cdot\cancel{4\pi})}=\frac{24}{60}=\frac{2}{5}\Longrightarrow 2+5=\boxed{7}.</cmath>
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~pinkpig
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==Solution 2==
 
Let the distance traveled by one revolution of the minute hand tip be <math>C</math>. Note that <math>m=24C</math>, <math>s = 60 \cdot 24 C</math>, and <math>h=2(\frac{C}{2}) = C</math>. Our desired expression becomes:
 
Let the distance traveled by one revolution of the minute hand tip be <math>C</math>. Note that <math>m=24C</math>, <math>s = 60 \cdot 24 C</math>, and <math>h=2(\frac{C}{2}) = C</math>. Our desired expression becomes:
 
<cmath>\frac{(24C)^2}{(C)(60 \cdot 24 C)}</cmath>
 
<cmath>\frac{(24C)^2}{(C)(60 \cdot 24 C)}</cmath>

Latest revision as of 10:10, 11 July 2022

Problem 4

A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of $m,$ the tip of the second hand travels a distance of $s,$ and the tip of the hour hand travels a distance of $h.$ The value of $\frac{m^2}{hs}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Solution 1

WLOG, assume that the length of the minute hand is 2. This means that the length of the second hand is 2 and the length of the hour hand is 1. In a day, there are 24 hours, which means that the minute hand travels $24\cdot4\pi$ each day. Also, since there are $24=2\cdot12$ hours, the hour hand travels $2\cdot2\pi=4\pi$ each day. Finally, there are $24\cdot60$ minutes in a day, which means that the second hand travels $24\cdot60\cdot4\pi$ each day. Thus, the final answer is \[\frac{(24\cdot4\pi)\cdot(24\cdot4\pi)}{(4\pi)\cdot(24\cdot60\cdot4\pi)}=\frac{(\cancel{24}\cdot\cancel{4\pi})\cdot(24\cdot\cancel{4\pi})}{(\cancel{4\pi})\cdot(\cancel{24}\cdot60\cdot\cancel{4\pi})}=\frac{24}{60}=\frac{2}{5}\Longrightarrow 2+5=\boxed{7}.\] ~pinkpig

Solution 2

Let the distance traveled by one revolution of the minute hand tip be $C$. Note that $m=24C$, $s = 60 \cdot 24 C$, and $h=2(\frac{C}{2}) = C$. Our desired expression becomes: \[\frac{(24C)^2}{(C)(60 \cdot 24 C)}\] \[=\frac{24^2}{60 \cdot 24}\] \[=\frac{2}{5}\]

This gives us an answer of $2 + 5 = \boxed{7}$.

~BigKahuna227

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