Difference between revisions of "2021 WSMO Accuracy Round Problems/Problem 4"
Bigkahuna227 (talk | contribs) (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...") |
|||
Line 2: | Line 2: | ||
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>. | ||
− | ==Solution== | + | ==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 <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> | ||
+ | ~pinkpig | ||
+ | |||
+ | ==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 the tip of the second hand travels a distance of
and the tip of the hour hand travels a distance of
The value of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
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 each day. Also, since there are
hours, the hour hand travels
each day. Finally, there are
minutes in a day, which means that the second hand travels
each day. Thus, the final answer is
~pinkpig
Solution 2
Let the distance traveled by one revolution of the minute hand tip be . Note that
,
, and
. Our desired expression becomes:
This gives us an answer of .
~BigKahuna227