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Two teams compete in a relay race, they're composed of 4 members, each of whom runs 100 meters. All of the members of team B runs at $b$ meters per second (mps). The first 3 members of team A can only run at 5$b$/6 (mps) while the 4th member runs at $a$ mps. Find $a$/$b$ if the race is a tie.

gt6989b
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Anthony
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2 Answers2

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The total time it takes team A to run the race is $$T_A=\frac{300 \text{ m}}{5b/6 \text{ ms}^{-1}}+\frac{100\text{ m}}{a \text{ ms}^{-1}}$$ The time it takes team B to run the race is $$T_B=\frac{400\text{ m}}{b\text{ ms}^{-1}}$$ Therefore, $$T_A=T_B \implies \frac{a}{b}=\frac{5}{2}$$ EDIT: My workings - $$\frac{300}{5b/6} + \frac{100}{a} = \frac{400}{b}$$ $$\frac{18}{5b} + \frac{1}{a} = \frac{4}{b}$$ $$\frac{18}{b} + \frac{5}{a}=\frac{20}{b}$$ $$\frac{2}{b}=\frac{5}{a}$$ It's straightforward from here.

K.defaoite
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  • Sorry, there was an error in my previous post. $a/b =1/6$ is wrong. The above answer is now correct. – K.defaoite Jun 11 '20 at 01:04
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    It's important to note that the $\implies$ conceals some simplification. – dan Jun 11 '20 at 01:19
  • Ok, I'll add my workings. – K.defaoite Jun 11 '20 at 01:20
  • "The team is composed of 4 members, each of whom runs 100 meters." – K.defaoite Jun 11 '20 at 01:37
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    @K. defaoite. You're right. It makes more practical sense, too. I'll delete my comment. – Deepak Jun 11 '20 at 01:46
  • Each of them runs 100m so why isn't it correct to write that the time team B takes is = 4 * (100/b) and team A = 3 * (100/5b/6) + 100/a ? Please help me understand – Anthony Jun 11 '20 at 15:28
  • @LazyJoeShow This is equivalent to what I wrote. All I did was multiply the 3 and the 4 into the fractions you wrote above. – K.defaoite Jun 11 '20 at 15:35
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    Oh right sorry I meant 300/3(5b/6) since there are 3 members with that speed and 400/4(b) since there are 4 members , thanks for taking the time to explain! – Anthony Jun 11 '20 at 15:46
  • What I've done is let d/r = d/r since it's a tie where d is the same 100m left with 1/r = 1/r team B combined rate = 4b while team A combined rate = 3* (5b/6) + a , solving this I get a/b = 3/2 . @K.defaoite please tell me whats wrong I appreciate it – Anthony Jun 11 '20 at 15:50
  • Just because the race is a tie doesn't mean both teams have to run the last 100 meters in the same amount of time. I could let Usain Bolt start running a 100 meter race and then wait till he's 0.0000035 meters from the finish line and then shine a beam of light at the start and it would still be a tie. – K.defaoite Jun 11 '20 at 15:55
  • If each of team B needs to cover 100m so total distance will be 400m and there are 4 members, so shouldn't the time be 400/4b? I think you haven't clarified my question enough or I'm just too dumb? – Anthony Jun 11 '20 at 16:20
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    Each member of team B runs 100 meters at a rate of $b$ meters per second. So the time it takes one member to run his portion is $\frac{100}{b}\text{s}$. If the team has $4$ members, then this needs to be repeated $4$ times, so the total time is $4\times \frac{100}{b}\text{s}=\frac{400}{b}\text{s}$, not $\frac{400}{4b}\text{s}$. – K.defaoite Jun 11 '20 at 16:29
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The total time for the tie race is

$$ \frac Sb = \frac {\frac {3S}4}{\frac56b}+ \frac{\frac S4}a $$

where $S$ is the total distance that cancels. Solve to obtain $\frac ab = \frac52$.

Quanto
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