Would Tripping Lose Boris The Race?

Probability Level 3

Alfred and Boris have a $70$ km race. Alfred runs at a constant velocity of $7$ km h $^{-1}$ , while Boris runs at a constant velocity of 10 km h $^{-1}$ . However, during the contest, Boris trips and subsequently runs at a constant velocity of 5 km h $^{-1}$ as his knee hurts.

We do not know when Boris tripped. If we assume that the position that Boris trips is random, then the probability that Alfred won the race can be expressed as $\frac ab$ where $a$ and $b$ are positive co-prime integers. Find the value of $a+b$

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The answer is 11.

1 solution

Nanayaranaraknas Vahdam
Apr 29, 2014

Since, Alfred's speed is constant throughout the race, it is known that he finishes the race in $\frac{70}{7}= 10$ hours. Let the distance from the starting line where Boris tripped be $x$ . Thus, the time taken by Boris is given by $\frac{x}{10} + \frac{70-x}{5}$ $=$ $14 - \frac{x}{10}$ . Now, for Alfred to win the race, Boris's time must be greater.

Thus, $14 -\frac{x}{10} > 10$ $\Rightarrow$ $x < 40$

So, $x \in ]0,40[$ The size of the interval compared to $]0,70[$ is given by $\frac{4}{7}$ . Thus, the answer is $\boxed{11}$

The answer is given comparing distance. I think the answer will be comparing time and the answer will be 7.

Yunus Ali - 7 years, 1 month ago

However, time is not constant for Boris. Thus, time cannot be used as an interval for when he trips.

Nanayaranaraknas Vahdam - 7 years, 1 month ago

I had the same question. what do you mean by 'time is not constant fro boris?'

ridima hasan - 7 years, 1 month ago

@Ridima Hasan – The time is not constant because it depends on the distance at which Boris trips. For example, if Boris trips at the beginning, the time taken is much longer than if he trips at the end. Thus, if time is taken, it is not constant and calculations are cumbersome, but ultimately boil down to distance intervals.

Nanayaranaraknas Vahdam - 7 years, 1 month ago

i did the same exact way

John Samuel - 7 years, 1 month ago

I m finding it difficult to understand (14-x/10) part....can anybudy please explain.? The question is awesome though!

Wrushabh Warshe - 6 years, 11 months ago

(x/10) - ((70-x) / 10) on simplifying will give you 14 - x/10

Anubhav Balodhi - 6 years, 5 months ago

great, I used the exact same approach ^_^

Anubhav Balodhi - 6 years, 5 months ago

Would Tripping Lose Boris The Race?

Image credit : Wikipedia Faud

The answer is 11.

1 solution

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