uc26 wrote:
Bunuel wrote:
Official Solution:
A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?
A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes
Let \(V_b\) denote the speed of the bus and \(V_c\) the speed of the cyclist. Then the distance between the buses is \(4(V_b + V_c) = 12(V_b - V_c)\) from where \(V_b = 2V_c\).
The interval between the buses is the ratio of the distance between the buses to the speed of the bus, or \(4\frac{V_b + V_c}{V_b} = 4\frac{\frac{3}{2}V_b}{V_b} = 6\) minutes.
Alternative Explanation
Let's say the distance between the buses is \(d\). We want to determine \(\text{Interval}=\frac{d}{b}\), where \(b\) is the speed of bus.
Let the speed of cyclist be \(c\).
Every 12 minutes a bus overtakes cyclist: \(\frac{d}{b-c}=12\), so \(d=12b-12c\);
Every 4 minutes cyclist meets an oncoming bus: \(\frac{d}{b+c}=4\), so \(d=4b+4c\);
From above we have that \(d=12b-12c=4b+4c\). Next, \(12b-12c=4b+4c\) simplifies to \(b=2c\), hence \(d=12b-12c=12b-6b=6b\).
\(\text{Interval}=\frac{d}{b}=\frac{6b}{b}=6\)
Answer: B
Hi
Bunuel,
I am having a hard time understand the above highlighted portion. My main concern is how is d used here in these two highlighted equations equal to the distance between the buses. As per my knowledge, in the first equation where we have a bus overtaking the cyclist every 12 mins, d1 (which is what i assumed) would mean the distance the overtaking bus covers to reach the cyclist. Likewise, in the second equation, where we have an oncoming bus every 4 mins, d2 would mean the distance the oncoming bus covers to reach the cyclist. How is d1 and d2 here the distance between the buses. In my head, d1 and d2 are the distances between the overtaking bus and the cyclist and between the oncoming bus and the cyclist respectively.
I hope i have been able to explain my doubt clearly. Please let me your thoughts?
Thanks.
Hello uc26
This is a tricky Problem Solving question with a tricky looking answer.
From your query I find you are having issues visualising the situation.
Please note that d is the distance between buses which are plying on the road in a particular direction. This is the same for incoming and overtaking buses.
The way you visualised it has another problem. The question does not talk about the distance between the points from where overtaking and oncoming buses start. Thus, you won't be able to correlate d1 and d2.
I hope you find this useful. I agree that the text of the question could be better worded.
Let me know if you have other questions. All the best
_________________
GMAT 760- V41 Q51; 620 to 760, V27 to V41
Kellogg admit- Class of 2023