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16 Sep 2014, 00:37
Kudos
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A man cycling along a road at a constant speed observes that a bus overtakes him every 12 minutes, while he meets an oncoming bus every 4 minutes. If the distance between each pair of consecutive buses, both those traveling in the same direction and those coming from the opposite direction, is the same, and the buses travel at the same constant speed, what is the time interval between two consecutive buses going in the same direction?
A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes
A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes
16 Sep 2014, 00:37
Kudos
Bookmarks
Official Solution:
A man cycling along a road at a constant speed observes that a bus overtakes him every 12 minutes, while he meets an oncoming bus every 4 minutes. If the distance between each pair of consecutive buses, both those traveling in the same direction and those coming from the opposite direction, is the same, and the buses travel at the same constant speed, what is the time interval between two consecutive buses going in the same direction?
A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes
Let's denote the distance between consecutive buses going in the same direction as \(d\) and the speed of the buses as \(b\). Our goal is to determine the time interval between two consecutive buses going in the same direction, which is equal to \(\text{time interval}=\frac{distance}{speed} =\frac{d}{b}\).
Let the speed of the cyclist be \(c\).
Every 12 minutes a bus overtakes the cyclist. When two objects move in the same direction, their relative speed is the difference between their individual speeds thus \(time= \frac{distance}{relative \ speed} =\frac{d}{b-c}=12\) minutes, which implies \(d=12(b-c)\).
Every 4 minutes the cyclist meets an oncoming bus. When two objects move in the opposite direction, their relative speed is the sum of their individual speeds thus \(time=\frac{distance}{relative \ speed} =\frac{d}{b+c}=4\) minutes, which implies \(d=4(b+c)\).
From the above equations, we have \(d=12(b-c)=4(b+c)\). Simplifying the equation, \(12b-12c=4b+4c\), we get \(b=2c\). Thus, \(d=12(b-c)=12(2c-c)=6b\).
Therefore, \(\text{time interval}=\frac{d}{b}=\frac{6b}{b}=6\).
Answer: B
A man cycling along a road at a constant speed observes that a bus overtakes him every 12 minutes, while he meets an oncoming bus every 4 minutes. If the distance between each pair of consecutive buses, both those traveling in the same direction and those coming from the opposite direction, is the same, and the buses travel at the same constant speed, what is the time interval between two consecutive buses going in the same direction?
A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes
Let's denote the distance between consecutive buses going in the same direction as \(d\) and the speed of the buses as \(b\). Our goal is to determine the time interval between two consecutive buses going in the same direction, which is equal to \(\text{time interval}=\frac{distance}{speed} =\frac{d}{b}\).
Let the speed of the cyclist be \(c\).
Every 12 minutes a bus overtakes the cyclist. When two objects move in the same direction, their relative speed is the difference between their individual speeds thus \(time= \frac{distance}{relative \ speed} =\frac{d}{b-c}=12\) minutes, which implies \(d=12(b-c)\).
Every 4 minutes the cyclist meets an oncoming bus. When two objects move in the opposite direction, their relative speed is the sum of their individual speeds thus \(time=\frac{distance}{relative \ speed} =\frac{d}{b+c}=4\) minutes, which implies \(d=4(b+c)\).
From the above equations, we have \(d=12(b-c)=4(b+c)\). Simplifying the equation, \(12b-12c=4b+4c\), we get \(b=2c\). Thus, \(d=12(b-c)=12(2c-c)=6b\).
Therefore, \(\text{time interval}=\frac{d}{b}=\frac{6b}{b}=6\).
Answer: B
General Discussion
31 May 2017, 07:48
Kudos
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I think this is a high-quality question and I agree with explanation.
