Two men, A and B run a 4 km race on a course 0.25 km round. If their speeds are in the ratio 5 : 4, how often does the winner pass the other?
Thrice
Analyzing the Circular Race Problem
This problem involves two runners, A and B, on a circular track. We are given the total race distance, the length of one lap, and the ratio of their speeds. We need to determine how many times the faster runner (A) passes the slower runner (B) during the race.
Understanding Passing in a Circular Race
In a circular race, one runner passes another when the faster runner completes exactly one more lap than the slower runner. This event occurs every time the relative distance between them is an integer multiple of the track length. The winner passes the other runner when they lap them.
Race Details and Calculations
The race is 4 km long, and the circular track is 0.25 km round. The speeds of A and B are in the ratio 5:4. Since the speed ratio is 5:4, A is faster and will be the winner.
First, let's find the total number of laps the winner completes.
Total laps = \(\frac{\text{Total Race Distance}}{\text{Length of one lap}}\)
Total laps = \(\frac{4 \text{ km}}{0.25 \text{ km/lap}} = \frac{4}{1/4} = 4 \times 4 = 16 \text{ laps}\).
So, runner A (the winner) completes 16 laps to finish the race.
Let the speeds of A and B be \(5v\) and \(4v\) respectively, where \(v\) is a unit of speed. The time taken by A to complete the race (16 laps or 4 km) is given by:
Time (T) = \(\frac{\text{Distance}}{\text{Speed}}\)
\(T = \frac{4 \text{ km}}{5v}\)
In this same time \(T\), runner B covers a distance of:
Distance covered by B = Speed of B \(\times\) Time
Distance covered by B = \(4v \times T = 4v \times \frac{4}{5v} = \frac{16}{5} \text{ km}\).
Now, let's find out how many laps B completes in this time:
Laps covered by B = \(\frac{\text{Distance covered by B}}{\text{Length of one lap}}\)
Laps covered by B = \(\frac{16/5 \text{ km}}{0.25 \text{ km/lap}} = \frac{16/5}{1/4} = \frac{16}{5} \times 4 = \frac{64}{5} = 12.8 \text{ laps}\).
So, when A finishes the race having completed 16 laps, B has completed 12.8 laps.
Determining the Number of Passes
A passes B every time A gains a full lap on B. The number of times A passes B is equal to the total number of full laps A has gained on B by the time A finishes the race.
Number of laps gained by A over B = Laps completed by A - Laps completed by B
Number of laps gained = \(16 \text{ laps} - 12.8 \text{ laps} = 3.2 \text{ laps}\).
Since A gains 3.2 laps on B by the time A finishes, A has completed gaining 1 full lap, 2 full laps, and 3 full laps. A is 0.2 laps into gaining the 4th lap when the race ends for A.
Therefore, A passes B 3 times during the race.
Alternatively, we can consider the relative speed. The relative speed of A with respect to B is \(5v - 4v = v\). The time it takes for A to gain one lap (0.25 km) on B is:
Time to gain 1 lap = \(\frac{\text{Distance of one lap}}{\text{Relative speed}} = \frac{0.25 \text{ km}}{v}\).
The total time of the race is \(T = \frac{4 \text{ km}}{5v}\).
The number of times A passes B is the total race time divided by the time it takes A to gain one lap:
Number of passes = \(\frac{\text{Total Race Time}}{\text{Time to gain 1 lap}} = \frac{4/(5v)}{0.25/v} = \frac{4}{5v} \times \frac{v}{0.25} = \frac{4}{5 \times 0.25} = \frac{4}{1.25} = \frac{4}{5/4} = 4 \times \frac{4}{5} = \frac{16}{5} = 3.2\).
Again, the number of times A fully laps B is the integer part of 3.2, which is 3.
The winner passes the other runner Thrice.
| Parameter | Value |
|---|---|
| Total Race Distance | 4 km |
| Length of one lap | 0.25 km |
| Speed Ratio (A:B) | 5:4 |
| Total Laps (Winner A) | 16 laps |
| Laps by B when A finishes | 12.8 laps |
| Laps Gained by A | 3.2 laps |
| Number of Passes | 3 (integer part of laps gained) |
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Circular Race Passing | Faster runner gains full laps on slower runner. |
| Laps Calculation | Total Distance / Lap Distance. |
| Speed Ratio | Ratio used to determine relative speeds and distances covered in the same time. |
| Relative Speed | Difference in speeds; useful for calculating time to gain a certain distance (like a lap). |
Additional Information: Relative Speed in Race Problems
Relative speed is a crucial concept in problems involving objects moving relative to each other. In a race on a circular track where runners move in the same direction, the relative speed is the difference between their speeds (\(v_A - v_B\)). The faster runner gains distance on the slower runner at this relative speed. To find how often the faster runner laps the slower one, you calculate the time it takes to gain one lap distance using the relative speed, and then see how many such time intervals fit into the total race time.
For example, if the relative speed is \(v_{rel}\) and the lap distance is \(L\), the time to gain one lap is \(T_{lap\_gain} = L / v_{rel}\). If the total race time is \(T_{race}\), the number of times the faster runner laps the slower one is approximately \(T_{race} / T_{lap\_gain}\), taking the integer part for the number of complete passes.
Similar Questions
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Important Questions from Relative Speed
In a circular race of 2500 m, a man and a woman start from a point towards opposite directions with speeds of 37 km/h and 35 km/h, respectively. After how much time from the start of the race will they meet for the first time?
- X and Yrun a 3 km race along a circular course of length 300 m. Their speeds are in the ratio 3 : 2. If they start together in the same direction, how many times would the first one pass the other (the start-off is not counted as passing)?
A train of length 600 meters passes another train of length 1000 meters moving in the opposite direction in 96 seconds. If the speed of both the trains is the same, then what is the speed of each train?
The distance between Delhi and Patna is $480$ km. A train starts from Delhi and travels towards Patna at a speed of $60$ kmph. Another train starts from Patna towards Delhi at a speed of $80$ kmph, but starts $1$ hour after the first train. In how much time, after the first train started, will they meet each other?
Ram traveling at the speed of 3 km/h reaches 15 minutes late. Had he walked at 4 km/h, he would have reached 15 minutes earlier. How much distance does Ram have to cover?
