Races
The topic of races in CAT (Common Admission Test) typically falls under the category of “Time, Speed, and Distance.” It’s a popular type of problem where competitors, usually running on a track or swimming in a pool, race against each other. Here’s an explanation of the key concepts you need to understand for solving race problems:
- Basics of Time, Speed, and Distance:
- Distance = Speed × Time
- When two or more entities (people, vehicles, etc.) are moving with different speeds, their relative speed comes into play.
- Types of Races:
- Simple Race: Two or more participants start from the same point and race towards a finish line. The one who reaches first wins.
- Head Start: One participant is given an advantage by starting ahead of the other(s). The challenge is often to calculate the time or distance by which one competitor would win or lose.
- Race with Equal Speeds: If two participants have the same speed, they will finish the race together if they start at the same time.
- Key Formulas and Concepts:
- Time Difference: If one participant wins by a certain time, you can calculate the time difference using the formula:
Time Difference =
- Distance Difference: If a participant wins by a certain distance, the distance difference can be calculated as: Distance Difference=Speed×Time Difference
- Winning Margin: In a race, the margin by which one participant wins over another can be in terms of time or distance.
- Relative Speed:
- When two participants are moving in the same direction, their relative speed is the difference between their speeds.
- When two participants are moving in opposite directions, their relative speed is the sum of their speeds.
- Race Problems with Head Start:
- If one participant starts a certain distance ahead, you can calculate the time or distance by which another participant would need to catch up or how much faster they need to be.
- Practice Tips:
- Understand the Problem Statement: Carefully read the problem to understand what is being asked.
- Identify Given Values: Note down the speeds, distances, head starts, etc., provided in the problem.
- Use the Right Formula: Choose the appropriate formula based on whether the question asks for time, distance, or speed.
- Practice Different Variations: Problems can vary, so practice different types to build familiarity with the various scenarios.
SOLVED EXAMPLES
- In a 200m race, if A gives B a start of 25 metres, then A wins the race by 10 seconds. Alternatively, if A gives B a start of 45 metres the race ends in a dead heat. How long does A take to run 200m?
- 100 seconds
- 112.5 seconds
- 77.5 seconds
- 87.5 seconds
Correct Answer – 77.5 seconds. Choice (3)
Explanatory Answer
A gives B a start of 25 metres and still wins the race by 10 seconds.
Alternatively, if A gives B a start of 45 metres, then the race ends in a dead heat.
Therefore, the additional 20 metres start given to B compensates for the 10 seconds.
i.e., B runs 20 metres in 10 seconds.
Hence, B will take 100 seconds to run 200 metres.
We know that A gives B a start of 45 metres. B will take 22.5 seconds to run this 45 metres as B runs 20 metres in 10 seconds or at the speed of 2 m/s.
Hence, A will take 22.5 seconds lesser than B or 100 – 22.5 = 77.5 seconds to complete the race.
- In a kilometre race, A can give B a start of 100 m or 15 seconds. How long does A take to complete the race?
- 150 seconds
- 165 seconds
- 135 seconds
- 66.67 seconds
- 180 seconds
Correct Choice is (3). Correct answer 135 seconds.
Explanatory Answer
In a 1000 metre race A gives B a start of 100 m or 15 seconds.
This essentially means that B takes 15 seconds to run 100 m.
Therefore, B will take 150 seconds to run the stretch of 1000 metres. (1000 m = 10 times 100 m and therefore the time taken will also be 10 times 15 seconds = 150 seconds).
As A gives B a start of 15 seconds, A obviously takes 15 seconds less than B to complete the race.
Hence, A will take 135 seconds to run the 1000 m.
- A gives B a start of 10 metres in a 100 metre race and still beats him by 1.25 seconds. How long does B take to complete the 100 metre race if A runs at the rate of 10 m/sec?
| (1) | 8 seconds | (2) | 10 seconds | (3) | 16.67 seconds | (4) | 12.5 seconds |
Correct Answer – (4)
Solution:
A gives B a start of 10 metres in a 100 metre race. This means that when A runs 100 metres, B runs only 90 metres.
Despite that start, A beats B by 1.25 seconds.
As A is running at the speed of 10 m/sec, he will take 10 seconds to complete the 100 metre race. And B takes 10 + 1.25 = 11.25 seconds to cover 90 metres.
Therefore, the speed at which B is running = = 8 m/sec.
Running at 8 m/sec, B will take = 12.5 seconds to complete the 100 metre race.
Hence the correct answer is (4).
- A hound is chasing its hare. The hound takes 4 leaps for every 6 leaps of the hare and the hound covers as much distance in 2 leaps as 3 leaps of the hare. Will the hound succeed in getting its food?
| (1) | Yes |
| (2) | In the 6th leap |
| (3) | Never |
| (4) | Cannot determine |
Correct Answer – (4)
Solution:
Distance covered in 2 leaps by hound = 3 leaps of the hare.
Distance covered in 1 leap of hound = 3/2 leaps of hare. —-(1)
4 leaps of hound : 6 leaps of hare—(2)
Using (1) and (2), we get
4*3/2 leaps of hound : 6 leaps of hare.
=> 1:1
If the hound and hare start simultaneously at the same point, the hound will catch the hare immediately. If not so, then the hound will never catch the hare as it was running at the same speed.
As it was not mentioned in the question that they start simultaneously from the same point or not, we can’t determine the answer. Therefore, the answer choice is (4).
- A can give B a start of 50 metres or 10 seconds in a kilometer race. How long does A take to complete the race?
| (1) | 200 seconds | (2) | 140 seconds | ||
| (3) | 220 seconds | (4) | 190 seconds |
Correct Answer – (4)
Solution:
A can give B a start of 50 metres or 10 seconds in a 1000 m race.
That is, B takes 10 seconds to run 50 metres.
Therefore, B will take (10/50) * 1000 = 200 seconds to run 1000 metres.
A who can give B a start of 10 seconds will take 10 seconds lesser to run the 1000m.
Hence, the time taken by A = 190 seconds.
