Race Problems: Formulas, Rules and Solved Examples
Race problems compare the speeds, times and distances covered by two or more runners. They use the basic relation speed = distance ÷ time and relative speed for runners moving in the same direction. Questions may involve a winning margin, head start, dead heat or the distance covered by the slower runner when the winner finishes.
What Is a Race Problem?
For a race of length L, if A and B run at speeds a and b, their finishing times are L ÷ a and L ÷ b respectively. When A is faster, the speed ratio is a:b and the time ratio is b:a. In a 200 m race, if A runs at 8 m/s and B at 6 m/s, their times are 25 seconds and 33⅓ seconds, respectively.
Race Formula & Tricks
Important Formulas
Use consistent units for distance and time, such as metres and seconds.
For a track of length L and speed v, time = L ÷ v.
Speed and time are inversely proportional when the race distance is the same.
This distance is measured when the winner completes the race of length L.
If A is faster than B, with speeds a and b, A's lead when A finishes is this distance.
If a runner receives a head start of h in an L-metre race and both finish together, their speed ratio follows this relation.
Quick Tricks
For the same race distance, reverse the speed ratio to obtain the time ratio.
When the winner completes the track, calculate how far the other runner travels using distance = speed × time. The difference from the track length is the winning margin.
A runner given h metres in an L-metre race has to cover only L − h metres. Use the distances covered in the same finishing time to form the speed ratio.
Race Concepts
Same-Start Race and Winning Margin
If A and B run a race of length L with speeds a and b, where a > b, the common time at A's finish is L/a. B covers bL/a. Therefore, A's margin is L − bL/a = L(a − b)/a. The speed ratio is also obtained from the distances covered in this common time.
Relative Speed in a Race
If the runners start together, the time taken by the faster runner to gain a distance d is d divided by their relative speed. If A is ahead by d metres after t seconds, then A's speed exceeds B's speed by d/t. For runners moving in opposite directions, relative speed is the sum of their speeds, but standard race comparisons usually use the difference.
Head Start in a Race
If A runs the full distance L and B starts h metres ahead, B covers L − h. Since both cover their respective distances in the same time, their speed ratio is A:B = L:(L − h). If the speed ratio and race length are known, the head start is h = L − L × (B's speed ÷ A's speed).
Race Won by Time
Let B take T seconds. Then A takes T − t seconds. Since both cover L metres, speed A:speed B = T:(T − t). To solve a question, use the given speed ratio to find T and then subtract t, or use the known time directly.
Dead Heat and Equal Speeds
If the runners cover the same distance L in equal times, then speed = L/time gives equal speeds. A head start can also produce a dead heat when the faster runner covers the full distance and the other runner covers only the remaining distance.
Race Video Lessons
Watch short topic-wise lessons for quick revision.
Relative Speed in Opposite Directions
Learn how to calculate relative speed when two objects move in opposite directions, and apply the method to time, speed and distance problems.
Practice Race Questions
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Race Quick Quiz
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Race Formula Revision
Use these direct relations while solving race questions.
- Speed = distance ÷ time, and time = distance ÷ speed.
- For the same distance, speed ratio and time ratio are reciprocals.
- In the same direction, relative speed = faster speed − slower speed.
- If A finishes an L-metre race and B runs at speed b while A runs at speed a, B covers Lb/a.
- Winning margin = L(a − b)/a when A is faster than B.
- With a head start of h in an L-metre race, the distance covered by the other runner is L − h.
- If A beats B by t seconds and B takes T seconds, speed A:speed B = T:(T − t).
- Keep distance and time units consistent before applying any race formula.
Race FAQs
If A and B have speed ratio 3:2, what is their time ratio in the same race?
Their time ratio is the inverse, 2:3. The slower runner takes more time for the same distance.
A wins a 500 m race against B by 100 m. What is the speed ratio A:B?
When A finishes, B covers 500 − 100 = 400 m. Therefore, speed ratio A:B = 500:400 = 5:4.
A runner completes a 300 m race in 30 seconds. What is the speed?
Speed = 300 ÷ 30 = 10 m/s.
A runs at 12 m/s and B at 9 m/s. How much time does A take to gain 60 m?
Relative speed = 12 − 9 = 3 m/s. Time = 60 ÷ 3 = 20 seconds.
In a 400 m race, B gets a 40 m head start and both finish together. What is the speed ratio A:B?
A covers 400 m and B covers 360 m in the same time. Hence, speed ratio A:B = 400:360 = 10:9.
A beats B by 8 seconds, and their speed ratio is 5:4. If A takes 32 seconds, how long does B take?
The time ratio is 4:5. Thus, B's time = 32 × 5/4 = 40 seconds, which is 8 seconds more.