Beaten By Problems in Race with Distance and Speed Tricks
Beaten By Problems in Race involve comparing the distances covered or times taken by two runners. The standard cases are when one runner beats another by a certain distance or by a certain number of seconds. Use speed ratios, time differences and relative speed to solve these questions accurately.
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What Are Beaten By Problems in Race?
For the same race, time is inversely proportional to speed. Hence, the time ratio of A and B is (L − d):L. If A beats B by t seconds, and A takes T seconds, then B takes T + t seconds and their speed ratio is (T + t):T.
Beaten By Problems in Race Video Lessons
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Race Problems: Head Start and Dead Heat
Learn to solve race problems involving head starts, dead heats, and beaten-by conditions using relative speed and distance relationships.
Beaten By Problems in Race Formula & Tricks
Important Formulas
If A wins a race of L metres by d metres, B has covered L − d metres when A finishes.
For the same distance, time is inversely proportional to speed.
This gives the time taken by B to complete the full race when A beats B by d metres.
If A finishes in T seconds and beats B by t seconds, B finishes in T + t seconds.
If A beats B by d metres in an L-metre race and by t seconds, this gives A's time.
Quick Tricks
When the winner finishes, subtract the winning margin from the total race length. Do not add the margin to the race length.
If the winner's time is T and the time margin is t, use winner:loser time = T:(T + t). Reverse this ratio to obtain the speed ratio.
When both the distance margin and time margin are given, use T = t(L − d) ÷ d to find the winner's time before calculating speeds.
Beaten By Problems in Race Concepts
Race Problems Based on Distance Margin
At the instant A reaches the finish line, B is d metres behind and has covered L − d metres. Since both runners are considered over the same time interval, their speeds are proportional to these distances.
Race Problems Based on Time Margin
Both runners cover the same race distance, so speed is inversely proportional to time. The runner taking fewer seconds has the greater speed.
Finding Time from Distance and Time Margins
The distance condition gives A:B speed = L:(L − d). Therefore, their time ratio is (L − d):L. The difference between these times is t, which leads to the formula.
Finding Speed or Race Length
If A:B speed = p:q and A's speed is v, then B's speed is vq ÷ p. If the race length is unknown but the distance margin and speed ratio are known, use L:(L − d) = p:q and solve for L.
Beaten By Problems in Race Data Table
Use the structured data below while following the explanation and examples.
Core Race-Margin Relations
Formulas for comparing two runners when one runner beats the other.
| Given condition | Winner's quantity | Loser's quantity | Required ratio |
|---|---|---|---|
| A beats B by d metres in an L-metre race | A covers L metres | B covers L − d metres | A:B speed = L:(L − d) |
| A beats B by t seconds | A takes T seconds | B takes T + t seconds | A:B speed = (T + t):T |
| A beats B by d metres and t seconds | A takes t(L − d) ÷ d seconds | B takes tL ÷ d seconds | A:B speed = L:(L − d) |
Note: L, d and distances must use the same unit. Time values must use the same unit.
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Beaten By Problems in Race: Revision Points
Use these direct rules to revise race-margin calculations.
- If A beats B by d metres in an L-metre race, B covers L − d metres when A finishes.
- Speed ratio A:B = L:(L − d).
- Time ratio A:B = (L − d):L for the same race.
- If A takes T seconds and beats B by t seconds, B takes T + t seconds.
- For a time margin, speed ratio A:B = (T + t):T.
- If both margins are given, winner's time = t(L − d) ÷ d.
- Keep all distances in one unit and all times in one unit before applying a formula.
Beaten By Problems in Race FAQs
What is the beaten by distance formula in a race?
If A beats B by d metres in an L-metre race, A:B speed = L:(L − d). For example, in a 100 m race with a 20 m margin, the speed ratio is 100:80 = 5:4.
If A beats B by 20 m in a 100 m race, what distance does B cover when A finishes?
B covers 100 − 20 = 80 m when A reaches the finish line.
A finishes a race in 16 seconds and beats B by 4 seconds. What is their speed ratio?
B takes 16 + 4 = 20 seconds. Since speed is inversely proportional to time, A:B speed = 20:16 = 5:4.
How do you find the winner's time when both race margins are given?
Use T = t(L − d) ÷ d, where L is the race length, d is the distance margin and t is the time margin. For L = 100, d = 20 and t = 4, T = 16 seconds.
If A beats B by 30 m in a 150 m race, what is the time ratio?
B covers 120 m when A finishes. Therefore, A:B time ratio = 120:150 = 4:5.
A beats B by 20 m in a race, and their speed ratio is 5:4. What is the race length?
Use L:(L − 20) = 5:4. Thus 4L = 5L − 100, giving L = 100 m.