Head Start in Race Problems with Speed and Distance

Head Start in Race problems compare two runners when one runner begins earlier or covers a shorter distance. The questions use speed, time, race distance and winning margin. This page explains the head start in race formula, distance and time cases, speed-ratio methods, solved examples and quick calculation tricks.

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What Is Head Start in a Race?

A head start is the distance or time advantage given to one runner before a race begins. If a faster runner gives a slower runner a head start and both finish together, their running distances and speeds determine the required advantage.

If the race distance is D and runner A gives runner B an x-metre start, A covers D metres while B covers D − x metres. For a tie, their speed ratio is vA:vB = D:(D − x). If B starts h seconds earlier, the time relation is D/vB = D/vA + h.

Head Start 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.

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Head Start in Race Formula & Tricks

Important Formulas

Distance head start
x = D(1 − vB/vA)

If faster runner A gives slower runner B a distance start in a race of length D, x is the required head start.

Speed ratio with a distance start
vA/vB = D/(D − x)

When A and B finish together after B receives an x-metre start, their speed ratio follows from the distances covered.

Head start from speed ratio
If vA:vB = m:n, then x = D(m − n)/m

Here m is the faster runner's ratio part and n is the slower runner's ratio part.

Time head start
h = D/vB − D/vA

If A and B finish together and B starts h seconds earlier, h equals B's running time minus A's running time.

Winning margin as head start
Head start = winning margin

If A beats B by y metres in a D-metre race, A can give B a y-metre start and both will finish together, assuming unchanged speeds.

Quick Tricks

Use the complement of the speed ratio

If the speed ratio of the faster and slower runners is m:n, the slower runner covers n/m of the race when the faster runner finishes. The head start is the remaining fraction, (m − n)/m, of the race.

Example: For a 500-metre race and speed ratio 5:4, head start = 500 × (5 − 4)/5 = 100 metres.
Convert a winning margin directly

In a race of fixed length, the distance by which the winner beats the other runner is the same distance that the winner can give as a head start for a tie.

Example: If A beats B by 30 metres in a 200-metre race, A can give B a 30-metre start and the result will be a tie.

Head Start in Race Concepts

Distance Head Start Between Two Runners

A distance head start is found by comparing the distance covered by the slower runner when the faster runner completes the race.

Suppose A is faster, the race length is D, and B receives x metres. At a tie, A covers D while B covers D − x. Therefore, vA/vB = D/(D − x), and x = D(1 − vB/vA).

Example: In a 400-metre race, A and B run at 5:4. A's head start for a tie is 400(1 − 4/5) = 80 metres.

Finding Head Start from a Speed Ratio

When the speed ratio is known, multiply the race distance by the difference of the ratio parts divided by the faster runner's ratio part.

If vA:vB = m:n, then in the time A covers D metres, B covers nD/m metres. The head start is D − nD/m = D(m − n)/m. This method applies when m > n.

Example: For a 600-metre race with speed ratio 7:5, head start = 600(7 − 5)/7 = 1200/7 metres, approximately 171.43 metres.

Time Head Start

A time head start is the difference between the slower runner's full running time and the faster runner's full running time.

For race distance D, speeds vA and vB, where A is faster, the time head start given to B is h = D/vB − D/vA. The slower runner starts h seconds earlier and both runners then finish together.

Example: For a 200-metre race, if A runs at 10 m/s and B at 8 m/s, h = 200/8 − 200/10 = 25 − 20 = 5 seconds.

Winning Margin and Equivalent Head Start

A winner's distance margin is equal to the distance head start the winner can give the other runner to produce a tie.

If A completes a D-metre race before B, B covers D − y metres when A finishes, where y is A's winning margin. Giving B y metres at the start makes B's effective distance D − y, producing the same finishing time.

Example: If A beats B by 50 metres in a 250-metre race, A can give B a 50-metre head start. The effective distance for B becomes 200 metres.

Head Start in Race Data Table

Use the structured data below while following the explanation and examples.

Head Start Formula Cases

Formulas for distance and time head starts in a race

Case Condition Formula
Distance start A gives B x metres in a D-metre race x = D(1 − vB/vA)
Speed-ratio method vA:vB = m:n x = D(m − n)/m
Time start B starts h seconds earlier h = D/vB − D/vA
Winning margin A beats B by y metres Equivalent head start = y metres

Note: A must be faster than B for A to give B a positive head start.

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Practice Head Start in Race Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Head Start in Race Revision Points

Use these direct rules and formulas for quick revision.

  • If A gives B x metres in a D-metre race, A covers D and B covers D − x.
  • For a tie, vA/vB = D/(D − x).
  • Distance head start: x = D(1 − vB/vA).
  • If the speed ratio is m:n, head start = D(m − n)/m.
  • Time head start: h = D/vB − D/vA.
  • The winner's distance margin equals the equivalent head start for a tie.
  • Use the faster runner's speed as the denominator when converting a speed ratio into a distance head start.

Head Start in Race FAQs

What is the head start in race formula when speeds are given?

If A is faster than B and the race distance is D, A's distance head start to B is x = D(1 − vB/vA).

A and B have speeds in the ratio 3:2 in a 300-metre race. What head start can A give B?

Head start = 300(3 − 2)/3 = 100 metres.

How is a time head start calculated?

For race distance D, the time head start is h = D/vB − D/vA, where A is faster and B starts earlier.

A runs at 12 m/s and B runs at 10 m/s in a 240-metre race. What time head start can A give B?

A's time is 240/12 = 20 seconds and B's time is 240/10 = 24 seconds. Therefore, A can give B a 4-second head start.

If A beats B by 40 metres in a 200-metre race, what head start can A give B for a tie?

A can give B a 40-metre head start. B will then need to cover only 160 metres.

Can the head start formula be negative?

A positive head start requires A to be faster than B. If vA is less than vB, the expression D(1 − vB/vA) is negative, indicating that A cannot give B a start and still tie.

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