AFCAT Numerical Ability · Speed, Time and Distance

Races: Beating, Head Starts and Dead Heats

Turn a result such as 'A beats B by 20 m' into a ratio of speeds, then use that ratio for a new race length, a head start, or a third runner.

All Speed, Time and Distance notes

Why this matters

No AFCAT paper has set a race question here, but a fifth of the page is HARD. Every question rests on one fact: in the same time, distances run are in the ratio of the speeds. Marks go on mixing up 'beats by 20 m' with 'beats by 20 s', and on adding the margins of a three runner race.

Concept 1 of 3: Beating by a distance or by a time

When A finishes, B is still some way short. In that same time A ran the whole race and B ran the race less the gap, so their speeds are in that ratio.

Definition

A beats B by $x$ m in a $D$ m race: $v_A : v_B = D : (D - x)$ .
A beats B by $t$ s: B's time is A's time plus $t$ .
'Beats by $x$ m or by $t$ s': B runs those last $x$ m in $t$ s, so $v_B = \tfrac xt$ .
A race of another length: B runs $\tfrac{D - x}{D}$ of the new distance while A runs all of it.

Speed ratio from a margin

\dfrac{v_A}{v_B} = \dfrac{D}{D - x}

Worked example

In a 400 m race A beats B by 40 m. By how much does A beat B in a 1,000 m race?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Speed, Time and Distance · Races

A can run

1

km in

4

minutes and B can run

1

km in

5

minutes. If they start together in a

3

km race, by how much distance will A beat B?

Beaten by a distance, or by a time

'Beats by 10 s' is a gap in time: B's time is 10 s more. It is not 10 m to subtract from the race.

Drill 13 more on beating by a distance or by a time

Concept 2 of 3: Head starts and dead heats

A head start shortens the slower runner's race. Work out where each runner is at the moment the first one finishes.

Definition

B starts $x$ m ahead: B has only $D - x$ to run.
While A runs $D$ , B runs $D \times \tfrac{v_B}{v_A}$ . Compare that with $D - x$ .
Dead heat (both finish together): B's start $= D - D \times \tfrac{v_B}{v_A}$ .
'Gives B a start of $x$ m and still beats him by $y$ m': B ran $D - x - y$ while A ran $D$ .
A race on a circular track with a head start works the same way.

Start for a dead heat

\text{start} = D\left(1 - \dfrac{v_B}{v_A}\right)

Worked example

In a 600 m race A runs at 6 m/s and B at 5 m/s. How much start should A give B for a dead heat?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Speed, Time and Distance · Races

A can run 200 meters in 20 seconds, and B can run the same distance in 25 seconds. If they start together in a 400-meter race, how much start should A give B to make it a dead heat?

Count the start once

With an 80 m start in an 800 m race, B has 720 m to run. When A wins by 30 m, B has run 690 m, not 770.

Drill 19 more on head starts and dead heats

Concept 3 of 3: Three runners: chaining the results

Turn each result into a ratio of speeds and chain them, the way two ratios are compounded. The margins themselves do not add.

Definition

A beats B by $x$ and B beats C by $y$ in $D$ m: when A finishes, C has run $\dfrac{(D - x)(D - y)}{D}$ .
A beats B by $x$ and C by $z$ in the same race: $A : B : C = D : (D - x) : (D - z)$ . Over $D$ m, B beats C by $D - \dfrac{D(D - z)}{D - x}$ .
Margins in time do add: A beats B by 2 s and B beats C by 3 s, so A beats C by 5 s.

C's distance when A finishes

\dfrac{(D - x)(D - y)}{D}

Worked example

In a 250 m race A beats B by 25 m and B beats C by 50 m. By how much does A beat C?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Speed, Time and Distance · Races

In a

100 \text{ m}

race, A beats B by

10 \text{ m}

and B beats C by

5 \text{ m}

. By how much does A beat C in the same race?

Distance margins do not add

Beating B by 25 m and B beating C by 50 m does not make 75 m: while A runs the race, B runs only 225 m, so C falls behind B by less than 50 m. The answer is 70 m.

Drill 12 more on three runners: chaining the results

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Beating by a distance or by a time
Speed ratio from a margin
$\dfrac{v_A}{v_B} = \dfrac{D}{D - x}$
Head starts and dead heats
Start for a dead heat
$\text{start} = D\left(1 - \dfrac{v_B}{v_A}\right)$
Three runners: chaining the results
C's distance when A finishes
$\dfrac{(D - x)(D - y)}{D}$

Watch out for (3)

Beaten by a distance, or by a time
→ Beating by a distance or by a time
Count the start once
→ Head starts and dead heats
Distance margins do not add
→ Three runners: chaining the results

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