CDS Mathematics · Time, Speed and Distance

Races

In a race the runners' speeds are in the ratio of the distances they cover in the same time; a start or a beat is just a difference in distance or time.

All Time, Speed and Distance notes CDS Mathematics strategy

Why this matters

Five PYQs, two of them HARD. Turn every 'A beats B by 150 m' into a speed ratio (1000 : 850), then combine ratios through a common runner. On a circular track, the winner laps the loser once for every full lap gained.

Concept 1 of 1: Starts, beats and laps

While the winner runs the full course, the loser runs the course minus the beat. Those two distances, run in the same time, are in the ratio of the speeds.

Definition

'A beats B by $x$ m in a $D$ m race': $v_A : v_B = D : (D - x)$ .
'A gives B a start of $s$ m': B runs only $D - s$ .
'Beats by $t$ seconds': B's time is A's time plus $t$ .
Circular track of length $c$ : the winner passes the other once for each full $c$ gained.
Chain ratios through a runner common to two races.

Beat by a distance

v_A : v_B = D : (D - x)

Worked example

In a

400

m race A beats B by

40

m. In a

500

m race, by how much would A beat B?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q23Moderate

Example 1 · Time, Speed and Distance · Races

In a 100 m race, A runs at a speed of

\dfrac{5}{3}

m/s. If A gives a start of 4 m to B and still beats him by 12 seconds, what is the speed of B ?

Count only complete laps

Gaining

3.2

laps means passing the other runner

3

times, not

3.2

and not

4

. A pass happens each time a WHOLE extra lap is gained.

Drill 4 more on starts, beats and laps

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 (1)

Starts, beats and laps
Beat by a distance
$v_A : v_B = D : (D - x)$

Watch out for (1)

Count only complete laps
→ Starts, beats and laps

Test yourself on Time, Speed and Distance

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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