Race problems are essentially applications of Time, Speed, and Distance. They involve two or more participants moving towards a common finish line. The key is understanding relative positions and timing.
A race takes place over a fixed Distance (D). Participants start at the same time (usually) and try to cover this distance in the minimum Time (T).
This means when the winner finishes the race (covers distance D), the loser is still X meters behind.
D meters.D − X meters (in the same time).Swinner / Sloser = D / (D − X)
This means both participants finish the race (cover distance D), but the loser takes T seconds more.
t.t + T.(D / Sloser) − (D / Swinner) = T
Sometimes, to make a race fair, a faster runner gives a slower runner an advantage.
“A gives B a start of x meters.”
D.D − x meters.“A gives B a start of t seconds.”
If the faster runner is Y times as fast (Sfast = Y × Sslow) and the slower gets a Z meter distance headstart, the course length
D for a dead heat comes from equal times:
D / Sfast = (D − Z) / Sslow
→ D/Y = D − Z →
D = Z × Y / (Y − 1)
Z, no
dead heat exists.
Example: SA = 6 × SB, slower gets Z = 300m → D = 300 × 6 / 5 = 360m. Check: fast
runs 360m, slow runs 60m (360−300); times
360/(6·SB) = 60/SB equal.
Races on a circular track involve relative speed concepts similar to trains.
If Runner A takes t1 seconds and Runner B takes
t2 seconds to complete one lap:
LCM(t1, t2) seconds.
If the track length is L:
| Direction | Condition | Time |
|---|---|---|
| Opposite directions | They meet when sum of distances = L |
L / (SA + SB)
|
| Same direction | The faster runner gains exactly one lap (L) over the slower one |
L / (SA − SB)
|
Often problems link multiple runners: “A beats B by 10m, B beats C by 20m.”
Example: Race 100m. A beats B by 10m (A runs 100, B runs 90). B beats C by 10m (B runs 100, C runs 90).
A/B = 100/90. B/C = 100/90.A/C = (100/90) × (100/90) = 100/81.| Symbol | Meaning |
|---|---|
D, L |
Total length of the race / track |
SA, SB
|
Speeds of runners |
t |
Time taken |
x |
Distance margin (start or beat) |
T |
Time margin |
In a 100m race, “A beats B by 10m”. How far has B run when A finishes?
A beats B by X meters in a race of length D. What is the ratio of their speeds?
“A beats B by 5 seconds”. If A's time is t, what is B's time?
On a circular track, A takes 60s per lap and B takes 80s per lap. After how long do they meet at the starting point again?
Two runners move in the SAME direction on a circular track of length L, with speeds S₁ > S₂. When do they first meet?
On a 200m track, two runners run in OPPOSITE directions at 10 m/s and 6 m/s. When do they first meet?
In a 100m race, A beats B by 10m and B beats C by 10m. By how much does A beat C?
Runner A is 6× as fast as B. Slower gets 300m headstart. What course length gives a dead heat?
Fixed distance D ; start together (usually)
Winner ⇒ least time over D ; Loser ⇒ more time / less distance in same time
Winner finishes D ; loser still x behind → loser ran D − x
Same time ⇒ S(winner)/S(loser) = D / (D − x)
Both finish D ; loser takes T extra
winner t ⇒ loser t + T ; (D/S(loser)) − (D/S(winner)) = T
a) Distance ⇒ "A gives B start of x m"
A runs D ; B runs D − x ; finish together → dead heat
b) Time ⇒ "A gives B start of t s"
B starts first ; A joins later → A must run faster
c) Speed ratio Y ⇒ dead heat ⇒ D = Z·Y/(Y−1) (slower gets Z)
(eg Y=6, Z=300 → D=360 ; faster 360, slower 60)
Meet at start again ⇒ LCM(t₁, t₂)
a) Opposite ⇒ distances sum to L ; t = L / (S_A + S_B)
b) Same ⇒ faster gains one full lap ; t = L / (S_A − S_B)
Margins × never add → chain ratios
a) A:B ⇒ distances in same time
b) B:C similarly ; combine → A:C
(eg 100m ; A beats B by 10 ; B beats C by 10 → A/B = 100/90 ; B/C = 100/90 → A/C = 100/81 → A beats C by 19 m)
D, L ⇒ race/track length ; S_A, S_B ⇒ speeds
t ⇒ time taken ; x ⇒ distance margin ; T ⇒ time margin
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