Early and Late Arrival Problems with Time Calculations
Early and Late Arrival Problems compare two travel speeds when a person reaches before or after the scheduled time. The method uses the same distance, different travel times and the relation between speed and time. This page covers the main formulas, unit conversions, solving steps and numerical examples used in quantitative aptitude.
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What Are Early and Late Arrival Problems?
Let the scheduled travel time be T, the early arrival be e, and the late arrival be l. If the corresponding speeds are v₁ and v₂, where v₁ is the faster speed, then the actual travel times are T − e and T + l. Since the distance is unchanged, v₁(T − e) = v₂(T + l). Therefore, T = (v₁e + v₂l) ÷ (v₁ − v₂), and distance = v₁v₂(e + l) ÷ (v₁ − v₂). Convert minutes into hours when speed is given in kilometres per hour.
Early and Late Arrival Problems Video Lessons
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Punctuality Problems in Time Speed Distance
Understand how to solve punctuality problems involving scheduled time and train approach, using relationships between time, speed, distance, early arrival and late arrival.
Early and Late Arrival Problems Formula & Tricks
Important Formulas
Use consistent units for speed and time. For speed in km/h, time must be in hours.
Here v₁ is the faster speed, v₂ is the slower speed, e is the early time and l is the late time.
This follows from v₁(T − e) = v₂(T + l), with e and l expressed in the same time unit.
The faster journey takes e less than the scheduled time, while the slower journey takes l more.
The scheduled time is the faster journey time plus the early duration, or the slower journey time minus the late duration.
Quick Tricks
The difference between the two actual travel times is e + l, not e − l. This gives the time difference needed when comparing the two speeds.
When speeds are in km/h, divide minutes by 60 before substitution. Keeping time units consistent prevents incorrect distance values.
For a fixed distance, write faster-speed distance equal to slower-speed distance: v₁(T − e) = v₂(T + l). This avoids assigning separate distances to the two journeys.
Early and Late Arrival Problems Concepts
Equal-Distance Equation
If the faster speed is v₁ and the slower speed is v₂, the respective travel times are T − e and T + l. Hence, v₁(T − e) = v₂(T + l). Expanding gives (v₁ − v₂)T = v₁e + v₂l.
Finding the Scheduled Time
The formula is obtained by rearranging the equal-distance equation. The faster speed must be paired with the early duration, and the slower speed must be paired with the late duration.
Finding the Distance
This formula is useful when both speeds and the early and late durations are given. The result has the distance unit associated with the speed, such as kilometres when speed is in km/h.
Recovering the Early or Late Duration
The faster journey takes less time than the scheduled time, so e = T − D/v₁. The slower journey takes more time, so l = D/v₂ − T. Both values must be non-negative for the stated early or late conditions.
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Early and Late Arrival Problems: Revision Points
Use these relations to solve fixed-distance arrival questions.
- For the same distance, v₁(T − e) = v₂(T + l).
- The difference between actual travel times is e + l.
- Scheduled time: T = (v₁e + v₂l) ÷ (v₁ − v₂).
- Distance: D = v₁v₂(e + l) ÷ (v₁ − v₂).
- Use the faster speed with T − e and the slower speed with T + l.
- Convert minutes to hours when speed is expressed in km/h.
- The faster speed must be greater than the slower speed for the standard formula.
Early and Late Arrival Problems FAQs
What is the formula for early and late arrival problems?
For faster speed v₁, slower speed v₂, early time e and late time l, scheduled time is T = (v₁e + v₂l) ÷ (v₁ − v₂). Distance is D = v₁v₂(e + l) ÷ (v₁ − v₂).
Why are the early and late times added?
The faster journey takes e less than the scheduled time and the slower journey takes l more. Therefore, the difference between their actual travel times is (T + l) − (T − e) = e + l.
A person travels at 60 km/h and reaches 15 minutes early, or at 45 km/h and reaches 5 minutes late. Find the scheduled time.
Convert the times to 1/4 hour and 1/12 hour. T = [60(1/4) + 45(1/12)] ÷ (60 − 45) = 1.25 hours, or 1 hour 15 minutes.
What distance is covered in the preceding example?
D = 60 × 45 × (1/4 + 1/12) ÷ 15 = 75 km.
How is 20 minutes converted for use with km/h?
20 minutes = 20/60 hour = 1/3 hour. Time must be expressed in hours when speed is given in kilometres per hour.
How can the late time be found when distance, slower speed and scheduled time are known?
Use l = D/v₂ − T. For example, if D = 90 km, v₂ = 45 km/h and T = 1.5 hours, then l = 90/45 − 1.5 = 0.5 hour, or 30 minutes.