Two Speed Problems: Formulas, Shortcuts and Examples

Two Speed Problems involve journeys or time intervals covered at two different speeds. The correct average speed depends on whether the distances or the times are equal. This page explains the two speed formula, total-distance methods, equal-distance and equal-time cases, percentage-based changes, and solved numerical examples.

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What Are Two Speed Problems?

Two Speed Problems calculate time, distance or average speed when motion takes place at two different speeds. Average speed is always calculated as total distance divided by total time.

For different distances or time intervals, use average speed = total distance ÷ total time rather than the ordinary average of the two speeds. For example, travelling at 40 km/h for 2 hours and 60 km/h for 3 hours gives total distance = 80 + 180 = 260 km and total time = 5 hours, so average speed = 260 ÷ 5 = 52 km/h.

Two Speed Problems Formula & Tricks

Important Formulas

General average speed
Average speed = Total distance ÷ Total time

Use this formula when the distances or time intervals are unequal.

Equal-distance two speed formula
Average speed = 2uv ÷ (u + v)

Here, u and v are the two speeds, and the two distances are equal.

Equal-time average speed
Average speed = (u + v) ÷ 2

This applies when the object travels at speeds u and v for equal amounts of time.

Time for a fixed distance
Time = Distance ÷ Speed

For equal distances, the time taken is inversely proportional to speed.

Percentage change in time
If speed increases by x%, time decreases by [x ÷ (100 + x)] × 100%; if speed decreases by x%, time increases by [x ÷ (100 − x)] × 100%

These formulas apply when the distance remains constant.

Quick Tricks

Use the harmonic mean for equal distances

For equal distances, do not take the ordinary average of the speeds. Multiply the two speeds by 2 and divide by their sum.

Example: For 30 km/h and 60 km/h over equal distances, average speed = 2 × 30 × 60 ÷ (30 + 60) = 40 km/h.
Compare time through inverse speed ratios

For the same distance, time varies inversely with speed. If speeds are in the ratio a:b, the corresponding times are in the ratio b:a.

Example: If speeds are 40 km/h and 60 km/h, their time ratio for the same distance is 60:40 = 3:2.
Use a common distance when no distance is given

For equal-distance questions, assume a convenient common distance such as the least common multiple of the speeds. This keeps the time calculations integral when possible.

Example: For speeds 20 km/h and 30 km/h, take a common distance of 60 km. The times are 3 hours and 2 hours, so average speed = 120 ÷ 5 = 24 km/h.

Two Speed Problems Concepts

Average Speed for Unequal Distances or Times

When the two parts have different distances or different time durations, average speed equals total distance divided by total time.

If distances d₁ and d₂ are covered at speeds u and v, then average speed = (d₁ + d₂) ÷ (d₁ ÷ u + d₂ ÷ v). If the time intervals are t₁ and t₂, then average speed = (ut₁ + vt₂) ÷ (t₁ + t₂).

Example: A vehicle travels 120 km at 40 km/h and 180 km at 60 km/h. Total time = 120 ÷ 40 + 180 ÷ 60 = 3 + 3 = 6 hours. Average speed = 300 ÷ 6 = 50 km/h.

Equal-Distance Two Speed Formula

For equal distances covered at speeds u and v, average speed is 2uv ÷ (u + v).

Let each distance be d. Total distance is 2d, while total time is d/u + d/v = d(u + v)/(uv). Therefore, average speed = 2d ÷ [d(u + v)/(uv)] = 2uv/(u + v). This value is less than the ordinary arithmetic mean when u and v are unequal.

Example: A person covers equal distances at 24 km/h and 36 km/h. Average speed = 2 × 24 × 36 ÷ 60 = 28.8 km/h.

Equal-Time Average Speed

When two speeds continue for equal amounts of time, average speed is the arithmetic mean, (u + v) ÷ 2.

If each speed lasts for time t, total distance = ut + vt = (u + v)t and total time = 2t. Hence average speed = (u + v)t ÷ 2t = (u + v) ÷ 2.

Example: A car travels for 2 hours at 50 km/h and 2 hours at 70 km/h. Total distance = 100 + 140 = 240 km, total time = 4 hours, and average speed = 60 km/h.

Speed Change and Time Change for Fixed Distance

For a fixed distance, speed and time are inversely proportional, so increasing speed reduces the time taken.

If the speed changes from u to v for the same distance d, the time changes from d/u to d/v. If speed increases by x%, the new speed is u(100 + x)/100, and the time reduction is x/(100 + x) of the original time. If speed decreases by x%, the time increase is x/(100 − x) of the original time.

Example: If speed increases by 25%, time decreases by 25/125 × 100% = 20%. If speed decreases by 20%, time increases by 20/80 × 100% = 25%.

Two Speed Problems Video Lessons

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11 Lessons
Lesson 1 of 11 Quick Revision

Average Speed and the Harmonic Mean Trap

Understand why average speed for equal distances uses the harmonic mean, and learn to avoid the common mistake of taking the simple arithmetic average of speeds.

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Practice Two Speed Problems Questions

Practise published questions related to this topic.

1After allowing a discount of 24%, the selling price of an article becomes ₹ 380. Find its marked price.→ 2Selling price and cost price of an article is Rs. 1000 and Rs. 800 respectively. What is the profit percentage?→ 3A bicycle is marked at ₹15,000. The shopkeeper offers a 10% discount, and the bicycle is sold with a 20% profit on cost price. Find the cost price of the bicycle.→ 4What is the square root of 1444?→ 5Simplify: (2³ × 3⁻² × 5⁴)/(6⁻¹ × 10²)→ 6Selling price of an article is Rs. 4000. If the value of the loss is Rs. 1000, then what is the loss percentage?→ 7Simplify: √(51 - √7 + √(11 - 4√7))→ 8Which of the following number is neither divisible by 3 nor by 4?→ 9A product priced at ₹800 is sold for ₹680. What is the percentage of discount offered?→ 10Surjeet buys 20 kg of sugar for ₹900. He sells 18 kg of sugar for ₹918. What is the percentage profit made by Surjeet?→

Two Speed Problems Quick Quiz

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Quick Revision Notes

Two Speed Problems: Quick Revision

Use the condition of the journey before selecting the average-speed formula.

  • Average speed = total distance ÷ total time.
  • For equal distances, average speed = 2uv ÷ (u + v).
  • For equal times, average speed = (u + v) ÷ 2.
  • For the same distance, time is inversely proportional to speed.
  • If speeds are in the ratio a:b for equal distances, times are in the ratio b:a.
  • For fixed distance, a speed increase of x% causes a time decrease of x/(100 + x) × 100%.
  • For fixed distance, a speed decrease of x% causes a time increase of x/(100 − x) × 100%.
  • Do not use the arithmetic mean of two speeds unless the corresponding time intervals are equal.

Two Speed Problems FAQs

What is the two speed formula for equal distances?

The formula is average speed = 2uv ÷ (u + v), where u and v are the two speeds.

What is the average speed for speeds 40 km/h and 60 km/h over equal distances?

Average speed = 2 × 40 × 60 ÷ (40 + 60) = 48 km/h.

When can the arithmetic average of two speeds be used?

It can be used when the object travels at the two speeds for equal amounts of time. The average speed is then (u + v) ÷ 2.

A vehicle travels 100 km at 50 km/h and 150 km at 75 km/h. What is its average speed?

Time taken = 100 ÷ 50 + 150 ÷ 75 = 2 + 2 = 4 hours. Average speed = 250 ÷ 4 = 62.5 km/h.

If speed increases by 20% for the same distance, what is the percentage decrease in time?

Time decreases by 20/(100 + 20) × 100% = 16⅔%.

For equal distances, which is greater: arithmetic mean or average speed?

The arithmetic mean is greater when the two speeds are unequal. Equal-distance average speed is the harmonic mean, 2uv/(u + v).

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