A person travels from X to Y a speed of 40 kmph and returns by increasing his speed by 50%. What is his average speed for both the trips ?

1. The Core TSD Relationship: Speed, Distance, and Time (basic)

At its heart, the relationship between Speed, Distance, and Time is the foundation of all motion-based problems in quantitative aptitude. Speed is defined as the distance covered by an object in a unit of time, such as one second, one minute, or one hour Science-Class VII, Measurement of Time and Motion, p.113. When we say a car is moving at 60 km/h, we are essentially saying that if it maintains this rate, it will cover 60 kilometers in exactly one hour. This relationship allows us to calculate any one of the three variables if the other two are known.

The core mathematical relationship can be expressed in three distinct ways depending on what we need to find:

To Find	Formula	Context Example
Speed	Distance ÷ Time	Calculating how fast a train travels 360 km in 4 hours Science-Class VII, Measurement of Time and Motion, p.118.
Distance	Speed × Time	Determining how far a bus goes in 2 hours at 50 km/h Science-Class VII, Measurement of Time and Motion, p.115.
Time	Distance ÷ Speed	Finding how long it takes to cover 150 meters at 15 m/s.

In the real world, objects rarely move at a perfectly constant speed. This is why we distinguish between Uniform Motion (covering equal distances in equal intervals of time) and Non-uniform Motion Science-Class VII, Measurement of Time and Motion, p.117. For most UPSC problems, unless stated otherwise, we assume uniform motion or use the concept of average speed, which is simply the total distance divided by the total time taken.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118

2. Unit Conversions and Dimensional Consistency (basic)

In the world of quantitative aptitude, a number without a unit is like a word without context—it lacks meaning. Unit conversion is the process of changing a measure from one unit to another (e.g., from kilometres to metres) while keeping the actual quantity the same. This is vital because, in any mathematical equation, you must maintain dimensional consistency. This means you can only add, subtract, or compare quantities if they are expressed in the same units. As taught in Science-Class VII . NCERT, Measurement of Time and Motion, p.113, the standard (SI) unit for speed is m/s, but we frequently encounter km/h in real-world scenarios like train or car travel.

To master conversions, you must understand the relationship between different scales. For instance, to convert speed from km/h to m/s, we use the factor 5/18. This is derived because 1 km = 1000m and 1 hour = 3600 seconds (1000/3600 simplifies to 5/18). Similar logic applies to economics, where convertibility allows us to exchange one currency for another at specific market rates Indian Economy, Nitin Singhania, India’s Foreign Exchange and Foreign Trade, p.498. Before solving any problem, always look at the units requested in the final answer and convert all your data points to match that system first.

Quantity	Common Unit A	Common Unit B	Conversion Rule
Distance	Kilometre (km)	Metre (m)	1 km = 1000 m
Time	Hour (h)	Second (s)	1 h = 3600 s
Speed	km/h	m/s	Multiply by 5/18

It is also important to recognize unitless quantities. These occur when you divide two quantities of the same dimension, causing the units to cancel out. A classic example is Relative Density, which is the ratio of the density of a substance to the density of water Science-Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.141. Since it is a ratio of identical units, the result is a pure number. Understanding when units cancel out and when they must be converted is the hallmark of a disciplined aspirant.

Sources: Science-Class VII . NCERT, Measurement of Time and Motion, p.113; Indian Economy, Nitin Singhania, India’s Foreign Exchange and Foreign Trade, p.498; Science-Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.141

3. Defining Average Speed: The 'Total' Perspective (intermediate)

In the world of quantitative aptitude, the most common trap students fall into is treating 'average speed' like a simple arithmetic average. If you drive at 40 km/h for one leg of a trip and 60 km/h for the next, your average speed is not necessarily 50 km/h. To understand why, we must look at the 'Total Perspective.' As defined in Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113, speed is the total distance covered divided by the total time taken. This 'Total' perspective is vital because, in reality, objects rarely move at a constant, uniform speed throughout a journey; they speed up, slow down, or even stop Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115.

The reason a simple average fails is that speed is inversely proportional to time for a fixed distance. If you travel a certain distance slowly, you spend more time at that lower speed. Since average speed is weighted by time, the result will always be 'pulled' closer to the slower speed. For instance, if a vehicle covers different distances in different time intervals, we must sum all distances and divide by the sum of all time intervals to find the true average Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119.

A very specific and frequent UPSC scenario involves a round trip or a journey divided into two equal distances. If the speed for the first half is 'a' and the second half is 'b', the average speed formula simplifies to the Harmonic Mean: 2ab / (a + b). This formula is derived directly from the 'Total Perspective' by setting the distance for each leg as 'D' and calculating Total Distance (2D) divided by Total Time (D/a + D/b).

Scenario	Calculation Method
Different distances and different times	(Total Distance₁ + Distance₂ + ...) / (Total Time₁ + Time₂ + ...)
Two equal distances with speeds 'a' and 'b'	2ab / (a + b)

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119

4. Relative Speed: Objects in Motion (intermediate)

To understand Relative Speed, we must first recognize that motion is never absolute; it is always measured from a specific point of reference. Imagine sitting on a merry-go-round turning anti-clockwise. To you, a stationary tree outside appears to be moving clockwise. This phenomenon, where objects appear to move in the opposite direction of your own motion, is the foundational intuition of relative velocity Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.170. In quantitative aptitude, relative speed is the speed of one moving body as observed from another moving body. It tells us how fast the distance between two objects is either increasing or decreasing. There are two primary scenarios you will encounter when two objects, like trains or cars, are in motion:

• Opposite Directions: When two objects move toward each other (or away from each other in opposite directions), they 'cooperate' to cover the distance. Their relative speed is the sum of their individual speeds (S₁ + S₂). This is why a train passing you from the opposite direction seems to zip by much faster than its actual speed.
• Same Direction: When one object is chasing another, the relative speed is the difference between their speeds (|S₁ - S₂|). The faster object must 'overcome' the speed of the slower one to close the gap.

Scenario	Relative Speed Formula	Effect on Distance
Moving toward each other	S₁ + S₂	Distance decreases rapidly
One chasing the other	S₁ - S₂	Distance decreases slowly

In practical terms, calculating these speeds accurately requires consistent units. As we see in railway operations, timing and distance between stations are critical for determining the speed of a train Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.114. Whether you are calculating the time it takes for two trains to cross or a police car to catch a thief, always identify the relative speed first before applying the standard formula: Distance = Relative Speed × Time.

Sources: Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.170; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.114

5. Speed-Time Inversity and Ratio Method (intermediate)

In quantitative aptitude, understanding the relationship between speed, time, and distance is fundamental. As defined in basic physics, Speed is the distance covered by an object in a unit of time (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113). When we keep the Distance (D) constant, a fascinating mathematical relationship emerges: speed and time become inversely proportional. This means if you increase your speed, the time taken to cover the same distance decreases proportionally, and vice versa.

The Ratio Method is the most efficient way to leverage this inversity. If the ratio of the speeds of two objects (or the same object in two different scenarios) is a : b, then the ratio of the time taken to cover the same distance will be b : a. This simple flip allows you to solve complex "late or early" arrival problems without ever needing to calculate the actual distance first. For instance, if a student increases their speed from 40 km/h to 60 km/h (a ratio of 2:3), the time taken will automatically shift from 3 units to 2 units.

Scenario	Speed Ratio	Time Ratio (Constant Distance)
Speed Increases (e.g., by 50%)	2 : 3	3 : 2
Speed Decreases (e.g., by 25%)	4 : 3	3 : 4

This concept is also the secret behind calculating Average Speed for a round trip. Since the distance going and coming back is the same, we cannot simply find the arithmetic average of the speeds. Because more time is spent traveling at the slower speed, the average is "pulled" toward the lower value. This is why we often use the harmonic mean formula or the ratio-based approach to ensure we account for the total distance divided by the total time taken (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115).

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115

6. Average Speed for Equal Distance Segments (Harmonic Mean) (exam-level)

At its core, average speed is defined as the total distance covered divided by the total time taken Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. While this sounds simple, a common pitfall in competitive exams occurs when a journey is split into equal distance segments but travelled at different speeds. Many students instinctively calculate the simple arithmetic mean (adding the speeds and dividing by two), but this is incorrect because the object spends more time travelling the segment where its speed is slower Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115. To find the true average speed when distances are equal (such as a round trip or two halves of a journey), we use the Harmonic Mean. If an object covers the first half of a distance at speed x and the second half (the same distance) at speed y, the formula for average speed is 2xy / (x + y). For example, if you drive to a destination at 40 km/h and return at 60 km/h, you aren't spending an equal amount of time at both speeds; you are spending more time on the slower 40 km/h leg. Therefore, the average speed will be pulled closer to the slower speed than the simple average of 50 would suggest.

Scenario	Calculation Method	Why?
Equal Time Intervals	Arithmetic Mean: (x + y) / 2	Distance varies, but time spent at each speed is the same.
Equal Distance Segments	Harmonic Mean: 2xy / (x + y)	Time varies; more time is spent at the slower speed.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115

7. Solving the Original PYQ (exam-level)

This question is a classic application of the Average Speed concept you just mastered. It brings together two building blocks: calculating percentage increases and applying the Harmonic Mean for constant distance. In your recent lessons, you learned that when a journey is divided into two equal distance segments, the average speed is not a simple arithmetic average because the time spent at each speed differs. To solve this, you first need to determine the return speed: a 50% increase on 40 kmph means 40 + (0.50 × 40), giving you a return speed of 60 kmph.

To arrive at the correct answer, think like a coach: since the distance from X to Y is the same as Y to X, you should use the formula 2xy/(x+y). Plugging in your values, you get (2 × 40 × 60) / (40 + 60), which simplifies to 4800 / 100 = 48 kmph. Alternatively, if you prefer the Total Distance / Total Time approach, assuming a distance of 120 km (the LCM of 40 and 60) makes the calculation seamless. The outbound trip takes 3 hours and the return takes 2 hours, resulting in 240 km / 5 hours, which confirms 48 kmph as the correct choice.

UPSC frequently uses distractors to test your conceptual clarity. Option (B) 45 kmph is the most common arithmetic mean trap, where students mistakenly just average the two speeds (40+50)/2 or (40+60)/2. Option (D) 50 kmph is another trap for those who confuse a "50% increase" with the number "50" itself. Remember, because you spend more time traveling at the slower speed (40 kmph) than the faster speed (60 kmph), the average speed must always be pulled closer to the lower value. This logical check immediately makes 48 kmph the most plausible answer compared to the simple average.