A train completes a journey with a few stoppages in between at an average speed of 40 km per hour. If the train had not stopped anywhere, it would have completed the journey at an average speed of 60 km per hour. On an average, how many minutes per hour does the train stop during the journey?

1. Fundamental Speed-Distance-Time Relationship (basic)

At its heart, speed is simply a measure of how fast an object moves. Specifically, speed is defined as the distance covered by an object in a unit of time — whether that unit is one second, one minute, or one hour Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. To master any quantitative aptitude problem involving motion, you must be comfortable with the three-way relationship between Speed (S), Distance (D), and Time (T). This relationship is expressed by the fundamental formula: Speed = Total Distance ÷ Total Time.

In the real world, objects rarely move at a perfectly steady pace. A car might slow down for traffic and speed up on a highway. This is called non-uniform motion. Because of this variation, we often calculate the Average Speed, which represents the total distance covered divided by the total time taken for the entire journey Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115, 119. If an object does move at a constant speed along a straight line, we call it uniform linear motion, meaning it covers equal distances in equal intervals of time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117.

Type of Motion	Characteristics	Example
Uniform	Constant speed; equal distance in equal time.	A light beam in a vacuum.
Non-Uniform	Changing speed; unequal distance in equal time.	A bus moving through city traffic.

One critical skill for competitive exams is unit conversion. Often, distance is given in kilometers and time in seconds, or speed in km/h and you need the answer in m/s. To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

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.119

2. Understanding Average Speed vs. Instantaneous Speed (intermediate)

In our journey to master quantitative aptitude, we must first distinguish between how an object moves at a specific moment versus its performance over a whole trip. Imagine you are driving a car: at one moment you might be cruising at 80 km/h, but a minute later, you could be at a complete standstill at a red light. The speed you see on your speedometer at any exact second is the instantaneous speed. However, for most competitive exams, we focus on the average speed, which 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). This calculation effectively "smooths out" all the accelerations, decelerations, and pauses into one single representative value.

In the real world, objects rarely move at a perfectly constant rate. This is known as non-uniform linear motion (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118). A critical concept to grasp here is how stoppages affect your results. Even if a train's engine is powerful enough to run at 100 km/h, if it stops at several stations along the way, the total time increases while the total distance remains the same. Consequently, the average speed for the entire journey will always be lower than the actual running speed of the train (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115).

Feature	Instantaneous Speed	Average Speed
Definition	Speed at a specific point in time.	Total distance ÷ Total time.
Stoppages	Is zero while the object is stopped.	Includes "stop time" in the calculation.
Consistency	Fluctuates in non-uniform motion.	A single value for the whole trip.

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.118

3. Proportionality between Speed and Time (intermediate)

To master quantitative aptitude, we must look beyond the basic formula of Speed = Distance / Time. At its core, this relationship tells us how physical quantities interact. When the distance remains constant, speed and time share an inverse proportionality. This means that if you double your speed, the time taken to cover that same distance is exactly halved. Mathematically, we express this as Speed ∝ 1/Time (when Distance is fixed).

In the context of motion, we define speed as the distance covered in a unit of time (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113). However, in real-world scenarios, objects rarely move at a perfectly constant rate. We often distinguish between uniform linear motion, where an object covers equal distances in equal intervals, and non-uniform motion, where speed fluctuates (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117). Understanding proportionality allows us to bridge the gap between these states—for instance, calculating how much "extra time" is added to a journey when the average speed drops due to external factors like traffic or stops.

When solving complex problems, remember that if the ratio of two speeds is x : y, the ratio of the time taken to cover the same distance will be y : x. This inverse relationship is the secret weapon for solving "stoppage" or "late/early" problems without needing to calculate the actual distance every time. As noted in common physics principles, the time taken can be found if both distance and speed are known (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115), but the ratio method often provides a much faster shortcut.

Variable Held Constant	Relationship between others	Proportionality Type
Distance	Speed and Time	Inverse (S ∝ 1/T)
Time	Speed and Distance	Direct (S ∝ D)
Speed	Distance and Time	Direct (D ∝ T)

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

4. Relative Speed and Train Mechanics (intermediate)

When we talk about Relative Speed, we are essentially looking at how fast one object moves in relation to another. Think of sitting on a merry-go-round; as you turn anti-clockwise, the trees outside appear to move clockwise. This shift in perspective is the essence of relative motion Science-Class VII, Earth, Moon, and the Sun, p.170. In the context of competitive exams, this translates into two primary rules: if two objects move in the opposite direction, their speeds are added (they close the gap faster); if they move in the same direction, the speed of the slower object is subtracted from the faster one.

Train Mechanics adds a layer of complexity because a train is not a mathematical "point"—it has significant length. When a train passes a stationary point (like a pole or a person), the distance it covers is equal to its own Length (L). However, when it crosses a platform or another train, the total distance covered is L₁ + L₂. We use the fundamental formula Speed = Distance / Time to solve these, but the "Speed" we use must be the Relative Speed if both objects are moving Science-Class VII, Measurement of Time and Motion, p.114.

A frequent and tricky application of these principles involves Stoppage Time. Often, a train's average speed is given in two scenarios: excluding stoppages (faster) and including stoppages (slower). To find the stoppage time per hour, you don't need complex algebra. Simply find the "lost distance" in one hour and calculate how much time the train would have taken to cover that distance at its original speed. For example, if a train drops from 60 km/h to 40 km/h due to stops, it "lost" 20 km of travel. At its full speed of 60 km/h, it would take 20/60 hours (or 20 minutes) to cover that gap. Thus, it stopped for 20 minutes every hour.

Scenario	Relative Speed Formula	Effective Distance
Opposite Directions	S₁ + S₂	L₁ + L₂
Same Direction	S₁ - S₂	L₁ + L₂
Crossing a Pole	S₁	L₁

Sources: Science-Class VII (NCERT), Earth, Moon, and the Sun, p.170; Science-Class VII (NCERT), Measurement of Time and Motion, p.114; Science-Class VII (NCERT), Measurement of Time and Motion, p.115

5. Unit Conversions and Time Fractions (basic)

In quantitative aptitude, mastering the relationship between different units of measurement is the bedrock of speed and time problems. As defined in the SI system, the standard unit for distance is the metre (m) and for time is the second (s); therefore, the standard unit of speed is m/s Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. However, in our daily lives and UPSC problems, we frequently encounter kilometres per hour (km/h). To convert between these, we use the factor of 5/18. Why? Because 1 km/h = 1000m / 3600s, which simplifies to 5/18 m/s. Understanding this allows you to compare a galloping horse (approx. 18 m/s) with a train moving at 72 km/h (which is also 20 m/s) with ease Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118.

Time itself is measured in intervals of 60: 60 seconds make 1 minute, and 60 minutes make 1 hour Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111. This sexagesimal system is also vital in geography, where we calculate local time based on longitude. Since the Earth rotates 360° in 24 hours, it covers 15° in one hour, or 1° every 4 minutes Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.11. Recognizing these fractions of time (like 4 minutes being 1/15th of an hour) is a crucial skill for solving complex aptitude questions.

A common application of time fractions is the "stoppage problem." When a vehicle's average speed decreases due to stops, the "lost" speed tells us how much time was spent standing still. If a train's speed drops from 60 km/h (fast) to 40 km/h (with stops), it "loses" 20 km of travel every hour. To find the stoppage time, we ask: How long would it have taken to cover that 20 km at the original speed? The answer is 20/60 = 1/3 of an hour. Converting this fraction to minutes (1/3 × 60) gives us 20 minutes of stoppage per hour.

Movement Type	Calculation Logic	Result Unit
Pure Running	Distance / Running Time	km/h (Excluding stops)
With Stoppages	Distance / (Running + Stop Time)	km/h (Including stops)
Stoppage Time	(Speed Diff / Original Speed) × 60	minutes per hour

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111, 113, 118; Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.11

6. Effective Speed with Stoppages (The Logic of Delay) (exam-level)

In the study of motion, 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). However, in real-world scenarios like train journeys or bus routes, vehicles rarely move at a constant speed without interruption. When we talk about Speed with Stoppages, we are effectively looking at how 'dead time' (time spent stationary) reduces the average speed over a journey. Even if a train maintains a high uniform motion while moving, the inclusion of stops makes the overall motion non-uniform (Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119), resulting in a lower effective speed for the entire trip. To master the logic of delay, we must understand that the difference between the 'speed excluding stoppages' (S₁) and the 'speed including stoppages' (S₂) represents the distance the vehicle could have traveled during the time it was stopped. For instance, if a bus has a speed of 60 km/h without stops but averages 45 km/h with stops, it 'lost' 15 km worth of distance in one hour. Since its actual running speed is 60 km/h, the time it would take to cover those 15 km is the stoppage time. Mathematically, the stoppage time per hour is calculated as:

Stoppage Time per hour = (Difference in Speeds) / (Speed excluding stoppages) When solving these problems, it is crucial to keep units consistent. While we often use km/h for large-scale journeys, some contexts like maritime navigation use knots (nautical miles per hour), where 1 knot is approximately 1.852 km/h (Physical Geography by PMF IAS, Tropical Cyclones, p.372). Always ensure your final answer for stoppage time is converted from a fraction of an hour into minutes (by multiplying by 60) to match common exam formats.

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.119; Physical Geography by PMF IAS, Tropical Cyclones, p.372

7. Solving the Original PYQ (exam-level)

This problem beautifully synthesizes the concepts of Average Speed and Time-Distance relationships. You’ve recently mastered the foundational formula Speed = Distance / Time; here, you must apply it to understand how stoppage time acts as a "theft" of travel time. To solve this efficiently, imagine the train traveling for exactly one hour without stopping. In that hour, it covers 60 km. However, when it includes stoppages, it only covers 40 km in that same hour. This means the train "lost" 20 km worth of distance because it was standing still. The key coaching insight is to ask: How long would it have taken the train to cover that missing 20 km if it hadn't stopped?

By calculating the time required to cover the 20 km difference at the original speed of 60 km/h, we calculate 20/60, which equals 1/3 of an hour. Converting this fraction into time, 1/3 × 60 minutes gives us exactly 20 minutes per hour, which is the correct answer (A). A helpful shortcut often used in the UPSC CSAT is the direct formula: (Difference in Speed / Speed without Stoppage) × 60. This logic helps you bypass complex algebraic variables and focus on the rate of time loss relative to the train's full potential.

UPSC often includes distractors like (C) 15 minutes or (D) 10 minutes to catch students who might mistakenly use the slower average speed (40 km/h) as the denominator or perform a hasty subtraction without considering the unit of time. Option (B) 18 minutes is a classic trap for those who make minor calculation errors under pressure. Always remember: the stoppage time must be calculated relative to the maximum potential speed (the non-stop speed), as that represents the true motion of the train when it is actually running. For further conceptual refinement, refer to Quantitative Aptitude for Competitive Examinations by R.S. Aggarwal.