Two trains leave New Delhi at the same time. One travels north at 60 kmph and the other travels south at 40 kmph. After how many hours will the trains be 150 km apart ?

1. Speed, Distance, and Time Fundamentals (basic)

Welcome to your first step in mastering Quantitative Aptitude! To understand how objects move, we start with the most fundamental concept: Speed. At its heart, speed is simply the measure of how much distance an object covers in a specific amount of time. As we see in Science-Class VII, Measurement of Time and Motion, p.113, comparing the distances moved by objects in a unit time (like one second or one hour) allows us to determine which one is moving faster.

The relationship between Speed, Distance, and Time is mathematically linked through a simple yet powerful formula. By knowing any two of these values, you can always find the third:

• Speed = Distance ÷ Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

In the real world, objects rarely move at the exact same pace throughout a journey. When an object moves along a straight line at a constant speed, we call it Uniform Motion Science-Class VII, Measurement of Time and Motion, p.117. However, if the speed changes—like a car slowing down for traffic—it is in Non-uniform Motion. In such cases, we often calculate the Average Speed by dividing the total distance covered by the total time taken Science-Class VII, Measurement of Time and Motion, p.119.

Finally, a crucial concept for competitive exams is Relative Speed. When two objects are moving, their speed relative to each other depends on their direction. If two objects move toward each other or away from each other (opposite directions), their relative speed is the sum of their individual speeds. Imagine two friends walking away from each other; the distance between them grows much faster than if only one was walking, because both are contributing to the increasing gap.

Sources: Science-Class VII, Measurement of Time and Motion, p.113; Science-Class VII, Measurement of Time and Motion, p.115; Science-Class VII, Measurement of Time and Motion, p.117; Science-Class VII, Measurement of Time and Motion, p.119

2. Unit Conversions and Dimensional Consistency (basic)

In the realm of quantitative aptitude, Unit Conversion is the bridge that allows us to compare different scales of measurement accurately. As we explore the study of motion, we find that the Standard International (SI) unit for speed is metres per second (m/s), which is logically derived from the units of length (metre) and time (second) Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113. However, for larger distances like the movement of trains or cars, we often use kilometres per hour (km/h). To solve any problem correctly, you must first ensure Dimensional Consistency—the principle that all variables in your formula (Distance, Speed, Time) must share a common unit family.

To master these conversions without confusion, we use the 5/18 Rule. Since 1 kilometre equals 1,000 metres and 1 hour equals 3,600 seconds (60 minutes × 60 seconds), the conversion factor is 1,000/3,600, which simplifies to 5/18. This is essential when comparing two different speeds, such as a galloping horse and a moving train, to see which is faster Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118.

Conversion Type	Operation	Example
km/h to m/s	Multiply by 5/18	90 km/h × (5/18) = 25 m/s
m/s to km/h	Multiply by 18/5	20 m/s × (18/5) = 72 km/h

Dimensional consistency acts as your personal "error checker." If you are calculating Time using the formula Time = Distance / Speed, your units must cancel out to leave only a time unit (like seconds or hours). If you try to divide 100 kilometres by 10 metres per second without converting, your answer will be numerically incorrect and dimensionally messy. Always align your units before you start the math!

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.118

3. Average Speed and Weighted Means (intermediate)

In our journey through quantitative aptitude, we must first master the fundamental definition of speed: 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. While we often speak of a bus moving at 50 km/h, in reality, vehicles rarely maintain a perfectly constant or uniform linear motion over long distances Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.117. They slow down for traffic, stop at stations, or speed up on highways. To account for these variations, we use the concept of 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.

A common mistake in competitive exams is treating average speed as a simple arithmetic average of the individual speeds. Instead, average speed is a weighted mean where the "weights" are the time durations spent at each speed. If you travel at 40 km/h for 4 hours and 60 km/h for only 1 hour, your average speed will be much closer to 40 than 60 because you spent more time at the slower speed. Therefore, always return to the first principle: Average Speed = (d₁ + d₂ + ...) / (t₁ + t₂ + ...).

Scenario	Condition	Simplified Formula
Equal Time Intervals	t₁ = t₂	(v₁ + v₂) / 2 (Arithmetic Mean)
Equal Distances	d₁ = d₂	2v₁v₂ / (v₁ + v₂) (Harmonic Mean)

When solving complex problems involving multiple segments, it is often helpful to calculate the time taken for each part of the journey separately using the relation Time = Distance / Speed Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115. Once you have the total distance and the sum of all time intervals, the average speed becomes a simple division. This approach ensures accuracy even when the object moves with non-uniform speed across different sections Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119.

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

4. Boats and Streams: Relative Motion in Fluids (intermediate)

To master the concept of Boats and Streams, we must first understand that motion in a fluid is never isolated; it is always influenced by the movement of the medium itself. When a boat moves in a river, its resultant speed depends on whether it is moving with the flow or against it. In geography, the speed of this water movement is often referred to as its "drift" FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.111. Just as Indian Railways aims to increase train speeds on specific corridors Indian Economy, Vivek Singh (7th ed. 2023-24), Infrastructure and Investment Models, p.413, a boat's speed is boosted or hindered by the environment it travels through.

There are two primary conditions of motion in this topic: Downstream and Upstream. While these terms are used in industry to describe stages of production—from raw material extraction (upstream) to final processing (downstream) Indian Economy, Vivek Singh (7th ed. 2023-24), Supply Chain and Food Processing Industry, p.363—in quantitative aptitude, they describe the direction of travel relative to the water current. If the speed of a boat in still water is u and the speed of the stream is v, their relationship is as follows:

Term	Direction	Resultant Speed Formula
Downstream	With the flow of the river	Speed (D) = u + v
Upstream	Against the flow of the river	Speed (U) = u - v

A fascinating application of relative speed occurs when two boats move toward each other on the same river—one going downstream and the other upstream. In this specific scenario, the speed of the stream (v) effectively cancels out. The relative speed of the two boats becomes simply the sum of their speeds in still water (u₁ + u₂). This allows us to use the fundamental relation where Time = Distance / Speed Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115 to find when or where they will meet.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.111; Indian Economy, Vivek Singh (7th ed. 2023-24), Infrastructure and Investment Models, p.413; Indian Economy, Vivek Singh (7th ed. 2023-24), Supply Chain and Food Processing Industry, p.363; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.115

5. Linear Races and Circular Motion Basics (intermediate)

To master quantitative aptitude, we must first distinguish between Linear Motion and Circular Motion. In linear motion, an object moves along a straight path—like a train on a track or a car on a highway. We define Uniform Linear Motion as an ideal state where an object covers equal distances in equal intervals of time Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.117. While real-world movement is often non-uniform, most competitive exam problems assume uniform speed to test your grasp of the core relationship: Distance = Speed × Time.

When two objects move simultaneously, we use the concept of Relative Speed. This is the speed of one object as observed from the other. The rules are intuitive yet vital:

• Opposite Directions: If two objects move toward each other (or away from each other), they "close" or "create" the gap faster. Therefore, we add their speeds (V₁ + V₂).
• Same Direction: If one object is chasing another, the gap closes only by the difference in their speeds. Therefore, we subtract their speeds (|V₁ - V₂|).

This logic applies whether people are running a race along a road or settlements are developing in a Linear Pattern along railway tracks or narrow valleys Geography of India, Majid Husain, Settlements, p.7.

Circular Motion introduces the complexity of the track's geometry. In a circular race, the "distance" for one full lap is the circumference. A critical point of fairness arises here: on a standard 400-meter track, the outer lanes have a larger radius than the inner lanes. If all runners started at the same linear line, those in the outer lanes would actually run a longer distance! This is why competitors in outer lanes are placed ahead at the start—to ensure every participant covers the exact same distance Democratic Politics-I, Political Science-Class IX, NCERT (Revised ed 2025), DEMOCRATIC RIGHTS, p.80. On a global scale, we see this principle in "Great Circles," which represent the shortest distance between two points on a sphere Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14.

Sources: Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.117; Geography of India, Majid Husain, Settlements, p.7; Democratic Politics-I, Political Science-Class IX, NCERT (Revised ed 2025), DEMOCRATIC RIGHTS, p.80; Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14

6. Relative Speed: Same vs. Opposite Directions (exam-level)

In our journey through quantitative aptitude, we have mastered the basic relationship where Speed = Distance ÷ Time Science-Class VII . NCERT(Revised ed 2025) | Measurement of Time and Motion | p.113. However, exam-level problems often involve two objects moving simultaneously. This brings us to the concept of Relative Speed — the speed of one object as observed from another moving object.

Think about sitting on a moving merry-go-round; objects around you appear to move in the opposite direction of your rotation Science-Class VII . NCERT(Revised ed 2025) | Earth, Moon, and the Sun | p.170. This intuitive sense of motion tells us that our own speed affects how we perceive the speed of others. In competitive exams, we simplify this into two primary scenarios based on the direction of travel:

Scenario	Relative Speed Formula	Logic
Opposite Directions (Moving towards or away)	Speed₁ + Speed₂	The distance between them changes faster because both are contributing to the gap.
Same Direction (Overtaking or Chasing)	Speed₁ - Speed₂	The distance changes slower because the trailing object has to "cancel out" the speed of the one ahead.

When solving these problems, the most critical step is to calculate the Relative Speed first, then substitute it into the standard formula: Time = Distance ÷ Relative Speed. For instance, if two trains are 360 km apart and moving toward each other Science-Class VII . NCERT(Revised ed 2025) | Measurement of Time and Motion | p.115, their combined speed determines how quickly that 360 km gap closes.

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

7. Solving the Original PYQ (exam-level)

This question is a textbook application of the Relative Speed concept you just mastered. In the CSAT, the UPSC often tests your ability to identify the direction of motion before applying any formula. Since one train travels North and the other South, they are moving in opposite directions. This means the gap between them increases at the combined rate of both speeds. By bridging your understanding of the Speed-Distance-Time formula with this additive rule, you can solve the problem efficiently as demonstrated in Quantitative Aptitude for Competitive Examinations.

To arrive at the solution, first calculate the relative speed by adding the individual speeds: 60 kmph + 40 kmph = 100 kmph. The problem asks for the time required for the trains to be 150 km apart. Using the fundamental relation Time = Distance / Speed, we calculate 150 / 100. After simplifying the fraction by dividing both terms by 50, we arrive at 3/2. Thus, the correct answer is (A) 3/2 hours. The key here is the immediate recognition that opposite vectors necessitate addition.

UPSC purposefully includes distractor options to catch common conceptual slips. For example, if a student mistakenly subtracted the speeds (an error of assuming same-direction motion), they would get a relative speed of 20 kmph, leading to 150/20 = 7.5, which is (D) 15/2. Option (C) 3/4 is a trap for those who might accidentally invert the formula to Speed / Distance. Always double-check your relative motion logic—opposite directions add, same directions subtract—to navigate these traps successfully.