Three men start together to travel the same way around a circular track of 11 km. Their speeds are 4, 5-5 and 8 kmph respectively. When will they meet at the starting point for the first time?

1. Foundations of Motion: Speed, Distance, and Time (basic)

Welcome to your first step in mastering quantitative aptitude! At its heart, the study of motion is about understanding the relationship between three fundamental variables: Speed, Distance, and Time. We define speed as the distance covered by an object in a unit of time, whether that unit is a second, a minute, or an hour Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.113. It is the most reliable way to determine which of two moving objects is faster; instead of looking at them, we calculate how much ground they cover in the same span of time.

The mathematical relationship between these three is the cornerstone of almost every aptitude problem involving motion. By knowing any two values, you can always find the third using these variations of the same core formula:

To Find...	Formula	Context
Speed	Total Distance ÷ Total Time	Determining how fast an object moves Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.113.
Distance	Speed × Time	Finding the total length of the path covered Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.115.
Time	Distance ÷ Speed	Calculating the duration of the journey Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.115.

In real-world scenarios, objects rarely move at a perfectly constant speed throughout their journey. For example, a bus might slow down at traffic lights and speed up on highways. In such cases, we use the term Average Speed, which is simply the total distance covered divided by the total time taken for the entire trip Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.113. When solving problems, always pay close attention to units. If distance is in kilometers (km) and time is in hours (h), speed is in km/h. If distance is in metres (m) and time is in seconds (s), speed is in m/s.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.115

2. Understanding Periodic Motion and Time Intervals (basic)

To master quantitative aptitude, we must first understand the rhythm of movement. Periodic motion is any motion that repeats itself at regular intervals of time. Think of a simple pendulum: as the metallic 'bob' swings from one side to the other and back to its starting point, it completes one oscillation Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.109. The time it takes to complete exactly one such cycle is known as its Time Period. This concept isn't just for clocks; it is the foundation for solving problems involving athletes running in circles or planets orbiting the sun. In the context of competitive exams, periodic motion often appears as 'circular track' problems. When a person runs around a track, their motion is periodic because they return to the starting point repeatedly. To find the 'Time Period' of one lap, we use the fundamental relationship: Time = Distance / Speed. If the runner maintains a constant speed, they are in uniform motion, and their lap time (time period) remains constant Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.118. Understanding the individual time period of different moving objects allows us to predict when they will synchronize. If three people start running at the same time but have different lap times, they will only meet at the starting point again when the total time elapsed is a common multiple of all their individual time periods.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.109; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.118

3. Number Systems: Multiples and the Lowest Common Multiple (LCM) (basic)

In our journey through quantitative aptitude, understanding Multiples is fundamental. A multiple of a number is simply the product of that number and any whole number (1, 2, 3, etc.). For instance, if a runner completes a lap every 4 minutes, they will be at the starting point at the 4th, 8th, 12th, and 16th minute. These are the multiples of 4. The Lowest Common Multiple (LCM) is the smallest positive integer that is divisible by each of the numbers in a set. Conceptually, the LCM represents the first moment in time when different cycles or "rhythms" coincide or sync up perfectly.

In many UPSC problems, you aren't just dealing with whole numbers, but with fractions—especially when calculating time using the formula Time = Distance / Speed, as discussed in Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 113. When you need to find the LCM of several fractions (to determine when multiple people on a circular track will meet at the start, for example), we use a specific mathematical rule:

For example, if three athletes have lap times of 11/4 hours, 2/1 hours, and 11/8 hours, we find their meeting time by taking the LCM of the numerators (11, 2, and 11), which is 22, and the HCF of the denominators (4, 1, and 8), which is 1. Thus, the LCM is 22/1, or 22 hours. This ensures that the time reached is a perfect multiple for all three participants, marking their simultaneous return to the starting point.

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

4. Circular Motion Dynamics and Track Problems (intermediate)

In our previous discussions, we looked at motion along a straight line, known as linear motion Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.116. However, track problems usually involve circular motion, where the path eventually returns to the starting point. The fundamental rule remains the same: Speed = Distance / Time Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.115. In a circular track problem, the 'distance' for one complete circuit is the length of the track. To master these problems, the first step is always to calculate the Lap Time for each individual—the time it takes for one person to complete exactly one round. To find when multiple people will meet specifically at the starting point, we are looking for a moment in time that is a common multiple of all their individual lap times. This is because Person A is at the start after 1 lap, 2 laps, 3 laps, and so on. They will all coincide at the start at the Least Common Multiple (LCM) of their respective lap times. If the lap times are simple integers, the LCM is straightforward. If they are fractions, we use a specific mathematical property: For example, if the track is 11 km and three people have speeds of 4 km/h, 5.5 km/h, and 8 km/h, their lap times would be 11/4, 11/5.5 (which is 2 or 2/1), and 11/8 hours Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.113. By finding the LCM of these three fractions, we determine the exact hour when their cycles synchronize at the starting line. This logic applies whether you are tracking athletes on a field or planets in orbit!

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113, 115, 116

5. Application of LCM: Synchronizing Recurring Events (intermediate)

In quantitative aptitude, synchronization refers to the exact moment when multiple independent events, each repeating at different intervals, occur at the same time. Think of the rhythmic tolling of school bells or the way athletes on a circular track eventually cross the starting line together. As we understand from Science-Class VII, Measurement of Time and Motion, p.113, different objects move at different speeds, meaning they will complete the same distance in different amounts of time. To find when they will 'sync up,' we must find a common multiple of their individual time periods.

The core mathematical tool for synchronization is the Least Common Multiple (LCM). If Runner A completes a lap every 4 minutes and Runner B every 6 minutes, they will meet at the start at the 12th minute (the LCM of 4 and 6). However, in intermediate problems, these time intervals are often fractions (e.g., 11/4 hours). When dealing with fractions, we use a specific formula to find the LCM:

To solve a complex synchronization problem, follow these steps:

1. Calculate the Time Period: If you are given distance and speed, first find the time for one cycle using the formula Time = Distance / Speed, as explored in Science-Class VII, Measurement of Time and Motion, p.118.
2. Standardize Units: Ensure all time intervals are in the same units (all seconds, all minutes, or all hours).
3. Find the LCM: Calculate the LCM of these time intervals. This result represents the first time all events will occur simultaneously again.

Runners on a circular track

Scenario	Goal	Mathematical Approach
Bells tolling/Lights flashing	Find the next simultaneous occurrence	LCM of the time intervals
Find when they meet at the starting point	LCM of the time taken to complete one full lap

Sources: Science-Class VII, Measurement of Time and Motion, p.113; Science-Class VII, Measurement of Time and Motion, p.118

6. The Arithmetic of Fractions: LCM and HCF of Fractions (exam-level)

In the realm of quantitative aptitude, we often encounter scenarios where we need to find the common multiple or highest common factor of values that aren't whole numbers—such as lap times on a track or intervals of light flashes. When dealing with fractions, the standard LCM and HCF rules require a specific adjustment to account for both the numerator (the part) and the denominator (the whole). As we explore in the context of motion and speed, time is often expressed as a fraction of distance over speed Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 113. To handle these, we apply the following two fundamental formulas:

• LCM of Fractions = (LCM of Numerators) / (HCF of Denominators)
• HCF of Fractions = (HCF of Numerators) / (LCM of Denominators)

There is one golden rule you must never forget: before applying these formulas, you must ensure all fractions are reduced to their lowest terms (simplest form). If you calculate the HCF or LCM using unsimplified fractions like 2/4 instead of 1/2, your result will often be mathematically incorrect for the set. For example, to find the LCM of 1/2 and 5/8, you take the LCM of the numerators (1 and 5 is 5) and divide it by the HCF of the denominators (2 and 8 is 2), giving you 5/2 or 2.5. These calculations are vital for solving 'meeting point' problems. When multiple objects move at different speeds, their individual lap times are fractions. The moment they all meet again at the starting point is simply the LCM of those fractional times. Understanding this allows you to bridge the gap between basic arithmetic and complex circular motion problems often seen in the UPSC CSAT.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.113

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamentals of Time, Speed, and Distance and the properties of LCM, this PYQ serves as the perfect synthesis of those concepts. In circular motion problems, the key is to recognize that for individuals to meet at the starting point, the elapsed time must be a perfect multiple of each individual's lap time. By applying the formula Time = Distance / Speed, as outlined in Science-Class VII . NCERT(Revised ed 2025), we first determine the time taken for one complete revolution: Man A takes 11/4 hours, Man B takes 2 hours (11/5.5), and Man C takes 11/8 hours.

To find the first instance they all coincide at the start, we must calculate the LCM of these fractional times: 11/4, 2/1, and 11/8. Remember the golden rule for fractions: the LCM is the LCM of the numerators (11, 2, 11) divided by the HCF of the denominators (4, 1, 8). The LCM of 11 and 2 is 22, and the HCF of 4, 1, and 8 is 1. This gives us a result of 22 hours. This logic ensures that by the 22nd hour, Man A has completed exactly 8 laps, Man B has completed 11 laps, and Man C has completed 16 laps, placing them all precisely back at the beginning.

In the UPSC CSAT, options are rarely random. Option (A) 11 hours is a calculation trap for students who might mistakenly find the LCM of the speeds or the distance itself without converting to time first. Option (D) 33 hours is a multiple trap; while they would indeed meet at the 33-hour mark if the math were different, the question specifically asks for the first time they meet. Always stay vigilant: calculate individual lap times first, then synchronize them using the LCM of fractions to avoid these common pitfalls.