CAT Quant Distance, Time and Speed: Two-End-and-Fro and Circular Track Concepts

CAT Quant Distance, Time and Speed questions test how well you apply basic motion concepts to different situations. Two-end-and-fro questions involve people moving between opposite or the same ends of a track, while circular-track questions use relative speed, round times and meeting points.

The key is to identify the movement pattern before applying a formula. This topic covers the common CAT Quant patterns for straight and circular tracks, including meeting points, repeated meetings, LCM-based questions and relative speed, with solved examples to help you understand each method.

What Is CAT Quant Distance, Time and Speed?

Distance, Time and Speed is a major topic in CAT Quant Arithmetic. In CAT, it mostly appears as motion on a straight track, where people go back and forth, and as motion on a circular track.

Everything starts with one base relation: Distance = Speed × Time. Every idea below is built on it. Draw the movement first, and the formula will do the rest.

How Do You Solve Two-End-and-Fro Questions When People Start from Opposite Ends?

In a two-end-and-fro motion, two people start from opposite ends of a track of length L. They meet, cross, touch the far ends, turn back and keep repeating. Questions ask about the 1st, 5th or nth meeting, the time, the distance walked or the place of the meeting.

For the first meeting, the two people together cover exactly L. After that, they need an extra 2L together for every new meeting. The table below shows the pattern.

Now you can follow a simple order for any question:

1. Find the combined distance.

2. Divide it in the ratio of their speeds. If both start together, the distance covered is in the same ratio as speed.

3. Track one person's distance along the track to find the place.

Example. The track is 250 m long. Two trains run at 10 m/s and 15 m/s from opposite ends. Find the time of the 5th meeting. The combined distance is 9 × 250 = 2250 m. The combined speed is 10 + 15 = 25 m/s, because their gap shrinks by the sum of their speeds. So time = 2250 ÷ 25 = 90 seconds.

The speed ratio is 2:3, so the two distances are 900 m and 1350 m. Take the first person's 900 m. He walks 250, 500 and 750 m, then 150 m back. He is 150 m from one end and 100 m from the nearest end. These two ways agree, since individual speed × time gives the same answer.

How Does Relative Speed Help in These Questions?

Relative speed is another way to see the same movement. Treat one person as standing still. If they move in opposite directions, the other person moves at the sum of the two speeds, which is 25 m/s in the example above.

The standing person is met first after L, and then after every extra 2L, so the pattern is the same. This method is very useful when the question does not give enough data for the usual combined-speed method.

What Happens When One Speed Is More Than Twice the Other?

This is the key special case. The normal L, 3L, 5L pattern holds only when the faster person is not more than twice as fast as the slower one. First, check which case you have: less than 2:1, exactly 2:1 or more than 2:1.

If the faster person is more than twice as fast, he finishes his trip and catches the slower person before the slower one finishes his first trip. So the second meeting is not at 3L, and you must draw what really happens.

Now look at the type where the track length is not given. Say the first meeting is 40 km from B and the second meeting is 30 km from A on the way back.

• The first meeting splits the track into 40 km and L − 40 km. The ratio of the two distances is the ratio of the speeds.

• In the normal case, the second meeting needs 3L together. So the person who walked 40 km in the first meeting walks 120 km by the second.

• That person went the full length and came back 30 km, so L + 30 = 120 and L = 90 km.

In the fast case, add up each person's real distance up to the second meeting and set the ratio equal to the speed ratio. This can give a quadratic equation, but the real task is drawing the right picture. If a question asks for the difference between the largest and smallest track length, solve both cases separately.

What Changes When Both Start from the Same End?

When both start from the same end, the pattern changes. The first meeting needs 2L together, and each next meeting needs another 2L. So the nth meeting needs 2nL combined distance.

Example. The track is 250 m, and the speeds are 4 and 6 m/s. For the 5th meeting, the combined distance is 10 × 250 = 2500 m. The combined speed is 10 m/s, so the time is 250 seconds. The two people cover 1000 m and 1500 m. Both are at an end of the track, so the distance from the nearest corner is zero.

When the total time is given instead, find the combined distance as combined speed × time. Then divide it by the distance needed per meeting. Suppose the track is 80 m, the speeds are 12 and 18 m/s, and the time is 100 seconds. The combined distance is 30 × 100 = 3000 m, and one meeting needs 2 × 80 = 160 m. Now 3000 ÷ 160 gives 18.75, so they meet 18 times. Count only the completed meetings.

How Do You Find Meeting at the Starting Point on a Circular Track?

On a circular track, the common question is when runners are back at the start together. Find each runner's time for one round, which is Track Length ÷ Speed. If all start together, the first time they are all at the start is the LCM of their round times.

Example. The track is 100 m, and the speeds are 5, 10 and 4 m/s. The round times are 20, 10 and 25 seconds. The LCM is 100 seconds. In that time, they run 5, 10 and 4 rounds.

Some points to remember:

• Clockwise or anticlockwise does not change the time of one round.

• On the same track, rounds are in the ratio of speeds.

• If the tracks are different, that speed rule fails. The rule that always works is that rounds are in the ratio of the reciprocals of the round times.

• Do not put in π or the real circumference. Make the ratio simple first, as the π cancels out.

CAT can also mix this with LCM and HCF. For example, the track lengths may form an increasing GP of whole numbers. Then the LCM is simply the largest term. For fractions, use the LCM of the numerators divided by the HCF of the denominators.

How Do You Find the First Meeting Point on a Circular Track?

The first meeting need not be at the start. Use relative speed here. In opposite directions, the speeds add, so time = L ÷ (a + b). In the same direction, the speeds subtract, so time = L ÷ (a − b).

The table below uses a 200 m track with speeds 5 m/s and 3 m/s.

For the nth meeting, the equation is (a + b) × t = nL for opposite directions and (a − b) × t = nL for the same direction. If a question says one person runs four more rounds than the other, write at − bt = 4L. Subtract here, whatever the direction, because it is about the distance gap and not about meeting. In same-direction questions, the difference in distance up to the first meeting equals L, which is a quick way to check your work.

How Many Unique Meeting Points Are There on a Circular Track?

If the speed ratio is m:n in simplest form, the number of different meeting points is m + n for opposite directions and m − n for the same direction. For 3:2, that gives 5 points in opposite directions and 1 point in the same direction.

To find the nth meeting point, divide n by the number of unique points and look at the remainder. With 8 points on a 100 m track, they are 12.5 m apart. The 65th meeting has remainder 1, so it is the first point. The 75th has remainder 3, so it is the third point.

Quick Summary of CAT Distance, Time and Speed Patterns

The table below puts all the main rules in one place. Use it for quick revision before a practice test.

For practice, solve CAT PYQs on this topic and always draw the track first. Note whether your mistake came from the wrong pattern, a wrong speed sign or a unit mix-up.

Distance, Time and Speed is easier when you practise a few full question sets and note which pattern each one uses. You can pair this topic with other CAT Quant Arithmetic notes and the CAT Quant Averages guide. A few CAT Quant mock tests and the PW CAT online course can help you check how quickly you spot the right pattern.