Section 12, arithmetic

Time, Speed & Distance

TSD is one of CAT’s most frequent arithmetic chapters and usually its trickiest. Nearly all of it runs on D = S × T, relative speed, and the habit of holding one quantity fixed so you can work in ratios.

How CAT asks it

Two people or trains start from different points, often at different times, and meet; a speed change makes someone late or early; average speed over several legs; trains passing poles, platforms, walkers and each other; boats in a current; races with head starts; runners on circular tracks; buses with stoppages.

Concepts you must know

  • D = S × T. Convert units first: km/h × 5/18 = m/s, and m/s × 18/5 = km/h.
  • Fixed distance: speed and time are inversely proportional. Fixed time: distance is proportional to speed.
  • Average speed = total distance ÷ total time. Equal times at speeds a and b give (a + b)/2.
  • Relative speed: add for opposite directions, subtract for the same direction.
  • Trains: passing a pole or a standing person takes length ÷ speed. Passing a platform or another train takes (sum of lengths) ÷ relative speed.
  • Boats: downstream = b + s and upstream = b − s, so b = (down + up)/2 and s = (down − up)/2.
  • Races: “A beats B by x m in an L m race” means B has run L − x when A finishes, so their speeds are in the ratio L : (L − x).
  • Circular track of length L, speeds a > b: first meeting after L/(a − b) in the same direction or L/(a + b) in opposite directions. First meeting at the start = LCM of the two lap times. With the speed ratio p : q in lowest terms, there are p − q distinct meeting points (same direction) or p + q (opposite).
  • On a straight track where both turn back at the ends, by the second meeting they have together covered 3 times its length.
  • Equal distances at speeds a and b, and the after-meeting rule (two start together towards each other and need t₁ and t₂ hours after meeting to finish; they met after √(t₁t₂) hours):
Average speed=2aba+b\text{Average speed} = \frac{2ab}{a+b}
S1S2=t2t1\frac{S_1}{S_2} = \sqrt{\frac{t_2}{t_1}}

Thought process

  1. Draw a line: mark the starting points, directions and start times.
  2. Convert all units to one system.
  3. Decide what is fixed and use ratios (fixed distance → inverse speeds).
  4. Two moving bodies → relative speed. One moving body → its own speed.
  5. Check the answer against the story: who should be faster, who should arrive first?

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Meeting point, often with one person starting later.
  2. 2After-meeting times → speed ratio through square roots.
  3. 3Late or early with a changed speed → distance or usual time.
  4. 4Average speed across legs.
  5. 5Trains: poles, platforms, walkers, other trains.
  6. 6Boats and streams.
  7. 7Races and head starts.
  8. 8Circular tracks: first meeting, first meeting at the start, number of meeting points.
  9. 9Ratio-only questions: “where do they meet?” needs only the speed ratio.
  10. 10Stoppages: minutes stopped per hour.
  11. 11To-and-fro trips and second meetings.

What the real papers show

In Cracku’s topic-wise TSD collection, the after-meeting square-root rule (pattern 2) appears in at least five slots between 2020 and 2024. Late/early trips (pattern 3) appear in 2017, 2020, 2024 and 2025; races (pattern 7) in 2017, 2019 and 2020; circular tracks (pattern 8) in 2019, 2020 and 2021; boats (pattern 6) in 2017, 2023 and 2025. An escalator question, a cousin of relative speed, appeared in 2022.

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1, late/early

    Walking at 4 km/h, Ravi reaches the station 10 minutes late; at 5 km/h he is 5 minutes early. How far is the station?

    Show solution

    The times differ by 15 minutes. Speeds 4 : 5 mean times 5 : 4, so 1 part = 15 minutes and the 4 km/h trip takes 75 minutes. Distance = 4 × 1.25. Answer: 5 km.

  2. Example 2, delayed start

    P and Q are 210 km apart. A car leaves P at 8:00 a.m. at 50 km/h; another leaves Q at 9:00 a.m. at 30 km/h towards P. When and where do they meet?

    Show solution

    By 9:00 the first car has covered 50 km. The 160 km gap closes at 80 km/h, taking 2 hours. Answer: 11:00 a.m., 150 km from P.

  3. Example 3, after-meeting times

    Two trains leave X and Y at the same time, heading towards each other. After meeting, the X-train takes 9 hours to reach Y and the Y-train takes 4 hours to reach X. Find the ratio of their speeds.

    Show solution

    Speed of X-train ÷ speed of Y-train = √(4/9) = 2/3, and they met after √(9 × 4) = 6 hours. Check with speeds 2 and 3: before meeting X covers 12 and Y covers 18; afterwards X needs 18/2 = 9 hours and Y needs 12/3 = 4. Answer: 2 : 3.

  4. Example 4, average speed

    A car goes from A to B at 40 km/h and returns at 60 km/h. Find its average speed.

    Show solution

    Equal distances: 2 × 40 × 60 ÷ 100. Answer: 48 km/h.

  5. Example 5, pole and platform

    A 240 m train passes a pole in 12 seconds. How long does it take to cross a 360 m platform?

    Show solution

    Speed = 20 m/s; it must cover 240 + 360 = 600 m. Answer: 30 seconds.

  6. Example 6, trains passing walkers

    A train overtakes a jogger running at 6 km/h in 9 seconds and a cyclist riding at 12 km/h, in the same direction, in 10 seconds. Find the train’s speed and length.

    Show solution

    Length = (v − 6) × 5/18 × 9 = (v − 12) × 5/18 × 10, so 9v − 54 = 10v − 120 and v = 66 km/h. Length = 60 × 5/18 × 9 = 150 m. Check: 54 km/h = 15 m/s, and 15 × 10 = 150. Answer: 66 km/h; 150 m.

  7. Example 7, boats

    A boat goes 30 km downstream in 2 hours and returns in 3 hours. Find the boat’s speed in still water and the stream’s speed.

    Show solution

    Downstream 15 km/h, upstream 10 km/h. Answer: boat 12.5 km/h; stream 2.5 km/h.

  8. Example 8, races

    In a 1,000 m race, A beats B by 100 m and B beats C by 100 m. By how much does A beat C?

    Show solution

    When A runs 1,000, B runs 900. C runs 90% of whatever B runs, so 810. Answer: 190 m.

  9. Example 9, circular track

    Two runners start together on a 600 m circular track at 5 m/s and 3 m/s in the same direction. When do they first meet, when do they first meet at the start, and at how many distinct points do they meet?

    Show solution

    Relative speed 2 m/s gives 600/2 = 300 s. Lap times are 120 s and 200 s, LCM 600 s. The speed ratio 5 : 3 gives 5 − 3 = 2 points (the start and the halfway mark). In opposite directions they would first meet after 600/8 = 75 s, at 5 + 3 = 8 points. Answer: 300 s; 600 s; 2 points.

  10. Example 10, stoppages

    Excluding stops, a bus averages 54 km/h; including stops, 45 km/h. For how many minutes per hour does it stop?

    Show solution

    It loses 9 km of travel each hour, which takes 9/54 hour at full speed. Answer: 10 minutes.

  11. Example 11, second meeting

    Two people start from opposite ends of a 1.6 km path at 5 km/h and 3 km/h, each turning back on reaching the far end. When and where do they meet for the second time?

    Show solution

    Together they cover 3 × 1.6 = 4.8 km at 8 km/h, taking 0.6 hours. The faster walker covers 3 km: 1.6 to the far end and 1.4 back, ending 0.2 km from his start. Check: the slower covers 1.8 km, reaching the faster walker’s end and coming 0.2 km back. Answer: after 36 minutes, 200 m from the faster walker’s start.

Traps

  • The average of two speeds is not the average speed, unless the times are equal.
  • Mixing metres with km/h.
  • In races, “beats by t seconds” means the loser needs t more seconds at his own speed.
  • “Meet at the start” on a circular track uses the LCM of lap times, not relative speed.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    A man travels at 30 km/h and returns at 20 km/h, taking 5 hours in all. Find the one-way distance.

    Show answer

    60 km: D/30 + D/20 = D/12 = 5.

  2. 2

    Trains 150 m and 100 m long run in opposite directions at 40 km/h and 50 km/h. How long do they take to cross each other?

    Show answer

    10 s: 250 m at 90 km/h = 25 m/s.

  3. 3

    A boat’s speed in still water is 15 km/h and the stream’s is 3 km/h. How long does it take to go 36 km downstream and come back?

    Show answer

    5 hours: 36/18 + 36/12.

  4. 4

    Walking at 3/4 of his usual speed, a man is 20 minutes late. What is his usual time?

    Show answer

    60 minutes: the time becomes 4/3 of usual, so the extra 1/3 is 20 minutes.

  5. 5

    A and B start towards each other from 100 km apart at 30 km/h and 20 km/h. Where do they meet?

    Show answer

    60 km from A’s start: they meet after 2 hours.

  6. 6

    Two runners start together on a 400 m circular track at 6 m/s and 4 m/s in opposite directions. When do they first meet, and when do they first meet at the start?

    Show answer

    40 s (400 ÷ 10) and 200 s (lap times of 200/3 s and 100 s; in 200 s they run exactly 3 and 2 laps).

  7. 7

    In a 200 m race, A beats B by 20 m or by 4 seconds. Find A’s time.

    Show answer

    36 s: B covers the last 20 m in 4 s, so B runs at 5 m/s and takes 40 s for the race; A finishes 4 s earlier.

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