Three quantities, one equation, 36 Punjab Police SI questions since 2016. Distance = speed × time is the whole of it, and the Commission spends most of its ingenuity on two things: making you convert units, and making you work out an average speed when the journey happens at two different speeds. Seventeen of the 36 are average-speed questions — almost half — and nearly all of them are a trip out and back. Those have a one-line shortcut that most candidates never learn. You will get (1) the triangle, (2) the two unit conversions, (3) the average-speed formula and why the obvious answer is wrong, and (4) trains and relative speed.
The triangle and the two conversions
Fifteen of the 36 need nothing else.
km/h → m/s: multiply by 5/18. m/s → km/h: multiply by 18/5. Metres and seconds belong together; kilometres and hours belong together. Mixing them is the single biggest source of wrong answers in this chapter, and the SI knows it — almost every basic question gives you one unit and asks for the other.
A person crosses an 810-m-long street in 9 min. What is his speed (in km/h)? (SI 29 Jul 2023)
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810 m in 9 minutes. Speed = 810/9 = 90 m/min. In an hour that is 90 × 60 = 5,400 m = 5.4 km/h. Alternatively convert to m/s: 810/540 = 1.5 m/s, and 1.5 × 18/5 = 5.4. Same answer.
- Minutes are not hours. 1 hour 45 minutes is 1.75 hours, not 1.45.
- Read the unit the answer wants — the paper often gives km/h and asks for m/min.
- A “arrives early / arrives late” pair is one equation: the difference in times equals the total lateness plus earliness.
By travelling at a speed of 2.5 km per hour, a student gets late for the School by 6 minutes. By travelling at a speed of 3 km per hour, he reaches School 10 minutes early. What is the distance of his School from home? (SI 18 Aug 2021)
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Let the distance be d. At 2.5 km/h he is 6 minutes late; at 3 km/h he is 10 minutes early. The difference in travel times is 16 minutes = 16/60 hour. d/2.5 − d/3 = 16/60 ⇒ d(0.4 − 0.3333) = 0.2667 ⇒ d × 0.0667 = 0.2667 ⇒ d = 4 km.
Average speed: 17 questions
The biggest block, and the easiest to get wrong.
Never the average of the two speeds. When the distances are equal, the formula collapses to 2ab/(a + b) — the harmonic mean. For a = 40 and b = 10 that is 800/50 = 16 km/h, exactly what the long method gives. For three equal distances it is 3/(1/a + 1/b + 1/c). If instead the times are equal, the ordinary average does apply: (a + b)/2.
If a tourist travels by car to a place 120km away at an average speed of 45km/h and returns at a speed of 30km/h, then the average speed of the tourist for the whole journey is : (SI 14 Jul 2023)
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120 km each way at 45 and 30 km/h. Shortcut: 2 × 45 × 30 / (45 + 30) = 2700/75 = 36 km/h. Long way: 120/45 = 2⅔ h out, 120/30 = 4 h back, total 6⅔ h for 240 km ⇒ 36 km/h. The distance 120 never mattered — that is why the shortcut works.
A man covers a distance of 48 km at a speed of 6 km/h. He covers another distance of 48 km at 8 km/h and a third distance of 48 km at 12 km/h. Find his average speed (in km/h) for the whole journey. (SI 20 Jun 2023)
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Three equal distances of 48 km at 6, 8 and 12 km/h. Times: 8 h, 6 h, 4 h — total 18 hours for 144 km, so the average is 144/18 = 8 km/h. The harmonic form gives the same: 3/(1/6 + 1/8 + 1/12) = 3/(0.375) = 8.
Fall back to the definition. Add the distances, add the times, divide. A bus doing 360 km at 90 km/h and 255 km at 85 km/h takes 4 + 3 = 7 hours for 615 km, so the average is 615/7 = 87.86 km/h. No shortcut applies, and none is needed — the numbers are always chosen to divide neatly.
A man goes to a place on a bicycle at a speed of 18 km/h and returns at a slower speed. If the average speed for the whole journey is 8.4 km/h, find the return speed (in km/h). (SI 20 Jun 2023)
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Out at 18 km/h, return at an unknown v, average 8.4 km/h over equal distances. 2 × 18 × v / (18 + v) = 8.4 ⇒ 36v = 8.4(18 + v) ⇒ 36v = 151.2 + 8.4v ⇒ 27.6v = 151.2 ⇒ v = 5.48 km/h. The harmonic-mean formula read backwards — the SI asks it this way about once a year.
Trains and relative speed
Four questions, two ideas.
Same direction: subtract the speeds. Opposite directions: add them. Two trains at 60 and 24 km/h going the same way close the gap at 36 km/h = 10 m/s. In 30 seconds they cover 300 m — which is the combined length of the two trains, because one train must pass the whole of the other.
Also asked onceTap to open
On a 600 km flight, an aircraft had to be flown slowly because of bad weather. For this journey its average speed fell by 100 km/h and the flight time increased by 30 minutes. What was the original duration of the flight? (SI 24 Aug 2021)
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600 km. Let the normal speed be s. 600/(s − 100) − 600/s = the extra time the stem gives. Solving the quadratic gives the paper's answer of 1½ hours for the increase.
Varun can cover three-fourths of a certain distance at 5 km/h and the remaining distance at 6 km/h in 1 hour 10 minutes. Find the total distance. (SI 19 Aug 2021)
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Let the distance be d. Three quarters at 5 km/h and a quarter at 6 km/h take 0.75d/5 + 0.25d/6 = 0.15d + 0.04167d = 0.19167d hours, and that equals 1 h 10 min = 7/6 h. d = (7/6)/0.19167 ≈ 6.08 km.
A bus covers a distance of 24 km in such a way that it changes its speed every 6 km, running at 25 km/h, 50 km/h, 75 km/h and 150 km/h. Find the average speed of the bus over the whole journey. (SI 2 Jul 2023)
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The bus runs 6 km at each of 25, 50, 75 and 150 km/h. Times: 6/25 + 6/50 + 6/75 + 6/150 = 0.24 + 0.12 + 0.08 + 0.04 = 0.48 hours for 24 km. Average = 24/0.48 = 50 km/h. Four equal distances → 4/(sum of reciprocals).
A, B & C are on a trip by a car. A drives during the first hour at an average speed of 50 km/hr. B drives during the next 2 hours at an average speed of 48 km/hr. C drives for the next 3 hours at an average speed of 52 km/hr. They reached their destination after exactly 6 hours. Their mean speed was (SI 2016)
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A drives 1 hour at 50, B drives 2 hours at 48, and C's leg completes the trip. Here the times are given, so the average is the time-weighted mean: total distance ÷ total time. The paper's answer is 50⅓ km/h.
A person travels equal distances at speeds of 2 km/h, 3 km/h and 4 km/h, respectively, and takes a total time of 52 minutes. The average speed is : (SI 14 Jul 2023)
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Equal distances at 2, 3 and 4 km/h taking 52 minutes in total. Average = 3/(1/2 + 1/3 + 1/4) = 3/(13/12) = 36/13 = 2¹⁰⁄₁₃ km/h. The 52 minutes is only there so you can find the distance if you want it.
One-page revision
Read only this the night before.
| Situation | What to use |
|---|---|
| Anything basic | D = S × T |
| km/h to m/s | × 5/18 |
| m/s to km/h | × 18/5 |
| Out and back, equal distance | 2ab/(a + b) |
| Three equal distances | 3/(1/a + 1/b + 1/c) |
| Equal times | (a + b)/2 |
| Unequal distances | total distance ÷ total time |
| Train passing a pole | distance = train length |
| Train crossing a bridge | distance = train + bridge |
| Two trains, same direction | relative speed = a − b |
| Two trains, opposite | relative speed = a + b |
Seven more equal-distance average-speed questionsTap to open
Robert covers a distance of P km by his car at a speed of 25 km/h and comes back at a speed of 40 km/h. What is his average speed during the travel? (SI 2 Jul 2023)
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The distance P cancels, so the answer does not depend on it: average = 2(25)(40)/(25 + 40) = 2000/65 = 400/13 = 30 10/13 km/h. Note how the answer sits below the plain average of 32.5 — it always does, because more time is spent at the slower speed.
A person travels 160 km at a speed of 64 km/h and the next 160 km at a speed of 80 km/h. Find the average speed for the first 320 km of the journey. (SI 14 Jul 2023)
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Equal distances of 160 km each, so use the harmonic mean: 2(64)(80)/(64 + 80) = 10,240/144 = 640/9 = 71.11 km/h. Checking the long way: 160/64 = 2.5 h and 160/80 = 2 h, so 320 km in 4.5 h = 71.11.
A person covers a certain distance from the starting point at a speed of 90 km/h and returns to the starting point at a speed of 70 km/h. Find the person's average speed for the whole journey. (SI 15 Jul 2023)
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2(90)(70)/(90 + 70) = 12,600/160 = 78.75 km/h.
The distance between two stations A and B is 778 km. A train covers the journey from A to B at a speed of 84 km/h and returns to A at a uniform speed of 56 km/h. Find the average speed of the train over the whole journey. (SI 20 Jun 2023)
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The 778 km is never needed — equal distances each way, so 2(84)(56)/(84 + 56) = 9,408/140 = 67.2 km/h. Whenever a there-and-back question hands you a distance, check whether it cancels before you start dividing by it.
A car driver drives 240 km at a speed of 35 km/h and returns at a speed of 25 km/h. Find the average speed (in km/h) for the entire journey (correct to two decimal places). (SI 29 Jun 2023)
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2(35)(25)/(35 + 25) = 1,750/60 = 175/6 = 29.17 km/h. The 240 km is spare information, included so that the long method also works.
A man makes his upward journey at a speed of 16 km/h and downward journey at a speed of 28 km/h. What is his average speed for the entire journey (correct to two decimal places)? (SI 29 Jul 2023)
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Up and down covers the same distance, so 2(16)(28)/(16 + 28) = 896/44 = 224/11 = 20.36 km/h.
A covers 5 km from home to office in 15 minutes and from office to home in 20 minutes. What is his speed (in km/h) during the complete round? (SI 9 Jul 2023)
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Here the times are given rather than the speeds, so go back to the definition: total distance ÷ total time. That is 10 km in 15 + 20 = 35 minutes. 35 minutes is 7/12 of an hour, so the speed is 10 ÷ 7/12 = 120/7 = 17.14 km/h. The harmonic-mean formula needs two speeds; when the paper hands you two times, the definition is quicker.
Unit-conversion questionsTap to open
A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour ? (SI 2016)
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600 m is 0.6 km and 5 minutes is 1/12 of an hour, so the speed is 0.6 × 12 = 7.2 km/h. Or use the standard factor: 600/300 = 2 m/s, and 2 × 18/5 = 7.2 km/h.
A person crosses a 500 metre long road in 4 minutes. What is his speed in km/h? (SI 20 Jun 2023)
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500 m = 0.5 km, and 4 minutes = 1/15 of an hour, so the speed is 0.5 × 15 = 7.5 km/h.
Irfan crossed a 250 m long bridge in 15 minutes. What is his speed in km/h? (SI 14 Jul 2023)
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250 m = 0.25 km, and 15 minutes = ¼ hour, so 0.25 ÷ 0.25 = 1 km/h.
A driver travelled between two cities 63 km apart from each other in 1 hour 45 minutes. Find the speed in metres per minute. (SI 2 Jul 2023)
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Convert both to the units asked for: 63 km = 63,000 m, and 1 hour 45 minutes = 105 minutes. 63,000 ÷ 105 = 600 m/min. Option A says “600 km/min”, which is the right number with the wrong unit — read the options as carefully as the stem.
R walks 5.25 km in 1 hour and 10 minutes. His speed in km/h is ______. (SI 29 Jul 2023)
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1 hour 10 minutes is 70 minutes = 7/6 of an hour. So the speed is 5.25 ÷ 7/6 = 5.25 × 6/7 = 4.5 km/h.
A boy was riding a bicycle at the speed of 20 kmph. How much time would he have taken to cover 10 km? (SI 16 Oct 2022)
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Time = distance ÷ speed = 10 ÷ 20 = ½ hour = 30 minutes.
How many minutes will Bindu take to cover a distance of 600 m, if she runs at a speed of 15 km/h? (SI 29 Jun 2023)
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At 15 km/h she covers 600 m = 0.6 km in 0.6/15 = 0.04 hour. In minutes that is 0.04 × 60 = 2.4 minutes, and 0.4 of a minute is 24 seconds, so 2 min 24 sec. Converting the decimal part of a minute into seconds is where option A's 2 min 48 sec comes from — that would be 2.8 minutes.
A train covers a certain distance in 1 hour 20 minutes, if its speed is 60 km/h. What should the speed of the train be if the same distance is to be covered in 50 minutes? (SI 19 Aug 2021)
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Find the distance first: 60 km/h for 1 hour 20 minutes = 60 × 4/3 = 80 km. To cover 80 km in 50 minutes = 5/6 hour, the speed must be 80 ÷ 5/6 = 96 km/h. Options A and D give the same numbers in m/sec, which would be absurdly fast — always check the unit on the answer.
The average speed of a trip out and back always sits nearer the slower leg, because the slow leg takes more time. If your answer is the plain average of the two speeds, it is wrong.