Solved Questions

How to Use These Questions

There are 36 questions here in mixed order. A train question can come right after one on average speed, so first decide what is moving: one thing, two things towards or after each other, or a boat or runner. After that the working is short.

Cover the answer and solve each one on paper with a timer. In the exam you will have 36 to 45 seconds for a question like these. If one takes you over a minute, find its method in Shortcuts & Speed Methods and do it again the next day.

About a third of the questions are on the basics: changing units, average speed, being late or early, speed and time ratios, stoppages and trips made partly on foot. Another third are on relative speed: people meeting, chases, overtaking and trains of every kind. The rest are on boats and streams, races and circular tracks.

The [Tier 2] questions need one extra step, but Tier 1 asks shorter versions of the same ideas, so do them too. Before you start each one, pick a single unit (km/h with hours, or m/s with seconds) and stay with it to the end.

Solved Examples

Question 1

A cyclist covers 13.5 km in 45 minutes. What is her speed in metres per second?

Answer. 45 minutes is $\frac{3}{4}$ of an hour, so her speed is 13.5 ÷ $\frac{3}{4}$ = 18 km/h. Then 18 × $\frac{5}{18}$ = 5 m/s.

Question 2

Towns A and B are 297 km apart. A car leaves A at 8 a.m. at 45 km/h, and another car leaves B at 9 a.m. at 63 km/h, coming towards A. When and where do they meet?

Answer. By 9 a.m. the first car has done 45 km, so 252 km are left. Now the gap closes at 45 + 63 = 108 km/h, which takes 252 ÷ 108 = $2\frac{1}{3}$ hours, that is 2 hours 20 minutes. So they meet at 11:20 a.m., and the first car has gone 45 + 45 × $\frac{7}{3}$ = 150 km from A. Check: the second car has done 63 × $\frac{7}{3}$ = 147 km, and 150 + 147 = 297.

Question 3

A man rows 44 km downstream in 4 hours and 21 km upstream in 3 hours. What is the speed of the stream?

Answer. Downstream he does 11 km/h and upstream 7 km/h. The stream is half the difference: 2 km/h. His speed in still water is half the sum, 9 km/h.

Question 4

The speeds of A and B are in the ratio 3 : 4. A takes 30 minutes more than B to cover a certain distance. How long does A take?

Answer. The distance is the same, so the times go the other way: 4 : 3. The gap is 1 part, which is 30 minutes. A takes 4 parts = 120 minutes = 2 hours.

Question 5

Two trains start at the same time, one from P towards Q and the other from Q towards P. After they cross, the first takes 2 hours 15 minutes to reach Q and the second takes 4 hours to reach P. If the second train runs at 54 km/h, what is the speed of the first?

Answer. First : second = $\sqrt{4 : 2.25}$ = 2 : 1.5 = 4 : 3. The first train needs less time after the crossing, so it is the faster one, and its speed is 54 × $\frac{4}{3}$ = 72 km/h. They crossed after $\sqrt{2.25 \times 4}$ = 3 hours, so P and Q are (72 + 54) × 3 = 378 km apart.

Question 6

Ramesh goes to a village at 14 km/h and comes back at 10 km/h. The whole trip takes 6 hours. How far is the village?

Answer. If the distance is D, then $\frac{D}{14} + \frac{D}{10} = 6$. Multiplying by 70, 5D + 7D = 420, so D = 35 km. Check: 2.5 hours going and 3.5 hours coming back.

Question 7

A boat goes at 14 km/h in still water, and the stream flows at 6 km/h. How long will it take to go 30 km downstream and come back?

Answer. Downstream it does 20 km/h, so that part takes 1.5 hours. Upstream it does 8 km/h, so 3.75 hours. In all, 5.25 hours, which is 5 hours 15 minutes. Using 14 km/h both ways would give $4\frac{2}{7}$ hours, which is too short.

Question 8

A thief steals a car at 3:10 p.m. and drives away at 42 km/h. The theft is found out at 3:50 p.m., and the owner chases him in another car at 56 km/h. When does he catch the thief?

Answer. In the first 40 minutes the thief gets 42 × $\frac{2}{3}$ = 28 km ahead. The owner gains 14 km/h on him, so he needs 28 ÷ 14 = 2 hours. He catches the thief at 5:50 p.m. Check: the thief has driven $2\frac{2}{3}$ hours at 42 km/h and the owner 2 hours at 56 km/h, and both come to 112 km.

Question 9

A car covers the first 40% of a trip at 32 km/h and the rest at 48 km/h. What is its average speed for the whole trip?

Answer. I take the trip as 100 km. The first 40 km take $\frac{40}{32} = \frac{5}{4}$ hours and the last 60 km take $\frac{60}{48} = \frac{5}{4}$ hours. So 100 km take $\frac{5}{2}$ hours, and the average is 100 × $\frac{2}{5}$ = 40 km/h. Both parts take the same time here, which is why the plain average of 32 and 48 also gives 40. That only happens when the times are equal.

Question 10

A man can row at 9 km/h in still water. It takes him twice as long to row upstream as to row the same distance downstream. What is the speed of the stream?

Answer. Twice the time for the same distance means half the speed, so his downstream speed is twice his upstream speed: 9 + s = 2(9 − s). That gives 3s = 9, so the stream flows at 3 km/h. Check: 12 km/h down and 6 km/h up.

Question 11

A policeman sees a thief 250 m ahead and starts chasing him. The thief runs at 9 km/h and the policeman at 14 km/h. How far will the thief have run when he is caught?

Answer. The gap closes at 5 km/h. While the gap closes by 250 m, the thief runs $\frac{9}{5}$ times as far, which is 250 × $\frac{9}{5}$ = 450 m. The policeman runs 250 m more than that, 700 m.

Question 12

A bus runs for 2 hours at 42 km/h and then for 4 hours at 54 km/h. What is its average speed for the whole journey?

Answer. It covers 84 + 216 = 300 km in 6 hours, so the average is 50 km/h. The plain average, 48 km/h, is wrong here: the bus spends twice as long at 54 km/h, so the average leans towards 54.

Question 13

A train 250 m long passes a man walking at 5 km/h in the same direction in 18 seconds. What is the speed of the train?

Answer. The relative speed is 250 ÷ 18 = $\frac{125}{9}$ m/s, which is $\frac{125}{9}$ × $\frac{18}{5}$ = 50 km/h. The man walks the same way, so the train runs at 50 + 5 = 55 km/h.

Question 14 [Tier 2]

A boat goes 24 km upstream and 36 km downstream in 6 hours. It goes 36 km upstream and 24 km downstream in 6 hours 30 minutes. Find the speed of the boat in still water and the speed of the stream.

Answer. Say the boat takes p hours for each km upstream and q hours for each km downstream. Then 24p + 36q = 6 and 36p + 24q = 6.5. Adding them gives 60(p + q) = 12.5, and taking one from the other gives 12(p − q) = 0.5. So p + q = $\frac{5}{24}$ and p − q = $\frac{1}{24}$, which give p = $\frac{1}{8}$ and q = $\frac{1}{12}$. The upstream speed is 8 km/h and the downstream speed 12 km/h, so the boat does 10 km/h in still water and the stream flows at 2 km/h.

Question 15

Walking at 5 km/h, a boy reaches school 8 minutes late. Jogging at 8 km/h, he reaches 10 minutes early. How far is the school?

Answer. One trip is late and one early, so the gap between the two times is 8 + 10 = 18 minutes = $\frac{3}{10}$ hour. Distance = $\frac{5 \times 8 \times \frac{3}{10}}{8 - 5}$ = 4 km. Check: 48 minutes at 5 km/h and 30 minutes at 8 km/h, and the right time is 40 minutes.

Question 16

A train passes a 120 m platform in 18 seconds and a 270 m platform in 24 seconds. How long is the train?

Answer. The extra 150 m of platform take 6 more seconds, so the speed is 25 m/s (90 km/h). In 18 s the train covers 450 m, which is the train plus 120 m, so the train is 330 m long. Check: 330 + 270 = 600 m in 24 s.

Question 17

In a 500 m race, A gives B a start of 40 m and still beats him by 20 m. What is the ratio of their speeds?

Answer. B starts 40 m ahead, so he has only 460 m to run, and he is still 20 m short when A finishes. So while A runs 500 m, B runs 440 m. A : B = 500 : 440 = 25 : 22.

Question 18

A train runs at $\frac{4}{7}$ of its usual speed and reaches 45 minutes late. What is its usual time for the trip?

Answer. At $\frac{4}{7}$ of the speed the time becomes $\frac{7}{4}$ of the usual time, so the extra time is $\frac{3}{4}$ of the usual time. $\frac{3}{4}$ of the usual time = 45 minutes, so the usual time is 60 minutes, that is 1 hour.

Question 19

Two trains, 175 m and 225 m long, run on parallel tracks in opposite directions. They pass each other completely in 16 seconds. One runs at 54 km/h. What is the speed of the other?

Answer. They clear 400 m in 16 s, so the relative speed is 25 m/s = 90 km/h. In opposite directions the speeds add, so the other train runs at 90 − 54 = 36 km/h.

Question 20

A plane flies 720 km. Because of a storm its speed is cut by 120 km/h, and the flight takes 1 hour longer. What is the usual time of the flight?

Answer. If the usual speed is x, then x(x − 120) = $\frac{720 \times 120}{1}$ = 86,400. I look for two numbers 120 apart with this product: 360 × 240 = 86,400. So the usual speed is 360 km/h and the usual time is 720 ÷ 360 = 2 hours. Check: at 240 km/h the flight takes 3 hours.

Question 21

In a 1.5 km race, A beats B by 75 m or 15 seconds. How long does A take to finish the race?

Answer. B runs the last 75 m in 15 s, so B's speed is 5 m/s. B needs 1,500 ÷ 5 = 300 s for the race, and A takes 15 s less: 285 s = 4 minutes 45 seconds.

Question 22 [Tier 2]

A train passes two men walking in the same direction as the train, one at 3 km/h and the other at 6 km/h, in 12 seconds and 13 seconds. Find the speed and the length of the train.

Answer. In m/s the men walk at $\frac{5}{6}$ and $\frac{5}{3}$. If the train does v m/s, its length is $12(v - \frac{5}{6}) = 13(v - \frac{5}{3})$. That gives 12v − 10 = 13v − $\frac{65}{3}$, so v = $\frac{35}{3}$ m/s, which is 42 km/h. Length = 12 × ($\frac{35}{3} - \frac{5}{6}$) = 12 × $\frac{65}{6}$ = 130 m. Check: 13 × ($\frac{35}{3} - \frac{5}{3}$) = 13 × 10 = 130.

Question 23

Without stoppages a bus runs at 70 km/h; with stoppages it averages 63 km/h. For how many minutes does it stop per hour?

Answer. The stops cost it 7 km/h. Stop time = $\frac{7}{70}$ × 60 = 6 minutes per hour. Check: in each hour it covers 63 km, which needs 54 minutes of running at 70 km/h.

Question 24

In a 300 m race, A beats B by 30 m. In another 300 m race, B beats C by 20 m. By how many metres will A beat C in a 300 m race?

Answer. When A runs 300 m, B runs 270 m. B : C = 300 : 280, so while B runs 270 m, C runs 270 × $\frac{280}{300}$ = 252 m. A beats C by 48 m, not 50.

Question 25

Walking to a fort and riding back takes 6 hours 45 minutes. Walking both ways takes 9 hours 30 minutes. How long would riding both ways take?

Answer. Walking one way is half of 9 hours 30 minutes, which is 4 hours 45 minutes. So riding one way takes 6 hours 45 minutes − 4 hours 45 minutes = 2 hours, and riding both ways takes 4 hours.

Question 26 [Tier 2]

Two trains pass a man standing on a platform in 30 seconds and 18 seconds. Running in opposite directions, they pass each other in 26 seconds. What is the ratio of their speeds?

Answer. Say the speeds are x and y. The lengths are 30x and 18y, so $\frac{30x + 18y}{x + y} = 26$. That gives 30x + 18y = 26x + 26y, so 4x = 8y and x : y = 2 : 1. The shortcut (26 − 18) : (30 − 26) = 8 : 4 gives the same.

Question 27

A runs $2\frac{1}{4}$ times as fast as B. If A gives B a start of 100 m, how long should the race be so that they finish together?

Answer. A : B = 9 : 4. For a dead heat, A runs the whole race L while B runs L − 100, so $\frac{L}{L - 100} = \frac{9}{4}$. That gives 4L = 9L − 900, so L = 180 m. Check: B runs 80 m while A runs 180 m, and 180 : 80 = 9 : 4.

Question 28

Car A runs at $\frac{5}{4}$ of the speed of car B and takes 1 hour 24 minutes less to cover 210 km. What is the speed of car A?

Answer. Speeds 5 : 4 means times 4 : 5. The gap of 1 part is 1 hour 24 minutes = 84 minutes, so B takes 420 minutes = 7 hours and A takes 5 hours 36 minutes. B's speed is 210 ÷ 7 = 30 km/h, so A's is 30 × $\frac{5}{4}$ = 37.5 km/h.

Question 29

Two cars leave the same place at the same time in the same direction at 52 km/h and 65 km/h. After how long will they be 39 km apart?

Answer. The gap grows at 65 − 52 = 13 km/h, so 39 km takes 39 ÷ 13 = 3 hours.

Question 30

Two runners start together from the same point on a circular track 1.8 km long and run in opposite directions at 7.2 km/h and 10.8 km/h. After how long do they meet for the first time?

Answer. In opposite directions the speeds add: 18 km/h, which is 5 m/s. Together they must cover one lap of 1,800 m, so they meet after 360 s = 6 minutes.

Question 31 [Tier 2]

Two hours after starting, a train has an accident and goes on at $\frac{3}{5}$ of its speed. It reaches 1 hour 20 minutes late. Had the accident happened 40 km further on, it would have been only 1 hour late. Find the usual speed of the train and the length of the trip.

Answer. At $\frac{3}{5}$ of the speed every stretch takes $\frac{5}{3}$ of its usual time, so the delay is $\frac{2}{3}$ of the usual time for the part after the accident. Moving the accident 40 km on cuts the delay by 20 minutes, so $\frac{2}{3}$ of the usual time for 40 km is 20 minutes. The usual time for 40 km is then 30 minutes, and the speed is 80 km/h. In the first case, $\frac{2}{3}$ of the usual time after the accident is 80 minutes, so that part usually takes 2 hours, which is 160 km. So the trip is 160 km before the accident (2 hours at 80 km/h) plus 160 km after it: 320 km.

Question 32

A man starts cycling from a village at 15 km/h. Two hours later his friend sets off after him on a motorbike and overtakes him 50 km from the village. What is the speed of the motorbike?

Answer. The cyclist takes 50 ÷ 15 = $3\frac{1}{3}$ hours to reach that point. The friend left 2 hours later, so he took $1\frac{1}{3}$ hours for the same 50 km. His speed is 50 ÷ $\frac{4}{3}$ = 37.5 km/h.

Question 33

A, B and C start together from the same point on a circular track and take 36 seconds, 48 seconds and 54 seconds to run one lap. After how long will all three be together at the starting point for the first time?

Answer. I need the LCM of 36, 48 and 54. Here 36 = $2^2 \times 3^2$, 48 = $2^4 \times 3$ and 54 = $2 \times 3^3$, so the LCM is $2^4 \times 3^3$ = 432. They are together at the start after 432 s = 7 minutes 12 seconds.

Question 34

A man drives from P to Q at 50 km/h. At what speed must he drive back so that his average speed for the round trip is 60 km/h?

Answer. Return speed = $\frac{60 \times 50}{2 \times 50 - 60} = \frac{3000}{40}$ = 75 km/h. Check: $\frac{2 \times 50 \times 75}{125}$ = 60. Treating 60 as the plain average gives 70 km/h, which is wrong.

Question 35

Two trains run in the same direction at 80 km/h and 62 km/h. The faster train passes a man sitting in the slower train in 24 seconds. How long is the faster train?

Answer. The man moves with the slower train, so the faster train gains on him at 18 km/h = 5 m/s. To pass him it clears only its own length: 5 × 24 = 120 m.

Question 36 [Tier 2]

A and B start together from the same point on a circular track and run at 9 m/s and 6 m/s. At how many different points on the track do they meet if they run in opposite directions? At how many if they run in the same direction?

Answer. First I cut the speeds down to lowest terms: 9 : 6 = 3 : 2. In opposite directions they meet at 3 + 2 = 5 points, and in the same direction at 3 − 2 = 1 point, which is the starting point itself. Don't take 9 + 6 = 15 points; cut the ratio down first. Why 5: going opposite ways they meet each time the two of them have covered one lap between them, and A does $\frac{3}{5}$ of that lap. So each meeting is $\frac{3}{5}$ of a lap past the last one, and only after 5 meetings are they back at the start.