All 8 problems from "Opposite-Direction Motion" with complete step-by-step worked solutions: recognising the type, the idea, the working steps and the conclusion. Use this page to review with your child after the animated lesson, or let them check their own working against it.
Every problem, fully solved
Two cyclists ride toward each other from towns 60 km apart. One rides at 12 km/h and the other at 18 km/h. In how many hours will they meet?
Key idea: Moving TOWARD each other: add the speeds into a closing speed.
Recognise
Two cyclists ride toward each other from towns 60 km apart, at 12 km/h and 18 km/h.
The idea
Together they close the gap at 12 + 18 km/h; divide the distance by that closing speed.
Steps
1
Closing speed: 12 + 18 = 30 km/h.
2
Time: 60 ÷ 30 = 2 hours.
So: So they meet in 2 hours.
A bus travels at 45 km/h. How many kilometres does it travel in 3 hours?
Key idea: Distance = speed × time — the one basic journey formula.
Recognise
A single bus travels at 45 km/h for 3 hours — no meeting, just distance.
The idea
Multiply the steady speed by the time travelled.
Steps
1
45 km/h means 45 km covered every hour.
2
45 × 3 = 135 km.
So: So it travels 135 km.
Two cars drive toward each other. Car A goes at 40 km/h and Car B at 50 km/h. How fast is the distance between them shrinking?
10 km/h
45 km/h
90 km/h
2000 km/h
Key idea: The gap between two approaching vehicles shrinks at the SUM of their speeds.
Recognise
Two cars drive toward each other at 40 km/h and 50 km/h — how fast is the distance between them shrinking?
The idea
Both cars eat into the gap at the same time, so their speeds work together.
Steps
1
40 + 50 = 90 km/h.
2
The gap shrinks by 90 km every hour.
So: So the distance between them is shrinking at 90 km/h.
A slow train leaves station A at 20 km/h. One hour later, a second train leaves station B, 140 km from A, travelling toward A. How many kilometres remain between the two trains when the second train starts?
Key idea: A head start shrinks the meeting distance first — subtract it before anything else.
Recognise
A slow train leaves at 20 km/h; one hour later a second train starts toward it from 140 km away.
The idea
The first train's head-start distance is already covered, so it comes off the gap.
Steps
1
Head start: 20 × 1 = 20 km.
2
Remaining distance: 140 − 20 = 120 km.
So: So 120 kilometres remain between the two trains.
After a head start, two cars are 120 km apart. They drive toward each other at 25 km/h and 35 km/h. In how many hours do they meet?
Key idea: Moving TOWARD each other: add the speeds into a closing speed.
Recognise
After a head start, two cars are 120 km apart and drive toward each other at 25 km/h and 35 km/h.
The idea
Closing speed first, then time = remaining distance ÷ closing speed.
Steps
1
Closing speed: 25 + 35 = 60 km/h.
2
Time: 120 ÷ 60 = 2 hours.
So: So they meet in 2 hours.
Two friends live 84 km apart. They cycle toward each other at 16 km/h and 12 km/h. They meet after ___ hours, and the first friend has cycled ___ km.
Key idea: One meeting, two questions: time first (closing speed), then each rider's own distance.
Recognise
Two friends live 84 km apart and cycle toward each other at 16 km/h and 12 km/h — the meeting time and the first friend's distance are both asked for.
The idea
Time comes from the closing speed; then the first friend's distance is his own speed × that time.
Steps
1
Closing speed: 16 + 12 = 28 km/h.
2
Time: 84 ÷ 28 = 3 hours.
3
First friend: 16 × 3 = 48 km.
4
Check: 48 + 12 × 3 = 48 + 36 = 84 km.
So: So they meet after 3 hours, and the first friend has cycled 48 km.
Two buses start from towns 210 km apart. The first bus leaves at 6:00 at 30 km/h. The second bus leaves the other town at 7:00 at 60 km/h, driving toward the first. At what hour do the two buses meet? (Give the hour only, e.g. 9 for 09:00.)
Key idea: Different start times: remove the head start, close the rest, then count the clock forward.
Recognise
Two buses start from towns 210 km apart: the first leaves at 6:00 at 30 km/h, the second at 7:00 at 60 km/h toward it — the meeting CLOCK time is asked for.
The idea
Head start off the distance, closing speed for what remains, then add the hours to 7:00.
Steps
1
Head start: 30 × 1 = 30 km, leaving 210 − 30 = 180 km.
2
Closing speed: 30 + 60 = 90 km/h.
3
Time after 7:00: 180 ÷ 90 = 2 hours.
So: So the two buses meet at hour 9.
Two cars are 150 km apart and drive toward each other. Car A starts from the left at 40 km/h and Car B at 60 km/h. How far from Car A's starting point do they meet?
40 km
60 km
90 km
100 km
Key idea: Meeting point from one side: time first, then THAT vehicle's speed × time.
Recognise
Two cars 150 km apart drive toward each other (A at 40 km/h, B at 60 km/h) — how far from Car A's starting point do they meet?
The idea
Find the meeting time from the closing speed, then ride with Car A only.
Steps
1
Meeting time: 150 ÷ (40 + 60) = 1.5 hours.
2
Car A travels: 40 × 1.5 = 60 km.
3
Careful: 90 km is Car B's distance — the common mix-up.
So: So they meet 60 km from Car A's starting point.
Learn it with the animated lesson
The problem
Research stations A and B are 120 km apart. A green car leaves A at 07:00 at 30 km/h. A purple car leaves B 1 hours later, at 08:00, traveling toward A at 60 km/h. When do they meet, and how far is the meeting point from A?
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Problem overview
Stations 120 km apart
AB = 120 km
The green car leaves station A at 07:00, traveling toward station B. The purple car is waiting at B and departs at 08:00.
Key idea
Toward each other — both cars shrink the gap together.
Solution steps
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