Problem
Train passing completely through a tunnel
A 60-yard train goes through a 300-yard tunnel. It travels a steady 15 yards every second. Timing starts when the front reaches the entrance. Find the time until the whole train is out.
Reasoning · Grade 6-1 Distance and Speed (2)
Your answer
Watch out Dividing the tunnel alone is short: the tail must clear too, so add the train's length.
How to solve
Strategy Draw a Diagram — The entire difficulty of this problem is the distance, not the arithmetic, so the plan is to draw the two moments that bookend the trip — front at the entrance, back at the exit — and read the travelled distance straight off the picture. Sliding a strip of paper (the train) along a drawn tunnel makes it impossible to forget the train's own length. Once the distance is settled, the problem breaks into two easy pieces: find the distance, then divide by the speed, with the units confirming the setup.
1STEP 1
Draw the starting moment
At the start the front is at the entrance.
Pinning down the exact instant the timing starts is half the battle in this kind of problem; drawing it removes any doubt about where the front and back are.
4.MD.A.2Draw A Diagram2STEP 2
Draw the finishing moment
At the end the tail clears the exit.
Two snapshots — start and finish — turn a moving situation into a plain measurement question about how far one point has shifted.
4.MD.A.2Draw A Diagram3STEP 3
Read off the distance travelled: tunnel plus train
The distance travelled is 360.
300 + 60 = 360 yards
Cut a 6 cm paper strip for the train and a 30 cm gap for the tunnel and slide the strip through. You can watch the strip travel one tunnel-length plus one strip-length before it is clear — a train is a long object, not a dot, and its length rides along in the total.
4.MD.A.2Create A Physical RepresentationFrom entering to fully leaving, the train covers the tunnel's length plus its own length.
Why?
The journey of the front of the train is the tunnel and the train laid end to end, with no gap and no overlap.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The train does not stretch as it moves, so its own length is the same at the finish as it was at the start.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
4STEP 4
Divide the distance by the speed
Dividing by the speed gives 24 seconds.
360 ÷ 15 = 24 seconds
Splitting 360 into 300 + 60 makes the division easy in your head: 300 / 15 = 20 and 60 / 15 = 4, so 20 + 4 = 24, no long division needed.
6.RP.A.3Identify Subproblems5STEP 5
Check by multiplying back and by watching the tail
Multiplying back gives 360.
15 × 24 = 360 ✓
Yards divided by yards-per-second leaves seconds, and getting the same 360 whether you track the front or the back confirms the distance was set up correctly rather than by luck.
6.RP.A.3Analyze The UnitsAnswer
24 seconds
360 ÷ 15 = 24
The units are right: 360 yards divided by 15 yards per second gives seconds. The size is right too, and it is easy to see how it should compare with the tempting wrong answer. Covering just the 300-yard tunnel would take 300 / 15 = 20 seconds, so the true answer must be a little more than 20 — and 24 is, by exactly the 60 / 15 = 4 seconds it takes the train to clear its own length. A useful bracket: the answer has to sit between the 20 seconds for the tunnel alone and the 24 seconds for tunnel-plus-train, and 'completely through' pushes it to the upper end. If the train were much shorter, say 15 yards, the answer would drop to 21 seconds, which is the pattern you would expect.
Takeaway
A train is not a dot: to get completely out of a tunnel it must travel the tunnel's length PLUS its own length.
- Draw the starting moment
- Draw the finishing moment
- Read off the distance travelled: tunnel plus train
- Divide the distance by the speed
- Check by multiplying back and by watching the tail