Repeating Events Time Problems Practice Test Worksheets
Repeating events questions ask you to count how many buses, trains, trams, or ferries arrive within a time window. The key is listing each service at its own interval and checking whether the start and end times are included.
For students who can count intervals, but get stuck when repeating events overlap after different cycle lengths.
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What Do Repeating Events Questions Look Like?
Repeating events questions ask you to track multiple scheduled services—such as trains, buses, or ferries—that each arrive at regular intervals. You are given a shared start time and asked to calculate the total number of arrivals within a specific time window. Here is a worked example.
Counting Arrivals Across Three Services
Question: Three train services pass through Waverley. The East Coast Freight arrives every 6 minutes, the Cross-Country arrives every 8 minutes, and the Southern Local arrives every 20 minutes.
If all three trains arrive at Waverley exactly at 12:00pm, how many trains in total will pass through Waverley from 12:30pm to 1:00pm, inclusive?
Worked Method
Step 1: Keep counting from the shared arrival at 12:00pm. The window starts 30 minutes later. The 6-minute service arrives at 12:30; the first arrivals inside the window for the other services are 12:32 and 12:40.
East Coast Freight (every 6 minutes): 12:30, 12:36, 12:42, 12:48, 12:54, 1:00 = 6 trains.
Cross-Country (every 8 minutes): 12:32, 12:40, 12:48, 12:56 = 4 trains.
Southern Local (every 20 minutes): 12:40, 1:00 = 2 trains.
Total: 6 + 4 + 2 = 12 trains.
Tip: Count scheduled arrivals at both endpoints, but do not assume every service arrives at the start of the window. Each timetable continues from 12:00pm.
Rules to Help You Solve Repeating Events Questions
Each service has its own rhythm. Using the structure from the example above, count one service at a time, then add the counts together.
⏱️ Rule 1: Start from the shared time
All services arrive or depart together at the stated starting time. Add each interval repeatedly from that time.
Example: An arrival at 10:00 repeating every 8 minutes gives 10:00, 10:08, 10:16, 10:24, 10:32, and so on. Count only those times inside the requested window.
🚦 Rule 2: Only count times inside the window
Ignore arrivals before the start of the window and after the end of the window.
Example: In 10:30 to 11:15, 10:24 is too early but 10:32 counts.
✅ Rule 3: Inclusive means count the endpoints
If the question says "inclusive", an arrival exactly at the start or end of the window counts.
Example: For a service scheduled at 10:30, 10:45, 11:00 and 11:15, all four arrivals count in a 10:30 to 11:15 inclusive window. Only count an endpoint if an arrival is scheduled then.
➕ Rule 4: Add service counts, not unique times
If two services arrive at the same clock time, they are still two arrivals.
Example: If two trains arrive at 11:00, that contributes 2 trains, not 1 time.
Two More Worked Examples
Here are two additional problems showing how to add multiple overlapping bus routes and how to handle simultaneous arrivals at the same scheduled minute.
Adding Several Services
Question: Four different bus routes stop at Eastside Hub. The Route C arrives every 4 minutes, the Route E arrives every 6 minutes, the Route H arrives every 10 minutes, and the Route I arrives every 20 minutes.
If all four buses arrive at Eastside Hub exactly at 4:00pm, how many buses in total will call at Eastside Hub from 5:00pm to 5:20pm, inclusive?
Worked Method
Start of the window: 5:00pm is 60 minutes after the shared arrival at 4:00pm. Since 60 is divisible by 4, 6, 10 and 20, all four routes have an arrival at 5:00pm.
Route C: 5:00, 5:04, 5:08, 5:12, 5:16, 5:20 = 6 buses.
Route E: 5:00, 5:06, 5:12, 5:18 = 4 buses.
Route H: 5:00, 5:10, 5:20 = 3 buses.
Route I: 5:00, 5:20 = 2 buses.
Total: 6 + 4 + 3 + 2 = 15 buses.
Tip: Use the shared arrival time to establish each schedule. Do not restart the intervals at the window’s beginning unless a service actually arrives then.
Same Time, Different Services
Question: Three train services pass through Piccadilly Station. The Intercity Fast arrives every 8 minutes, the East Coast Freight arrives every 15 minutes, and the City Commuter arrives every 20 minutes.
If all three trains arrive at Piccadilly Station exactly at 10:00am, how many trains in total will pass through Piccadilly Station from 10:30am to 11:15am, inclusive?
Worked Method
Intercity Fast (every 8 minutes): 10:32, 10:40, 10:48, 10:56, 11:04, 11:12 = 6 trains.
East Coast Freight (every 15 minutes): 10:30, 10:45, 11:00, 11:15 = 4 trains.
City Commuter (every 20 minutes): 10:40, 11:00 = 2 trains.
Total: 6 + 4 + 2 = 12 trains.
Tip: Do not merge matching times. At 11:00, two different trains arrive, so both are counted.
