This activity is based on finding the cheapest train tickets for journeys with stopping stations, where one ticket can be replaced by split tickets (using the stopping points).

The concept is straightforward but gets more complex as the number of stopping stations increases. Two examples are given for journeys for specified trains, one a single ticket and other a return ticket. Students are asked to find the cheapest way and the percentage gain that can be achieved. They are also set the challenge of generalising the result and finding the number of price comparisons needed.

Overview of task

This activity is based on finding the cheapest train tickets for journeys with stopping stations, where one ticket can be replaced by split tickets (using the stopping points).

The concept is straightforward but gets more complex as the number of stopping stations increases. Two examples are given for journeys for specified trains, one a single ticket and other a return ticket. Students are asked to find the cheapest way and the percentage gain that can be achieved. They are also set the challenge of generalising the result and finding the number of price comparisons needed.

Strands

Financial Maths; Algebra and Graphs (for the extension)

Prior knowledge

Only requires familiarity with basic financial calculations, including percentage changes. The extension requires generalising a sequence to find the *n*th term.

Relevance to Core Maths qualifications

- AQA
- C&G
- Eduqas
- Pearson / Edexcel
- OCR

Suggested approaches

Some background to train travel might be needed and a simple example of just one intermediate station might be helpful when introducing the problem.

Resources/documentation

The activity is fully resourced through the handout but you or your students could find more examples by going to the National Rail Enquiries website at:

Relevant digital technologies

The calculations, including the extension, could be explored using a spreadsheet.

Possible extensions

The extension is given on the handout; there are a number of ways of tackling this extension, but the simplest strategy is to build up from 2, 3, 4,… stations and recognise the structure of the sequence.

Acknowledgement

This resource was developed for Core Maths by David Burghes and Russell Geach (CIMT, Plymouth University).