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A Long Time at the Till

Age 14 to 18 Challenge Level:

This problem involves considering, comparing and assessing different ways to solve a very difficult 'background' problem. There are three different parts to this unusual and thought-provoking task.


Background problem
Note: solving the background problem is very involved and not the main focus of this task!
A mathematician goes into a supermarket and buys four items. It has been a while since she has used a calculator and she multiplies the cost (in pounds, using the decimal point for the pence) instead of adding them. At the checkout she says, "So that's £7.11" and the checkout man, correctly adding the items, agrees.

The mathematician very, very slowly puts the items into her bag whilst thinking and tapping away on her calculator. She eventually says "I believe that the prices of four items with this property is unique".
 
Spend a few minutes trying this problem yourself to get a feel for its mathematical structure.
 
Please note: Although it involves only the basic properties of numbers, the background problem is very difficult and time-consuming to solve directly. Now there's a challenge ....  

Main problem
Read carefully the two solutions provided in the hints tab. How do your attempts at the first part compare to, or differ from, these two solutions? Which of the two solutions do you prefer? Why?
 
Follow up task 
If you were now to be given related problems with £7.11 replaced by £7.12 or £7.13 or £7.14 how would you now choose to proceed? Can you assess in advance which of these problems will probably be harder or easier? Can you efficiently solve any of these problems with the benefit of hindsight?
 

This problem has been adapted from the book "Sums for Smart Kids" by Laurie Buxton, published by BEAM Education. This book is out of print but can still be found on Amazon.