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Searching for Mean(ing)

Age 11 to 16 Challenge Level:

Khue from British Vietnamese International School in Vietnam said:

An easier way of finding a mean weight for a combination of weights that is [the] numbers weights add up to a multiple of 5.

These diagrams illustrate Khue's reasoning:

or 

In fact, there are combinations of 3 kg and 8 kg weights which have means equal to 3, 4, 5, 6, 7 and 8 kg. Click to see them all.

 mean = 3
 mean = 4
 mean = 5
 mean = 6
 mean = 7
 mean = 8

 

Mahdi from Mahatma Gandhi International School in India used algebra to extend the problem to other pairs of weights: