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Most of you sent us the right answer to the
first question of the problem ('How many of 3kg and 8kg weights
would you need for the average (mean) of the weights to be 6kg?').
Some found it by trial and error, for example, Ellen from
Shincliffe Primary:
It is all about trial and error. I thought I had the answer when I
had fourof 3kg weights and threeof 8kg weights. But then I noticed
that I had seven weights. Then I kept my three of 8kg which is 24kg
and all I had to do was add 2 of 3kg weights. And I had my
answer.
3 x 8kg + 2 x 3kg =30, and 30/5 = 6kg.
Andy (Garden International School) pointed
out:
The weight averages are from 3 to 8
Rosie from St Bartholomew's Cof E Primary
School gave us the answers to some wholenumber averages in between
3 and 8:
For an average of 7, you would need 4 of 8kg weights and 1 of 3kg
weight.
For an average of 6, you would need 9 of 8kg weights and 6 of 3kg
weights.
For an average of 5, you would need 2 of 8kg weights and 3 of 3kg
weights.
A general solution was provided by Hyeon
(British School Muscat):
Imagine that the lighter weight is a and the heavier weight is
b.
As long as a < b, the smallest average you can get is a and the
biggest average you can get is b.
It is possible to get every single number in between. There are
[(ba)+1] averages.
Hyeon also noticed something important from
the results of her trial with different weights:
If the total number of averages is odd, then 1 of a and 1 of b
would give the middle average weight.
The amount of a and the amount of b used should add up to a factor
or a multiple of the difference between a and b for the average to
be a whole number.
Here are some of her trials:
For the 1kg and 5 kg weights
1 kg 
5 kg 
Total 
Average 
1 
0 
1 
1 
3 
1 
8 
2 
1 
1 
6 
3 
1 
3 
16 
4 
0 
1 
5 
5

For the 17 kg and 57 kg weights
17 kg 
57 kg 
Total 
Average 
9 
1 
210 
21 
13 
27 
1760 
44 
1 
7 
1020 
52 
Although we received many answers to
this problem, most of the answers simply stated the results rather
than providing a general strategy for finding them. However,
Anurag, from Queen Elizabeth's Grammar School in Horncastle, did
draw some general conclusions. You can see his working
here