<s>[INST] Finding the total number of objects(stones). I have $N$ stones. Then the stones are arranged in ascending order of weights. If I remove three stones that are heaviest, then the total weight of the stones decreases by $35$%. Now if I remove the three lightest stones, the total weight of the stones further decreases by a factor of $\frac{5}{13}$.
Find the value of $N$.

What is tried:Let the total weight be $S$.
After removing three stones that are heaviest the total weight is $0.65 S$ and after further removing three stones that are lightest the total weight is $0.40 S$.
Now using average weights we have:
$$\frac{0.65}{N-3} < \frac1N \text{ and } \frac{0.65}{N-3} < \frac{0.4}{N-6}$$
from which we can conculde that $N>8$ and $N< 11$. So we are having two possible values of $N$ which are $9$ or $10$.

But how to narrow it down between $9$ and $10$ which one is correct?
Source: Homework Question.
 [/INST] For another approach: The three heaviest stones weigh $\frac{7}{20} S$. The three lightest stones weigh $\frac{5}{20} S$. The $N-6$ other stones weigh $\frac{8}{20} S$. The averages of these three groups must be in order:
$$ \frac{5 S}{3 \cdot 20} < \frac{8 S}{(N-6)20} < \frac{7 S}{3 \cdot 20} $$
$$ \frac{24}{5} > N-6 > \frac{24}{7} $$
There's only one integer in that range. $N=10$.
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