Solution:

Assuming 6 legs for each insect, 16 insects will have 96 legs.
But, there are 104 − 96 = 8 extra legs.

It is important to note that spiders have 8 legs and flies have 6 legs. So, there are 2 extra legs per spider on the web.
Thus, the 8 extra legs belong to 4 spiders.

Alternative Solution through Equations:

Let s be the number of spiders and f be the number of flies. Then,

s + f = 16; and
8s + 6f = 104.

On solving the above two equations, we get

8s + 6 (16 − s) = 104; or
2s = 104 − (6 × 16).
Thus, s = 4.

Food for thought:

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