5 chickens can lay 10 eggs in 20 days. How long does it take 18 chickens to lay 100 eggs?
Let $c,g,d$ be the number of chickens, eggs, days, respectively.
$c$ is inversely proportional to $d:\quad cd=x$
$g$ is directly proportional to $d:\quad g=yd$
Combining the two equations: $$\quad cg = xy,$$ that is, $c$ is inversely proportional to $g,$ in other words, for the same time period if we need more eggs we need to decrease the number of chickens. But this defies common sense. Where is my mistake?
You understood your equation wrongly. The amount of eggs you want to have as a result is not $eggs$, but $xy$. The variable $eggs$ represent the amount of eggs a chicken can lay in the whole time span (and NOT the total amount of laid eggs), you could interpret it as $eggs / chicken$, which can make it clearer.
It is correct that if a chicken can lay more eggs, the amount of chicken can be reduced, if we want to keep the total amount of eggs the same.