Combining proportionality equations fails to solve "5 chickens can lay 10 eggs in 20 days" problem

Question

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?

Answer 1

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.

Answer 2

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 $d:\quad cd=x$

$c$ is inversely proportional to $d$ while the value of $g$ is being fixed, so $x$ is actually a function of $g.$

$g$ is directly proportional to $d:\quad g=yd$

Similarly, $y$ is actually a function of $c.$

Combining the two equations: $$\quad cg = xy,$$ that is, $c$ is inversely proportional to $g$

With the above corrections, it can then be shown that while the value of $d$ is being fixed, $c$ is actually directly proportional to $g.$

The linked proof also justifies directly writing the problem's joint-proportionality equation $$\frac{cd}g=K,$$ which is a hop away from the required answer 56 days.