Step 1 :The problem is asking for the number of small and large shirts the manager will order given that he orders 2 small shirts and 3 large shirts for every 4 medium shirts. Since he plans to order 24 medium shirts, we can set up a proportion to find out how many small and large shirts he will order.
Step 2 :Let's denote the number of medium shirts as \(medium\_shirts\), the ratio of small to medium shirts as \(small\_to\_medium\), and the ratio of large to medium shirts as \(large\_to\_medium\).
Step 3 :We know that \(medium\_shirts = 24\), \(small\_to\_medium = 0.5\), and \(large\_to\_medium = 0.75\).
Step 4 :We can calculate the number of small shirts as \(small\_shirts = medium\_shirts \times small\_to\_medium = 24 \times 0.5 = 12.0\).
Step 5 :Similarly, we can calculate the number of large shirts as \(large\_shirts = medium\_shirts \times large\_to\_medium = 24 \times 0.75 = 18.0\).
Step 6 :Final Answer: The manager will order \(\boxed{12}\) small shirts and \(\boxed{18}\) large shirts.