The question in the book is ultimatley trying to find the Cost of Goods Sold for the company. There is no Work in Progress.

Increase in Raw materials Inventory... $7500
Decrease in Finished Goods inventory... 17500
Raw materials purchased... 215,000
Direct-labor Payroll... 100,000
Factory Overhead... 150,000
Freight-OUT... 22,500

I read that Freight-In would be linked to the products and computed as part of the costs, but Freight-Out would be listed separately as part of shipping and expensed in the period.
I know that to find the COGS, I would need to find the Cost of Goods Manufactured and add/ subtract the change in finished Goods.
If I had a Beginning Inventory for Raw Materials then I would add the Raw Materials Purchased to get the Raw materials Available. Then I would add Payroll, and overhead to find the cost of Goods Manufactured. However, I don't have the Beginning Inventory number, just the increase in Inventory.
My Math...
COGM = $215,000 + $100,000 + $150,000 = $465,000
- Decrease Finished Goods = $ 17,500
= COGS = $447,500

Even with adding the increase to COGM (=$472500) the ending answer (=$455,000) is not the correct answer. If I had been give the correct answer I might be able to figure this one out, but I don't have a correct anwer to work with. HELP!

The question says that raw materials inventory has increased by 7,500, meaning that raw materials consumed are less by 7,500. Hence raw materials consumed is 215,000 - 7,500 = 207.500.

The second error in your answer is that you have decreased the cost of goods manufactured to give effect to change in finished goods inventory. It has to be added. Based on this correct figure would be $475,000.