In searching the Internet on a method of how to do this, all examples include beginning inventory and purchases to determine average inventory. I know ending inventory for one year is beginning inventory for the next, but when you don't have information on purchases, how does one arrive at finding the answer? This is a homework question. The answer I am trying to arrive at is Inventory Turnover, but you need to know what the Average Inventory is in order to divide it into the Costs of Goods Sold. Thanks.

The average inventory is the beginning inventory + ending inventory divided by 2.

In other words, purchases aren't involved in finding average inventory. You are mixing up two things you are seeing.

In a periodic inventory system, purchases are used in determining the ending inventory. However, once that is determined, the ending inventory shows up on the balance sheet and you've already got your number. So you don't need purchases if you already have ending inventory.

In searching the Internet on a method of how to do this, all examples include beginning inventory and purchases to determine average inventory. I know ending inventory for one year is beginning inventory for the next, but when you don't have information on purchases, how does one arrive at finding the answer? This is a homework question. The answer I am trying to arrive at is Inventory Turnover, but you need to know what the Average Inventory is in order to divide it into the Costs of Goods Sold. Thanks.

As any one would understand the problem, you are required to calculate inventory turnover and the only information available is the COGS and ending inventory.

Granted that inventory turnover is usually calculated by using average inventory, but where the amount of beginning inventory is not given then it is prudent to use the ending inventory. You do not need to bother any more with the non availability of beginning inventory in these conditions.