I am having touble how to work with the retail inventory method to estimate the companies year end inventory.

i have the following information but i just do not see it with even the example in the book.
It is in a 5x3 matrix of colimns and rows. what is in the parenthesis is the first one at cost and the second one at retail

Title (at cost)(at retail)
jan. 1 beginning inventory (81670)(114610)
cost of goods purchased (492250)(751730)
sales (n/a)(786120)
sales returns (n/a)(4480)

and it says in the back of the book the inventory is 55902 cost. how did they get that?

To estimate the year-end inventory using the retail inventory method, you need to follow a specific calculation process. Here's how you can do it using the information you provided:

Step 1: Calculate the Cost-to-Retail Ratio (CTR)
CTR = (Cost of Goods Purchased at Cost) / (Goods Purchased at Retail)

In your case:
CTR = 492250 / 751730

Step 2: Calculate the Estimated Retail Value of Inventory (ERV)
ERV = (Cost of Goods Available for Sale at Retail) – (Total Sales at Retail) + (Sales Returns at Retail)

In your case:
ERV = (114610 + 751730) - (786120) + (4480)

Step 3: Calculate the Estimated Cost of Inventory (ECI)
ECI = ERV x CTR

In your case:
ECI = ERV x CTR

Once you have performed the above calculations, you will obtain the estimated cost of the company's year-end inventory.

Here's how you can calculate it based on the information provided:

CTR = 492250 / 751730 ≈ 0.654

ERV = (114610 + 751730) - (786120) + (4480) ≈ 316700

ECI = 316700 x 0.654 ≈ 207162.8 (rounded to the nearest dollar)

Therefore, the estimated cost of the company's year-end inventory is approximately $207,162.