The data below regards The Bag House, a store that sells bags.

Budgeted Sales (credit) : For January, February, and March, are $420,000, $310,000, and $280,000.
Budgeted Collections : 60% in month of sale, 30% in following month, and 10% uncollectible
Cost of Goods sold : 50% of sales
Purchases of Merchandise: 30% in month prior to sale, and 70% in month of sale
Payment for Purchases : 80% in month of purchase and 20% in following month
Selling and Admin Costs : $32,500 paid in cash monthly
Depreciation of Equipment : $21,500 monthly
Question : What is the cash disbursement for purchases of merchandise in February?
A) 165,500 B) 159,100 C) 149,750 D) 153,000 E) None
Solution shows 159,100 but I don't get their steps as to how:
They started with budgeted sales for February of 310,000 and then showed COGS of $155,000 (1/2 of 310,000 I know that), and then they add ending inventory of $42,000 (I don't understand where this comes from), and then they subtract beginning inventory of 46,500 ( I don't get where this comes from, i.e. the math, I know it comes from January), to get required purchases of 150,500. Prior to this they showed the same thing for January, with budgeted sales of 420,000, then COGS of 210,000, then they added ending inventory of 46,500 (don't get where it came from) and then they subtract beginning inventory of 63,000 to get required purchases of 193,500.

I understand the end steps : 0.8 x February Required Purchases + 0.2 x January Required Purchases where I plug in the above numbers to get $159,100 but again I don't get how to calculate the beginning and ending inventory for January and the ending inventory for February.

Thanks for your help in advance.

logic if you purchase 30% in the month prior to sale this must be in inventory ie; 30% of 210000 = 63000 you have all the numbers to do the calculations