Caricolor Limited operates a paint manufacturing plant, housing two distinct departments: the mixing department and the quality control department. The paint undergoes sequential processing in both of these departments. The company's process$-$costing system is structured around two primary cost categories: direct materials and conversion costs. Direct materials are introduced at the initial stages of the process, while conversion costs are gradually incurred throughout the process. After completion of the mixing department's tasks on the paint, the product is promptly forwarded to the quality control department. Caricolor employs the weighted$-$average method for process costing to calculate its manufacturing costs. Data for the mixing department for October $2020$ are as follows:

Physical Units

$($Gallons of Paint$)$

Costs

Work $in$ process, October $1^x,16,000,$$$2,800,000,$$$901,400$

Started during October $2020,44,000$

Completed during October $202051,000$

Work in process, October ${31}^y,9,000$

Total costs added during October $2020$

$$9,400,000$

$$5,111,000$

x Degree of completion: direct materials, $100\%$; conversion costs, $60\%.$

y Degree of completion: direct materials, $100\%$; conversion costs, $70\%.$

Required:

$1.$ Compute equivalent units in the mixing department, for each cost category?

$2.$ Summarize total mixing department costs for October $2020$ for each cost category, and calculate the cost per equivalent unit?

$3.$ Assign total costs to units completed and transferred out and to units in ending work in process?

$4.$ By reference to the above data, explain the concept of equivalent units in process costing?