Chapter 21, Problem 21.5BPR

1.

To determine

Sales mix: It refers to the relative distribution of the total sales among the number of products sold by a company. In other words, it is expressed as a percentage of units sold for each product with respect to the total units sold for all the products.

Break-even Point: It refers to a point in the level of operations at which a company experiences its revenues generated is equal to its costs incurred. Thus, when a company reaches at its break-even point, it reports neither an income nor a loss from operations. The formula to calculate the break-even point in sales units is as follows:

$\text{Break-even}\text{point}\text{in}\text{Sales}\left(\text{units}\right)=\frac{\text{Fixed}\text{Costs}}{\text{Contribution}\text{Margin}\text{per}\text{unit}}$

To determine: the break-even point in sales units for the overall Product E for the current year.

Explanation of Solution

Determine the break-even point in sales units for the overall Product E.

Fixed cost =$46,800

Contribution margin per unit =$10.40 per unit (3)

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}]=\frac{\text{Fixed}\text{Costs}}{\text{Contribution}\text{Margin}\text{per}\text{unit}}\\ =\frac{\$46,800}{\$10.40}\\ =4,500\text{units}\end{array}$

Working notes:

Note: For break-even analysis, the Product-12” Pizza and Product-13” Pizza are considered as the components of one overall company’s Product E.

Determine the selling price per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Selling}\text{price}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}12"\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}12"\text{Pizza}\end{array}\right)+\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}16"\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}16"\text{Pizza}\end{array}\right)\\ \text{=}\left(\text{\$12}\text{per}\text{unit}\times 30\%\right)+\left(\text{\$15}\text{per}\text{unit}\times 70\text{\%}\right)\\ =\$3.60\text{per}\text{unit}+\$10.50\text{per}\text{unit}\\ \text{=\$14}\text{.10}\text{per}\text{unit}\end{array}$ (1)

Determine the variable cost per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Variable}\text{cost}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{12"}\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}\text{12"}\text{Pizza}\end{array}\right)+\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{16"}\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}\text{16"}\text{Pizza}\end{array}\right)\\ \text{=}\left(\text{\$3}\text{per}\text{unit}\times 30\%\right)+\left(\text{\$4}\text{per}\text{unit}\times 7\text{0\%}\right)\\ =\$0.90\text{per}\text{unit}+\$2.80\text{per}\text{unit}\\ \text{=\$3}\text{.70}\text{per}\text{unit}\end{array}$ (2)

Determine the unit contribution margin of Product E.

$\begin{array}{c}\begin{array}{l}\text{Unit}\text{Contribution}\\ \text{Margin}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\end{array}\right)-\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\end{array}\right)\\ \text{=\$14}\text{.10}\text{per}\text{unit}\left(1\right)-\text{\$3}\text{.70}\text{per}\text{unit}\left(2\right)\\ \text{=\$10}\text{.40}\text{per}\text{unit}\end{array}$ (3)

Conclusion

Therefore, the break-even point in sales units for the overall Product E for the current year is 4,500 units.

2.

To determine

the break-even sales (units) for Product-12” Pizza and Product-16” Pizza.

Explanation of Solution

Determine the break-even point in sales units:

For Product-12” Pizza

Break-even point in sales units for Product E =4,500 units (refer Part a)

Sales Mix for Product Laptops =30%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{12"}\text{Pizza}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\\ \text{12"}\text{Pizza}\end{array}\right)\\ =4,500\text{units}\times 3\text{0\%}\\ \text{=1,350}\text{units}\end{array}$

For Product-16” Pizza

Break-even point in sales units for Product E =4,500 units (refer Part a)

Sales Mix for Product Tablets =70%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{16"}\text{Pizza}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\text{16"}\text{Pizza}\end{array}\right)\\ =4,500\text{units}\times 7\text{0\%}\\ \text{=3,150}\text{units}\end{array}$

Conclusion

Therefore, the break-even point in sales units for the Product 12” Pizza is 1,350 units and for the Product 16” Pizza is 3,150 units.

3.

To determine

To compare: the break-even point with that in Part (1).

Explanation of Solution

The break-even point calculated in (1) with a sales mix of 50% 12” Pizza and 50% 16” Pizza is 4,680 units. It is more than the break-even point of 4,500 units calculated in Part 1.

The reason for the difference is the sales mix which is allocated at a lower percentage to the 12” Pizza (50%) and 16” Pizza (50%) in the present case. It resulted in the lower contribution margin per unit of $10.00 per unit than in Part 1 ($10.40 per unit). Thus, it increases the break-even point of sales (units) in the present case.

Working notes:

Determine the break-even point in sales units for the overall Product E.

Fixed cost =$46,800

Contribution margin per unit =$10.00 per unit (7)

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}]=\frac{\text{Fixed}\text{Costs}}{\text{Contribution}\text{Margin}\text{per}\text{unit}}\\ =\frac{\$46,800}{\$10.00}\\ =4,680\text{units}\end{array}$ (4)

Note: For break-even analysis, the Product-12” Pizza and Product-16” Pizza are considered as the components of one overall company’s Product E.

Determine the selling price per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Selling}\text{price}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}\text{12"}\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}\text{12"}\text{Pizza}\end{array}\right)+\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}\text{16"}\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}\text{16"}\text{Pizza}\end{array}\right)\\ \text{=}\left(\text{\$12}\text{per}\text{unit}\times 50\%\right)+\left(\text{\$15}\text{per}\text{unit}\times 50\text{\%}\right)\\ =\$6\text{per}\text{unit}+\$7.50\text{per}\text{unit}\\ \text{=\$13}\text{.50}\text{per}\text{unit}\end{array}$ (5)

Determine the variable cost per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Variable}\text{cost}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{12"}\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}\text{12"}\text{Pizza}\end{array}\right)+\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{16"}\text{Pizza}\\ \text{×}\text{sales}\text{mix}\text{of}\text{16"}\text{Pizza}\end{array}\right)\\ \text{=}\left(\text{\$3}\text{per}\text{unit}\times 50\%\right)+\left(\text{\$4}\text{per}\text{unit}\times 5\text{0\%}\right)\\ =\$1.50\text{per}\text{unit}+\$2\text{per}\text{unit}\\ \text{=\$3}\text{.50}\text{per}\text{unit}\end{array}$ (6)

Determine the unit contribution margin of Product E.

$\begin{array}{c}\begin{array}{l}\text{Unit}\text{Contribution}\\ \text{Margin}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\end{array}\right)-\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\end{array}\right)\\ \text{=\$13}\text{.50}\text{per}\text{unit}\left(5\right)-\text{\$3}\text{.50}\text{per}\text{unit}\left(6\right)\\ \text{=\$10}\text{.00}\text{per}\text{unit}\end{array}$ (7)

Determine the break-even point in sales units:

For Product-12” Pizza

Break-even point in sales units for Product E =4,680 units (4)

Sales Mix for Product 12” Pizza =50%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{12"}\text{Pizza}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\\ \text{12"}\text{Pizza}\end{array}\right)\\ =4,680\text{units}\times 5\text{0\%}\\ \text{=2,340}\text{units}\end{array}$ (8)

For Product-16” Pizza

Break-even point in sales units for Product E =4,680 units (4)

Sales Mix for Product 16” Pizza =50%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{16"}\text{Pizza}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\text{16"}\text{Pizza}\end{array}\right)\\ =4,680\text{units}\times 5\text{0\%}\\ \text{=2,340}\text{units}\end{array}$ (9)