Chapter 6, Problem 5PA

1.

To determine

Calculate the break-even point in sales units for the overall Product E for the current year.

Explanation of Solution

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:

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

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

Fixed cost =$2,498,600

Contribution margin per unit =$620 per unit (3)

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}]=\frac{\text{Fixed}\text{Costs}}{\text{Contribution}\text{Margin}\text{per}\text{unit}}\\ =\frac{\$2,498,600}{\$620}\\ =4,030\text{units}\end{array}$

Note: For break-even analysis, the Product-Laptops and Product-Tablets are considered as the components of one overall company’s Product E.

Working note (1):

Determine the selling price per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Selling}\text{price}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}\text{Laptops}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Laptops}\end{array}\right)+\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}\text{Tablets}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Tablets}\end{array}\right)\\ \text{=}\left(\text{\$1,600}\text{per}\text{unit}\times 40\%\right)+\left(\text{\$850}\text{per}\text{unit}\times 60\text{\%}\right)\\ =\$640\text{per}\text{unit}+\$510\text{per}\text{unit}\\ \text{=\$1,150}\text{per}\text{unit}\end{array}$

Working note (2):

Determine the variable cost per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Variable}\text{cost}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{Laptops}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Laptops}\end{array}\right)+\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{Tablets}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Tablets}\end{array}\right)\\ \text{=}\left(\text{\$800}\text{per}\text{unit}\times 40\%\right)+\left(\text{\$350}\text{per}\text{unit}\times 6\text{0\%}\right)\\ =\$320\text{per}\text{unit}+\$210\text{per}\text{unit}\\ \text{=\$530}\text{per}\text{unit}\end{array}$

Working note (3):

Determine the unit contribution margin of Product E.

$\begin{array}{c}\begin{array}{l}\text{Unit}\text{Contribution}\\ \text{Margin}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\end{array}\right)-\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\end{array}\right)\\ \text{=\$1,150}\text{per}\text{unit}\left(1\right)-\text{\$530}\text{per}\text{unit}\left(2\right)\\ \text{=\$620}\text{per}\text{unit}\end{array}$

Conclusion

Therefore, the break-even point in sales units for the overall Product E for the current year is 4,030 units.

2.

To determine

Calculate the break-even sales (units) for Product-Laptops and Product-Tablets.

Explanation of Solution

Determine the break-even point in sales units:

For Product-Laptops

Break-even point in sales units for Product E =4,030 units (refer Part a)

Sales Mix for Product Laptops =40%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{Laptops}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\\ \text{Laptops}\end{array}\right)\\ =4,030\text{units}\times 4\text{0\%}\\ \text{=1,612}\text{units}\end{array}$

For Product-Tablets

Break-even point in sales units for Product E =4,030 units (refer Part a)

Sales Mix for Product Tablets =60%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{Tablets}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\text{Tablets}\end{array}\right)\\ =4,030\text{units}\times 6\text{0\%}\\ \text{=2,418}\text{units}\end{array}$

Conclusion

Therefore, the break-even point in sales units for the Product Laptops is 1,612 units and for the Product Tablets are 2,418 units.

3.

To determine

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% laptops and 50% tablets is 3,844 units. It is less than the break-even point of 4,030 units calculated in Part 1.

The reason for the difference is the sales mix which is allocated at a higher percentage to the laptops (50%) and tablets (50%) in the present case. It resulted in the higher contribution margin per unit of $650 per unit than in Part 1 ($620 per unit). Thus, it decreases the break-even point of sales (units) in the present case.

Working note (4):

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

Fixed cost =$2,498,600

Contribution margin per unit =$620 per unit (7)

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}]=\frac{\text{Fixed}\text{Costs}}{\text{Contribution}\text{Margin}\text{per}\text{unit}}\\ =\frac{\$2,498,600}{\$650}\\ =3,844\text{units}\end{array}$

Note: For break-even analysis, the Product-Laptops and Product-Tablets are considered as the components of one overall company’s Product E.

Working note (5):

Determine the selling price per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Selling}\text{price}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}\text{Laptops}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Laptops}\end{array}\right)+\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\text{of}\text{Tablets}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Tablets}\end{array}\right)\\ \text{=}\left(\text{\$1,600}\text{per}\text{unit}\times 50\%\right)+\left(\text{\$850}\text{per}\text{unit}\times 50\text{\%}\right)\\ =\$800\text{per}\text{unit}+\$425\text{per}\text{unit}\\ \text{=\$1,225}\text{per}\text{unit}\end{array}$

Working note (6):

Determine the variable cost per unit of Product E.

$\begin{array}{c}\begin{array}{l}\text{Variable}\text{cost}\text{per}\\ \text{unit}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{Laptops}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Laptops}\end{array}\right)+\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\text{of}\text{Tablets}\\ \text{×}\text{sales}\text{mix}\text{of}\text{Tablets}\end{array}\right)\\ \text{=}\left(\text{\$800}\text{per}\text{unit}\times 50\%\right)+\left(\text{\$350}\text{per}\text{unit}\times 5\text{0\%}\right)\\ =\$400\text{per}\text{unit}+\$175\text{per}\text{unit}\\ \text{=\$575}\text{per}\text{unit}\end{array}$

Working note (7):

Determine the unit contribution margin of Product E.

$\begin{array}{c}\begin{array}{l}\text{Unit}\text{Contribution}\\ \text{Margin}\text{of}\text{Product}\text{E}\end{array}]\text{=}\left(\begin{array}{l}\text{Selling}\text{price}\\ \text{per}\text{unit}\end{array}\right)-\left(\begin{array}{l}\text{Variable}\text{cost}\\ \text{per}\text{unit}\end{array}\right)\\ \text{=\$1,225}\text{per}\text{unit}\left(5\right)-\text{\$575}\text{per}\text{unit}\left(6\right)\\ \text{=\$650}\text{per}\text{unit}\end{array}$

Working note (8):

Determine the break-even point in sales units:

For Product-Laptops:

Break-even point in sales units for Product E =3,844 units (4)

Sales Mix for Product Laptops =50%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{Laptops}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\\ \text{Laptops}\end{array}\right)\\ =3,844\text{units}\times 5\text{0\%}\\ \text{=1,922}\text{units}\end{array}$

Working note (9):

For Product-Tablets:

Break-even point in sales units for Product E =3,844 units (4)

Sales Mix for Product Tablets =50%

$\begin{array}{c}\begin{array}{l}\text{Break-even}\text{point}\text{in}\\ \text{Sales}\left(\text{units}\right)\text{for}\\ \text{Product}\text{Tablets}\end{array}]=\left(\begin{array}{l}\text{Break-even}\text{point}\text{in}\text{Sales}\\ \left(\text{units}\right)\text{for}\text{Product}\text{E}\end{array}\right)\times \left(\begin{array}{l}\text{Sales}\text{mix}\text{for}\\ \text{Product}\text{Tablets}\end{array}\right)\\ =3,844\text{units}\times 5\text{0\%}\\ \text{=1,922}\text{units}\end{array}$