Chapter 16, Problem 42P

Good Scent, Inc., produces two colognes: Rose and Violet. Of the two, Rose is more popular. Data concerning the two products follow:

The company uses a conventional costing system and assigns overhead costs to products using direct labor hours. Annual overhead costs follow. They are classified as fixed or variable with respect to direct labor hours.

Required:

1. 1. Using the conventional approach, compute the number of cases of Rose and the number of cases of Violet that must be sold for the company to break even.
2. 2. Using an activity-based approach, compute the number of cases of each product that must be sold for the company to break even.

To determine

Ascertain the break-even point using the conventional approach.

Explanation of Solution

Contribution Margin Ratio: The contribution margin ratio shows the amount of difference in the actual sales value and the variable expenses in percentage. This margin indicates that percentage which is available for sale above the fixed costs and the profit. The formula for variable cost ratio is shown below:

$\text{Contribution Margin ratio = }\frac{\text{Contribution Margin}}{\text{Sales}}$

Break-Even Point: The break-even point refers to the point of sales at which the firm neither earns a profit nor suffers a loss. It is also known as the point of sales or sales value at which the firm recovers the entire cost of fixed and variable nature.

Break-Even in sales revenue: The break-even in sales revenue refers to the sales volume required to cover the fixed and variable costs and left out with neither profit nor loss.

Compute the package contribution margin units:

Input	Price (A)	Unit Variable cost (B)	Unit Contribution margin $\left(\text{C = A}-\text{B}\right)$	Sales Mix (D)	Package Unit Contribution margin $\left(\text{E = C × D}\right)$
Rose	$100.00	$67.92	$32.08	5	$160.40
Violet	$80.00	$56.60	$23.40	1	$23.40
Package Total					$183.80

Table (1)

Compute the break-even packages:

$\begin{array}{c}\text{Break-even packages }=\frac{\text{Total fixed cost}}{\text{Package Contribution margin}}\\ =\frac{\$550,000}{\$183.80}\\ =\text{2,992 packages }\left(\text{rounded}\right)\end{array}$

The number of break-even packages is 2,992.

Compute the break-even for Rose:

$\begin{array}{c}\text{Break-even for Rose }=\text{Sales mix }\times \text{Break-even packages}\\ =5\times \text{ 2,992}\\ =14,960\end{array}$

The number of break-even for Rose is 14,960.

Compute the break-even for Violet:

$\begin{array}{c}\text{Break-even for Violets }=\text{Sales mix }\times \text{Break-even packages}\\ =1\times \text{ 2,992}\\ =\text{2,992}\end{array}$

The number of break-even for Violet is 2,992.

Working Notes:

Compute the unit variable cost for rose:

$\begin{array}{c}\begin{array}{l}\text{Unit Variable }\\ \text{Cost for Rose }\end{array}\}=\left[\begin{array}{l}\left(\begin{array}{l}\text{Material cost}\\ \text{ per case}\end{array}\right)\text{+}\left(\begin{array}{l}\text{Direct labor }\\ \text{cost per case}\end{array}\right)\text{+}\\ \left[\left(\begin{array}{l}\text{Direct Labor}\\ \text{ hour}\end{array}\right)\times \frac{\text{Direct labor hours}}{\text{Expected sales}}\right]\end{array}\right]\\ =\$50+\$10+\left[\left(\frac{\text{\$462,000}}{\text{42,000}}\right)\times \left(\frac{\text{36,000}}{\text{50,000}}\right)\right]\\ =\$60+\left(\$11\times 0.72\right)\\ =\$67.92\end{array}$

The variable cost per unit for rose is $67.92.

Compute the unit variable cost for violet:

$\begin{array}{c}\begin{array}{l}\text{Unit Variable }\\ \text{Cost for Violet}\end{array}\}=\left[\begin{array}{l}\left(\begin{array}{l}\text{Material cost}\\ \text{ per case}\end{array}\right)\text{+}\left(\begin{array}{l}\text{Direct labor }\\ \text{cost per case}\end{array}\right)\text{+}\\ \left[\left(\begin{array}{l}\text{Direct Labor}\\ \text{ hour}\end{array}\right)\times \frac{\text{Direct labor hours}}{\text{Expected sales}}\right]\end{array}\right]\\ =\$43+\$7+\left[\left(\frac{\text{\$462,000}}{\text{42,000}}\right)\times \left(\frac{\text{6,000}}{\text{10,000}}\right)\right]\\ =\$50+\left(\$11\times 0.60\right)\\ =\$56.60\end{array}$

The variable cost per unit for violet is $56.60.

To determine

Compute the break-even point and the incremental profit using the activity based costing.

Explanation of Solution

Compute the unit-based variable cost per unit:

Particulars	Rose	Violet
Unit-based variable costs:
Prime costs	$60.00	$50.00
Benefits	$3.43	$2.86
Machine costs	$4.03	$6.05
Total	$67.46	$58.91
Sales Mix	$(\times )$5	$(\times )$1
Package Cost	$337.30	$58.91

Table (2)

Compute the total package cost:

$\begin{array}{c}\text{Total Package Cost }=\text{Package }\mathrm{C}\text{ost of Rose}+\text{Package Cost of Violet}\\ =\text{\$337}\text{.30}+\text{\$ 58}\text{.91}\\ =\text{\$396}\text{.21}\end{array}$

The total package cost is $396.21 (X1).

Compute the benefits cost per unit for rose:

$\begin{array}{c}\begin{array}{l}\text{Unit Benefits }\\ \text{Cost for Rose }\end{array}\}=\left[\left(\frac{\text{Direct Labor benefits}}{\text{Total labor hours}}\right)\times \left(\frac{\text{Direct labor hours}}{\text{Expected sales}}\right)\right]\\ =\left[\left(\frac{\text{\$200,000}}{\text{42,000}}\right)\times \left(\frac{\text{36,000}}{\text{50,000}}\right)\right]\\ =\$4.762\times 0.72\\ =\$3.43\left(\text{rounded}\right)\end{array}$

The benefits cost per unit for rose is $3.43.

Compute the benefits cost per unit for violet:

$\begin{array}{c}\begin{array}{l}\text{Unit Benefits }\\ \text{Cost for Violet }\end{array}\}=\left[\left(\frac{\text{Direct Labor benefits}}{\text{Total labor hours}}\right)\times \left(\frac{\text{Direct labor hours}}{\text{Expected sales}}\right)\right]\\ =\left[\left(\frac{\text{\$200,000}}{\text{42,000}}\right)\times \left(\frac{\text{6,000}}{\text{10,000}}\right)\right]\\ =\$4.762\times 0.60\\ =\$2.86\left(\text{rounded}\right)\end{array}$

The benefits cost per unit for violet is $2.86.

Compute the unit machine cost for rose:

$\begin{array}{c}\begin{array}{l}\text{Unit Machine }\\ \text{Cost for Rose }\end{array}\}=\left[\left(\frac{\text{Direct Machine Costs}}{\text{Total Machine hours}}\right)\times \left(\frac{\text{Direct Machine hours}}{\text{Expected sales}}\right)\right]\\ =\left[\left(\frac{\text{\$262,000}}{\text{13,000}}\right)\times \left(\frac{\text{10,000}}{\text{50,000}}\right)\right]\\ =\$20.15\times 0.20\\ =\$4.03\left(\text{rounded}\right)\end{array}$

The machine cost per unit for rose is $4.03.

Compute the unit machine cost for violet:

$\begin{array}{c}\begin{array}{l}\text{Unit Variable }\\ \text{Cost for Violet}\end{array}\}=\left[\left(\frac{\text{Direct Machine Costs}}{\text{Total Machine hours}}\right)\times \left(\frac{\text{Direct Machine hours}}{\text{Expected sales}}\right)\right]\\ =\left[\left(\frac{\text{\$262,000}}{\text{13,000}}\right)\times \left(\frac{\text{3,000}}{\text{10,000}}\right)\right]\\ =\$20.15\times 0.30\\ =\$6.05\left(\text{rounded}\right)\end{array}$

The machine cost per unit for violet is $6.05.

Compute the non-unit-based variable overhead cost per unit:

Particulars	Calculations	Amount ($)
Non-unit-based variable costs:
Receiving (X2)	$\left(\text{\$225,000/75}\right)$	$3,000
Packing (X3)	$\left(\text{\$125,000/150}\right)$	$833.33

Table (2)

Compute the sales and variable cost per unit:

Particulars	Calculations	Amount ($)
Sales	$\left[\left(\text{\$100}\times \text{5}\right)+\left(\$80\times 1\right)\right]$	$580.00
Variable cost		$396.20
Contribution margin		$183.80

Table (2)

CVP analysis:

Let,

X1 = Number of packages

X2 = Number of receiving orders

X3 = Number of packing orders

$\begin{array}{c}\text{Packages = }\frac{\text{Fixed Cost+ Non-unit-based variable costs}}{\text{Contribution Margin}}\\ \text{=}\frac{\text{(\$200,000 + \$3,000X2 + \$833}\text{.33X3)}}{\text{\$183}\text{.79}}\\ =\frac{\text{(\$200,000 + \$3,000}\left(75\right)\text{ + \$833}\text{.33}\left(150\right)\text{)}}{\text{\$183}\text{.79}}\\ =\frac{\text{\$550,000}}{\text{\$183}\text{.79}}\end{array}$

$\text{Packages = 2,992 packages }\left(\text{rounded}\right)$

The number of break-even packages is 2,992.

Compute the break-even for Rose:

$\begin{array}{c}\text{Break-even for Rose }=\text{Sales mix }\times \text{Break-even packages}\\ =5\times \text{ 2,992}\\ =14,960\end{array}$

The number of break-even for Rose is 14,960.

Compute the break-even for Violet:

$\begin{array}{c}\text{Break-even for Violets }=\text{Sales mix }\times \text{Break-even packages}\\ =1\times \text{ 2,992}\\ =\text{2,992}\end{array}$

The number of break-even for Violet is 2,992.

Thus, the break-even point for Roses and Violets are same under both the requirements.