Chapter 20, Problem 5E

a.

To determine

Compute the missing amount using contribution margin per unit.

Explanation of Solution

Contribution Margin:

The process or theory which is used to judge the benefit given by each unit of the goods produced is called as contribution margin.

The contribution margin is the difference between the selling price and the cost of the product.

Formula to compute the unit contribution margin:

$\text{Unit Contribution Margin}=\text{Unit Sales Price}-\text{Per Unit Variable Cost}$

	Sales	Variable Costs		Contribution Margin Ratio per Unit	Fixed Costs	Operating Income	Units Sold
(1)	$200,000	$120,000		$20	$55,000	$25,000	4,000
(2)	180,000	105,000		15	45,000	30,000	5,000
(3)	600,000	360,000		30	150,000	90,000	8,000

Table (1)

Working notes:

Calculate the amount of sales:

Step 1: Calculate the unit variable cost:

$\begin{array}{c}\text{Unit variable cost = }\left(\frac{\text{Total variable cost}}{\text{Units sold}}\right)\\ \text{= }\left(\frac{\text{\$120,000}}{\text{4,000}}\right)\\ =\$30\end{array}$

Step 2: Calculate the unit sales price:

$\begin{array}{c}\text{Unit Contribution Margin}=\text{Unit Sales Price}-\text{Per Unit Variable Cost}\\ \text{\$20=Sales Price}-\$30\\ \text{Unit Sales Price}=\text{\$20}+\$30\\ \text{Unit Sales Price}=\text{\$50}\end{array}$

Step 3: Calculate the amount of sales using contribution margin per unit:

$\begin{array}{c}\text{Amount of sales =Units Sold}\times \text{Unit Sales Price}\\ \text{=4,000}\times \text{\$50}\\ \text{=\$200,000}\end{array}$

Calculate the amount of variable costs using contribution margin per unit:

$\begin{array}{c}\text{Contribution margin per unit = }\left(\frac{\text{Sales}-\text{Varaible cost}}{\text{Units sold}}\right)\\ \$15=\left(\frac{\text{\$180,000}-\text{Varaible cost}}{\text{5,000}}\right)\\ \text{Varaible cost}=\text{\$180,000}-\$15\times \text{5,000}\\ \text{Varaible cost}=\text{\$180,000}-\$75,000\\ =\$105,000\end{array}$

Calculate the amount of variable costs using contribution margin per unit:

$\begin{array}{c}\text{Contribution margin per unit = }\left(\frac{\text{Sales}-\text{Varaible cost}}{\text{Units sold}}\right)\\ \$30=\left(\frac{\text{\$600,000}-\text{Varaible cost}}{\text{8,000}}\right)\\ \text{Varaible cost}=\text{\$600,000}-\$15\times 8\text{,000}\\ \text{Varaible cost}=\text{\$600,000}-\$240,000\\ =\$360,000\end{array}$

Calculate contribution margin ratio per unit:

Step 1: Calculate unit variable cost.

$\begin{array}{c}\text{Unit variable cost = }\left(\frac{\text{Total variable cost}}{\text{Units sold}}\right)\\ \text{= }\left(\frac{\text{\$105,000}}{\text{5,000}}\right)\\ =\$21\end{array}$

Step 2: Calculate the unit sales price.

$\begin{array}{c}\text{Unit sales cost = }\left(\frac{\text{Total sales cost}}{\text{Units sold}}\right)\\ \text{= }\left(\frac{\text{\$180,000}}{\text{5,000}}\right)\\ =\$36\end{array}$

Step 3: Calculate contribution margin ratio per unit:

$\begin{array}{c}\text{Unit Contribution Margin}=\text{Unit Sales Price}-\text{Per Unit Variable Cost}\\ \text{=\$36}-\$21\\ =\text{\$15}\end{array}$

Calculate the amount of fixed costs:

Step 1: Calculate contribution margin.

$\begin{array}{c}\text{Contribution Margin}=\text{Sales}-\text{Variable cost}\\ \text{=}\$200,000-\$120,000\\ =\text{\$80,000}\end{array}$

Step 2: Calculate the amount of fixed costs.

$\begin{array}{c}\text{Operating income}=\text{Contribution margin}-\text{Fixed cost}\\ \text{\$25,000=\$}80,000-\text{Fixed cost}\\ \text{Fixed cost}=\text{\$}80,000-\text{\$25,000}\\ \text{=\$55,000}\end{array}$

Calculate the amount of units sold using contribution margin per unit:

$\begin{array}{c}\text{Contribution margin per unit = }\left(\frac{\text{Sales}-\text{Varaible cost}}{\text{Units sold}}\right)\\ \$30=\left(\frac{\text{\$600,000}-\text{\$360,000}}{\text{Units sold}}\right)\\ \text{Units sold}=\left[\frac{\left(\text{600,000}-\text{\$360,000}\right)}{\$30}\right]\\ \text{Units sold}=\frac{\$240,000}{\$30}\\ =\$8,000\end{array}$

b.

To determine

Compute the missing amount using contribution margin ratio.

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.

$\begin{array}{c}\text{Contribution margin ratio = }\left[\frac{\begin{array}{l}\left(\text{Sales}-\text{Variable cost}\right)\\ \text{Contribution margin}\end{array}}{\text{Sales}}\right]\times 100\\ \end{array}$

	Sales	Variable Costs		Margin Ratio (%)	Fixed Costs	Operating Income
(1)	$900,000	$720,000		20%	$85,000	$95,000
(2)	600,000	360,000		40%	165,000	75,000
(3)	500,000	350,000		30%	90,000	60,000

Table (1)

Working notes:

Calculate the contribution margin ratio:

$\begin{array}{c}\text{Contribution margin ratio = }\left[\frac{\left(\text{Sales}-\text{Variable cost}\right)}{\text{Sales}}\right]\times 100\\ =\left[\frac{\left(\text{\$900,000}-\$720,000\right)}{\$900,000}\right]\times 100\\ =20\%\end{array}$

Calculate the amount of fixed costs:

Step 1: Calculate contribution margin.

$\begin{array}{c}\text{Contribution Margin}=\text{Sales}-\text{Variable cost}\\ \text{=}\$900,000-\$720,000\\ =\text{\$180,000}\end{array}$

Step 2: Calculate the amount of fixed costs.

$\begin{array}{c}\text{Operating income}=\text{Contribution margin}-\text{Fixed cost}\\ \text{\$95,000=\$1}80,000-\text{Fixed cost}\\ \text{Fixed cost}=\text{\$1}80,000-\text{\$95,000}\\ \text{=\$85,000}\end{array}$

Calculate the amount of variable cost using contribution margin ratio:

$\begin{array}{c}\text{Contribution margin ratio = }\left[\frac{\left(\text{Sales}-\text{Variable cost}\right)}{\text{Sales}}\right]\times 100\\ 40\%=\left[\frac{\left(\text{\$600,000}-\text{Variable cost}\right)}{\$600,000}\right]\times 100\\ \text{Variable cost}=\left[\text{\$600,000}-\left(\$600,000\times 40\%\right)\right]\\ \text{Variable cost}=\left(\$600,000-\$240,000\right)\\ =\$360,000\end{array}$

Calculate the amount of fixed costs:

Step 1: Calculate contribution margin.

$\begin{array}{c}\text{Contribution Margin}=\text{Sales}-\text{Variable cost}\\ \text{=}\$600,000-\$360,000\\ =\text{\$240,000}\end{array}$

Step 2: Calculate the amount of fixed costs.

$\begin{array}{c}\text{Operating income}=\text{Contribution margin}-\text{Fixed cost}\\ \text{\$75,000=\$}240,000-\text{Fixed cost}\\ \text{Fixed cost}=\text{\$}240,000-\text{\$75,000}\\ \text{=\$165,000}\end{array}$

Calculate the amount of sales using contribution margin ratio:

Step 1: Calculate the amount of fixed costs.

$\begin{array}{c}\text{Operating income}=\text{Contribution margin}-\text{Fixed cost}\\ \text{\$60,000=Contribution margin}-\text{\$90,000}\\ \text{Contribution margin}=\text{\$}60,000-\text{\$90,000}\\ \text{=\$150,000}\end{array}$

Step 2: Calculate the amount of sales using contribution margin ratio:

$\begin{array}{c}\text{Contribution margin ratio = }\left[\frac{\text{Contribution margin}}{\text{Sales}}\right]\times 100\\ 30\%=\left[\frac{\$150,000}{\text{Sales}}\right]\times 100\\ \text{Sales}=\frac{\$150,000}{.3}\\ \text{Sales}=\$500,000\end{array}$

Calculate the amount of variable cost using contribution margin ratio:

$\begin{array}{c}\text{Contribution margin ratio = }\left[\frac{\left(\text{Sales}-\text{Variable cost}\right)}{\text{Sales}}\right]\times 100\\ 30\%=\left[\frac{\left(\text{\$500,000}-\text{Variable cost}\right)}{\$500,000}\right]\times 100\\ \text{Variable cost}=\left[\text{\$500,000}-\left(\$500,000\times 30\%\right)\right]\\ \text{Variable cost}=\left(\$500,000-\$150,000\right)\\ =\$350,000\end{array}$