Chapter 7, Problem 12OQ

a)

Summary Introduction

To calculate: The break-even point in dollars of sales and in chairs and bar stools when the sales mix is 1:1.

Break-even point:

It is the point in a business at which the total costs and total revenues are equal. It is the position after which the firm begins to see profits.

Explanation of Solution

Given information:

Fixed cost (FC)     = $20,000

Chairs (c):

Price (P)         = $50

Variable cost (VC)     = $25

Bar stools (b):

Price (P)         = $50

Variable cost (VC)     = $20

Calculation of break-even point in chairs and stools:

$\begin{array}{c}\text{Let FC be the fixed cost}\text{.}\\ {\text{Let P}}_{\text{i}}\text{ be the price for product 'i'}\text{.}\\ {\text{Let VC}}_{\text{i}}\text{ be the variable cost for product 'i'}\text{.}\\ {\text{Let V}}_{\text{i}}\text{ be the volume for product 'i'}\text{.}\\ \text{Where 'i' is the chair}\left(\text{c}\right)\text{ and bar stool }\left(\text{b}\right)\text{.}\end{array}$

$\begin{array}{c}\text{FC}=\left(\left({\text{P}}_{\text{c}}-{\text{VC}}_{\text{c}}\right)\times {\text{V}}_{\text{c}}\right)+\left(\left({\text{P}}_{\text{b}}-{\text{VC}}_{\text{b}}\right)\times {\text{V}}_{\text{b}}\right)\\ 20,000=\left(\left(50-25\right)\times \text{V}\right)+\left(\left(50-20\right)\times \text{V}\right)\left[{\text{Since, V=V}}_{\text{c}}{\text{=V}}_{\text{b}}\right]\\ 20,000=25\text{V}+30\text{V}\\ \text{V}=\frac{20,000}{55}\\ =364\text{ chairs and bar stools}\text{.}\end{array}$

Calculation of break-even point in dollars:

$\begin{array}{c}{\text{BEP}}_{\text{dollars}}=\left(\text{Cost of chair}\times \text{Number of units}\right)+\left(\text{Cost of bar stool}\times \text{Number of units}\right)\\ =\left(\$50\times 364\right)+\left(\$50\times 364\right)\\ =18,200+18,200\\ =\$36,400\end{array}$

b)

Summary Introduction

To calculate: The break-even point in dollars of sales and in chairs and bar stools when the sales mix is 1:4.

Explanation of Solution

Given information:

Fixed cost (FC)     = $20,000

Chairs (c):

Price (P)         = $50

Variable cost (VC)     = $25

Bar stools (b):

Price (P)         = $50

Variable cost (VC)     = $20

Calculation of break-even point in chairs and stools:

$\begin{array}{c}\text{Let FC be the fixed cost}\text{.}\\ {\text{Let P}}_{\text{i}}\text{ be the price for product 'i'}\text{.}\\ {\text{Let VC}}_{\text{i}}\text{ be the variable cost for product 'i'}\text{.}\\ {\text{Let V}}_{\text{i}}\text{ be the volume for product 'i'}\text{.}\\ \text{Where 'i' is the chair}\left(\text{c}\right)\text{ and bar stool }\left(\text{b}\right)\text{.}\end{array}$

$\begin{array}{c}\text{FC}=\left(\left({\text{P}}_{\text{c}}-{\text{VC}}_{\text{c}}\right)\times {\text{V}}_{\text{c}}\right)+\left(\left({\text{P}}_{\text{b}}-{\text{VC}}_{\text{b}}\right)\times {\text{V}}_{\text{b}}\right)\\ 20,000=\left(\left(50-25\right)\times \text{V}\right)+\left(\left(50-20\right)\times 4\text{V}\right)\left[{\text{Since, V=V}}_{\text{c}}=0.{\text{25V}}_{\text{b}}\right]\\ 20,000=25\text{V}+120\text{V}\\ {\text{V}}_{\text{Chairs}}=\frac{20,000}{145}\\ =138\text{ chairs}\\ {\text{V}}_{\text{Bar stools}}=138\times 4\\ =552\text{ bar stools}\end{array}$

Calculation of break-even point in dollars:

$\begin{array}{c}{\text{BEP}}_{\text{dollars}}=\left(\text{Cost of chair}\times \text{Number of units}\right)+\left(\text{Cost of bar stool}\times \text{Number of units}\right)\\ =\left(\$50\times 138\right)+\left(\$50\times 552\right)\\ =6,900+27,600\\ =\$34,500\end{array}$