Cost structure

You are currently trying to decide between two cost structures for your business: one with a greater proportion of short-term fixed costs and another more heavily weighted to variable costs. Estimated revenue and cost data for each alternative are as follows:  

Cost Structure:	Selling price per unit	Variable cost per unit	Short-term fixed costs per year
Alternative 1	100	75	60000
Alternative 2	100	70	65000

Required:  
1. What sales volume, in units, is needed for the total costs in each cost-structure alternative to be the same?  
2. Suppose your profit goal for the coming year is 5% of sales (i. E., operating profit/sales = 5%). What is the sales level in units needed under each alternative to achieve this goal?
3. Suppose again that your profit goal for the coming year is 5% of sales. What is sales volume in dollars needed under each alternative to achieve this goal?

Correct answer:

a =  1000
p₁ =  2405
p₂ =  2170

Step-by-step explanation:

$$\begin{gathered} 75\cdot a+60000=70\cdot a+65000 \\ 5a=5000 \\ a=\frac{5000}{5}=1000 \\ a=1000 \end{gathered}$$

$$\begin{gathered} p=5\%=\frac{5}{100}=\frac{1}{20}=0.05 \\ (100-75)\cdot p_1-60000=p\cdot p_1 \\ (100-75)\cdot p_1-60000=0.05\cdot p_1 \\ 24.95p_1=60000 \\ {p}_1=\frac{60000}{24.95}=2404.80961924 \\ {p}_1=2404.809619=2405 \end{gathered}$$

$$\begin{gathered} (100-70)\cdot p_2-65000=p\cdot p_2 \\ (100-70)\cdot p_2-65000=0.05\cdot p_2 \\ 29.95p_2=65000 \\ {p}_2=\frac{65000}{29.95}=2170.28380634 \\ {p}_2=2170.283806=2170 \end{gathered}$$

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