# 2.2 Revenue and Cost Functions

## LEARNING OBJECTIVES

• Solve cost-volume-profit analysis problems using the revenue and cost function approach.

Most businesses are “for-profit” businesses, meaning that they operate to make money.  There are no guarantees in business, and the future is always uncertain. What if you have to pay your employees higher wages?  What if the cost of the raw material you use in production decreases?  What if you sell more or less items than expected?  Successful business managers plan for the future and perform many “what-if” scenarios to answer these types of questions. This section develops a model for calculating total net income based on total revenues and total costs. The model allows managers to analyze various scenarios and determine the impact on profitability.

However, simply looking at the fixed costs, variable costs, potential revenues, contribution margins, and typical net income is not enough. Ultimately, to have any chance of staying in business all costs in a business need to be recovered through sales. Do you know how many units have to be sold to pay your bills? The answer to this question helps assess the feasibility of your business idea.

## Revenue and Cost Functions

Recall from the previous section that total costs ($TC$) are the total costs (fixed and variable) incurred by a business and total revenue ($TR$) is the amount of money received from the sales of the items produced and sold.

$\begin{eqnarray*}TC & = & FC+VC \times x \\ \\ TR & = & S \times x \end{eqnarray*}$

where

• $TC$ is the total costs.  The total costs is the sum of all of costs incurred by the business.
• $FC$ is the fixed costs.  These are the costs that do not change regardless of the level of production.
• $VC$ is the variable cost per unit. The unit variable cost is the typical or average variable cost associated with an individual unit of output.
• $TR$ is the total revenue. This is how much money or gross income the sale of the product at a certain output level brings into the organization.
• $S$ is the selling price per unit. The unit selling price of the product.
• $x$ is the number of units produced. This is the number of units produced or sold or the total output that incurred the total variable costs.

Because the number of units, $x$, varies, we can think of the total costs and total revenue formulas as functions of the number of units, $x$.

## Net Income using Revenue and Cost Functions

The net income ($NI$) is the difference between the total revenue and the total costs.

$\displaystyle{NI=TR-TC}$

Note that many companies use the terms net earnings or net profit instead of the term net income.  Net income is based on a certain level of output.  Here, we assume that the number of units produced or purchased exactly matches the number of units that are output or sold by the company.  In other words, there is no inventory, and its associated costs, to consider.

Substituting in the functions for $TR$ and $TC$, we can express the net income $NI$ in terms of the number of units $x$.

$\displaystyle{NI=S \times x-(FC+VC \times x)}$

where

• $NI$ is the net income. The amount of money left over after all costs have been paid is the net income. If the number is positive, then the business is profitable. If the number is negative, then the business suffers a loss.
• $S$ is the selling price per unit. The unit selling price of the product.
• $x$ is the number of units produced. This is the number of units produced or sold or the total output that incurred the total variable costs.
• $FC$ is the fixed costs.  These are the costs that do not change regardless of the level of production.
• $VC$ is the variable cost per unit. The unit variable cost is the typical or average variable cost associated with an individual unit of output.

There are three scenarios to consider.

• Total revenue is greater than total costs ($TR \gt TC$).  In this case the net income is positive ($NI \gt 0$) and the business makes a profit.
• Total revenue is less than total costs ($TR \lt TC$). In this case the net income is negative ($NI \lt 0$) and the business incurs a loss.
• Total revenue equals total cost ($TR=TC$).  In this case the net income is zero ($NI=0$).  This is the break-even point.

## EXAMPLE

A company’s monthly fixed costs for producing an item are $12,000. The variable cost per unit is$10.  The company sells the item for $25 each. 1. What is the net income if the company produces and sells 1,000 units next month? Did they make a profit or loss at this level of output? 2. What is the net income if the company produces and sells 500 units next month? Did they make a profit or loss at this level of output? 3. How many units must be produced and sold for a profit of$2,400?
4. How many units must be produced and sold for a loss of $2,850 Solution: Step 1: The given information is $\begin{eqnarray*} FC & = & 12,000 \\ VC & = & \10 \\ S & = & \25 \end{eqnarray*}$ The total revenue function is $\begin{eqnarray*} TR & = & S \times x \\ & = & 25 \times x \end{eqnarray*}$ The total cost function is $\begin{eqnarray*} TC & = & FC +VC \times x \\ & = & 12,000+10 \times x\end{eqnarray*}$ Step 2: Calculate the net income for $x=1,000$. $\begin{eqnarray*} TR & = & 25 \times 1,000 \\ & = & \25,000 \\ \\ TC & = & 12,000+10 \times 1,000 \\ & = & \22,000 \\ \\ NI & = & TR-TC \\ & = & 25, 000-22,000 \\ & = & \3,000 \end{eqnarray*}$ At 1,000 units, the net income is$3,000.  Because the net income is positive, the company made a profit at 1,000 units.

Step 3:  Calculate the net income for $x=500$.

$\begin{eqnarray*} TR & = & 25 \times 500 \\ & = & \12,500 \\ \\ TC & = & 12,000+10 \times 500 \\ & = & \17,000 \\ \\ NI & = & TR-TC \\ & = & 12, 500-17,000 \\ & = & -\4,500 \end{eqnarray*}$

At 500 units, the net income is -$4,500. Because the net income is negative, the company incurs a loss at 500 units. Step 4: Calculate the units for $NI=2,400$. Because this is a profit the net income is positive. $\begin{eqnarray*} NI & = & S \times -(FC+VC \times x) \\ 2,400 & = & 25 \times x-(12,000+10 \times x) \\ 2,400 & = & 25 \times x-12,000-10 \times x \\ 14,400 & = & 15 \times x \\ \frac{14,400}{15} & = & x \\ 960 & = & x \end{eqnarray*}$ A profit of$2,400 occurs when 960 units are produced and sold.

Step 5:  Calculate the units for $NI=-2,850$. Because this is a loss the net income is negative.

$\begin{eqnarray*} NI & = & S \times -(FC+VC \times x) \\ -2,850 & = & 25 \times x-(12,000+10 \times x) \\ -2,850 & = & 25 \times x-12,000-10 \times x \\ 9,150 & = & 15 \times x \\ \frac{9,150}{15} & = & x \\ 610 & = & x \end{eqnarray*}$

A loss of $2,850 occurs when 610 units are produced and sold. ### NOTES 1. In the previous example, a loss occurred at 500 units and a profit occurred at 1,000 units. This means that somewhere between 500 units and 1,000 units is a point where the net income is 0. 2. Remember, net income is positive when you have a profit and net income is negative when you have a loss. ## TRY IT A furniture manufacturer sells tables for$300 each.  The variable cost per table is $120. The monthly fixed costs are$5,000.

1. What is the net income if 30 tables are produced next month?
2. How many tables must be produced and sold to incur a loss of $3,200 next month? Click to see Solution 1. Net income for 30 tables. $\begin{eqnarray*} NI & = & 300 \times 30-(5,000+120 \times 30) \\ & = & 9,000-5,000-3,600 \\ & = & \400 \end{eqnarray*}$ 2. Units for loss of$300.

$\begin{eqnarray*} NI & = & 300 \times x-(5,000+120 \times x) \\ -3,200 & = & 300 \times x-5,000-120 \times x \\ 1,800 & = & 180 \times x \\ 10 & = & x \end{eqnarray*}$

## EXAMPLE

You run a small business that produces custom-ordered t-shirts.  You have fixed costs of $2,700 per month. Each t-shirt has a variable cost of$25.  You sell each t-shirt for $40. 1. What is the net income for a month in which you produce and sell 150 shirts? Would you incur a profit or loss at this level of production? 2. You decide to increase your selling price to$45 per shirt.  What is the net income for 150 shirts with this higher selling price? Are you making a profit or loss?
3. One of your suppliers has lowered their prices.  This decreases the variable cost per shirt to $18. What is the net income for 150 shirts with this lower variable cost? Are you making a profit or loss? (Assume the original$40 selling price).

Solution:

Step 1:  The given information is

$\begin{eqnarray*} FC & = & 2,700 \\ VC & = & \25 \\ S & = & \40 \end{eqnarray*}$

The total revenue function is

$\begin{eqnarray*} TR & = & S \times x \\ & = & 40 \times x \end{eqnarray*}$

The total cost function is

$\begin{eqnarray*} TC & = & FC +VC \times x \\ & = & 2,700+25 \times x\end{eqnarray*}$

Step 2:  Calculate the net income for $x=150$.

$\begin{eqnarray*} TR & = & 40 \times 150 \\ & = & \6,000 \\ \\ TC & = & 2,700+25 \times 150 \\ & = & \6,450 \\ \\ NI & = & TR-TC \\ & = & 6,000-6,450 \\ & = & -\450 \end{eqnarray*}$

At 150 shirts, the net income is -$450. Because the net income is negative, you incur a loss at 150 shirts. Step 3: Calculate the net income for $x=150$ with $S=\45$. $\begin{eqnarray*} TR & = & 45 \times 150 \\ & = & \6,750 \\ \\ TC & = & 2,700+25 \times 150 \\ & = & \6,450 \\ \\ NI & = & TR-TC \\ & = & 6,750-6,450 \\ & = & \350 \end{eqnarray*}$ With the selling price per shirt at$45, a profit of $350 is made when 150 shirts are produced and sold. Step 4: Calculate the net income for $x=150$ with $VC=\18$. $\begin{eqnarray*} TR & = & 40 \times 150 \\ & = & \6,000 \\ \\ TC & = & 2,700+18 \times 150 \\ & = & \5,400 \\ \\ NI & = & TR-TC \\ & = & 6,000-5,400 \\ & = & \600 \end{eqnarray*}$ With the variable cost per shirt at$18, a profit of $600 is made when 150 shirts are produced and sold. ## Break-Even Analysis with Revenue and Cost Functions If you are starting your own business and head to the bank to initiate a start-up loan, one of the first questions the banker will ask you is your break-even point. You calculate this number through break-even analysis, which is the analysis of the relationship between costs, revenues, and net income with the sole purpose of determining the point at which total revenue equals total cost ($TR=TC$). This break-even point is the level of output (in units or dollars) at which all costs are paid but no profits are earned, resulting in a net income equal to zero. The purpose of break-even analysis is to determine the point at which total cost equals total revenue. To find the break-even point in terms of the number of units sold ($x$), solve for $x$ in the equation where $TR=TC$. That is the break-even point is the value of $x$ so that $\displaystyle{S \times x=FC+VC \times x}$ Because total revenue equals total cost at the break-even point, the net income at the break-even point is zero. ## EXAMPLE A company’s monthly fixed costs for producing an item are$12,000.  The variable cost per unit is $10. The company sells the item for$25 each.

1. Calculate the number of units the company should produce and sell to break-even.
2. What is the revenue at the break-even point?
3. Suppose the fixed costs increase by $1,500 with no changes to the variable costs or selling price. What is the new break-even point? 4. Suppose the selling price decreases by$3 with no changes to the variable costs or fixed costs.  What is the new break-even point?
5. Suppose the variable costs increase by 50% with no changes to the fixed costs or selling price.  What is the new break-even point?

Solution:

Step 1:  The given information is

$\begin{eqnarray*} FC & = & 12,000 \\ VC & = & \10 \\ S & = & \25 \end{eqnarray*}$

The total revenue function is

$\begin{eqnarray*} TR & = & S \times x \\ & = & 25 \times x \end{eqnarray*}$

The total cost function is

$\begin{eqnarray*} TC & = & FC +VC \times x \\ & = & 12,000+10 \times x\end{eqnarray*}$

Step 2:  Calculate the break-even point.

$\begin{eqnarray*} TR & = & TC \\ 25 \times x & = & 12,000+10 \times x \\ 25 \times x -10 \times x & = & 12,000 \\ 15 \times x & = & 12,000 \\ x & = & \frac{12,000}{15} \\ x & = & 800 \end{eqnarray*}$

The break-even point is 800 units.

Step 3:  Calculate the revenue for $x=800$.

$\begin{eqnarray*} TR & = & 25 \times x \\ & = & 25 \times 800 \\ & = & \20,000 \end{eqnarray*}$

The revenue at the break-even point is $20,000. Step 4: Calculate the new break-even point when $FC=12,000+1,500=\13,500$. $\begin{eqnarray*} TR & = & TC \\ 25 \times x & = & 13,500+10 \times x \\ 25 \times x -10 \times x & = & 13,500 \\ 15 \times x & = & 13,500 \\ x & = & \frac{13,500}{15} \\ x & = & 900 \end{eqnarray*}$ When the fixed costs increase by$1,500, the break-even point is 900 units.

Step 5:  Calculate the new break-even point when $S=25-3=\22$.

$\begin{eqnarray*} TR & = & TC \\ 22 \times x & = & 12,000+10 \times x \\ 22 \times x -10 \times x & = & 12,000 \\ 12 \times x & = & 12,000 \\ x & = & \frac{12,000}{12} \\ x & = & 1,000 \end{eqnarray*}$

When the selling price decreases by $3, the break-even point is 1,000 units. Step 6: Calculate the new break-even point when $VC=10+0.5 \times 10=\15$. $\begin{eqnarray*} TR & = & TC \\ 25 \times x & = & 12,000+15 \times x \\ 25 \times x -15 \times x & = & 12,000 \\ 10 \times x & = & 12,000 \\ x & = & \frac{12,000}{10} \\ x & = & 1,200 \end{eqnarray*}$ When the variable costs increase by 50%, the break-even point is 1,200 units. ### NOTE When you calculate the break-even units, the value of $x$ you get when you solve $S \times x=FC+VC \times x$ may be a decimal. For example, a break-even point might be 324.39 units. How should you handle the decimal? A partial unit cannot be sold, so the rule is always to round the level of output up to the next integer, regardless of the decimal. Why? The main point of a break-even analysis is to show the point at which you have recovered all of your costs. If you round 324.39 down, you are 0.39 units short of recovering all of your costs. In the long-run, you always operate at a loss, which ultimately puts you out of business. If you round the level of output up to 325, all costs are covered and a tiny dollar amount, as close to zero as possible, is left over as profit. At least at this level of output you can stay in business. ## EXAMPLE You run a small business that produces custom-ordered t-shirts. You have fixed costs of$2,700 per month.  Each t-shirt has a variable cost of $25. You sell each t-shirt for$40.  Because this is only a part-time business, you can produce a maximum of 500 shirts each month.

1. How many t-shirts do you need to produce and sell each month to break-even?
2. What is the break-even point as a percentage of the maximum capacity?
3. You decide to increase your selling price to $45 per shirt. What is the new break-even point? 4. Your fixed costs decrease by$300 with no changes to the selling price or variable costs.  What is the new break-even point?
5. Your variable costs increase by $5 with no changes to the selling price or fixed costs. What is the new break-even point? Solution: Step 1: The given information is $\begin{eqnarray*} FC & = & 2,700 \\ VC & = & \25 \\ S & = & \40 \end{eqnarray*}$ The total revenue function is $\begin{eqnarray*} TR & = & S \times x \\ & = & 40 \times x \end{eqnarray*}$ The total cost function is $\begin{eqnarray*} TC & = & FC +VC \times x \\ & = & 2,700+25 \times x\end{eqnarray*}$ Step 2: Calculate the break-even point. $\begin{eqnarray*} TR & = & TC \\ 40 \times x & = & 2,700+25 \times x \\ 40 \times x -25 \times x & = & 2,700 \\ 15 \times x & = & 2,700 \\ x & = & \frac{2,700}{15} \\ x & = & 180 \end{eqnarray*}$ The break-even point is 180 units. Step 3: Calculate the break-even point as a percentage of the maximum capacity of 500 units. $\begin{eqnarray*}\mbox{Break-even as a Percent of Capacity} & = & \frac{\mbox{Break-Even Point}}{\mbox{Maximum Capacity}} \times 100\% \\ & = & \frac{180}{500} \times 100\% \\ & = & 36 \% \end{eqnarray*}$ The break-even point is 36% of the maximum capacity. Step 4: Calculate the new break-even point when $S=45$. $\begin{eqnarray*} TR & = & TC \\ 45 \times x & = & 2,700+25 \times x \\ 45 \times x -25 \times x & = & 2,700 \\ 20 \times x & = & 2,700 \\ x & = & \frac{2,700}{20} \\ x & = & 135 \end{eqnarray*}$ When the selling price is$45, the break-even point is 135 units.

Step 5:  Calculate the new break-even point when $FC=2,700-300=\2,400$.

$\begin{eqnarray*} TR & = & TC \\ 40 \times x & = & 2,400+25 \times x \\ 40 \times x -25 \times x & = & 2,400 \\ 15 \times x & = & 2,400 \\ x & = & \frac{2,400}{15} \\ x & = & 160 \end{eqnarray*}$

When the fixed costs decreases by $300, the break-even point is 160 units. Step 6: Calculate the new break-even point when $VC=25+5=\30$. $\begin{eqnarray*} TR & = & TC \\ 40 \times x & = & 2,700+30 \times x \\ 40 \times x -30 \times x & = & 2,700 \\ 10 \times x & = & 2,700 \\ x & = & \frac{2,700}{10} \\ x & = & 270 \end{eqnarray*}$ When the variable costs increase by$5, the break-even point is 270 units.

## TRY IT

A furniture manufacturer sells tables for $300 each. The variable cost per table is$120.  The monthly fixed costs are $5,000. The manufacturer can produce a maximum of 38 tables a month. 1. How many tables must be produced and sold to break-even? 2. What is the break-even revenue? 3. What is the break-even point as a percent of capacity? 4. Suppose the fixed costs increase by$1,300 with no changes to the variables costs or selling price.  What is the new break-even point?
5. Suppose the selling price decreases by 10% with no changes to the fixed costs or variable costs.  What is the new break-even point?
Click to see Solution

1. Break-even point.

$\begin{eqnarray*} 300 \times x & = & 5,000+120 \times x \\ 180 \times x& = & 5,000 \\ x & = & 27.777\ldots \\ & \rightarrow & 28 \end{eqnarray*}$

2. Break-even revenue.

$\begin{eqnarray*} TR & = & 300 \times 28 \\ & = & \8,400 \end{eqnarray*}$

3. Percent capacity.

$\begin{eqnarray*}\mbox{Break-even as a Percent of Capacity} & = & \frac{28}{38} \times 100\% \\ & = & 73.68 \% \end{eqnarray*}$

4. Break-even point with $FC=\6,300$.

$\begin{eqnarray*} 300 \times x & = & 6,300+120 \times x \\ 180 \times x& = & 6,300 \\ x & = & 35 \end{eqnarray*}$

5. Break-even point with $S=\330$.

$\begin{eqnarray*} 330 \times x & = & 5,000+120 \times x \\ 210 \times x& = & 5,000 \\ x & = & 23.8095\ldots \\ & \rightarrow & 24 \end{eqnarray*}$

### NOTE

You need to be very careful with the interpretation and application of a break-even number. In particular, the break-even point must have a point of comparison, and it does not provide information about the viability of the business.

Break-Even Points Need to Be Compared.

The break-even number by itself, whether in units or revenue, is meaningless. You need to compare it against some other quantity (or quantities) to determine the feasibility of the number you have produced. The other number needs to be some baseline that allows you to grasp the scope of what you are planning. This baseline could include, but is not limited to, industry sales, number of competitors fighting for market share in your industry, or production capacity of your business.

For example, suppose the break-even point for your internet sales business is 179 units per month. Is that good? Imagine you have eight competitors in this business.  If the market can only support a total of 1,000 sales each month, then split evenly between the nine businesses (the eight competitors plus yourself) would mean 111 sales each, which is much lower than the break-even point.  To just pay your bills, you would have to sell almost 61% higher than the even split and achieve a 17.9% market share, which is fairly unlikely.

Break-Even Points Are Not Green Lights.

A break-even point alone cannot tell you to do something, but it can tell you not to do something. In other words, break-even points can put up red lights, but at no point does it give you the green light. In the above scenario, your break-even point of 179 units put up a whole lot of red lights because it does not seem feasible to obtain. However, what if your industry sold 10,000 units instead of 1,000 units? Your break-even point would now be a 1.79% market share (179 units out of 10,000 units), which certainly seems realistic and attainable. This does not mean “Go for it,” however. It just means that from a strictly financial point of view breaking even seems possible. Whether you can actually sell that many units depends on a whole range of factors beyond just a break-even number.

## Exercises

1. In the current period, Blue Mountain Packers in Salmon Arm, British Columbia, had fixed costs of $228,000 and a total cost of$900,000 while maintaining a level of output of 6,720 units. Next period sales are projected to rise by 20%. What total cost should Blue Mountain Packers project?
Click to see Answer

$1,034,000 2. Fred runs a designer candle-making business out of his basement. He sells the candles for$15 each, and every candle costs him $6 to manufacture. If his fixed costs are$2,300 per month, what is his projected net income or loss next month, for which he forecasts sales of 225 units?
Click to see Answer

-$275 3. Gayle is thinking of starting her own business. Total fixed costs are$19,000 per month and unit variable costs are estimated at $37.50. From some preliminary studies that she completed, she forecasts sales of 1,400 units at$50 each, 1,850 units at $48 each, 2,500 units at$46 each, and 2,750 units at $44 each. What price would you recommend Gayle set for her products? Click to see Answer$46

4. What level of output would generate a net income of $15,000 if a product sells for$24.99, has unit variable costs of $9.99, and total fixed costs of$55,005?
Click to see Answer

4,667

5. Franklin has started an ink-jet print cartridge refill business. He has invested $2,500 in equipment and machinery. The cost of refilling a cartridge including labor, ink, and all other materials is$4. He charges $14.95 for his services. How many cartridges does he need to refill to break even? Click to see Answer 229 6. Hasbro manufactures a line of children’s pet toys. If it sells the toy to distributors for$2.30 each while variable costs are $0.75 toy, how many toys does it need to sell to recover the fixed cost investment in these toys of$510,000? What total revenue would this represent?
Click to see Answer

329,033, $756,775.90 7. If the break-even point is 15,000 units, the selling price is$95, and the unit variable cost is $75, what are the company’s total fixed costs? Click to see Answer$300,000

8. Louisa runs a secretarial business part time in the evenings. She takes dictation or handwritten minutes and converts them into printed word-processed documents. She charges $5 per page for her services. Including labor, paper, toner, and all other supplies, her unit variable cost is$2.50 per page. She invested $3,000 worth of software and equipment to start her business. How many pages will she need to output to break even? Click to see Answer 1200 9. Ashley rebuilds old laptops as a home hobby business. Her variable costs are$125 per laptop and she sells them for $200. She has determined that her break-even point is 50 units per month. Determine her net income for a month in which she sells 60 units. Click to see Answer$750

10. Shardae is starting a deluxe candy apple business. The cost of producing one candy apple is $4.50. She has total fixed costs of$5,000. She is thinking of selling her deluxe apples for $9.95 each. 1. Determine her unit break-even point at her selling price of$9.95.
2. Shardae thinks her price might be set too high and lowers her price to $8.95. Determine her new break-even point. 3. An advertising agency approaches Shardae and says people would be willing to pay the$9.95 if she ran some “upscale” local ads. They would charge her $1,000. Determine her break-even point. 4. If she wanted to maintain the same break-even units as determined in a., what would the price have to be to pay for the advertising? Click to see Answer a. 918; b. 1,124; c. 1,101; d.$11.04

11. Robert is planning a wedding social for one of his close friends. Costs involve $865 for the hall rental,$135 for a liquor license, $500 for the band, and refreshments and food from the caterer cost$10 per person. If he needs to raise $3,000 to help his friend with the costs of his wedding, what price should he charge per ticket if he thinks he can fill the social hall to its capacity of 300 people? Click to see Answer$25

12. École Van Belleghem is trying to raise funds to replace its old playground equipment with a modern, child-safe structure. The Blue Imp playground equipment company has quoted the school a cost of $49,833 for its 20m by 15m megastructure. To raise the funds, the school wants to sell Show ‘n’ Save books. These books retail for$15.00 each and cost $8.50 to purchase. How many books must the school sell to raise funds for the new playground? Click to see Answer 7,667 13. Suppose the selling price per unit is$100, the variable costs per unit is $60 and the fixed costs are$250,000
1. Calculate the break-even point in both units and revenue.
2. Suppose the fixed costs are reduced by 15% and the unit variable costs rise by 5%.  What is the new break-even point and break-even revenue?
3. Suppose the unit variable costs are reduced by 10% and the fixed costs rise by 5%.  What is the new break-even point and break-even revenue?
4. Suppose the total fixed costs are reduced by 20% and the unit variable costs rise by 10%.  What is the new break-even point and break-even revenue?
5. Suppose the unit variable costs are reduced by 15% and the fixed costs rise by 15%.  What is the new break-even point and break-even revenue?
Click to see Answer

a. 6,250, $625,000; b. 5,744,$574,400; c. 5,707, $570,700; d. 5,883,$588,300; e. 5,868, \$586,800

4.6: Cost-Revenue-Net Income Analysis” from Introduction to Business Math by Margaret Dancy is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

4.7: Break-Even Analysis” from Introduction to Business Math by Margaret Dancy is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

5.1: Cost-Revenue-Net Income Analysis” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.

5.2: Break-Even Analysis” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.