Profit, Revenue, and Cost Problems on the Digital SAT

Quick Answer: Profit, revenue, and cost problems require you to apply the fundamental business formula: Profit = Revenue - Cost. A great tip is to use the Desmos calculator to graph revenue and cost functions to quickly find the break-even point where they intersect.

pie title Common Errors in Problem-Solving & Data Analysis
    "Misreading axes or scales" : 35
    "Confusing mean vs median" : 22
    "Failing to convert units" : 18
    "Misinterpreting margin of error" : 15
    "Other calculation errors" : 10

What Are Profit, Revenue, and Cost Problems?

Profit, revenue, and cost problems are a staple of the Problem-Solving & Data Analysis domain on the College Board Digital SAT. These questions test your ability to model real-world business scenarios using linear equations and inequalities. To succeed, you must understand the relationship between three core concepts: money coming in, money going out, and what is left over.

The foundational formula is Profit = Revenue - Cost. Revenue is calculated by multiplying the selling price by the number of units sold. Cost is usually split into two parts: fixed costs (like rent or equipment, which don't change) and variable costs (like materials, which scale with production). Building these equations often requires a solid grasp of /sat/math/unit-rates to properly calculate variable costs and revenue per item.

In many cases, these problems ask you to find the "break-even point"—the exact number of units a business must sell so that revenue equals cost (meaning profit is $0$). Understanding how these elements scale together is closely related to concepts of /sat/math/direct-and-inverse-variation, making this a highly integrative topic on the exam.

Step-by-Step Method

1. Step 1: Identify the variables. Read the problem carefully to determine the selling price per unit, the variable cost per unit, and the fixed costs. Let $x$ represent the number of units.
2. Step 2: Build the Revenue function. Write the revenue equation as $R(x) = \text{price} \cdot x$.
3. Step 3: Build the Cost function. Write the cost equation as $C(x) = \text{fixed costs} + (\text{variable cost} \cdot x)$.
4. Step 4: Set up the Profit function. Use the formula $P(x) = R(x) - C(x)$. Be extremely careful to use parentheses around the entire cost function so you distribute the negative sign properly.
5. Step 5: Solve for the target. Set your equation to the desired outcome. If looking for the break-even point, set $P(x) = 0$ (or $R(x) = C(x)$). If looking for a specific profit target, set $P(x)$ equal to that number and solve for $x$.

Desmos Shortcut

The built-in Desmos Calculator is an incredibly powerful tool for these problems. If a question asks for the break-even point, you don't necessarily have to solve it algebraically.

Simply type your revenue function as one line (e.g., y = 25x) and your cost function as another line (e.g., y = 10x + 1500). Zoom out on the graph until you see where the two lines cross. Click the intersection point; the x-coordinate is the number of units needed to break even, and the y-coordinate is the dollar amount of revenue/cost at that point.

Worked Example

Question: A local bakery sells custom cakes for $45 each. The bakery has fixed monthly operating costs of$1,200. The ingredients and packaging for each cake cost $15. How many cakes must the bakery sell in a single month to earn a profit of exactly$900?

A) 40 B) 50 C) 70 D) 90

Solution:

First, set up the Revenue and Cost functions where $x$ is the number of cakes. Revenue: $R(x) = 45x$ Cost: $C(x) = 1200 + 15x$

Next, set up the Profit function using $P(x) = R(x) - C(x)$: $P(x) = 45x - (1200 + 15x)$

$P(x) = 30x - 1200$

The question asks for the number of cakes needed to make a profit of $900. Set$P(x)$to$900$: $900 = 30x - 1200$

Add $1200$ to both sides: $2100 = 30x$

Divide by $30$: $x = 70$

The bakery must sell 70 cakes.

C

Common Traps

1. Forgetting to distribute the negative sign — When subtracting the cost function from the revenue function, you must subtract all costs. Based on Lumist student data, 15% of algebra errors involve forgetting to distribute negative signs across parentheses. If you write $P(x) = 45x - 1200 + 15x$ instead of $45x - (1200 + 15x)$, you will accidentally add your variable costs to your profit instead of subtracting them.

2. Failing to align units — Sometimes a problem will give the fixed cost in thousands of dollars (e.g., $1.5$ representing $1,500) but the selling price in standard dollars. Our data shows that 18% of problem-solving errors involve not converting units before calculating rates. Always ensure your revenue and cost functions are operating on the exact same scale before combining them.