Question 3·Hard·Linear Equations in One Variable
A tech company produces a smart sensor for $120 per unit. Each unit is sold online for $200, but the online platform charges the company a fee equal to 8% of the sales price for each unit sold. In addition, the company spends $6 per unit on advertising. The company’s net profit is the total sales revenue minus all platform fees, advertising costs, and production costs. If the company reports a net profit of $15,960 from selling the sensor, which equation can be used to determine the number of units, , that were sold?
For word problems about net profit, first separate revenue from each type of cost and decide whether each amount is per unit or a one-time (fixed) amount. Translate percentage fees into expressions like rate * price, then compute net profit per unit as revenue per unit minus all per-unit costs. Finally, multiply this per-unit profit by the variable representing the number of units and set the result equal to the given total profit, watching signs carefully to ensure that all costs are subtracted, not added.
Hints
List all money amounts mentioned
Write down separately: the selling price per unit, the production cost per unit, the advertising cost per unit, and the fee percentage. Which of these are money coming in, and which are money going out?
Interpret the 8% fee
The platform charges a fee equal to 8% of the sales price of each unit. How can you represent this fee in dollars using the price ?
Build the per-unit profit, then scale by g
First find an expression for the profit from selling just 1 unit by taking revenue minus all per-unit costs. Then, because each of the g units gives the same profit, multiply that expression by and set it equal to the total net profit, .
Desmos Guide
Compute each per-unit amount
In separate Desmos lines, type 200, 0.08*200, 120, and 6 to see the revenue per unit, fee per unit, and each cost per unit. This helps you keep track of which numbers are income and which are expenses.
Find the net profit per unit in Desmos
On a new line, define profit_per_unit = 200 - 0.08*200 - 120 - 6. Check that this value is positive but less than 200, which makes sense for a profit after subtracting all per-unit costs.
Express total profit as a function of g
Define P(g) = profit_per_unit * g. This function represents total net profit for selling g units. Compare the structure of P(g) to the answer choices: it should match the option where the expression for per-unit profit (revenue minus all per-unit costs) is multiplied by g and set equal to 15960.
Step-by-step Explanation
Identify revenue and each per-unit cost
From the problem:
- Revenue per unit: dollars (the selling price).
- Platform fee per unit: of , which is .
- Production cost per unit: dollars.
- Advertising cost per unit: dollars. All of these costs apply to every unit sold.
Write the net profit per unit
Net profit per unit is revenue per unit minus all costs per unit:
- Start with (revenue per unit).
- Subtract the platform fee: .
- Subtract the production cost: .
- Subtract the advertising cost: .
So the net profit for one unit is represented by the expression
Relate net profit per unit to total net profit
If the net profit for one unit is , then the total net profit for units is that per-unit profit multiplied by .
The company reports a total net profit of , so we set
This matches answer choice A.