Question 93·Hard·Linear Inequalities in One or Two Variables
Lila rents a booth at a craft fair for $100 to sell handmade bracelets. Her material cost is $3 per bracelet. She sells each bracelet for $10, but the fair takes a 5% commission on her sales revenue.
What is the minimum number of bracelets Lila must sell to earn a profit of at least $600?
For profit-and-loss inequality problems, start by defining the variable clearly and separating fixed costs (like a booth fee) from per-item revenues and costs. Compute the net profit per item by adjusting the selling price for any percentages (tax, commission, discounts) and then subtracting per-item costs. Write a total profit expression as (net profit per item) × (number of items) minus fixed costs, set it greater than or equal to the required profit, and solve the linear inequality. Finally, because you usually cannot sell a fraction of an item, round up to the nearest whole number when the inequality gives a non-integer result.
Hints
Define the unknown and list the costs
Let be the number of bracelets Lila sells. What are her fixed costs (paid no matter what) and what are her variable costs (that depend on )?
Account for the commission correctly
Lila sells each bracelet for , but the fair takes of the sales revenue. How much money from each sale does Lila actually keep after this commission?
Find profit per bracelet before the booth fee
Once you know how much Lila keeps from each sale, subtract the material cost to get her net profit per bracelet. Then write an expression for her total profit including the booth fee.
Set up and solve the inequality
Use your profit expression and the phrase "at least $600" to write an inequality, then solve for . Remember that must be a whole number of bracelets.
Desmos Guide
Enter the profit function
In a Desmos expression line, type P(x) = (0.95*10 - 3)*x - 100 or simply P(x) = 6.5x - 100 to represent Lila’s profit as a function of the number of bracelets .
Graph the profit and profit goal
On the next line, type y = P(x) to see Lila’s profit graph, and on another line, type y = 600 to show a horizontal line representing a $600 profit.
Find the minimum integer number of bracelets
Use the intersection tool or tap where the graphs of y = P(x) and y = 600 meet to read the -value where the profit first reaches . Then identify the smallest whole number of bracelets greater than or equal to this -value.
Step-by-step Explanation
Define the variable and identify all parts of the profit
Let be the number of bracelets Lila sells.
Break down the money involved:
- Fixed cost: She pays a booth fee of no matter what.
- Material cost: She spends per bracelet, so total material cost is .
- Sales price: She sells each bracelet for , so her total sales revenue is .
- Commission: The fair takes of her sales revenue, so the commission is .
Find Lila’s net profit per bracelet
First find how much Lila actually keeps from each sale after the commission:
- Commission per bracelet: dollars.
- Money Lila keeps from each sale: dollars.
Now subtract the material cost per bracelet:
- Net profit per bracelet (before the booth fee) is dollars.
So, ignoring the fixed booth fee for a moment, Lila earns in profit for each bracelet she sells.
Write an expression for total profit and set up the inequality
Total profit equals total net profit from bracelets minus the fixed booth fee.
- Total net profit from bracelets: .
- Subtract the booth fee: profit .
We want Lila’s profit to be at least dollars, so write the inequality:
Now solve this inequality for .
Solve the inequality and interpret the result
Solve the inequality step by step:
Compute the fraction:
Lila cannot sell a fraction of a bracelet, and the profit must be at least , so she must sell the smallest whole number greater than or equal to , which is .
Therefore, the correct answer is 108.