A booth fee, per-item costs, and a sales commission call for a linear inequality for profit. Find the money kept from sales first: the commission is a percent of sales revenue, not profit. Subtract the costs and require the result to meet the target. Use a Desmos regression to find the cutoff, then choose the first whole-number count that works.
Hints
- Hint 1
A commission is the fair’s share of sales. If bracelets sell for $10 each, what amount does the fair take 5% of? Lila keeps the other 95% of that amount.
- Hint 2
Profit is money kept from sales minus costs. The $3 material cost repeats for every bracelet, but the $100 booth fee is paid once. How can you write the profit for bracelets?
- Hint 3
An inequality using “at least” allows the cutoff and everything above it. Find the cutoff where profit equals the target, then choose the first whole number that works. You can’t sell part of a bracelet.
Step-by-step
Model profit and find the cutoff
Step 1Find the sales money Lila keeps
Let be the number of bracelets. Selling them for $10 each brings in dollars. A commission is the fair’s share of those sales, so subtract its 5% from the full 100%:
Lila keeps dollars from sales. The commission comes out of sales revenue, not profit after costs.
- Step 2
Write the profit requirement
Subtract $3 for each bracelet and the $100 booth fee once. Profit means the money left after those costs. “At least $600” means the profit can equal or exceed $600, so
Multiplying the booth fee by would charge it again for every bracelet.
- Step 3
Find where profit reaches the target
The cutoff is where profit equals exactly $600. Type in Desmos; tells it to find the value of that makes both sides equal. Under PARAMETERS, Desmos shows . Selling more bracelets raises profit, so the allowed counts are at or above that cutoff.
- Step 4
Choose and check the first whole count
Bracelets come in whole numbers, so test the counts on either side of the cutoff. Add and in Desmos. It shows $595.50 for , which falls short, and $602 for , which meets the target. The minimum is bracelets. Choice B.