Question 300·Hard·Algebra

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A manufacturing company produces and sells gadgets. The company earns $45 in revenue for each gadget sold but spends $18 in variable costs per gadget and has fixed weekly operating costs of $k$ dollars. The company’s weekly profit must be at least $20\%$ and at most $35\%$ of its weekly revenue.

Let $g$ be the number of gadgets the company produces and sells in a week.

Which inequality represents all possible values of $g$ that satisfy the company’s goal?

A profit goal with a lower and upper percentage becomes a compound inequality: profit must lie between two parts of revenue. Write revenue, subtract both kinds of cost to get profit, and use Desmos to calculate the percent targets per gadget. The key trap is taking the percentages of profit or cost instead of revenue; if you multiply the inequality chain by a negative, flip both signs.

Hints

Hint 1
Profit means revenue minus all costs. The fixed weekly cost is one payment of $k$ , not $kg$ . How would you write revenue and each kind of cost for $g$ gadgets?
Hint 2
A percent base is the amount a percent is taken of. Both profit targets use revenue as their base. What expression gives weekly revenue for $g$ gadgets?
Hint 3
A compound inequality puts profit between two limits. When you multiply all three parts by a negative number, both inequality signs must flip.

Step-by-step

Build and solve the profit inequality
Step 1
Write weekly profit
Revenue is $45g$ , variable costs are $18g$ , and the fixed cost $k$ is paid once. Profit is revenue minus both costs:
$P=45g-(18g+k).$
Step 2
Find profit per gadget before fixed costs
Type $45-18$ in Desmos. It shows $27$ , the amount earned per gadget before paying $k$ . Combine the $g$ -terms:
$P=27g-k.$
Step 3
Put profit between the two targets
“At least” and “at most” both include equality. Both percentages are taken of revenue, which is $45g$ , so the compound inequality is:
$20\%\text{ of }(45g)\le P\le35\%\text{ of }(45g).$
Step 4
Calculate the percent targets
Type $20\%\operatorname{of}45$ and $35\%\operatorname{of}45$ in Desmos. It shows $9$ and $15.75$ . For $g$ gadgets, multiply each by $g$ ; then substitute profit from the earlier step:
$9g\le27g-k\le15.75g.$
Step 5
Move the gadget earnings out of the middle
Subtract $27g$ from all three parts. For the coefficients, type $27-9$ and $27-15.75$ in Desmos; it shows $18$ and $11.25$ . The fraction button shows $11.25=\frac{45}{4}$ . Subtract:
$-18g\le-k\le-\frac{45}{4}g.$
Step 6
Flip both comparisons
Multiply every part by $-1$ . Multiplying by a negative reverses the order, so both inequality signs flip:
$18g\ge k\ge\frac{45}{4}g.$
Step 7
Find the lower bound
From $18g\ge k$ , divide by positive $18$ , which keeps the sign pointing the same way:
$g\ge\frac{k}{18}.$
Step 8
Find the upper bound and combine
From $k\ge\frac{45}{4}g$ , divide by the positive $\frac{45}{4}$ :
$\frac{4k}{45}\ge g.$
Both bounds must hold, so $\frac{k}{18}\le g\le\frac{4k}{45}$ . Since $g$ counts gadgets, use whole-number values in that range. Choice D.

Question 300·Hard·Algebra

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