Water Bottle Fundraiser: Break-Even With and Without a Contribution
The Riverside Middle School robotics club wants to sell custom water bottles at the spring festival. The print shop charges a one-time $85 setup fee for the club's design plus $4.50 for each bottle printed, and the club plans to sell every bottle for $9.00. The club's faculty advisor has $60 of leftover activity-fund money that she will put toward the setup fee, so the club itself covers only the remainder of that fee. The order form is due Friday and the shop accepts any whole number of bottles. A club member, Priya, has written on the planning sheet: 9n + 85 = 4.5n, and reports, 'I solved it and got a negative number, so the club cannot make a profit.' The treasurer wants the smallest whole number of bottles that avoids a loss both before and after the advisor's contribution, and she also wants to know the profit from a 50-bottle order.
1. Ignoring the advisor's contribution for now, write an equation for break-even and find the smallest whole number of bottles that avoids a loss. Show why rounding to the nearest whole number would be wrong.
Read the worked answer
Let n be the number of bottles. Money in = 9n; money out = 85 + 4.5n.
Break-even: 9n = 85 + 4.5n. Subtract 4.5n from both sides: 4.5n = 85, so n = 85 ÷ 4.5 = 18.888…
Check both sides of the boundary. At 18 bottles: sales 162, cost 85 + 81 = 166, a loss of $4. At 19 bottles: sales 171, cost 85 + 85.50 = 170.50, a small profit of $0.50.
You cannot print 18.9 bottles, and 18 is a loss, so the answer is 19 - round up, not to the nearest whole number. The fractional answer tells you that 18 bottles falls short, which is exactly what a rounding decision has to respect.
2. How does the $60 contribution change the break-even point? Support the answer with a check of the two nearest whole numbers.
Read the worked answer
The contribution lowers the fixed cost the club must recover: 85 − 60 = 25.
New equation: 9n = 25 + 4.5n, so 4.5n = 25 and n = 5.555…, giving 6 bottles.
Check: at 5 bottles, sales 45 against cost 25 + 22.50 = 47.50, a loss of $2.50. At 6 bottles, sales 54 against cost 25 + 27 = 52, a profit of $2.
The break-even drops from 19 bottles to 6 because the contribution removes $60 from a fixed cost that was the whole obstacle. The $4.50 margin earned on each bottle is unchanged; only the amount that must be recovered up front shrinks.
3. Diagnose Priya's equation 9n + 85 = 4.5n and explain what each side should represent. Then compute the profit on a 50-bottle order, including the advisor's contribution.
Read the worked answer
Priya placed the setup fee on the same side as the sales revenue: 9n + 85 = 4.5n. Solving it gives 4.5n = −85 and n = −18.9, a negative number of bottles, which is the signal that the equation is built wrong rather than that profit is impossible.
Break-even compares two separate quantities, so each side must measure the same kind of thing - dollars:
- Money in = 9n
- Money out = 25 + 4.5n (after the $60 contribution)
Setting them equal gives n ≈ 5.6, so 6 bottles.
At 50 bottles: sales = 9(50) = 450; cost = 25 + 4.5(50) = 25 + 225 = 250; profit = 450 − 250 = $200. A useful habit is checking that both sides of any equation carry the same units.