Filling the Concession Cooler: Two Recipes, One Scarce Supply
Next Saturday the Northside Soccer Boosters run the concession stand at a regional tournament, and the drink cooler is their biggest seller. The cooler holds exactly 80 cups. The recipe card taped inside the lid reads: 2 cups frozen concentrate to 3 cups cold water. A second card, from a different supplier, reads: 3 cups concentrate to 5 cups water. Before the tournament the crew has 26 cups of concentrate in the storage room, and there is no time for a second shopping run. A volunteer named Dana has already written on the whiteboard: '2/3 of the cooler is concentrate, so 2/3 × 80 is about 53 cups concentrate.' Two other volunteers are arguing about which recipe makes the stronger drink, and the crew leader wants the cooler filled as close to the top as the concentrate supply allows. The stand opens at 8:00 a.m.
1. Dana's whiteboard note is wrong. Explain the error, then give the correct number of cups of concentrate and of water needed to fill the 80-cup cooler using the first recipe (2 concentrate : 3 water).
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The error. In a 2 : 3 recipe the 3 counts only the water, not the whole drink. The drink is 2 + 3 = 5 equal parts, so concentrate is 2/5 of the cooler, not 2/3.
Correct amounts. Concentrate = 2/5 × 80 = 32 cups; water = 3/5 × 80 = 48 cups. Check: 32 + 48 = 80 cups, and 32 : 48 simplifies to 2 : 3 (divide both by 16).
Dana's 53⅓ cups would leave only 26⅔ cups of water, a ratio of about 2 : 1 - far too strong. A quick ratio check catches the mistake before anyone mixes a batch.
2. Which recipe makes the more concentrated drink? Justify with numbers rather than pointing to which card lists more concentrate.
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Compare concentrate as a share of the finished drink, which is what a taster actually notices.
- Recipe 1: 2/(2 + 3) = 2/5 = 0.40, so 40% concentrate.
- Recipe 2: 3/(3 + 5) = 3/8 = 0.375, so 37.5% concentrate.
Recipe 1 is stronger. The trap is that recipe 2 lists the larger concentrate amount (3 cups), but it also pairs it with 5 cups of water instead of 3, so each cup of finished drink holds less concentrate. An equivalent check: in 40 cups of drink, recipe 1 uses 16 cups of concentrate and recipe 2 uses 15.
3. Using all 26 cups of concentrate and no more, which recipe lets the crew fill the cooler closer to full? Find the concentrate and water amounts and the fraction of the cooler filled for the better choice.
Read the worked answer
Divide the supply by the concentrate share of each recipe.
- Recipe 1: 26 ÷ (2/5) = 26 × 5/2 = 65 cups of drink, so water = 65 − 26 = 39 cups. That fills 65/80 = 81.25% of the cooler.
- Recipe 2: 26 ÷ (3/8) = 26 × 8/3 = 208/3 = 69⅓ cups of drink, so water = 69⅓ − 26 = 43⅓ cups. That fills 69⅓ ÷ 80, about 86.7%.
Recipe 2 stretches the scarce concentrate further (8/3 ≈ 2.67 cups of drink per cup of concentrate, versus 2.5) and gains about 10⅔ cups. The trade-off is that it is the milder drink, so the crew must weigh a fuller cooler against a weaker taste. Neither recipe fills the cooler completely.