Punch for the Spring Assembly
The cafeteria at Ridgeway Middle School must serve punch to 480 students at a spring assembly, pouring 200 mL into each cup. The punch is mixed from frozen concentrate and water at a ratio of 1 part concentrate to 4 parts water by volume. The kitchen has two purchasing options: 750 mL bottles of concentrate at $4.25 each, or 2.5 L jugs at $11.40 each. The manager has promised the principal that the punch will cost under $100. The assistant suggests simply buying 20 bottles because "that is about how many we always buy." Deliveries arrive only in whole containers, and opened concentrate cannot be stored overnight, so any excess is wasted.
1. How much concentrate is actually required, and which container should the manager purchase to keep the total under $100?
Read the worked answer
Total punch needed: 480 × 200 mL = 96,000 mL = 96 L.
Concentrate: the ratio 1:4 means concentrate is 1 of every 5 parts, so concentrate = 96 ÷ 5 = 19.2 L = 19,200 mL.
Bottles (750 mL): 19,200 ÷ 750 = 25.6 → 26 bottles → 26 × $4.25 = $110.50 (over budget).
Jugs (2.5 L): 19.2 ÷ 2.5 = 7.68 → 8 jugs → 8 × $11.40 = $91.20 (under budget).
Unit-price check: bottles cost 4.25 ÷ 0.75 = $5.67/L; jugs cost 11.40 ÷ 2.5 = $4.56/L.
Decision: buy 8 jugs. The 20-bottle suggestion would supply only 15 L of concentrate (15 L of punch concentrate vs. 19.2 L needed) and would still cost $85 - cheaper, but it under-mixes the punch.
2. On the day of the assembly, staff fill cups to 250 mL instead of 200 mL. How much concentrate is actually consumed, what does it cost, and what happens to the $100 promise?
Read the worked answer
New total volume: 480 × 250 mL = 120,000 mL = 120 L.
Concentrate: 120 ÷ 5 = 24 L = 24,000 mL.
Jugs: 24 ÷ 2.5 = 9.6 → 10 jugs (whole containers only) → 10 × $11.40 = $114.00.
Budget impact: $114.00 − $100 = $14.00 over the promised limit (and $22.80 more than the planned $91.20).
Note: the leftover 1 L in the tenth jug cannot be saved, so it is waste. The overrun is caused by a 25% increase in pour size (250 ÷ 200 = 1.25), which raises concentrate demand from 19.2 L to 24 L - a 25% increase that crosses a whole-container boundary from 8 to 10 jugs.
3. Suppose the manager refuses to buy extra concentrate and keeps the original 96 L of punch. If cups are still filled to 250 mL, how many students get punch, and what percent of the 480 students go without?
Read the worked answer
Cups available: 96,000 mL ÷ 250 mL = 384 cups.
Students unserved: 480 − 384 = 96 students.
Percent unserved: 96 ÷ 480 = 0.20 = 20%.
Interpretation: the 96 L batch is fixed, so every 50 mL added to a cup removes one cup from the line. Going from 200 mL to 250 mL is a 25% increase in pour size, which should reduce the count by the reciprocal factor: 480 × (200 ÷ 250) = 384. One in five students would be turned away, so the manager must either dilute further (changing the ratio), serve 200 mL cups, or buy the two extra jugs.