Scaling a Recipe and Comparing Unit Prices
A caterer is making a punch recipe that calls for 3 parts juice to 2 parts sparkling water by volume, plus 0.25 liters of simple syrup per 5 liters of the juice-water mixture. She needs to produce exactly 30 liters of finished punch (juice + water + syrup combined). She can buy juice in 4-liter bottles for $9.60 each or in 1.5-liter bottles for $3.90 each, and she can buy only whole bottles of either size. She wants to buy the cheapest combination that still gives her enough juice.
1. Determine how many liters of juice, water, and syrup are needed for exactly 30 liters of punch.
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Let the juice-water mixture be 5 parts total (3 juice + 2 water). Syrup is 0.25 L per 5 L of mixture, i.e., syrup = 0.05 × mixture. Total punch = mixture + syrup = mixture + 0.05·mixture = 1.05·mixture. Set 1.05·mixture = 30 → mixture = 30 ÷ 1.05 ≈ 28.571 L. Syrup = 30 − 28.571 = 1.429 L (≈ 1.43 L). Juice = (3/5) × 28.571 = 17.143 L. Water = (2/5) × 28.571 = 11.429 L. Check: 17.143 + 11.429 + 1.429 ≈ 30.0 ✓
2. Compare the unit price of the two juice bottle sizes and find the cheapest whole-bottle purchase that provides at least the needed juice.
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Unit prices:
- 4-L bottle: 9.60 ÷ 4 = $2.40 per liter
- 1.5-L bottle: 3.90 ÷ 1.5 = $2.60 per liter The 4-L bottle is cheaper per liter, but she needs at least 17.143 L and can only buy whole bottles. Try combinations (need ≥ 17.143 L):
- 5 × 4 L = 20 L, cost 5 × 9.60 = $48.00
- 4 × 4 L = 16 L (too little) plus 1 × 1.5 L = 17.5 L, cost 4(9.60) + 3.90 = 38.40 + 3.90 = $42.30
- 3 × 4 L = 12 L + 4 × 1.5 L = 18 L, cost 3(9.60) + 4(3.90) = 28.80 + 15.60 = $44.40
- 2 × 4 L = 8 L + 7 × 1.5 L = 18.5 L, cost 19.20 + 27.30 = $46.50 Cheapest that reaches ≥ 17.143 L is 4 four-liter bottles + 1 one-and-a-half-liter bottle = 17.5 L for $42.30. The judgment: the lower unit price does not automatically give the cheapest total once whole-bottle rounding is required.