Scaling the Cafeteria Recipe: Two Methods, One Shopping List
The cafeteria at Ortega Middle School is preparing its rice-and-bean entrée for a district showcase with 132 expected guests. The base recipe serves 24 and calls for 2¾ cups of dry rice, 3⅓ quarts of milk, and 1⅛ cups of cooking oil per batch. Kitchen policy is to prepare 8% more food than the guest count to cover seconds and spills. Two staff members build the order differently: Mr. Okafor scales by 132 ÷ 24 = 5.5 batches and orders for exactly 5.5 batches, while Ms. Lin first increases the guest count by 8% and then converts to batches. Dry rice is sold in 5-pound bags, and the kitchen's yield chart lists 2½ cups of dry rice per pound. Milk is sold in half-gallon containers. The manager also asks whether the 40-minute bake time should be scaled by the same factor.
1. How many 5-pound bags of rice does each method require? Show the conversions and state whether the two shopping lists agree.
Read the worked answer
Mr. Okafor (scale batches directly)
- Rice: 5.5 × 2.75 cups = 15.125 cups
- One bag yields 5 lb × 2.5 cups/lb = 12.5 cups
- 15.125 ÷ 12.5 = 1.21 → 2 bags
Ms. Lin (inflate guests first)
- 132 × 1.08 = 142.56 guest-equivalents
- 142.56 ÷ 24 = 5.94 batches
- 5.94 × 2.75 = 16.335 cups
- 16.335 ÷ 12.5 = 1.307 → 2 bags
Judgment: the lists agree at 2 bags, but that is an artifact of rounding up to whole bags, not evidence the methods are equivalent. Ms. Lin's plan carries about 1.2 cups of buffer; Mr. Okafor's carries essentially none. The correct general procedure is to inflate the demand (8%) and then round purchases up to whole containers.
2. Apply both methods to the milk (3⅓ quarts per batch; sold by the half-gallon). Do the container counts differ, and which plan would you choose? Justify with the size of the leftover margin.
Read the worked answer
Mr. Okafor: 5.5 × 3⅓ = 5.5 × 3.333 = 18.33 qt. Each half-gallon = 2 qt, so 18.33 ÷ 2 = 9.17 → 10 containers (20 qt).
Ms. Lin: 5.94 × 3.333 = 19.8 qt. 19.8 ÷ 2 = 9.9 → 10 containers (20 qt).
Both require 10 containers, so the purchase is identical. The difference is safety margin:
- Mr. Okafor leaves 20 − 18.33 = 1.67 qt spare.
- Ms. Lin leaves 20 − 19.8 = 0.2 qt spare.
Judgment call: choose Ms. Lin's method because it satisfies the written 8% policy and produces a defensible audit trail. But the 0.2 qt cushion is thin enough that a spill or an over-pour consumes it, so a prudent buyer would note the risk and consider an 11th container if the kitchen has storage and the budget allows.
3. The manager asks whether the 40-minute bake time should be multiplied by 5.94 as well. Answer and explain the reasoning that distinguishes quantities that scale this way from those that do not.
Read the worked answer
No. The 5.94 factor is an extensive quantity - it counts how much material is needed. Bake time is an intensive quantity: it is the time for one pan at a given temperature, and it does not depend on how many pans exist.
- Scaling the recipe to ~6 batches means baking about 6 pans, not baking one pan for 6 × 40 = 240 minutes.
- Each pan still bakes about 40 minutes.
- The total kitchen time does grow - pans must share oven space, so if the oven holds 2 pans, the club needs about 3 sequential rounds ≈ 120 minutes.
Reasoning to teach: a scale factor may be applied to amounts (cups, quarts, pounds) and to counts of discrete items, but not to rates, temperatures, densities, or per-unit times. Those are held constant while the number of units changes.