Package Rounding and Proportional Reasoning in a School Kitchen
A school catering team must prepare 375 servings of pasta. Their recipe yields 24 servings and lists 1.8 kg pasta, 900 g sauce, and 2.4 L stock. Pasta is sold in 5 kg bags for AED 42, sauce in 2 kg tubs for AED 28, and stock in 1 L cartons for AED 6.50. The team leader says to multiply every ingredient by 375 ÷ 24 = 15.625 and buy exactly that amount, because rounding to whole packages wastes too much. An assistant instead rounds 375 ÷ 24 to 16 and multiplies all quantities by 16. Storage limits pasta to 80 kg, and the team must spend under AED 900. The kitchen also has only 40 L of refrigerator space for stock.
1. Using whole packages, determine the minimum purchase and total cost for each method. Does either method meet the storage and budget constraints?
Read the worked answer
For 375 servings the scale factor is 375 ÷ 24 = 15.625. Required amounts are pasta 1.8 × 15.625 = 28.125 kg, sauce 0.9 × 15.625 = 14.0625 kg, and stock 2.4 × 15.625 = 37.5 L.
Whole packages for the first method:
- Pasta: ceil(28.125 ÷ 5) = 6 bags, giving 30 kg and costing 6 × 42 = AED 252.
- Sauce: ceil(14.0625 ÷ 2) = 8 tubs, giving 16 kg and costing 8 × 28 = AED 224.
- Stock: ceil(37.5 ÷ 1) = 38 cartons, costing 38 × 6.50 = AED 247. Total = AED 723, with 30 kg pasta and 38 L stock.
For scale factor 16: pasta 28.8 kg, sauce 14.4 kg, stock 38.4 L. This gives 6 pasta bags, 8 sauce tubs, and 39 stock cartons, costing 252 + 224 + 39 × 6.50 = AED 729.50, with 39 L stock.
Both methods are under AED 900, under 80 kg pasta, and within 40 L stock space. The first method buys less stock, but the leader cannot buy the exact fractional ingredient amounts.
2. The team leader claims the assistant's rounding increases every ingredient by less than 3%. Check the claim before package rounding and after package rounding for stock, then explain which comparison is more useful for purchasing.
Read the worked answer
Before package rounding, the increase is 16 ÷ 15.625 = 1.024, or 2.4%, so the under-3% claim holds at ingredient level. After package rounding, stock rises from 38 to 39 cartons, an increase of 1/38 = 2.63%, still under 3%; pasta and sauce package counts do not change.
So the claim is numerically true here, but it is a poor purchasing test. A 2.4% ingredient increase is not meaningful when packages force jumps of 5 kg, 2 kg, or 1 L. The comparison also ignores absolute leftovers and total cost. For purchasing, the team should compare package-level quantities, storage use, and total spend, not just percentage changes in ingredient amounts.