Scaling the Lentil Soup
A cafeteria's lentil soup recipe yields 24 portions and calls for 3¾ cups of lentils, 2⅔ cups of broth, and 1¼ teaspoons of salt. The kitchen now needs 36 portions. On hand: 8 cups of lentils, 6 cups of broth, and plenty of salt. Broth is bought in 32-fluid-ounce cartons (1 cup = 8 fluid ounces). Lentils are sold in 1-pound bags that contain about 2¼ cups each and cost $2.40 per bag; the kitchen will not open a bag it does not need for this batch.
1. Scale the recipe to 36 portions and state whether the kitchen's on-hand lentils and broth are enough.
Read the worked answer
Scale factor: 36 ÷ 24 = 1.5
| Ingredient | Recipe (24) | × 1.5 | Needed (36) |
|---|---|---|---|
| Lentils | 3¾ c | 5⅝ c | 5⅝ c |
| Broth | 2⅔ c | 4 c | 4 c |
| Salt | 1¼ tsp | 1⅞ tsp | 1⅞ tsp |
Working the fractions: 3.75 × 1.5 = 5.625 = 5⅝; 2⅔ × 1.5 = 4; 1.25 × 1.5 = 1.875 = 1⅞.
Check against stock: lentils 5⅝ ≤ 8 ✓; broth 4 ≤ 6 ✓; salt is not a constraint. Both ingredients are sufficient, with 2⅜ cups of lentils and 2 cups of broth left over.
2. How many 32-fluid-ounce cartons of broth should be purchased for this batch, given that the kitchen wants a 10% safety margin?
Read the worked answer
Convert units first. 4 cups are needed, and 1 cup = 8 fl oz, so 4 × 8 = 32 fl oz - exactly one carton.
Add the safety margin: 32 × 1.10 = 35.2 fl oz.
One carton holds 32 fl oz, which is 3.2 fl oz short of the margin, so one carton does not meet the requirement.
Answer: 2 cartons (64 fl oz), leaving 28.8 fl oz after the batch.
The judgment call: 4 cups matches one carton exactly on paper, which looks like a clean answer. But an exact match means any spill, a recipe correction, or a slightly short pour leaves the batch short, and the kitchen has no reserve. The 10% margin is what forces the second carton, not the base arithmetic.
3. What is the cost of the lentils for this batch, and why is it not simply 5⅝ cups times a per-cup price?
Read the worked answer
Cups per bag: 2¼ cups.
Bags needed: 5⅝ ÷ 2¼ = 5.625 ÷ 2.25 = 2.5 bags.
Since the kitchen will not open a bag it does not need, it must buy whole bags: 3 bags.
Cost: 3 × $2.40 = $7.20
Why not a per-cup price: lentils are only sold in whole 1-pound bags, so cost jumps in steps rather than rising smoothly with the amount used. Charging 5⅝ cups at a per-cup rate would understate the real cost.
Check the step: 2 bags give 4½ cups, which is less than 5⅝, so 2 bags are not enough. 3 bags give 6¾ cups, leaving about 1⅛ cups of lentils on the shelf. That leftover is a real cost of this batch even though it is not used in it.