Recipe Scaling With Fractions
A caterer has a sauce recipe written for 8 servings that calls for 3/4 cup of stock, 2/3 cup of cream, and 1/2 cup of wine. A client needs 20 servings, so the caterer must scale every ingredient by the factor 20/8 = 5/2. The kitchen measures in quarter-cup and third-cup scoops only, and the caterer wants each scaled amount expressed both as an improper fraction and as a mixed number to plan scoop counts. One cook worries that multiplying 2/3 by 5/2 will produce an amount that cannot be measured with the available scoops, so the caterer must also check measurability against a 1/12-cup precision.
1. Scale each ingredient by 5/2 and express the results as improper fractions and mixed numbers.
Read the worked answer
Stock: (3/4) × (5/2) = 15/8 = 1 7/8 cups. Cream: (2/3) × (5/2) = 10/6 = 5/3 = 1 2/3 cups. Wine: (1/2) × (5/2) = 5/4 = 1 1/4 cups.
Check by cross-canceling: (2/3)×(5/2) → the 2s cancel to give 5/3. Correct. Sum sanity: original total = 3/4 + 2/3 + 1/2 = 9/12 + 8/12 + 6/12 = 23/12; scaled by 5/2 gives 115/24 ≈ 4.79 cups. Scaled parts: 15/8 + 5/3 + 5/4 = 45/24 + 40/24 + 30/24 = 115/24. Consistent.
2. Are all three scaled amounts measurable to 1/12-cup precision? Show the check.
Read the worked answer
An amount is measurable to 1/12 cup if, in twelfths, it is a whole number.
Stock 15/8 cups = 15/8 × 12/12 = 180/96... convert directly: 15/8 = 22.5/12, i.e., 22.5 twelfths - NOT a whole number of twelfths (it is 22 1/2 twelfths). So 1 7/8 cups is not expressible in 1/12-cup increments; it needs 1/24 precision (it equals 45/24).
Cream 5/3 = 20/12 → exactly 20 twelfths. Measurable. Wine 5/4 = 15/12 → exactly 15 twelfths. Measurable.
So the cook's concern is partly right: the cream is fine, but the stock (1 7/8) lands on a half-twelfth and cannot be hit exactly with 1/12-cup precision - the caterer should either accept a 1/24-cup scoop or round to 1 11/12 or 1 7/8 with a note.