Granola Bars: Partial Batches, Ingredient Rounding, and Budget
A school bake sale chair has a granola-bar recipe that makes 24 bars using 3/4 cup honey, 2 1/4 cups rolled oats, 1/2 cup sliced almonds, and 1/3 cup oil. The sale needs at least 150 bars, and the chair can make full or half batches but not smaller. Honey sells in 16-ounce bottles (2 cups per bottle) for $9.50. Oats sell in 5-cup containers for $3.25. Almonds sell in 2-cup bags for $7.00. Oil sells in 3-cup bottles for $4.00. The chair has a $60 budget for these four ingredients and wants to avoid buying more than necessary, but also does not want to run short. A helper suggests making 6 batches because 6 × 24 = 144 is “close enough,” while another suggests 7 batches to be safe.
1. What is the minimum number of batches-allowing half batches-that produces at least 150 bars?
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Six full batches make 6 × 24 = 144 bars, which is 6 short, so 6 is not enough. A half batch makes 12 bars. Six and a half batches make 144 + 12 = 156 bars, which meets the requirement. Therefore the minimum is 6.5 batches. The helper’s 6-batch suggestion fails the “at least 150” condition even though 144 is close.
2. Scale the ingredients for 6.5 batches and determine the exact amounts needed.
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Multiply each recipe amount by 6.5. Honey: 3/4 × 6.5 = 0.75 × 6.5 = 4.875 cups. Oats: 2 1/4 = 2.25; 2.25 × 6.5 = 14.625 cups. Almonds: 0.5 × 6.5 = 3.25 cups. Oil: 1/3 × 6.5 = 6.5/3 ≈ 2.1667 cups. These are the amounts that must be available; purchases will be rounded up to package sizes.
3. List the minimum packages to buy, compute the total cost, and state whether the $60 budget is enough. Compare this with the 7-batch suggestion.
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Honey: 4.875 cups ÷ 2 cups per bottle = 2.4375, so 3 bottles. Oats: 14.625 ÷ 5 = 2.925, so 3 containers. Almonds: 3.25 ÷ 2 = 1.625, so 2 bags. Oil: 2.1667 ÷ 3 = 0.722, so 1 bottle. Cost = 3 × $9.50 + 3 × $3.25 + 2 × $7.00 + 1 × $4.00 = $28.50 + $9.75 + $14.00 + $4.00 = $56.25. The budget is enough, with $3.75 left. Seven batches would make 168 bars but require 15.75 cups of oats; 3 containers hold only 15 cups, so a fourth container is needed. That raises the cost to $59.50-still under $60, but with more leftovers and less margin. The 6.5-batch plan is the better fit.