The Youth Group Muffin Sale
The Youth Achievers group plans a muffin sale to raise funds for a trip. Their recipe makes 24 muffins and uses 300 g flour, 150 g sugar, 2 eggs, 120 mL milk, 100 g butter and 2 teaspoons baking powder. They have 2 kg flour, 1 kg sugar, 12 eggs, 1 L milk, 500 g butter and plenty of baking powder. One oven is available for 3 hours. Each tray holds 12 muffins and takes 25 minutes to bake; trays cannot be baked together, and there is no time for extra batches beyond the 3 hours. They hope to sell 150 muffins at $2.50 each. A parent says, 'Just multiply the recipe by 6 and you will have 144 muffins, which is close enough.' Another says, 'We have enough flour and sugar, so we can make 150 easily.'
1. Scale the recipe for 150 muffins and identify which ingredients are insufficient.
Read the worked answer
Multiplier = 150 ÷ 24 = 6.25.
Flour: 300 × 6.25 = 1,875 g (have 2,000 g, enough). Sugar: 150 × 6.25 = 937.5 g (have 1,000 g, enough). Eggs: 2 × 6.25 = 12.5, so 13 eggs (have 12, short 1). Milk: 120 × 6.25 = 750 mL (have 1,000 mL, enough). Butter: 100 × 6.25 = 625 g (have 500 g, short 125 g). Baking powder: 2 × 6.25 = 12.5 tsp (plenty).
The insufficient ingredients are eggs and butter.
2. Given the ingredients and oven time, what is the maximum number of whole trays of 12 muffins they can bake? Show the limiting factor.
Read the worked answer
Check each limit in whole trays. One tray = 12 muffins, so the recipe per tray is half of the 24-muffin recipe: 150 g flour, 75 g sugar, 1 egg, 60 mL milk, 50 g butter.
Butter: 500 ÷ 50 = 10 trays = 120 muffins. Eggs: 12 ÷ 1 = 12 trays = 144 muffins. Flour: 2,000 ÷ 150 = 13.33 trays. Sugar: 1,000 ÷ 75 = 13.33 trays. Milk: 1,000 ÷ 60 = 16.67 trays. Oven: 3 hours = 180 minutes; 180 ÷ 25 = 7.2, so 7 full trays = 84 muffins.
The oven is the limiting factor, so the maximum is 84 muffins (7 trays).
3. The group wants to reach 150 muffins. Suggest a practical combination of changes (buying ingredients and/or changing baking arrangements) that would allow exactly 150 muffins, and calculate the extra ingredient quantities needed beyond what they have.
Read the worked answer
To reach 150 muffins, they need 150 ÷ 12 = 12.5 trays, i.e. 12 full trays plus one tray of 6.
Extra ingredients beyond what they have: eggs need 13, have 12, so buy 1 egg; butter needs 625 g, have 500 g, so buy 125 g. Flour, sugar, milk and baking powder are sufficient.
Oven time is still the problem: 13 trays × 25 minutes = 325 minutes on one oven, which is too long. With a second oven, 13 trays can be baked in 7 rounds of 25 minutes = 175 minutes (2 h 55 min), within the 3-hour limit.
Practical plan: buy 1 egg and 125 g butter, borrow a second oven, and bake 13 trays (156 muffins), selling 150 and keeping 6 for the group.