Cupcake Booth: Parentheses, Leftovers, and the Cheaper Carton
The fifth grade is running the cupcake booth at the spring carnival. Six classrooms each donate 24 cupcakes, and the cafeteria adds 38 more. Cupcakes are packed in boxes that hold 10 each, and each box costs $4.25 from the supplier. A student writes the expression 6 × 24 + 38 ÷ 10, evaluates it as 147.8, and announces that the booth needs "about 148 boxes." The booth chair knows that number cannot be right, but she is not sure where the expression went wrong. She needs a correct expression, the actual number of boxes to buy, and the total cost. She also insists that every cupcake must end up in a box - none can be sold loose - and she wants to know how many cupcakes sit in the last, partly filled box. A supplier flyer also offers a larger box holding 24 cupcakes for $9.50.
1. Write a correct expression for the number of boxes, evaluate it, and identify the error in the student's expression.
Read the worked answer
The total number of cupcakes must be found first, then divided by 10: (6 × 24 + 38) ÷ 10.
Evaluate: 6 × 24 = 144; 144 + 38 = 182; 182 ÷ 10 = 18.2.
The student's error is order of operations. Without parentheses, multiplication and division are done left to right before addition, so 38 ÷ 10 = 3.8 is computed first and then added to 144, giving 147.8. That result mixes "number of cupcakes" with "number of boxes" in a single number, which is why 148 boxes is absurd - the booth has only 182 cupcakes, so 148 boxes of 10 would hold 1,480.
A magnitude check catches it immediately: 182 cupcakes at 10 per box must land between 18 and 19 boxes.
2. How many boxes must the booth buy, how much will they cost, and how many cupcakes are in the last box?
Read the worked answer
18 boxes hold 18 × 10 = 180 cupcakes, leaving 2 cupcakes unboxed. Since no cupcake may be sold loose, the booth must buy a 19th box, so 19 boxes.
Cost: 19 × $4.25. Compute 20 × $4.25 = $85.00, then subtract one box: $85.00 − $4.25 = $80.75.
The 19th box holds the 2 leftover cupcakes, so it has 8 empty spaces. Note the difference between 18.2 and 19: the decimal 0.2 is not "two more boxes," it is a fraction of a box that still requires a whole box to be bought. Rounding down to 18 would leave 2 cupcakes with nowhere to go.
3. Using the flyer's 24-cupcake box at $9.50 each, determine whether the booth saves money, and state one drawback of switching.
Read the worked answer
182 ÷ 24 = 7 with remainder 14, since 7 × 24 = 168 and 182 − 168 = 14. The 14 extra cupcakes still need a box, so 8 large boxes.
Cost: 8 × $9.50 = $76.00. Compare with $80.75 for the small boxes: the large boxes save $4.75.
Capacity check: 8 × 24 = 192 spaces for 182 cupcakes, so 10 spaces sit empty - more wasted space than the small-box plan, which had 19 × 10 = 190 spaces and 8 empty. The drawback is real but small: the booth pays for 10 unused slots yet still spends less money overall.
The decision rule is to compare total cost after rounding up to whole boxes, never cost per box alone, because the 24-count box only looks cheaper per cupcake if the last box is full.