AMC 10 · 2014 · #3

Grade 5 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
fraction-multiplicationdecimal-arithmeticorder-of-operations identify-subproblemsdimensional-analysis ↑ Prerequisites: fraction-multiplication
📏 Medium solution 💡 1 insight
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Problem
Bridget bakes 48 loaves of bread. She sells half of them in the morning at $2.50 each; in the afternoon she sells two-thirds of what is left, at half price; in the late afternoon she sells all the remaining loaves at $1 each. Each loaf costs her $0.75 to make. Find her profit for the day, in dollars.

Pick an answer.

(A)
24
(B)
36
(C)
44
(D)
48
(E)
52

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The day happens in three separate selling times, and there is also a cost to subtract at the end. Tool #7 (Identify Subproblems) handles this by treating each time-of-day as its own small money problem: figure out how many loaves sell and at what price, then add the three amounts. Tool #8 (Analyze the Units) keeps the two kinds of numbers straight — how many loaves (a count) versus how many dollars (money) — so the fraction steps act on loaves and the price steps turn loaves into dollars. At the end, cost is subtracted from revenue to get profit.

1STEP 1

Morning sale

Half of the 48 loaves — 24 of them — sell at $2.50 each, so the morning brings in $60 and leaves 24 loaves.

1/2 × 48 = 24 loaves, 24 × 2.50=2.50 = 60
2STEP 2

Afternoon sale

Two-thirds of those 24 is 16 loaves, sold at the half price of $1.25, so the afternoon adds $20 and leaves 8 loaves.

2/3 × 24 = 16 loaves, 16 × 1.25=1.25 = 20
3STEP 3

Late-afternoon sale and total revenue

The last 8 loaves go for $1 each, giving $8, so the three sales total $88 in revenue.

8 × 1=1 = 8; total revenue = 60 + 20 + 8 = $ 88
4STEP 4

Subtract cost to get profit

All 48 loaves cost $0.75 each to bake, or $36 in all, so the profit is 88 minus 36, which is $52 — choice (E).

cost = 48 × 0.75=0.75 = 36; profit = 88 - 36 = $ 52 → (E)
Answer
52
Check the loaf count balances: 24 in the morning, 16 in the afternoon, 8 late — that is 24+16+8 = 48, exactly the number baked, so no loaf is lost or double-counted. The revenue $88 is more than the cost $36, so a positive profit is expected, and $52 sits right at the top of the answer choices. Note 88 - 36 = 52 matches, and picking $88 (forgetting cost) would wrongly land off the list, while $36 is the cost itself, a common trap answer (B).
💡Key takeaway

Break a busy word problem into one small money question per time of day, add up what you took in, then subtract what it cost to find the real profit.

  • Morning sale
  • Afternoon sale
  • Late-afternoon sale and total revenue
  • Subtract cost to get profit

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