AMC 10 · 2014 · #3
Easy mode Grade 5Bridget bakes 48 loaves of bread. In the morning she sells half of them for \textdollar2.50 each. In the afternoon she sells two thirds of what is left, but these loaves are no longer fresh, so she charges half price. In the late afternoon she sells every loaf that is still left for \textdollar1 each. Each loaf costs her \textdollar0.75 to make. Her profit is the money she takes in minus the money she spent making the bread. How many dollars is her profit for the day?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Bridget bakes $48$ loaves. She sells half in the morning at $\textdollar 2.50$ each; then two-thirds of what is left in the afternoon at half price; then all the rest in the late afternoon at $\textdollar 1$ each. Each loaf costs her $\textdollar 0.75$ to make. Find her profit for the day, in dollars.
Givens: She bakes $48$ loaves total; Morning: sells $\tfrac{1}{2}$ of the loaves at $\textdollar 2.50$ each; Afternoon: sells $\tfrac{2}{3}$ of what remains at half price; Late afternoon: sells every remaining loaf at $\textdollar 1$ each; Each loaf costs $\textdollar 0.75$ to make; Answer choices: (A) $24$, (B) $36$, (C) $44$, (D) $48$, (E) $52$
Unknowns: The profit for the day, in dollars (money taken in minus money spent making the loaves)
Understand
Restated: Bridget bakes $48$ loaves. She sells half in the morning at $\textdollar 2.50$ each; then two-thirds of what is left in the afternoon at half price; then all the rest in the late afternoon at $\textdollar 1$ each. Each loaf costs her $\textdollar 0.75$ to make. Find her profit for the day, in dollars.
Givens: She bakes $48$ loaves total; Morning: sells $\tfrac{1}{2}$ of the loaves at $\textdollar 2.50$ each; Afternoon: sells $\tfrac{2}{3}$ of what remains at half price; Late afternoon: sells every remaining loaf at $\textdollar 1$ each; Each loaf costs $\textdollar 0.75$ to make; Answer choices: (A) $24$, (B) $36$, (C) $44$, (D) $48$, (E) $52$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units
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.
Execute — Answer: E
5.NBT.B.7 Step 1 Morning sale
- She sells half of the $48$ loaves in the morning, which is $24$ loaves, each at $\textdollar 2.50$.
- Multiply the count by the price: $24\times 2.50 = 60$.
- So the morning brings in $\textdollar 60$, and $48-24 = 24$ loaves are left.
💡 Count how many loaves sell first, then turn that count into dollars by multiplying by the price.
4.NF.B.4 Step 2 Afternoon sale
- Of the $24$ loaves left, she sells two-thirds: $\tfrac{2}{3}\times 24 = 16$ loaves.
- These go at half price, and half of $\textdollar 2.50$ is $\textdollar 1.25$.
- So the afternoon brings in $16\times 1.25 = \textdollar 20$, and $24-16 = 8$ loaves are left.
💡 "Two-thirds of what's left" acts on the $24$ loaves now on hand, not on the original $48$.
4.OA.A.3 Step 3 Late-afternoon sale and total revenue
- The $8$ remaining loaves each sell for $\textdollar 1$, giving $8\times 1 = \textdollar 8$.
- Add the three selling times to get all the money she takes in: $60 + 20 + 8 = \textdollar 88$.
💡 Revenue is just the three dollar amounts stacked together, once every loaf is accounted for.
5.NBT.B.7 Step 4 Subtract cost to get profit
- All $48$ loaves cost $\textdollar 0.75$ each to make, so the total cost is $48\times 0.75 = \textdollar 36$.
- Profit is money in minus money spent: $88 - 36 = \textdollar 52$.
- So the answer is (E).
💡 Profit is what's left of the money you collected after paying for what it cost to make everything.
5.NBT.B.7 She sells half of the $48$ loaves in the morning, which is $24$ loaves, each at 4.NF.B.4 Of the $24$ loaves left, she sells two-thirds: $\tfrac{2}{3}\times 24 = 16$ loav 4.OA.A.3 The $8$ remaining loaves each sell for $\textdollar 1$, giving $8\times 1 = \tex 5.NBT.B.7 All $48$ loaves cost $\textdollar 0.75$ each to make, so the total cost is $48\t Review
Reasonableness: 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 $\textdollar 88$ is more than the cost $\textdollar 36$, so a positive profit is expected, and $\textdollar 52$ sits right at the top of the answer choices. Note $88 - 36 = 52$ matches, and picking $\textdollar 88$ (forgetting cost) would wrongly land off the list, while $\textdollar 36$ is the cost itself, a common trap answer (B).
Alternative: Track it as a running table instead of separate pieces. Start: $48$ loaves. After morning: $24$ loaves left, $\textdollar 60$ collected. After afternoon: $8$ loaves left, $\textdollar 60+\textdollar 20 = \textdollar 80$ collected. After late afternoon: $0$ loaves left, $\textdollar 80+\textdollar 8 = \textdollar 88$ collected. Subtract the $\textdollar 36$ cost once at the end: $88-36 = \textdollar 52$, the same answer (E).
CCSS standards used (min grade 5)
5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Multiplying loaf counts by dollar-and-cents prices ($24\times 2.50$, $16\times 1.25$, $48\times 0.75$) and subtracting cost from revenue ($88-36$).)4.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a whole number (Taking $\tfrac{1}{2}$ of $48$ loaves and $\tfrac{2}{3}$ of the $24$ loaves left to find how many sell in each part of the day.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Organizing the three selling times into one running total and combining the dollar amounts into total revenue.)
⭐ 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.
⭐ 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.
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