AMC 10 · 2005 · #1
Easy mode Grade 4A scout troop buys 1000 candy bars. The store sells them at 5 bars for 2 dollars, so that is what the troop pays. Later the troop sells all 1000 bars, at 2 bars for 1 dollar. Profit is the money they take in from selling minus the money they paid to buy. How much profit did they make, in dollars?
Pick an answer.
AMC 10 2005 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: A troop buys $1000$ candy bars at the price of five for $2$ dollars, then sells all $1000$ of them at the price of two for $1$ dollar. Find the profit in dollars, where profit is the money taken in from selling minus the money paid to buy.
Givens: They buy $1000$ candy bars; Buying price: five bars for $2$ dollars; Selling price: two bars for $1$ dollar; All $1000$ bars are sold; Answer choices: (A) $100$, (B) $200$, (C) $300$, (D) $400$, (E) $500$
Unknowns: The profit in dollars: total money from selling minus total money spent buying
Understand
Restated: A troop buys $1000$ candy bars at the price of five for $2$ dollars, then sells all $1000$ of them at the price of two for $1$ dollar. Find the profit in dollars, where profit is the money taken in from selling minus the money paid to buy.
Givens: They buy $1000$ candy bars; Buying price: five bars for $2$ dollars; Selling price: two bars for $1$ dollar; All $1000$ bars are sold; Answer choices: (A) $100$, (B) $200$, (C) $300$, (D) $400$, (E) $500$
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
Both prices are quoted per group of bars, not per single bar, so Tool #8 (Analyze the Units) keeps the two units straight: how many groups of bars, and how many dollars each group is worth. Tool #7 (Identify Subproblems) splits the work into three clean pieces — money spent buying, money taken in selling, then the difference. Tool #3 (Eliminate Possibilities) guards the main trap: choice (E) $500$ is exactly the selling total, which is what you get if you forget to subtract the buying cost.
Execute — Answer: A
4.NBT.B.6 Step 1 Money spent buying
- The bars are bought in groups of five, so split $1000$ bars into groups of $5$: $1000\div5=200$ groups.
- Each group costs $2$ dollars, so the total spent is $200\times2=400$ dollars.
💡 Count how many price-groups the bars make, then pay the group price that many times.
4.NBT.B.6 Step 2 Money taken in selling
- The bars are sold two at a time, so split $1000$ bars into groups of $2$: $1000\div2=500$ groups.
- Each group sells for $1$ dollar, so the total taken in is $500\times1=500$ dollars.
💡 Same idea as buying, but now the group is a pair and each pair brings in a dollar.
4.OA.A.3 Step 3 Profit is selling minus buying
- Profit is the money taken in minus the money spent: $500-400=100$ dollars.
- Note that $500$ by itself is only the selling total — that is the trap answer (E); the profit is what is left after paying back the $400$ it cost to buy the bars.
- So the profit is $100$ dollars, choice (A).
💡 Profit is what stays in your pocket after you pay back what you spent.
4.NBT.B.6 The bars are bought in groups of five, so split $1000$ bars into groups of $5$: 4.NBT.B.6 The bars are sold two at a time, so split $1000$ bars into groups of $2$: $1000\ 4.OA.A.3 Profit is the money taken in minus the money spent: $500-400=100$ dollars. Note Review
Reasonableness: Check the per-bar prices as a sanity test: buying costs $2\div5=\$0.40$ a bar and selling brings $1\div2=\$0.50$ a bar, so each bar earns $0.50-0.40=\$0.10$ of profit. Across $1000$ bars that is $1000\times0.10=\$100$, matching the grouped calculation. The gain per bar is small and positive, so a total near $\$100$ is sensible; the larger choices would need the troop to more than double its money, which the $10$-cent margin per bar does not support. Choice (E) $500$ is the selling total with the cost ignored — a classic slip that the profit definition rules out.
Alternative: Work per single bar from the start: buying price is $\$0.40$ per bar and selling price is $\$0.50$ per bar, a profit of $\$0.10$ per bar. Multiply by all $1000$ bars: $1000\times\$0.10=\$100$, landing on (A) in one multiplication.
CCSS standards used (min grade 4)
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Dividing $1000$ bars into groups of $5$ ($200$ groups) and groups of $2$ ($500$ groups), then multiplying by each group price to get the $\$400$ cost and $\$500$ revenue.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Combining the buying total and selling total by subtraction, $500-400=100$, to answer the profit question.)
⭐ Turn a per-group price into a total by counting the groups, then remember profit is what you sell for minus what you paid — not just the money you took in.
⭐ Turn a per-group price into a total by counting the groups, then remember profit is what you sell for minus what you paid — not just the money you took in.
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