AMC 10 · 2005 · #1
Grade 4 arithmeticA scout troop buys 1000 candy bars at a price of five for 2 dollars. They sell all the candy bars at the price of two for 1 dollar. What was their profit, 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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