AMC 10 · 2005 · #4
Easy mode Grade 4A store sells windows for $100 each. Right now, every time you pay for four windows, you get one more window free. Dave wants seven windows and Doug wants eight windows. How many dollars will they save if they buy all their windows in one order together, instead of each buying their own?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Windows cost $100 each. The store's deal gives one free window for every four you buy. Dave needs seven windows and Doug needs eight. Find how many more dollars they save by buying all their windows in one combined order instead of buying separately.
Givens: Each window normally costs $100$ dollars; The deal: for each purchase of four windows, one extra window is free; Dave needs $7$ windows and Doug needs $8$ windows; Answer choices: (A) $100$, (B) $200$, (C) $300$, (D) $400$, (E) $500$
Unknowns: The extra dollars saved when the two boys combine their orders instead of ordering separately
Understand
Restated: Windows cost $100 each. The store's deal gives one free window for every four you buy. Dave needs seven windows and Doug needs eight. Find how many more dollars they save by buying all their windows in one combined order instead of buying separately.
Givens: Each window normally costs $100$ dollars; The deal: for each purchase of four windows, one extra window is free; Dave needs $7$ windows and Doug needs $8$ windows; Answer choices: (A) $100$, (B) $200$, (C) $300$, (D) $400$, (E) $500$
Plan
Primary tool: #16 Change Focus / Count the Complement
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
Instead of tracking dollars, Tool #16 (Change Focus) turns the whole problem into one count: how many free windows does each plan earn? Each free window is worth $100$, so counting free windows is enough. Tool #7 (Identify Subproblems) splits the work into three clean pieces — free windows Dave earns, free windows Doug earns, and free windows the combined order earns. Tool #3 (Eliminate Possibilities) guards the main trap: the combined order earns three free windows worth $300$, which is choice (C), but the question asks only for the extra saving over buying separately.
Execute — Answer: A
4.NBT.B.6 Step 1 Count Dave's free windows alone
- A free window is earned only for a full group of four that you pay for.
- Dave needs $7$ windows.
- He pays for one group of $4$ and gets $1$ free, giving $5$ windows; the last $2$ windows he just buys outright, and $2$ is not a full group of four, so it earns nothing more.
- Dave alone earns $1$ free window.
💡 Only complete groups of four hand you a free window; leftovers under four give nothing.
4.NBT.B.6 Step 2 Count Doug's free windows alone
- Doug needs $8$ windows.
- He pays for one group of $4$ and gets $1$ free, giving $5$ windows; then he buys the remaining $3$ outright, and $3$ is not a full group of four, so it earns nothing more.
- Doug alone earns $1$ free window.
- Buying separately, the two boys together earn $1 + 1 = 2$ free windows.
💡 Each boy's leftover after his first group is too small to trigger a second free window.
4.NBT.B.6 Step 3 Count free windows for the combined order
- Together they need $7 + 8 = 15$ windows.
- Now the leftovers combine: they pay for three groups of $4$ (that is $12$ paid windows) and get $3$ free, giving exactly $12 + 3 = 15$ windows.
- Combining lets the two short leftovers ($2$ and $3$) join into full groups of four, so the combined order earns $3$ free windows.
💡 Pooling the wasted leftovers lets them add up to whole groups of four that each unlock another free window.
4.OA.A.3 Step 4 Take the difference to get the extra saving
- The combined order earns $3$ free windows while ordering separately earns only $2$, so combining earns $3 - 2 = 1$ extra free window.
- Each window is worth $100$ dollars, so the extra saving is $1 \times 100 = 100$ dollars.
- Watch the trap: the combined order's total discount is $3 \times 100 = 300$ dollars, which is choice (C), but the question asks only how much more they save by combining, so the answer is $100$ dollars, choice (A).
💡 The reward for combining is exactly the one extra free window that the pooled leftovers unlock.
4.NBT.B.6 A free window is earned only for a full group of four that you pay for. Dave nee 4.NBT.B.6 Doug needs $8$ windows. He pays for one group of $4$ and gets $1$ free, giving $ 4.NBT.B.6 Together they need $7 + 8 = 15$ windows. Now the leftovers combine: they pay for 4.OA.A.3 The combined order earns $3$ free windows while ordering separately earns only $ Review
Reasonableness: The whole saving comes from free windows, and each is worth $100$, so any answer must be a whole-number multiple of $100$ — every choice fits that, but the size must be small. Separately the boys waste leftovers of $2$ and $3$ windows, which together make $5$, enough for exactly one more free window when pooled — so the extra gain is a single free window, $100$ dollars. Larger choices like $300$ or $500$ would require several extra free windows, which the small leftovers cannot produce. Choice (C) $300$ is the combined order's whole discount, not the extra amount over separate buying, so it is the natural wrong turn.
Alternative: Compute the actual cash each plan pays. Separately Dave pays for $6$ windows and Doug pays for $7$, a total of $13$ paid windows costing $13 \times \$100 = \$1300$. Combined they pay for $12$ windows costing $12 \times \$100 = \$1200$. The difference $\$1300 - \$1200 = \$100$ is the extra saving, again choice (A).
CCSS standards used (min grade 4)
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Splitting each window count ($7$, $8$, and $15$) into full groups of four plus a leftover to count how many free windows each plan earns.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Combining the free-window counts by subtraction ($3-2=1$) and multiplying by the $\$100$ window price to find the extra saving.)
⭐ Count the free windows, not the dollars: buying together lets the wasted leftovers team up into one more full group of four, and that single extra free window is the $100$ dollars you save.
⭐ Count the free windows, not the dollars: buying together lets the wasted leftovers team up into one more full group of four, and that single extra free window is the $100$ dollars you save.
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