AMC 10 · 2005 · #5

Grade 4 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
multiplesmulti-digit-arithmetic complementary-countingidentify-subproblems ↑ Prerequisites: multiplesmulti-digit-arithmetic
📏 Short solution 💡 2 insights
📘 View easy version →
Problem
Windows cost 100 dollars 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.

Pick an answer.

(A)
100
(B)
200
(C)
300
(D)
400
(E)
500

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

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.

1STEP 1

Count Dave's free windows alone

Dave needs 7: one paid group of four earns 1 free, and the leftover 2 is too small for another group. Dave alone earns 1 free window.

7 = 4_pay + 1_free + 2_pay → 1 free
2STEP 2

Count Doug's free windows alone

Doug needs 8: again one paid group of four earns 1 free, and his leftover 3 is still short. Separately they earn 2 free windows in all.

8 = 4 + 1 + 3 → 1 free; 1 + 1 = 2 free (separate)
3STEP 3

Count free windows for the combined order

Together they need 15 = 12 paid + 3 free: the stranded 2 and 3 join into a full group of four, so the combined order earns 3 free windows.

15 = 3 × 4 + 3 = 12 paid + 3 free → 3 free
4STEP 4

Take the difference to get the extra saving

Combining gains 3 - 2 = 1 extra free window, worth 100 dollars — choice (A). The 300 trap is the combined order's whole discount.

(3 - 2) × 100=100 =100 → (A)
Answer
100
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.
💡Key takeaway

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 Dave's free windows alone
  • Count Doug's free windows alone
  • Count free windows for the combined order
  • Take the difference to get the extra saving

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