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2005 AMC 10A Problem 5

2005 AMC 10A Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

A store normally sells windows at $\$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?

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

Answer

Key insight

Count free windows: 7 and 8 alone each earn one free window, but 15 together earns three, so pooling gains exactly one free window.

Solution

Under the deal, every group of $5$ windows costs only $4$ , so a buyer gets one free window per full group of five.

Separately: Dave's $7$ windows contain one full group of five, so he pays for $6$ . Doug's $8$ windows also contain one group of five, so he pays for $7$ . Together that is $13$ paid windows, or $\$1300$ .

Together: $15$ windows contain three groups of five, so they pay for $12$ windows, or $\$1200$ .

The saving is $1300 - 1200 = 100$ dollars, which is exactly one extra free window.

The answer is $\boxed{\textbf{(A)}\ 100}$ .

Why this works

"Buy four, get one free" means free windows come in whole groups of five, and leftover windows below a full group earn nothing. Dave's two leftovers and Doug's three leftovers are wasted separately but combine into a full group when pooled. Look for leftovers that sum to a complete group.

The trap

Computing what they pay together ($1200) or separately ($1300) and reporting a total instead of the difference.

Common mistakes

Computing what they pay together ( $1200) or separately ($ 1300) and reporting a total instead of the difference.
Giving Doug two free windows for $8$ windows (as if "buy 4 get 1" applied to $8 = 4 + 4$ paid windows); the free window counts toward the eight he needs.

Techniques

Set up the equation/formula and compute; no special trick needed