2005 AMC 10A Problem 5

P1-10~2 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Each group of 5 windows gets one free. Alone, 7 and 8 each earn one free window; together, 15 earns three. Buying together gains one extra free window.

Solution

"Buy four, get one free" means every full group of $5$ windows costs only $4$ . So count how many full groups of five each purchase contains.

Dave alone, $7$ windows. That is one full group of $5$ plus $2$ extra. He gets $1$ free window and pays for $6$ : $\$600$ .
Doug alone, $8$ windows. That is one full group of $5$ plus $3$ extra. He gets $1$ free window and pays for $7$ : $\$700$ .
Apart, in total: $600 + 700 = \$1300$ .
Together, $15$ windows. That is exactly three groups of $5$ . They get $3$ free windows and pay for $12$ : $\$1200$ .
Savings: $1300 - 1200 = \$100$ .

The saving is exactly one extra free window. Dave's $2$ leftover windows and Doug's $3$ leftover windows earned nothing on their own, but together they make a full group of five.

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

Why this works

A "get one free" deal only pays off for complete groups. Leftovers that do not fill a group are wasted. When people combine orders, their leftovers can join up into a new full group, and that is where the saving comes from. Look for leftovers that add up to a full group.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(A)}\ 100}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)