2005 AMC 10A Problems/Problem 5

Problem

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?

$\textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D) } 400\qquad \textbf{(E) } 500$

Solution

The store's offer means that every $5$ th window is free, so Dave would get $\left\lfloor\frac{7}{5}\right\rfloor=1$ free window and Doug would also get $\left\lfloor\frac{8}{5}\right\rfloor=1$ free window.

This is a total of $1+1 = 2$ free windows.

Together, by contrast, they would get $\left\lfloor\frac{8+7}{5}\right\rfloor = \left\lfloor\frac{15}{5}\right\rfloor=3$ free windows.

Hence they get $3-2=1$ additional window if they purchase the windows together, so the amount (in dollars) that they save is $1\cdot100= \boxed{\textbf{(A) } 100}$ .

Video Solution 1

https://youtu.be/ECBXQ1cE2SI

Video Solution 2

https://youtu.be/tsai14EkXgQ

~Charles3829

2005 AMC 10A (Problems • Answer Key • Resources)
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2005 AMC 10A Problems/Problem 5

Contents

Problem

Solution

Video Solution 1

Video Solution 2

See also