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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?

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

Solution

The store's offer means that every $5$th window is free.

Dave would get $\lfloor\frac{7}{5}\rfloor=1$ free window.

Doug would get $\lfloor\frac{8}{5}\rfloor=1$ free window.

This is a total of $2$ free windows.

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

So they get $3-2=1$ additional window if they purchase the windows together.

Therefore they save $1\cdot100=100\Rightarrow A$

See Also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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