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==Problem== | ==Problem== | ||
| − | A store normally sells windows at <math>100</math> 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 store normally sells windows at <math>\$100</math> 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? |
| − | <math> \ | + | <math> |
| + | \textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D) } 400\qquad \textbf{(E) } 500 | ||
| + | </math> | ||
==Solution== | ==Solution== | ||
| − | The store's offer means that every <math>5</math>th window is free. | + | The store's offer means that every <math>5</math>th window is free, so Dave would get <math>\left\lfloor\frac{7}{5}\right\rfloor=1</math> free window and Doug would also get <math>\left\lfloor\frac{8}{5}\right\rfloor=1</math> free window. |
| − | + | This is a total of <math>1+1 = 2</math> free windows. | |
| − | + | Together, by contrast, they would get <math>\left\lfloor\frac{8+7}{5}\right\rfloor = \left\lfloor\frac{15}{5}\right\rfloor=3</math> free windows. | |
| − | + | Hence they get <math>3-2=1</math> additional window if they purchase the windows together, so the amount (in dollars) that they save is <math>1\cdot100= \boxed{\textbf{(A) } 100}</math>. | |
| − | + | ==Video Solution 1== | |
| + | https://youtu.be/ECBXQ1cE2SI | ||
| − | + | ==Video Solution 2== | |
| + | https://youtu.be/tsai14EkXgQ | ||
| − | + | ~Charles3829 | |
==See also== | ==See also== | ||
Latest revision as of 17:03, 1 July 2025
Problem
A store normally sells windows at 
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?
Solution
The store's offer means that every 
th window is free, so Dave would get 
free window and Doug would also get 
free window.
This is a total of 
free windows.
Together, by contrast, they would get 
free windows.
Hence they get 
additional window if they purchase the windows together, so the amount (in dollars) that they save is 
.
Video Solution 1
Video Solution 2
~Charles3829
See also
| 2005 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
