Difference between revisions of "2005 AMC 12A Problems/Problem 4"
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== Problem == | == Problem == | ||
| − | A store normally sells windows at | + | 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 much will they save if they purchase the windows together rather than separately? |
<math> | <math> | ||
(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500 | (\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500 | ||
</math> | </math> | ||
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== Solution == | == Solution == | ||
For <math>n</math> windows, the store offers a discount of <math>100 \cdot \left\lfloor\frac{n}{5}\right\rfloor</math> ([[floor function]]). Dave receives a discount of <math>100 \cdot \left\lfloor \frac{7}{5}\right \rfloor = 100</math> and Doug receives a discount of <math>100 \cdot \left\lfloor \frac{8}{5}\right\rfloor = 100</math>. These amount to <math>200</math> dollars in discounts. Together, they receive a discount of <math>100 \cdot \left\lfloor \frac{15}{5} \right\rfloor = 300</math>, so they save <math>300-200=100\ \mathrm{(A)}</math>. | For <math>n</math> windows, the store offers a discount of <math>100 \cdot \left\lfloor\frac{n}{5}\right\rfloor</math> ([[floor function]]). Dave receives a discount of <math>100 \cdot \left\lfloor \frac{7}{5}\right \rfloor = 100</math> and Doug receives a discount of <math>100 \cdot \left\lfloor \frac{8}{5}\right\rfloor = 100</math>. These amount to <math>200</math> dollars in discounts. Together, they receive a discount of <math>100 \cdot \left\lfloor \frac{15}{5} \right\rfloor = 300</math>, so they save <math>300-200=100\ \mathrm{(A)}</math>. | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 12:05, 7 August 2019
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 much will they save if they purchase the windows together rather than separately?
Solution
For 
windows, the store offers a discount of 
(floor function). Dave receives a discount of 
and Doug receives a discount of 
. These amount to 
dollars in discounts. Together, they receive a discount of 
, so they save 
.
See also
| 2005 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
