Difference between revisions of "2014 AMC 10A Problems/Problem 11"
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For coupon <math>1</math> to give a greater price reduction than the other coupons, we must have <math>.1x>20\implies x>200</math> and <math>.1x>.18x-18\implies.08x<18\implies x<225</math>. | For coupon <math>1</math> to give a greater price reduction than the other coupons, we must have <math>.1x>20\implies x>200</math> and <math>.1x>.18x-18\implies.08x<18\implies x<225</math>. | ||
| − | From the first inequality, the listed price must be greater than <math>\textdollar200</math>, so answer choices <math>\textbf{(A)}</math> and <math>\textbf{(B)}</math> are eliminated. | + | The only choice that satisfies such conditions From the first inequality, the listed price must be greater than <math>\textdollar200</math>, so answer choices <math>\textbf{(A)}</math> and <math>\textbf{(B)}</math> are eliminated. |
From the second inequality, the listed price must be less than <math>\textdollar225</math>, so answer choices <math>\textbf{(D)}</math> and <math>\textbf{(E)}</math> are eliminated. | From the second inequality, the listed price must be less than <math>\textdollar225</math>, so answer choices <math>\textbf{(D)}</math> and <math>\textbf{(E)}</math> are eliminated. | ||
Revision as of 17:05, 20 September 2019
- The following problem is from both the 2014 AMC 12A #8 and 2014 AMC 10A #11, so both problems redirect to this page.
Problem
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: 
off the listed price if the listed price is at least 
Coupon 2: 
off the listed price if the listed price is at least 
Coupon 3: 
off the amount by which the listed price exceeds
For which of the following listed prices will coupon 
offer a greater price reduction than either coupon 
or coupon 
?
Solution 1
Let the listed price be 
. Since all the answer choices are above , we can assume 
. Thus the discounts after the coupons are used will be as follows:
Coupon 1: 
Coupon 2: 
Coupon 3: 
For coupon to give a greater price reduction than the other coupons, we must have 
and 
.
The only choice that satisfies such conditions From the first inequality, the listed price must be greater than 
, so answer choices 
and 
are eliminated.
From the second inequality, the listed price must be less than 
, so answer choices 
and 
are eliminated.
The only answer choice that remains is 
.
Solution 2 (Using The Answers)
For coupon to be the most effective, we want 10% of the price to be greater than 20. This clearly occurs if the value is over 200. For coupon 1 to be more effective than coupon 3, we want to minimize the value over 200, so is the smallest number over 200.
See Also
| 2014 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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 | ||
| 2014 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 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. 
