Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2014 AMC 10A Problems/Problem 11"
(Added solution)
Line 18: Line 18:
  
 
==Solution==
 
==Solution==
 +
Let the listed price be <math>x</math>. Since all the answer choices are above <math>&#036;100</math>, we can assume <math>x > 100</math>. Thus the prices after coupons will be as follows:
 +
 +
Coupon 1: <math>x-10\%\cdot x=90\%\cdot x</math>
 +
 +
Coupon 2: <math>x-20</math>
 +
 +
Coupon 3: <math>x-18\%\cdot(x-100)=82\%\cdot x+18</math>
 +
 +
For coupon <math>1</math> to give a better price reduction than the other coupons, we must have <math>90\%\cdot x < x-20</math> and <math>90\%\cdot x < 82\%\cdot x+18</math>.
 +
 +
From the first inequality, <math>90\%\cdot x+(-90\%\cdot x) +(20)< x-20+(-90\%\cdot x)+(20)\Rightarrow 20 < 10\%\cdot x\Rightarrow 200 < x</math>.
 +
 +
From the second inequality, <math>90\%\cdot x +(-82\%\cdot x)< 82\%\cdot x+18+(-82\%\cdot x)\Rightarrow 8\%\cdot x < 18\Rightarrow x < 225</math>.
 +
 +
The only answer choice that satisfies these constraints is <math>\boxed{\textbf{(C) }\textdollar219.95}</math>
  
 
==See Also==
 
==See Also==

Revision as of 23:15, 6 February 2014

Problem

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: $10\%$ off the listed price if the listed price is at least $\textdollar50$

Coupon 2: $\textdollar 20$ off the listed price if the listed price is at least $\textdollar100$

Coupon 3: $18\%$ off the amount by which the listed price exceeds $\textdollar100$

For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?

$\textbf{(A) }\textdollar179.95\qquad \textbf{(B) }\textdollar199.95\qquad \textbf{(C) }\textdollar219.95\qquad \textbf{(D) }\textdollar239.95\qquad \textbf{(E) }\textdollar259.95\qquad$

Solution

Let the listed price be $x$. Since all the answer choices are above $&#036;100$, we can assume $x > 100$. Thus the prices after coupons will be as follows:

Coupon 1: $x-10\%\cdot x=90\%\cdot x$

Coupon 2: $x-20$

Coupon 3: $x-18\%\cdot(x-100)=82\%\cdot x+18$

For coupon $1$ to give a better price reduction than the other coupons, we must have $90\%\cdot x < x-20$ and $90\%\cdot x < 82\%\cdot x+18$.

From the first inequality, $90\%\cdot x+(-90\%\cdot x) +(20)< x-20+(-90\%\cdot x)+(20)\Rightarrow 20 < 10\%\cdot x\Rightarrow 200 < x$.

From the second inequality, $90\%\cdot x +(-82\%\cdot x)< 82\%\cdot x+18+(-82\%\cdot x)\Rightarrow 8\%\cdot x < 18\Rightarrow x < 225$.

The only answer choice that satisfies these constraints is $\boxed{\textbf{(C) }\textdollar219.95}$

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_10A_Problems/Problem_11&oldid=59132"