2014 AMC 10A Problems/Problem 11

Revision as of 23:16, 25 November 2020 by Sugar rush (talk | contribs)
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: $10\%$ 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 1

Let the listed price be $x$. Since all the answer choices are above $\textdollar100$, we can assume $x > 100$. Thus the discounts after the coupons are used will be as follows:

Coupon 1: $x\times10\%=.1x$

Coupon 2: $20$

Coupon 3: $18\%\times(x-100)=.18x-18$


For coupon $1$ to give a greater price reduction than the other coupons, we must have $.1x>20\implies x>200$ and $.1x>.18x-18\implies.08x<18\implies x<225$.

The only choice that satisfies such conditions is $\boxed{\textbf{(C)}\ \textdollar219.95}$

Solution 2 (Using The Answers)

For coupon $1$ 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 $\boxed{\textbf{(C) }\textdollar219.95}$ is the smallest number over 200.

Video Solution One

https://youtu.be/aGEB3ykW1BM

~savannahsolver

Video Solution Two

https://youtu.be/rJytKoJzNBY

See Also

2014 AMC 10A (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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. AMC logo.png

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