2014 AMC 10A Problems/Problem 11

Revision as of 14:15, 9 February 2014 by Flamedragon (talk | contribs) (Solution 2: Using the Answer Choices)
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 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 $\boxed{\textbf{(C) }\textdollar219.95}$

Solution 2: Using the Answer Choices

When you look at the answer choices, you know that we can immediately take out A and B because when you take off 20 dollars from them, it is more than 10%. We know C is the right answer because it is the first one that is larger than $200, which is the line in which taking off$20 is less than 10%.

(Solution by William Chi)

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

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