Difference between revisions of "2010 AMC 10B Problems/Problem 11"

(Made ending a bit clearer, x and y aren't the amount by which the price exceeds $100 (p), they're the actual price)
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<math>A \geq C \Rightarrow 0.15p + 15 \geq 0.25p \Rightarrow p \leq 150</math>
 
<math>A \geq C \Rightarrow 0.15p + 15 \geq 0.25p \Rightarrow p \leq 150</math>
  
We see here that the greatest possible value for <math>p</math> is <math> 150 = y </math> and the smallest is <math> 100 = x </math>
+
We see here that the greatest possible value for <math>p</math> is <math> 150 </math>, thus <math> y = 100 + 150 = 250 </math> and the smallest value for p is <math> 100 </math> so <math> x = 100 + 100 = 200 </math>.
  
The difference between <math>y</math> and <math>x</math> is <math>y - x = 150 - 100 = \boxed{\textbf{(A)}\ 50}</math>
+
The difference between <math>y</math> and <math>x</math> is <math>y - x = 250 - 200 = \boxed{\textbf{(A)}\ 50}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2010|ab=B|num-b=10|num-a=12}}
 
{{AMC10 box|year=2010|ab=B|num-b=10|num-a=12}}

Revision as of 13:33, 4 February 2012

Problem

A shopper plans to purchase an item that has a listed price greater than $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $\textdollar 100$.
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$?

$\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 80  \qquad \textbf{(E)}\ 100$

Solution

Let the listed price be $(100 + p)$, where $p > 0$

Coupon A saves us: $0.15(100+p) = (0.15p + 15)$

Coupon B saves us: $30$

Coupon C saves us: $0.25p$

Now, the condition is that A has to be greater than or equal to either B or C which give us the following inequalities:

$A \geq B \Rightarrow 0.15p + 15 \geq 30 \Rightarrow p \geq 100$

$A \geq C \Rightarrow 0.15p + 15 \geq 0.25p \Rightarrow p \leq 150$

We see here that the greatest possible value for $p$ is $150$, thus $y = 100 + 150 = 250$ and the smallest value for p is $100$ so $x = 100 + 100 = 200$.

The difference between $y$ and $x$ is $y - x = 250 - 200 = \boxed{\textbf{(A)}\ 50}$

See Also

2010 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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All AMC 10 Problems and Solutions
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