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

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A shopper plans to purchase an item that has a listed price greater than <math>\textdollar 100</math> and can use any one of the three coupons. Coupon A gives <math>15\%</math> off the listed price, Coupon B gives <math>\textdollar 30</math> off the listed price, and Coupon C gives <math>25\%</math> off the amount by which the listed price exceeds
 
A shopper plans to purchase an item that has a listed price greater than <math>\textdollar 100</math> and can use any one of the three coupons. Coupon A gives <math>15\%</math> off the listed price, Coupon B gives <math>\textdollar 30</math> off the listed price, and Coupon C gives <math>25\%</math> off the amount by which the listed price exceeds
 
<math>\textdollar 100</math>. <br/>
 
<math>\textdollar 100</math>. <br/>
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Let <math>x</math> and <math>y</math> be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B and C. What is <math>y - x</math>?
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Let <math>x</math> and <math>y</math> be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is <math>y - x</math>?
  
 
<math>\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 80  \qquad \textbf{(E)}\ 100</math>
 
<math>\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 80  \qquad \textbf{(E)}\ 100</math>
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Coupon C saves us: <math>0.25p</math>
 
Coupon C saves us: <math>0.25p</math>
  
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Now, the condition is that A has to be greater than or equal to either B or C which give us the following inequalities:
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Now, the condition is that A has to be greater than or equal to either B or C which gives us the following inequalities:
  
 
<math>A \geq B \Rightarrow 0.15p + 15 \geq 30 \Rightarrow p \geq 100</math>
 
<math>A \geq B \Rightarrow 0.15p + 15 \geq 30 \Rightarrow p \geq 100</math>
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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>
  
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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>.  
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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 <math>p</math> 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 = 250 - 200 = \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>
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 +
==Video Solution==
 +
https://youtu.be/vYXz4wStBUU
 +
 +
~IceMatrix
  
 
==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}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:02, 11 July 2021

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 gives 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 $y - x = 250 - 200 = \boxed{\textbf{(A)}\ 50}$

Video Solution

https://youtu.be/vYXz4wStBUU

~IceMatrix

See Also

2010 AMC 10B (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

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