Difference between revisions of "2014 AMC 12B Problems/Problem 2"

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Orvin went to the store with just enough money to buy <math> 30 </math> balloons. When he arrived he discovered that the store had a special sale on balloons: buy <math> 1 </math> balloon at the regular price and get a second at <math> \frac{1}{3} </math> off the regular price. What is the greatest number of balloons Orvin could buy?
 
Orvin went to the store with just enough money to buy <math> 30 </math> balloons. When he arrived he discovered that the store had a special sale on balloons: buy <math> 1 </math> balloon at the regular price and get a second at <math> \frac{1}{3} </math> off the regular price. What is the greatest number of balloons Orvin could buy?
  
<math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}}\ 38\qquad\textbf{(E)}\ 39 </math>
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<math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 39 </math>
  
==Solution==
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==Solution 1==
  
 
If every balloon costs <math>n</math> dollars, then Orvin has <math>30n</math> dollars.  For every balloon he buys for <math>n</math> dollars, he can buy another for <math> \frac{2n}{3}</math> dollars.  This means it costs him <math> \frac{5n}{3}</math> dollars to buy a bundle of <math>2</math> balloons.  With <math>30n</math> dollars, he can buy <math>\frac{30n}{\frac{5n}{3}} = 18</math> sets of two balloons, so the total number of balloons he can buy is <math>18\times2  \implies \boxed{\textbf{(C)}\ 36 }</math>
 
If every balloon costs <math>n</math> dollars, then Orvin has <math>30n</math> dollars.  For every balloon he buys for <math>n</math> dollars, he can buy another for <math> \frac{2n}{3}</math> dollars.  This means it costs him <math> \frac{5n}{3}</math> dollars to buy a bundle of <math>2</math> balloons.  With <math>30n</math> dollars, he can buy <math>\frac{30n}{\frac{5n}{3}} = 18</math> sets of two balloons, so the total number of balloons he can buy is <math>18\times2  \implies \boxed{\textbf{(C)}\ 36 }</math>
  
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==Solution 2==
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Similar to solution 1, but quicker.
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We see the fraction <math>\frac13,</math> so we let each balloon cost <math>\$3</math> WLOG. Thus, Orvin has <math>\$3\cdot30=\$90.</math>
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Each pair of balloons (one full-priced and one a third off) costs <math>\$3+\left(1-\frac13\right)\cdot\$3=\$3+\$2=\$5.</math> Therefore, Orvin can buy <math>\frac{\$90}{\$5}=18</math> pairs of balloons, at maximum. <math>18\cdot2=36</math> balloons.
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~Technodoggo
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==Video Solution 1 (Quick and Easy)==
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https://youtu.be/vkItt-jxrIs
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~Education, the Study of Everything
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== See also ==
 
{{AMC12 box|year=2014|ab=B|num-b=1|num-a=3}}
 
{{AMC12 box|year=2014|ab=B|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:41, 30 May 2023

Problem

Orvin went to the store with just enough money to buy $30$ balloons. When he arrived he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?

$\textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 39$

Solution 1

If every balloon costs $n$ dollars, then Orvin has $30n$ dollars. For every balloon he buys for $n$ dollars, he can buy another for $\frac{2n}{3}$ dollars. This means it costs him $\frac{5n}{3}$ dollars to buy a bundle of $2$ balloons. With $30n$ dollars, he can buy $\frac{30n}{\frac{5n}{3}} = 18$ sets of two balloons, so the total number of balloons he can buy is $18\times2  \implies \boxed{\textbf{(C)}\ 36 }$

Solution 2

Similar to solution 1, but quicker. We see the fraction $\frac13,$ so we let each balloon cost $$3$ WLOG. Thus, Orvin has $$3\cdot30=$90.$ Each pair of balloons (one full-priced and one a third off) costs $$3+\left(1-\frac13\right)\cdot$3=$3+$2=$5.$ Therefore, Orvin can buy $\frac{$90}{$5}=18$ pairs of balloons, at maximum. $18\cdot2=36$ balloons. ~Technodoggo

Video Solution 1 (Quick and Easy)

https://youtu.be/vkItt-jxrIs

~Education, the Study of Everything

See also

2014 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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All AMC 12 Problems and Solutions

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