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Difference between revisions of "2009 AMC 10A Problems/Problem 8"

(New page: A senior ticket costs <dollar/><math>6.00</math>, so a regular ticket costs <math>6 * \frac{1}{\frac{3}{4}}\:=\:6*\frac{4}{3}\:=\:8</math> dollars. Therefore children's tickets cost half t...)
 
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A senior ticket costs <dollar/><math>6.00</math>, so a regular ticket costs <math>6 * \frac{1}{\frac{3}{4}}\:=\:6*\frac{4}{3}\:=\:8</math> dollars. Therefore children's tickets cost half that, or <dollar/><math>4.00</math>, so we have:
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== Problem 8 ==
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Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a <math>50</math>% discount as children. The two members of the oldest generation receive a <math>25\%</math> discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <math>\$6.00</math>, is paying for everyone. How many dollars must he pay?
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 +
<math>
 +
\mathrm{(A)}\ 34
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\qquad
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\mathrm{(B)}\ 36
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\qquad
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\mathrm{(C)}\ 42
 +
\qquad
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\mathrm{(D)}\ 46
 +
\qquad
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\mathrm{(E)}\ 48
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</math>
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 +
== Solution ==
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 +
A senior ticket costs <math>\$6.00</math>, so a regular ticket costs <math>6 \cdot \frac{1}{\frac{3}{4}}\:=\:6\cdot\frac{4}{3}\:=\:8</math> dollars. Therefore children's tickets cost half that, or <math>\$4.00</math>, so we have:
  
 
<math>2(6+8+4)\:=\:36</math>
 
<math>2(6+8+4)\:=\:36</math>
  
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So Grandfather Wen pays <dollar/><math>36</math>, or <math>B</math>.
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So Grandfather Wen pays <math>\$36</math>, or <math>\fbox{B}</math>.
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==Solution 2==
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Using average, we know that assuming the middle-aged people's ticket cost x dollars, <math>2\cdot(100\%x+75\%x+50\%x)</math>. We average this into 2*(75%x+75%x+75%x). We know that 75%x=6.00, which means <math>6\cdot6</math> is the answer, or 36. -RealityWrites
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== See Also ==
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{{AMC10 box|year=2009|ab=A|num-b=7|num-a=9}}
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{{MAA Notice}}

Latest revision as of 15:53, 28 July 2023

Problem 8

Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$% discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $$6.00$, is paying for everyone. How many dollars must he pay?

$\mathrm{(A)}\ 34 \qquad \mathrm{(B)}\ 36 \qquad \mathrm{(C)}\ 42 \qquad \mathrm{(D)}\ 46 \qquad \mathrm{(E)}\ 48$

Solution

A senior ticket costs $$6.00$, so a regular ticket costs $6 \cdot \frac{1}{\frac{3}{4}}\:=\:6\cdot\frac{4}{3}\:=\:8$ dollars. Therefore children's tickets cost half that, or $$4.00$, so we have:

$2(6+8+4)\:=\:36$

So Grandfather Wen pays $$36$, or $\fbox{B}$.

Solution 2

Using average, we know that assuming the middle-aged people's ticket cost x dollars, $2\cdot(100\%x+75\%x+50\%x)$. We average this into 2*(75%x+75%x+75%x). We know that 75%x=6.00, which means $6\cdot6$ is the answer, or 36. -RealityWrites

See Also

2009 AMC 10A (Problems • Answer Key • Resources)
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 10 Problems and Solutions

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